Finite Group Databasebeta

← #26228306

Subgroups — #26228306

Order 12032.

728 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689254C211yesmaximal, characteristic, complemented
2C2×#15689184C211yesmaximal, characteristic
3#20244325C211yesmaximal, characteristic, complemented
4#20244112C211yesmaximal, characteristic, complemented
5#20244107C211yesmaximal, characteristic, complemented
6#20244107C211yesmaximal, characteristic, complemented
7#20244003C211yesmaximal, characteristic
8#20244003C211yesmaximal, characteristic
9#20243996C211yesmaximal, characteristic
10#20243996C211yesmaximal, characteristic
11#20243954C211yesmaximal, characteristic, complemented
12#20243947C211yesmaximal, characteristic, complemented
13#20243372C211yesmaximal, characteristic
14#20243263C211yesmaximal, characteristic, complemented
15C47×#2396C211yesmaximal, characteristic
16#15689254—24no—
17#15689184—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×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C22⋊(C47⋊C8)C2211yescharacteristic
30C2×#13093935C2211yescharacteristic
31C2×#13093935C2211yescharacteristic
32C2×#13093935C2211yescharacteristic
33C2×C4×C47⋊C8C2211yescharacteristic
34C2×C4×C47⋊C8C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36#15688878C2211yescharacteristic
37#15688878C2211yescharacteristic
38#15688878C2211yescharacteristic
39#15688878C2211yescharacteristic
40#15688878C2211yescharacteristic
41#15688878C2211yescharacteristic
42#15688878C2211yescharacteristic
43#15688878C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46C2×C4×Q8×C47C2211yescharacteristic
47C2×#117×C47C2211yescharacteristic
48C47×#400C411yescharacteristic
49C47×#399C411yescharacteristic
50C47×#398C411yescharacteristic
51C47×#398C411yescharacteristic
52C47×#396C411yescharacteristic
53C47×#396C411yescharacteristic
54C47×#395C411yescharacteristic
55C47×#393C411yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C22⋊C4×C47C2211yescharacteristic
60C4×C22⋊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—
75C4⋊(C47⋊C8)—41no—
76C4⋊(C47⋊C8)—41no—
77C22⋊(C47⋊C8)—22no—
78C22⋊(C47⋊C8)—22no—
79C22⋊(C47⋊C8)—41no—
80C22⋊(C47⋊C8)—41no—
81#13093935—22no—
82#13093935—41no—
83#13093935D414yes—
84C4×C47⋊C8—22no—
85C4×C47⋊C8—22no—
86C4×C47⋊C8—41no—
87C22×Q8×C47C2×C411yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C4×Q8×C47—22no—
90C4×Q8×C47—22no—
91#117×C47—22no—
92#117×C47—22no—
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×C22⋊C4×C47C2×C411yescharacteristic
109C2×C22⋊C4×C47C2×C411yescharacteristic
110C2×C42×C47C2311yescharacteristic
111C2×C42×C47C2311yescharacteristic
112C2×C42×C47C2311yescharacteristic
113C2×C42×C47C2311yescharacteristic
114C2×C42×C47C2311yescharacteristic
115C2×C42×C47C2311yescharacteristic
116#95×C47C2×C411yescharacteristic
117#95×C47C2×C411yescharacteristic
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
127C2×Q8×C47—22no—
128C2×Q8×C47—22no—
129C2×Q8×C47—41no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47C22×C411yescharacteristic
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×C47C2411yescharacteristic
150C4⋊C4×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—41no—
159C4⋊C4×C47—41no—
160C4⋊C4×C47—41no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—22no—
163C22⋊C4×C47—22no—
164C22⋊C4×C47—22no—
165C22⋊C4×C47—41no—
166C22⋊C4×C47—41no—
167C42×C47—21no—
168C42×C47—21no—
169C42×C47—21no—
170C42×C47—21no—
171C42×C47—21no—
172C42×C47—21no—
173C42×C47—21no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47C2×D412yes—
178C42×C47#4112yes—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
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⋊C8C2×D412yes—
202C2×C47⋊C8C2×D412yes—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47C23×C411yescharacteristic
211Q8×C47—44no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
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—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
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—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47C4×D412yes—
252C2×C4×C47C4×D412yes—
253C2×C4×C47#13112yes—
254C2×C4×C47#13112yes—
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47H(5,𝔽2)11yescharacteristic
258C2×C4×C472−1+411yescharacteristic
259C2×C4×C472−1+411yescharacteristic
260C2×C4×C472−1+411yescharacteristic
261C47⋊C8—22no—
262C47⋊C8—22no—
263C47⋊C8—41no—
264C47⋊C8—41no—
265C47⋊C8—41no—
266C47⋊C8—41no—
267C47⋊C8—41no—
268C47⋊C8—41no—
269#22109—471nomaximal, Sylow
270C22×C47—41no—
271C22×C47—41no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47—41no—
277C22×C47C2×C4×D411yescharacteristic
278C22×C47#51911yescharacteristic
279C22×C47#52011yescharacteristic
280C22×C47C2×#13111yescharacteristic
281C22×C47C2×#13111yescharacteristic
282C22×C47#56811yescharacteristic
283C22×C47#56811yescharacteristic
284C4×C47—22no—
285C4×C47—22no—
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—41no—
291C4×C47—41no—
292C4×C47—41no—
293C4×C47—41no—
294C4×C47—41no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47#53611yescharacteristic
298C4×C47#53711yescharacteristic
299C4×C47#54111yescharacteristic
300C4×C47#54211yescharacteristic
301C2×#446—471no—
302C2×#433—471no—
303#2396D4711yescharacteristic
304#1984—471no—
305#1940—471no—
306#1939—471no—
307#1939—471no—
308#1924—471no—
309#1924—471no—
310#1923—471no—
311#1923—471no—
312#1917—471no—
313#1916—471no—
314#1798—471no—
315#1771—471no—
316C2×C47—81no—
317C2×C47#232311yescharacteristic
318C2×C47#293311yescharacteristic
319C2×C47#293511yescharacteristic
320C2×C47#298611yescharacteristic
321C2×C47#298711yescharacteristic
322C2×C47#299211yescharacteristic
323C2×C47#299311yescharacteristic
324C2×C4×Q8C2×D4711yescharacteristic
325C2×#117C2×D4711yescharacteristic
326#446—944no—
327#433—944no—
328C2×C4⋊C8—471no—
329C2×C4⋊C8—471no—
330C2×C4⋊C8—471no—
331C2×C4⋊C8—471no—
332C2×C4⋊C8—471no—
333C2×C4⋊C8—471no—
334C2×C4⋊C8—471no—
335C2×C4⋊C8—471no—
336C2×C22⋊C8—471no—
337C2×C22⋊C8—471no—
338C2×C22⋊C8—471no—
339C2×C22⋊C8—471no—
340C2×#97—471no—
341C2×#97—471no—
342C2×#97—471no—
343C2×C4×C8—471no—
344C2×C4×C8—471no—
345C2×C4×C8—471no—
346#400Dic4711yescharacteristic
347#399Dic4711yescharacteristic
348#398Dic4711yescharacteristic
349#398Dic4711yescharacteristic
350#396Dic4711yescharacteristic
351#396Dic4711yescharacteristic
352#395Dic4711yescharacteristic
353#393Dic4711yescharacteristic
354C4×C4⋊C4C2×D4711yescharacteristic
355C4×C4⋊C4C2×D4711yescharacteristic
356C4×C4⋊C4C2×D4711yescharacteristic
357C4×C22⋊C4C2×D4711yescharacteristic
358C4×C22⋊C4C2×D4711yescharacteristic
359#336—471no—
360#336—471no—
361#336—471no—
362#336—471no—
363#336—471no—
364#336—471no—
365#336—471no—
366#336—471no—
367#336—471no—
368#336—471no—
369C47#2210911yescharacteristic, complemented, Sylow
370C22×Q8C2×Dic4711yescharacteristic
371C23×C4C22×D4711yescharacteristic
372C22×C8—471no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8—471no—
376C22×C8—471no—
377C22×C8—471no—
378C22×C8—471no—
379C22×C8—471no—
380C4×Q8—22no—
381C4×Q8—22no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4C2×Dic4711yescharacteristic
385C2×C4⋊C4C2×Dic4711yescharacteristic
386C2×C4⋊C4C2×Dic4711yescharacteristic
387C2×C4⋊C4C2×Dic4711yescharacteristic
388C2×C4⋊C4C2×Dic4711yescharacteristic
389C2×C4⋊C4C2×Dic4711yescharacteristic
390C2×C4⋊C4C2×Dic4711yescharacteristic
391C2×C4⋊C4C2×Dic4711yescharacteristic
392C2×C4⋊C4C2×Dic4711yescharacteristic
393C2×C4⋊C4C2×Dic4711yescharacteristic
394C2×C4⋊C4C2×Dic4711yescharacteristic
395C2×C22⋊C4C2×Dic4711yescharacteristic
396C2×C22⋊C4C2×Dic4711yescharacteristic
397C2×C22⋊C4C2×Dic4711yescharacteristic
398C2×C22⋊C4C2×Dic4711yescharacteristic
399C2×C22⋊C4C2×Dic4711yescharacteristic
400C2×C22⋊C4C2×Dic4711yescharacteristic
401C2×C42C22×D4711yescharacteristic
402C2×C42C22×D4711yescharacteristic
403C2×C42C22×D4711yescharacteristic
404C2×C42C22×D4711yescharacteristic
405C2×C42C22×D4711yescharacteristic
406C2×C42C22×D4711yescharacteristic
407C4⋊C8—942no—
408C4⋊C8—942no—
409C4⋊C8—942no—
410C4⋊C8—942no—
411C4⋊C8—942no—
412C4⋊C8—942no—
413C4⋊C8—1881no—
414C4⋊C8—1881no—
415C22⋊C8—942no—
416C22⋊C8—942no—
417C22⋊C8—1881no—
418C22⋊C8—1881no—
419#97—474no—
420#97—942no—
421#97—1881no—
422C4×C8—942no—
423C4×C8—942no—
424C4×C8—1881no—
425#95C2×Dic4711yescharacteristic
426#95C2×Dic4711yescharacteristic
427#95C2×Dic4711yescharacteristic
428#95C2×Dic4711yescharacteristic
429#95C2×Dic4711yescharacteristic
430#95C2×Dic4711yescharacteristic
431#95C2×Dic4711yescharacteristic
432#95C2×Dic4711yescharacteristic
433#95C2×Dic4711yescharacteristic
434#95C2×Dic4711yescharacteristic
435C24C22×Dic4711yescharacteristic
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—41no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4C22×Dic4711yescharacteristic
446C22×C4C22×Dic4711yescharacteristic
447C22×C4C22×Dic4711yescharacteristic
448C22×C4C22×Dic4711yescharacteristic
449C22×C4C22×Dic4711yescharacteristic
450C22×C4C22×Dic4711yescharacteristic
451C22×C4C22×Dic4711yescharacteristic
452C22×C4C22×Dic4711yescharacteristic
453C22×C4C22×Dic4711yescharacteristic
454C22×C4C22×Dic4711yescharacteristic
455C22×C4C22×Dic4711yescharacteristic
456C22×C4C22×Dic4711yescharacteristic
457C22×C4C22×Dic4711yescharacteristic
458C22×C4C23×D4711yescharacteristic
459C2×C8—472no—
460C2×C8—472no—
461C2×C8—941no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
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—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42D4×D4712yes—
511C42#1062535212yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C23×Dic4711yescharacteristic
520Q8—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
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—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
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—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4D4×Dic4712yes—
561C2×C4D4×Dic4712yes—
562C2×C4#1309403112yes—
563C2×C4#1309403112yes—
564C2×C4C2×D4×D4711yescharacteristic
565C2×C4C2×#1062535211yescharacteristic
566C2×C4#1309407011yescharacteristic
567C2×C4#1309407111yescharacteristic
568C2×C4#1309407211yescharacteristic
569C2×C4#1309407211yescharacteristic
570C8—942no—
571C8—942no—
572C8—1881no—
573C8—1881no—
574C8—1881no—
575C8—1881no—
576C8—1881no—
577C8—1881no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×D4×Dic4711yescharacteristic
586C22#1568960111yescharacteristic
587C22#1568960611yescharacteristic
588C22C2×#1309403111yescharacteristic
589C22C2×#1309403111yescharacteristic
590C22#1568989011yescharacteristic
591C22#1568989011yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4#1568969611yescharacteristic
606C4#1568970311yescharacteristic
607C4#1568973311yescharacteristic
608C4#1568974611yescharacteristic
609C2—81no—
610C2#2024607211yescharacteristic
611C2#2025211511yescharacteristic
612C2#2025213211yescharacteristic
613C2#2025249011yescharacteristic
614C2#2025249411yescharacteristic
615C2#2025252711yescharacteristic
616C2#2025253711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#2396C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenter#2396D47
hypocenterC47#22109
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #2396

Complements

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

NH ≅ G/Ntype
1. C2×#15689254C2semidirect
3. #20244325C2semidirect
4. #20244112C2semidirect
5. #20244107C2semidirect
6. #20244107C2semidirect
11. #20243954C2semidirect
12. #20243947C2semidirect
14. #20243263C2semidirect
369. C47#22109semidirect

Maximal subgroups

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

subgroupindex
C2×#156892542
C2×#156891842
#202443252
#202441122
#202441072
#202440032
#202439962
#202439542
#202439472
#202433722
#202432632
C47×#23962
#2210947