Finite Group Databasebeta

← #26228302

Subgroups — #26228302

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×#15689251C211yesmaximal, characteristic, complemented
2#20246656C211yesmaximal, characteristic
3#20244324C211yesmaximal, characteristic, complemented
4#20244108C211yesmaximal, characteristic, complemented
5#20244106C211yesmaximal, characteristic, complemented
6#20244103C211yesmaximal, characteristic, complemented
7#20244004C211yesmaximal, characteristic
8#20243998C211yesmaximal, characteristic
9#20243997C211yesmaximal, characteristic
10#20243991C211yesmaximal, characteristic
11#20243949C211yesmaximal, characteristic, complemented
12#20243942C211yesmaximal, characteristic, complemented
13#20243372C211yesmaximal, characteristic
14#20243260C211yesmaximal, characteristic, complemented
15C2×C47×#433C211yesmaximal, characteristic
16C2×C4×Dic94C2211yescharacteristic
17#15689251—24no—
18C2×C4⋊(C47⋊C8)C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×Dic47⋊C8C2211yescharacteristic
21C2×Dic47⋊C8C2211yescharacteristic
22C2×Dic47⋊C8C2211yescharacteristic
23C2×Dic47⋊C8C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2211yescharacteristic
25C2×C22⋊(C47⋊C8)C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30#15689062C411yescharacteristic
31#15689059C411yescharacteristic
32#15689056C411yescharacteristic
33#15689056C411yescharacteristic
34#15689046C411yescharacteristic
35#15689046C411yescharacteristic
36#15689043C411yescharacteristic
37#15689035C411yescharacteristic
38C4×Dic47⋊C4C2211yescharacteristic
39C4×Dic47⋊C4C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C4×C22⋊Dic47C2211yescharacteristic
42C4×C22⋊Dic47C2211yescharacteristic
43#15688878C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688877C2211yescharacteristic
46#15688877C2211yescharacteristic
47#15688877C2211yescharacteristic
48#15688877C2211yescharacteristic
49#15688877C2211yescharacteristic
50#15688877C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C47×#433—24no—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×C4⋊C8×C47C2211yescharacteristic
57C2×C22⋊C8×C47C2211yescharacteristic
58C2×C22⋊C8×C47C2211yescharacteristic
59C2×#97×C47C2211yescharacteristic
60C2×C4×C8×C47C2211yescharacteristic
61C22×Dic94C2×C411yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C4×Dic94—22no—
67C4×Dic94—22no—
68C2×C4⋊Dic47C2×C411yescharacteristic
69C2×C4⋊Dic47C2×C411yescharacteristic
70C2×C4⋊Dic47C2×C411yescharacteristic
71C2×Dic47⋊C4C2×C411yescharacteristic
72C2×Dic47⋊C4C2×C411yescharacteristic
73C2×Dic47⋊C4C2×C411yescharacteristic
74C2×Dic47⋊C4C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×C22⋊Dic47C2×C411yescharacteristic
78C2×C22⋊Dic47C2×C411yescharacteristic
79C2×C22⋊Dic47C2×C411yescharacteristic
80C2×C22⋊Dic47C2×C411yescharacteristic
81C2×C4×Dic47C2311yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C4⋊(C47⋊C8)—22no—
86C4⋊(C47⋊C8)—22no—
87Dic47⋊C8—22no—
88Dic47⋊C8—22no—
89Dic47⋊C8—41no—
90Dic47⋊C8—41no—
91C22⋊(C47⋊C8)—41no—
92C22⋊(C47⋊C8)—41no—
93#13093934—22no—
94#13093934—41no—
95C8×Dic47—22no—
96C8×Dic47—41no—
97C47⋊#95C2×C411yescharacteristic
98C47⋊#95C2×C411yescharacteristic
99C47⋊#95C2×C411yescharacteristic
100C47⋊#95C2×C411yescharacteristic
101C47⋊#95C2×C411yescharacteristic
102C47⋊#95C2×C411yescharacteristic
103C47⋊#95C2×C411yescharacteristic
104C47⋊#95C2×C411yescharacteristic
105C47⋊#95C2×C411yescharacteristic
106C47⋊#95C2×C411yescharacteristic
107C23×C4×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112#117×C47—22no—
113#117×C47—22no—
114C2×C4⋊C4×C47C2×C411yescharacteristic
115C2×C4⋊C4×C47C2×C411yescharacteristic
116C2×C22⋊C4×C47C2×C411yescharacteristic
117C2×C22⋊C4×C47C2×C411yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120C4⋊C8×C47—22no—
121C4⋊C8×C47—22no—
122C22⋊C8×C47—22no—
123C22⋊C8×C47—22no—
124#97×C47D414yes—
125C4×C8×C47—22no—
126C24×C47C22×C411yescharacteristic
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C22×Dic47C22×C411yescharacteristic
144C22×Dic47C22×C411yescharacteristic
145C22×Dic47C22×C411yescharacteristic
146C22×Dic47C22×C411yescharacteristic
147C2×Dic94—22no—
148C2×Dic94—22no—
149C2×Dic94—41no—
150C2×C8×C47—21no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47C2×D412yes—
161C2×C8×C47C2×D412yes—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C22⋊C4×C47—22no—
165C22⋊C4×C47—22no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47C2×D412yes—
169C42×C47#4112yes—
170C22⋊Dic47—22no—
171C22⋊Dic47—22no—
172C22⋊Dic47—41no—
173C22⋊Dic47—41no—
174C4⋊Dic47—22no—
175C4⋊Dic47—22no—
176C4⋊Dic47—41no—
177Dic47⋊C4—22no—
178Dic47⋊C4—22no—
179Dic47⋊C4—22no—
180Dic47⋊C4—22no—
181Dic47⋊C4—41no—
182Dic47⋊C4—41no—
183C4×Dic47—21no—
184C4×Dic47—21no—
185C4×Dic47—21no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188C4×Dic47—21no—
189C4×Dic47—21no—
190C4×Dic47—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—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47C23×C411yescharacteristic
211C2×C4×C47—21no—
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—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47C4×D412yes—
229C2×C4×C47C4×D412yes—
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47H(5,𝔽2)11yescharacteristic
233C2×C4×C472−1+411yescharacteristic
234C2×C4×C472−1+411yescharacteristic
235C2×C4×C472−1+411yescharacteristic
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47#13112yes—
259C2×Dic47#13112yes—
260Dic94—44no—
261C8×C47—22no—
262C8×C47—22no—
263C8×C47—41no—
264C8×C47—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—41no—
287C4×C47—41no—
288C4×C47—41no—
289C4×C47#53611yescharacteristic
290C4×C47#53711yescharacteristic
291C4×C47#54111yescharacteristic
292C4×C47#54211yescharacteristic
293Dic47—22no—
294Dic47—22no—
295Dic47—41no—
296Dic47—41no—
297Dic47—41no—
298Dic47—41no—
299Dic47—41no—
300Dic47—41no—
301C2×#446—471no—
302C2×#433D4711yescharacteristic
303#2396—471no—
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×Q8—471no—
325C2×#117C2×D4711yescharacteristic
326#446—944no—
327#433—24no—
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⋊C8C2×D4711yescharacteristic
335C2×C4⋊C8C2×D4711yescharacteristic
336C2×C22⋊C8—471no—
337C2×C22⋊C8—471no—
338C2×C22⋊C8C2×D4711yescharacteristic
339C2×C22⋊C8C2×D4711yescharacteristic
340C2×#97—471no—
341C2×#97—471no—
342C2×#97C2×D4711yescharacteristic
343C2×C4×C8—471no—
344C2×C4×C8—471no—
345C2×C4×C8C2×D4711yescharacteristic
346#400—471no—
347#399—471no—
348#398—471no—
349#398—471no—
350#396—471no—
351#396—471no—
352#395—471no—
353#393—471no—
354C4×C4⋊C4—471no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—471no—
357C4×C22⋊C4—471no—
358C4×C22⋊C4—471no—
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×Q8—471no—
371C23×C4C22×D4711yescharacteristic
372C22×C8—471no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8—471no—
376C22×C8C22×D4711yescharacteristic
377C22×C8C22×D4711yescharacteristic
378C22×C8C22×D4711yescharacteristic
379C22×C8C22×D4711yescharacteristic
380C4×Q8—942no—
381C4×Q8—942no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4C4×D4711yescharacteristic
394C2×C4⋊C4C4×D4711yescharacteristic
395C2×C22⋊C4—471no—
396C2×C22⋊C4—471no—
397C2×C22⋊C4—471no—
398C2×C22⋊C4—471no—
399C2×C22⋊C4C4×D4711yescharacteristic
400C2×C22⋊C4C4×D4711yescharacteristic
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—471no—
405C2×C42C22×D4711yescharacteristic
406C2×C42C22×D4711yescharacteristic
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8—942no—
410C4⋊C8—942no—
411C4⋊C8—942no—
412C4⋊C8—942no—
413C4⋊C8—1881no—
414C4⋊C8—1881no—
415C22⋊C8—22no—
416C22⋊C8—22no—
417C22⋊C8—1881no—
418C22⋊C8—1881no—
419#97—942no—
420#97—1881no—
421#97C22⋊D4714yes—
422C4×C8—22no—
423C4×C8—942no—
424C4×C8—1881no—
425#95—471no—
426#95—471no—
427#95—471no—
428#95—471no—
429#95—471no—
430#95—471no—
431#95—471no—
432#95—471no—
433#95—471no—
434#95—471no—
435C24C2×C4×D4711yescharacteristic
436C2×Q8—942no—
437C2×Q8—942no—
438C2×Q8—1881no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—471no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—471no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4C2×C4×D4711yescharacteristic
454C22×C4C2×C4×D4711yescharacteristic
455C22×C4C2×C4×D4711yescharacteristic
456C22×C4C2×C4×D4711yescharacteristic
457C22×C4C2×C4×D4711yescharacteristic
458C22×C4C23×D4711yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
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×C8C2×C22⋊D4712yes—
482C2×C8C2×C22⋊D4712yes—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—942no—
486C4⋊C4—942no—
487C4⋊C4—942no—
488C4⋊C4—942no—
489C4⋊C4—942no—
490C4⋊C4—942no—
491C4⋊C4—1881no—
492C4⋊C4—1881no—
493C4⋊C4—1881no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—942no—
497C22⋊C4—942no—
498C22⋊C4—1881no—
499C22⋊C4—1881no—
500C42—21no—
501C42—21no—
502C42—941no—
503C42—941no—
504C42—941no—
505C42—941no—
506C42—941no—
507C42—941no—
508C42—941no—
509C42—941no—
510C42#1062535012yes—
511C42C2×C22⋊D4712yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C22×C4×D4711yescharacteristic
520Q8—1884no—
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—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—472no—
539C2×C4—472no—
540C2×C4—941no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—941no—
562C2×C4C4×C22⋊D4712yes—
563C2×C4C4×C22⋊D4712yes—
564C2×C4C22×C22⋊D4711yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C2×C4#1309407011yescharacteristic
567C2×C4#1309407111yescharacteristic
568C2×C4#1309407111yescharacteristic
569C2×C4#1309407111yescharacteristic
570C8—22no—
571C8—22no—
572C8—41no—
573C8—41no—
574C8—1881no—
575C8—1881no—
576C8—1881no—
577C8—1881no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×C4×C22⋊D4711yescharacteristic
586C22#1568960011yescharacteristic
587C22#1568960311yescharacteristic
588C22#131×D4711yescharacteristic
589C22#131×D4711yescharacteristic
590C22#1568988811yescharacteristic
591C22#1568988811yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—942no—
598C4—942no—
599C4—1881no—
600C4—1881no—
601C4—1881no—
602C4—1881no—
603C4—1881no—
604C4—1881no—
605C4#1568969411yescharacteristic
606C4#1568970111yescharacteristic
607C4#1568973511yescharacteristic
608C4#1568974411yescharacteristic
609C2—81no—
610C2#2024607011yescharacteristic
611C2#2025211011yescharacteristic
612C2#2025213011yescharacteristic
613C2#2025248711yescharacteristic
614C2#2025249211yescharacteristic
615C2#2025252211yescharacteristic
616C2#2025253111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC2×C47×#433C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenterC2×#433D47
hypocenterC47#22109
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#15689251C2semidirect
3. #20244324C2semidirect
4. #20244108C2semidirect
5. #20244106C2semidirect
6. #20244103C2semidirect
11. #20243949C2semidirect
12. #20243942C2semidirect
14. #20243260C2semidirect
369. C47#22109semidirect

Maximal subgroups

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

subgroupindex
C2×#156892512
#202466562
#202443242
#202441082
#202441062
#202441032
#202440042
#202439982
#202439972
#202439912
#202439492
#202439422
#202433722
#202432602
C2×C47×#4332
#2210947