Finite Group Databasebeta

← #26233860

Subgroups — #26233860

Order 12032.

780 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689184C211yesmaximal, characteristic
2C2×#15689180C211yesmaximal, characteristic
3#20244112C211yesmaximal, characteristic, complemented
4#20244107C211yesmaximal, characteristic, complemented
5#20243954C211yesmaximal, characteristic, complemented
6#20243954C211yesmaximal, characteristic, complemented
7#20243954C211yesmaximal, characteristic, complemented
8#20243947C211yesmaximal, characteristic, complemented
9#20243459C211yesmaximal, characteristic
10#20243456C211yesmaximal, characteristic
11#20243372C211yesmaximal, characteristic
12#20243263C211yesmaximal, characteristic, complemented
13#20243257C211yesmaximal, characteristic, complemented
14#20243154C211yesmaximal, characteristic
15C47×#2296C211yesmaximal, characteristic
16#15689184—24no—
17#15689180—24no—
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2211yescharacteristic
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×#13093935C2211yescharacteristic
30C2×#13093935C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32C2×C4×C47⋊C8C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34#15688878C2211yescharacteristic
35#15688878C2211yescharacteristic
36#15688878C2211yescharacteristic
37#15688878C2211yescharacteristic
38#15688878C2211yescharacteristic
39#15688878C2211yescharacteristic
40#15688878C2211yescharacteristic
41#15688878C2211yescharacteristic
42#15688878C2211yescharacteristic
43#15688878C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46C2×#117×C47C2211yescharacteristic
47C2×#117×C47C2211yescharacteristic
48C22×C42×C47C2211yescharacteristic
49C47×#389C411yescharacteristic
50C47×#389C411yescharacteristic
51C47×#389C411yescharacteristic
52C47×#389C411yescharacteristic
53C47×#385C411yescharacteristic
54C47×#385C411yescharacteristic
55C47×#385C411yescharacteristic
56C47×#385C411yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C4×C4⋊C4×C47C2211yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C4⋊(C47⋊C8)—22no—
70C4⋊(C47⋊C8)—22no—
71C4⋊(C47⋊C8)—22no—
72C4⋊(C47⋊C8)—22no—
73C4⋊(C47⋊C8)—22no—
74C4⋊(C47⋊C8)—22no—
75C22⋊(C47⋊C8)—22no—
76C22⋊(C47⋊C8)—22no—
77C22⋊(C47⋊C8)—22no—
78C22⋊(C47⋊C8)D414yes—
79#13093935—22no—
80#13093935—22no—
81#13093935—22no—
82#13093935—22no—
83C4×C47⋊C8—22no—
84C4×C47⋊C8—22no—
85C23×C4×C47C2×C411yescharacteristic
86C23×C4×C47C2×C411yescharacteristic
87C23×C4×C47C2311yescharacteristic
88#117×C47—22no—
89#117×C47—22no—
90#117×C47—22no—
91#117×C47—22no—
92C2×C4⋊C4×C47C2×C411yescharacteristic
93C2×C4⋊C4×C47C2×C411yescharacteristic
94C2×C4⋊C4×C47C2×C411yescharacteristic
95C2×C4⋊C4×C47C2×C411yescharacteristic
96C2×C4⋊C4×C47C2×C411yescharacteristic
97C2×C4⋊C4×C47C2×C411yescharacteristic
98C2×C4⋊C4×C47C2×C411yescharacteristic
99C2×C4⋊C4×C47C2×C411yescharacteristic
100C2×C4⋊C4×C47C2×C411yescharacteristic
101C2×C4⋊C4×C47C2×C411yescharacteristic
102C2×C4⋊C4×C47C2×C411yescharacteristic
103C2×C4⋊C4×C47C2×C411yescharacteristic
104C2×C22⋊C4×C47C2×C411yescharacteristic
105C2×C22⋊C4×C47C2×C411yescharacteristic
106C2×C22⋊C4×C47C2×C411yescharacteristic
107C2×C22⋊C4×C47C2×C411yescharacteristic
108C2×C42×C47—22no—
109C2×C42×C47—22no—
110C2×C42×C47C2×C411yescharacteristic
111C2×C42×C47C2×C411yescharacteristic
112C2×C42×C47C2311yescharacteristic
113C2×C42×C47C2311yescharacteristic
114C2×C42×C47C2311yescharacteristic
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118#95×C47C2×C411yescharacteristic
119#95×C47C2×C411yescharacteristic
120#95×C47C2×C411yescharacteristic
121#95×C47C2×C411yescharacteristic
122#95×C47C2×C411yescharacteristic
123#95×C47C2×C411yescharacteristic
124#95×C47C2×C411yescharacteristic
125#95×C47C2×C411yescharacteristic
126C24×C47C22×C411yescharacteristic
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—41no—
135C22×C4×C47—41no—
136C22×C4×C47—41no—
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C22×C411yescharacteristic
139C22×C4×C47C22×C411yescharacteristic
140C22×C4×C47C22×C411yescharacteristic
141C22×C4×C47C22×C411yescharacteristic
142C22×C4×C47C22×C411yescharacteristic
143C22×C4×C47C22×C411yescharacteristic
144C22×C4×C47C22×C411yescharacteristic
145C22×C4×C47C22×C411yescharacteristic
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C22×C411yescharacteristic
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C2×D412yes—
151C22×C4×C47#4112yes—
152C22×C4×C47C2411yescharacteristic
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—41no—
160C4⋊C4×C47—41no—
161C4⋊C4×C47—41no—
162C4⋊C4×C47—41no—
163C4⋊C4×C47—41no—
164C4⋊C4×C47—41no—
165C22⋊C4×C47—22no—
166C22⋊C4×C47—22no—
167C22⋊C4×C47—22no—
168C22⋊C4×C47—22no—
169C42×C47—22no—
170C42×C47—22no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—21no—
181C42×C47#4112yes—
182C42×C47#4112yes—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8C2×D412yes—
206C2×C47⋊C8C2×D412yes—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47C4×D412yes—
214C23×C47C23×C411yescharacteristic
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—21no—
251C2×C4×C47—21no—
252C2×C4×C47—21no—
253C2×C4×C47—21no—
254C2×C4×C47—44no—
255C2×C4×C47—44no—
256C2×C4×C47#11712yes—
257C2×C4×C47#11712yes—
258C2×C4×C47C4×D412yes—
259C2×C4×C47#13112yes—
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C472−1+411yescharacteristic
265C2×C4×C472−1+411yescharacteristic
266C47⋊C8—22no—
267C47⋊C8—22no—
268C47⋊C8—41no—
269C47⋊C8—41no—
270C47⋊C8—41no—
271C47⋊C8—41no—
272C47⋊C8—41no—
273C47⋊C8—41no—
274#22605—471nomaximal, Sylow
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47—41no—
281C22×C47—41no—
282C22×C47C2×#11711yescharacteristic
283C22×C47C2×C4×D411yescharacteristic
284C22×C47#51911yescharacteristic
285C22×C47C2×#13111yescharacteristic
286C22×C47#56811yescharacteristic
287C22×C47#56811yescharacteristic
288C22×C47#56811yescharacteristic
289C4×C47—22no—
290C4×C47—22no—
291C4×C47—22no—
292C4×C47—22no—
293C4×C47—41no—
294C4×C47—41no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47—41no—
298C4×C47—41no—
299C4×C47—41no—
300C4×C47—41no—
301C4×C47—41no—
302C4×C47#54711yescharacteristic
303C4×C47#54711yescharacteristic
304C4×C47#55211yescharacteristic
305C4×C47#55211yescharacteristic
306C2×#433—471no—
307C2×#432—471no—
308#2296D4711yescharacteristic
309#1940—471no—
310#1939—471no—
311#1917—471no—
312#1917—471no—
313#1917—471no—
314#1916—471no—
315#1818—471no—
316#1817—471no—
317#1798—471no—
318#1771—471no—
319#1770—471no—
320#1750—471no—
321C2×C47—42no—
322C2×C47#234411yescharacteristic
323C2×C47#292311yescharacteristic
324C2×C47#292511yescharacteristic
325C2×C47#293011yescharacteristic
326C2×C47#293311yescharacteristic
327C2×C47#298511yescharacteristic
328C2×C47#298711yescharacteristic
329C2×#117C2×D4711yescharacteristic
330C2×#117C2×D4711yescharacteristic
331C22×C42C2×D4711yescharacteristic
332#433—944no—
333#432—944no—
334C2×C4⋊C8—471no—
335C2×C4⋊C8—471no—
336C2×C4⋊C8—471no—
337C2×C4⋊C8—471no—
338C2×C4⋊C8—471no—
339C2×C4⋊C8—471no—
340C2×C22⋊C8—471no—
341C2×C22⋊C8—471no—
342C2×C22⋊C8—471no—
343C2×C22⋊C8—471no—
344C2×#97—471no—
345C2×#97—471no—
346C2×#97—471no—
347C2×#97—471no—
348C2×C4×C8—471no—
349C2×C4×C8—471no—
350#389Dic4711yescharacteristic
351#389Dic4711yescharacteristic
352#389Dic4711yescharacteristic
353#389Dic4711yescharacteristic
354#385Dic4711yescharacteristic
355#385Dic4711yescharacteristic
356#385Dic4711yescharacteristic
357#385Dic4711yescharacteristic
358C4×C4⋊C4C2×D4711yescharacteristic
359C4×C4⋊C4C2×D4711yescharacteristic
360C4×C4⋊C4C2×D4711yescharacteristic
361C4×C4⋊C4C2×D4711yescharacteristic
362#336—471no—
363#336—471no—
364#336—471no—
365#336—471no—
366#336—471no—
367#336—471no—
368#336—471no—
369#336—471no—
370#336—471no—
371#336—471no—
372#336—471no—
373#336—471no—
374C47#2260511yescharacteristic, complemented, Sylow
375C23×C4C2×Dic4711yescharacteristic
376C23×C4C2×Dic4711yescharacteristic
377C23×C4C22×D4711yescharacteristic
378C22×C8—471no—
379C22×C8—471no—
380C22×C8—471no—
381C22×C8—471no—
382C22×C8—471no—
383C22×C8—471no—
384C22×C8—471no—
385C22×C8—471no—
386#117—22no—
387#117—22no—
388#117—22no—
389#117—22no—
390C2×C4⋊C4C2×Dic4711yescharacteristic
391C2×C4⋊C4C2×Dic4711yescharacteristic
392C2×C4⋊C4C2×Dic4711yescharacteristic
393C2×C4⋊C4C2×Dic4711yescharacteristic
394C2×C4⋊C4C2×Dic4711yescharacteristic
395C2×C4⋊C4C2×Dic4711yescharacteristic
396C2×C4⋊C4C2×Dic4711yescharacteristic
397C2×C4⋊C4C2×Dic4711yescharacteristic
398C2×C4⋊C4C2×Dic4711yescharacteristic
399C2×C4⋊C4C2×Dic4711yescharacteristic
400C2×C4⋊C4C2×Dic4711yescharacteristic
401C2×C4⋊C4C2×Dic4711yescharacteristic
402C2×C22⋊C4C2×Dic4711yescharacteristic
403C2×C22⋊C4C2×Dic4711yescharacteristic
404C2×C22⋊C4C2×Dic4711yescharacteristic
405C2×C22⋊C4C2×Dic4711yescharacteristic
406C2×C42—22no—
407C2×C42—22no—
408C2×C42C2×Dic4711yescharacteristic
409C2×C42C2×Dic4711yescharacteristic
410C2×C42C22×D4711yescharacteristic
411C2×C42C22×D4711yescharacteristic
412C2×C42C22×D4711yescharacteristic
413C2×C42C22×D4711yescharacteristic
414C2×C42C22×D4711yescharacteristic
415C2×C42C22×D4711yescharacteristic
416C4⋊C8—942no—
417C4⋊C8—942no—
418C4⋊C8—942no—
419C4⋊C8—942no—
420C4⋊C8—942no—
421C4⋊C8—942no—
422C22⋊C8—474no—
423C22⋊C8—942no—
424C22⋊C8—942no—
425C22⋊C8—942no—
426#97—942no—
427#97—942no—
428#97—942no—
429#97—942no—
430C4×C8—942no—
431C4×C8—942no—
432#95C2×Dic4711yescharacteristic
433#95C2×Dic4711yescharacteristic
434#95C2×Dic4711yescharacteristic
435#95C2×Dic4711yescharacteristic
436#95C2×Dic4711yescharacteristic
437#95C2×Dic4711yescharacteristic
438#95C2×Dic4711yescharacteristic
439#95C2×Dic4711yescharacteristic
440C24C22×Dic4711yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4D4×D4712yes—
452C22×C4#1062535212yes—
453C22×C4C22×Dic4711yescharacteristic
454C22×C4C22×Dic4711yescharacteristic
455C22×C4C22×Dic4711yescharacteristic
456C22×C4C22×Dic4711yescharacteristic
457C22×C4C22×Dic4711yescharacteristic
458C22×C4C22×Dic4711yescharacteristic
459C22×C4C22×Dic4711yescharacteristic
460C22×C4C22×Dic4711yescharacteristic
461C22×C4C22×Dic4711yescharacteristic
462C22×C4C22×Dic4711yescharacteristic
463C22×C4C22×Dic4711yescharacteristic
464C22×C4C22×Dic4711yescharacteristic
465C22×C4C22×Dic4711yescharacteristic
466C22×C4C23×D4711yescharacteristic
467C2×C8—472no—
468C2×C8—472no—
469C2×C8—941no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42#1062535012yes—
520C42#1062535012yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23D4×Dic4712yes—
528C23C23×Dic4711yescharacteristic
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4#1309398312yes—
571C2×C4#1309398312yes—
572C2×C4D4×Dic4712yes—
573C2×C4#1309403112yes—
574C2×C4C2×D4×D4711yescharacteristic
575C2×C4C2×#1062535211yescharacteristic
576C2×C4C2×#1062535011yescharacteristic
577C2×C4C2×#1062535011yescharacteristic
578C2×C4#1309407211yescharacteristic
579C2×C4#1309407211yescharacteristic
580C8—942no—
581C8—942no—
582C8—1881no—
583C8—1881no—
584C8—1881no—
585C8—1881no—
586C8—1881no—
587C8—1881no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×#1309398311yescharacteristic
596C22C2×D4×Dic4711yescharacteristic
597C22#1568960111yescharacteristic
598C22C2×#1309403111yescharacteristic
599C22#1568989011yescharacteristic
600C22#1568989011yescharacteristic
601C22#1568989011yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4#1568977411yescharacteristic
616C4#1568977411yescharacteristic
617C4#1568980811yescharacteristic
618C4#1568980811yescharacteristic
619C2—42no—
620C2#2024625711yescharacteristic
621C2#2025203911yescharacteristic
622C2#2025205411yescharacteristic
623C2#2025209211yescharacteristic
624C2#2025211511yescharacteristic
625C2#2025248311yescharacteristic
626C2#2025249411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#2296C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenter#2296D47
hypocenterC47#22605
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2296

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20244112C2semidirect
4. #20244107C2semidirect
5. #20243954C2semidirect
6. #20243954C2semidirect
7. #20243954C2semidirect
8. #20243947C2semidirect
12. #20243263C2semidirect
13. #20243257C2semidirect
374. C47#22605semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156891842
C2×#156891802
#202441122
#202441072
#202439542
#202439472
#202434592
#202434562
#202433722
#202432632
#202432572
#202431542
C47×#22962
#2260547