Finite Group Databasebeta

← #37576052

Subgroups — #37576052

Order 24448.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259294C211yesmaximal, characteristic, complemented
2C2×#20259293C211yesmaximal, characteristic, complemented
3C2×#20259291C211yesmaximal, characteristic, complemented
4C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic191C211yesmaximal, characteristic, complemented
6#27132586C211yesmaximal, characteristic, complemented
7#27132579C211yesmaximal, characteristic, complemented
8#27132577C211yesmaximal, characteristic, complemented
9#27132571C211yesmaximal, characteristic, complemented
10#27132566C211yesmaximal, characteristic, complemented
11#27132551C211yesmaximal, characteristic, complemented
12#27132543C211yesmaximal, characteristic, complemented
13#27132539C211yesmaximal, characteristic, complemented
14#27132538C211yesmaximal, characteristic, complemented
15C2×#120×C191C211yesmaximal, characteristic, complemented
16C22×C22⋊D191C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C22×C22⋊D191C2211yescharacteristic
19C22×C22⋊D191C2211yescharacteristic, complemented
20C23×Dic191C2211yescharacteristic
21C23×Dic191C2211yescharacteristic
22C22×C4×D191C2211yescharacteristic, complemented
23#20259294—24no—
24#20259293—24no—
25#20259291—24no—
26(C22⋊D191)⋊C4—24no—
27C2×C4⋊Dic191C2211yescharacteristic, complemented
28C2×C4⋊Dic191C2211yescharacteristic, complemented
29C2×Dic191⋊C4C2211yescharacteristic, complemented
30C2×Dic191⋊C4C2211yescharacteristic, complemented
31C2×C22⋊Dic191—24no—
32C2×C22⋊Dic191C2211yescharacteristic
33C2×C22⋊Dic191C2211yescharacteristic
34C2×C22⋊Dic191C2211yescharacteristic
35C2×C22⋊Dic191C2211yescharacteristic
36C2×C22⋊Dic191C2211yescharacteristic
37C2×C22⋊Dic191C2211yescharacteristic
38C2×C22⋊Dic191C2211yescharacteristic
39C2×C22⋊Dic191C2211yescharacteristic, complemented
40C2×C22⋊Dic191C2211yescharacteristic, complemented
41C2×C22⋊Dic191C2211yescharacteristic, complemented
42C2×(C2×D191)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D191)⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic191C2211yescharacteristic, complemented
45C191⋊#95C2211yescharacteristic, complemented
46C191⋊#95C2211yescharacteristic, complemented
47C191⋊#95C2211yescharacteristic, complemented
48C191⋊#95C2211yescharacteristic, complemented
49C25×C191C2211yescharacteristic
50C22×D4×C191C2211yescharacteristic
51C22×D4×C191C2211yescharacteristic
52C22×D4×C191C2211yescharacteristic
53#120×C191—24no—
54C2×C22⋊C4×C191C2211yescharacteristic
55C2×C22⋊C4×C191C2211yescharacteristic
56C2×C22⋊C4×C191C2211yescharacteristic
57C23×D191C2311yescharacteristic
58C2×C22⋊D191—22no—
59C2×C22⋊D191—22no—
60C2×C22⋊D191—22no—
61C2×C22⋊D191—22no—
62C2×C22⋊D191—22no—
63C2×C22⋊D191—22no—
64C2×C22⋊D191—22no—
65C2×C22⋊D191—22no—
66C2×C22⋊D191—41no—
67C2×C22⋊D191—41no—
68C2×C22⋊D191—41no—
69C2×C22⋊D191D414yescomplemented
70C22×Dic191—22no—
71C22×Dic191—22no—
72C22×Dic191—22no—
73C22×Dic191—22no—
74C22×Dic191—41no—
75C22×Dic191—41no—
76C22×Dic191C2311yescharacteristic
77C22×Dic191C2311yescharacteristic
78C22×Dic191C2311yescharacteristic
79C22×Dic191C2311yescharacteristic
80C22×Dic191C2311yescharacteristic
81C22×Dic191C2311yescharacteristic
82C22×Dic191C2311yescharacteristic
83C2×C4×D191—22no—
84C2×C4×D191—22no—
85C2×C4×D191—41no—
86C4⋊Dic191—22no—
87C4⋊Dic191—41no—
88Dic191⋊C4—22no—
89Dic191⋊C4—22no—
90C22⋊Dic191—22no—
91C22⋊Dic191—22no—
92C22⋊Dic191—22no—
93C22⋊Dic191—22no—
94C22⋊Dic191—22no—
95C22⋊Dic191—22no—
96C22⋊Dic191—22no—
97C22⋊Dic191—22no—
98C22⋊Dic191—41no—
99C22⋊Dic191D414yescomplemented
100(C2×D191)⋊C4—22no—
101(C2×D191)⋊C4—22no—
102C4×Dic191—22no—
103C24×C191—22no—
104C24×C191—22no—
105C24×C191—22no—
106C24×C191—22no—
107C24×C191—22no—
108C24×C191—41no—
109C24×C191—41no—
110C24×C191C2311yescharacteristic
111C24×C191C2311yescharacteristic
112C24×C191C2311yescharacteristic
113C24×C191C2311yescharacteristic
114C2×D4×C191—22no—
115C2×D4×C191—22no—
116C2×D4×C191—22no—
117C2×D4×C191—22no—
118C2×D4×C191—22no—
119C2×D4×C191—22no—
120C2×D4×C191—22no—
121C2×D4×C191—41no—
122C2×D4×C191—41no—
123C22×C4×C191C2311yescharacteristic
124C22×C4×C191C2311yescharacteristic
125C22×C4×C191C2311yescharacteristic
126C22⋊C4×C191—22no—
127C22⋊C4×C191—22no—
128C22⋊C4×C191D414yescomplemented
129C22×D191—21no—
130C22×D191—21no—
131C22×D191—21no—
132C22×D191—21no—
133C22×D191—21no—
134C22×D191—21no—
135C22×D191C2×D412yes—
136C22⋊D191—28no—
137C22⋊D191—44no—
138C22⋊D191—44no—
139C22⋊D191—44no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—21no—
144C2×Dic191—21no—
145C2×Dic191—21no—
146C2×Dic191—21no—
147C2×Dic191—21no—
148C2×Dic191—21no—
149C2×Dic191—21no—
150C2×Dic191—21no—
151C2×Dic191—21no—
152C2×Dic191—21no—
153C2×Dic191—21no—
154C2×Dic191—21no—
155C2×Dic191—21no—
156C2×Dic191—21no—
157C2×Dic191—21no—
158C2×Dic191—44no—
159C2×Dic191—44no—
160C2×Dic191C2×D412yes—
161C2×Dic191C2×D412yes—
162C2×Dic191C2×D412yes—
163C4×D191—44no—
164C23×C191—21no—
165C23×C191—21no—
166C23×C191—21no—
167C23×C191—21no—
168C23×C191—28no—
169C23×C191—21no—
170C23×C191—21no—
171C23×C191—21no—
172C23×C191—21no—
173C23×C191—21no—
174C23×C191—21no—
175C23×C191—21no—
176C23×C191—21no—
177C23×C191—21no—
178C23×C191—21no—
179C23×C191—21no—
180C23×C191—21no—
181C23×C191—21no—
182C23×C191—21no—
183C23×C191—21no—
184C23×C191—21no—
185C23×C191—21no—
186C23×C191—21no—
187C23×C191—21no—
188C23×C191—44no—
189C23×C191—44no—
190C23×C191—44no—
191C23×C191—44no—
192C23×C191—44no—
193C23×C191—44no—
194C23×C191C2×D412yes—
195C23×C191C2×D412yes—
196C23×C191C2×D412yes—
197C23×C191#4112yes—
198C23×C191#4112yes—
199C23×C191C2411yescharacteristic
200D4×C191—44no—
201D4×C191—44no—
202D4×C191—44no—
203C2×C4×C191—21no—
204C2×C4×C191—21no—
205C2×C4×C191—21no—
206C2×C4×C191—21no—
207C2×C4×C191—21no—
208C2×C4×C191—21no—
209C2×C4×C191—21no—
210C2×C4×C191C2×D412yes—
211C2×C4×C191C2×D412yes—
212C22×C191—22no—
213C22×C191—22no—
214C22×C191—22no—
215C22×C191—22no—
216C22×C191—22no—
217C22×C191—22no—
218C22×C191—22no—
219C22×C191—22no—
220C22×C191—22no—
221C22×C191—22no—
222C22×C191—22no—
223C22×C191—22no—
224C22×C191—416no—
225C22×C191—41no—
226C22×C191—41no—
227C22×C191—41no—
228C22×C191—41no—
229C22×C191—41no—
230C22×C191—41no—
231C22×C191—41no—
232C22×C191—41no—
233C22×C191—41no—
234C22×C191—41no—
235C22×C191—41no—
236C22×C191—41no—
237C22×C191—41no—
238C22×C191—41no—
239C22×C191—41no—
240C22×C191#12114yescomplemented
241C22×C191C22×D411yescharacteristic
242C22×C191C22×D411yescharacteristic
243C22×C191C22×D411yescharacteristic
244C22×C191C2×#4111yescharacteristic
245C22×C191C2×#4111yescharacteristic
246C22×C191H(5,𝔽2)11yescharacteristic
247C22×C191H(5,𝔽2)11yescharacteristic
248C2×D191—22no—
249C2×D191—22no—
250C2×D191—22no—
251C2×D191—41no—
252C2×D191—41no—
253C2×D191—41no—
254C2×D191—41no—
255C4×C191—22no—
256C4×C191—22no—
257C4×C191—41no—
258Dic191—22no—
259Dic191—22no—
260Dic191—22no—
261Dic191—41no—
262Dic191—41no—
263Dic191—41no—
264Dic191—41no—
265C2×C191—24no—
266C2×C191—42no—
267C2×C191—42no—
268C2×C191—81no—
269C2×C191C2×#12111yescharacteristic
270C2×C191#52511yescharacteristic
271C2×C191#53811yescharacteristic
272C2×C191D4211yescharacteristic
273C2×C191D4211yescharacteristic
274C2×C191#54611yescharacteristic
275C2×C191#54611yescharacteristic
276D191—42no—
277C191#242411yescharacteristic, complemented, Sylow
278#2424—1911nomaximal, Sylow
279C2×#121—1911no—
280C2×#121—1911no—
281C2×#120—1911no—
282C2×#120D19111yescharacteristic, complemented
283C2×C4×D4—1911no—
284C22×C22⋊C4—1911no—
285#396—1911no—
286#394—1911no—
287#394—1911no—
288#392—1911no—
289#390—1911no—
290#388—1911no—
291#386—1911no—
292#386—1911no—
293#385—1911no—
294C25C2×D19111yescharacteristic
295C22×D4—1911no—
296C22×D4—1911no—
297C22×D4—1911no—
298C22×D4—1911no—
299C22×D4C2×D19111yescharacteristic
300C22×D4C2×D19111yescharacteristic
301C22×D4C2×D19111yescharacteristic
302C23×C4—1911no—
303C23×C4—1911no—
304C23×C4—1911no—
305#121—3824no—
306#121—3824no—
307#120—24no—
308#120—3824no—
309C4×D4—3824no—
310C2×C4⋊C4—1911no—
311C2×C4⋊C4—1911no—
312C2×C4⋊C4—1911no—
313C2×C4⋊C4—1911no—
314C2×C22⋊C4—3824no—
315C2×C22⋊C4—1911no—
316C2×C22⋊C4—1911no—
317C2×C22⋊C4—1911no—
318C2×C22⋊C4—1911no—
319C2×C22⋊C4—1911no—
320C2×C22⋊C4—1911no—
321C2×C22⋊C4—1911no—
322C2×C22⋊C4—1911no—
323C2×C22⋊C4—1911no—
324C2×C22⋊C4—1911no—
325C2×C22⋊C4—1911no—
326C2×C22⋊C4—1911no—
327C2×C22⋊C4C2×D19111yescharacteristic
328C2×C22⋊C4C2×D19111yescharacteristic
329C2×C22⋊C4C2×D19111yescharacteristic
330C2×C42—1911no—
331#95—1911no—
332#95—1911no—
333#95—1911no—
334#95—1911no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—1911no—
343C24C22×D19111yescharacteristic
344C24C22×D19111yescharacteristic
345C24C22×D19111yescharacteristic
346C24C22×D19111yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—3822no—
357C2×D4—3822no—
358C2×D4—3822no—
359C2×D4—3822no—
360C2×D4—3822no—
361C2×D4—3822no—
362C2×D4—3822no—
363C2×D4—3822no—
364C2×D4—1914no—
365C2×D4—7641no—
366C2×D4—7641no—
367C2×D4—7641no—
368C22×C4—3822no—
369C22×C4—3822no—
370C22×C4—3822no—
371C22×C4—3822no—
372C22×C4—3822no—
373C22×C4—3822no—
374C22×C4—1911no—
375C22×C4—1911no—
376C22×C4—1911no—
377C22×C4—1911no—
378C22×C4—1911no—
379C22×C4—1911no—
380C22×C4—1911no—
381C22×C4—7641no—
382C22×C4—7641no—
383C22×C4—7641no—
384C22×C4C22×D19111yescharacteristic
385C22×C4C22×D19111yescharacteristic
386C22×C4C22×D19111yescharacteristic
387C4⋊C4—3822no—
388C4⋊C4—3822no—
389C4⋊C4—3822no—
390C4⋊C4—7641no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—3822no—
394C22⋊C4—3822no—
395C22⋊C4—3822no—
396C22⋊C4—3822no—
397C22⋊C4—3822no—
398C22⋊C4—3822no—
399C22⋊C4—3822no—
400C22⋊C4—3822no—
401C22⋊C4—3822no—
402C22⋊C4—3822no—
403C22⋊C4—1914no—
404C22⋊C4—7641no—
405C22⋊C4C22⋊D19114yescomplemented
406C42—3822no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—28no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—3821no—
438C23—3821no—
439C23—3821no—
440C23—3821no—
441C23—3821no—
442C23—3821no—
443C23—1912no—
444C23D4×D19112yes—
445C23D4×D19112yes—
446C23C2×C22⋊D19112yes—
447C23#1569028612yes—
448C23#1569028612yes—
449C23C23×D19111yescharacteristic
450D4—44no—
451D4—44no—
452D4—44no—
453D4—3828no—
454D4—7644no—
455D4—7644no—
456D4—7644no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—3821no—
465C2×C4—3821no—
466C2×C4—3821no—
467C2×C4—3821no—
468C2×C4—3821no—
469C2×C4—3821no—
470C2×C4—3821no—
471C2×C4—3821no—
472C2×C4—3821no—
473C2×C4—3821no—
474C2×C4—3821no—
475C2×C4—3821no—
476C2×C4—3821no—
477C2×C4—3821no—
478C2×C4—3821no—
479C2×C4—3821no—
480C2×C4—3821no—
481C2×C4—3821no—
482C2×C4—1912no—
483C2×C4—1912no—
484C2×C4—1912no—
485C2×C4—7644no—
486C2×C4—7644no—
487C2×C4—7644no—
488C2×C4C2×C22⋊D19112yes—
489C2×C4C2×C22⋊D19112yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—3822no—
519C22—3822no—
520C22—3822no—
521C22—7641no—
522C22—7641no—
523C22—7641no—
524C22—7641no—
525C22#2025929314yescomplemented
526C22C2×D4×D19111yescharacteristic
527C22C2×D4×D19111yescharacteristic
528C22C22×C22⋊D19111yescharacteristic
529C22C2×#1569028611yescharacteristic
530C22C2×#1569028611yescharacteristic
531C22#2025936711yescharacteristic
532C22#2025936711yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—3822no—
537C4—3822no—
538C4—3822no—
539C4—7641no—
540C4—7641no—
541C4—7641no—
542C4—7641no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—81no—
547C2—7642no—
548C2C2×#2025929311yescharacteristic
549C2#2713316911yescharacteristic
550C2#2713324811yescharacteristic
551C2#2713329511yescharacteristic
552C2D4×C22⋊D19111yescharacteristic
553C2#2713330011yescharacteristic
554C2#2713330711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC2×#120×C191C2
hypercenterC2×#120D191
hypocenterC191#2424
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#20259294C2semidirect
2. C2×#20259293C2semidirect
3. C2×#20259291C2semidirect
4. C2×(C22⋊D191)⋊C4C2semidirect
5. C22×C22⋊Dic191C2semidirect
6. #27132586C2semidirect
7. #27132579C2semidirect
8. #27132577C2semidirect
9. #27132571C2semidirect
10. #27132566C2semidirect
11. #27132551C2semidirect
12. #27132543C2semidirect
13. #27132539C2semidirect
14. #27132538C2semidirect
15. C2×#120×C191C2semidirect
19. C22×C22⋊D191C22semidirect
22. C22×C4×D191C22semidirect
27. C2×C4⋊Dic191C22semidirect
28. C2×C4⋊Dic191C22semidirect
29. C2×Dic191⋊C4C22semidirect
30. C2×Dic191⋊C4C22semidirect
39. C2×C22⋊Dic191C22semidirect
40. C2×C22⋊Dic191C22semidirect
41. C2×C22⋊Dic191C22semidirect
42. C2×(C2×D191)⋊C4C22semidirect
43. C2×(C2×D191)⋊C4C22semidirect
44. C2×C4×Dic191C22semidirect
45. C191⋊#95C22semidirect
46. C191⋊#95C22semidirect
47. C191⋊#95C22semidirect
48. C191⋊#95C22semidirect
69. C2×C22⋊D191D4semidirect
99. C22⋊Dic191D4semidirect
128. C22⋊C4×C191D4semidirect
240. C22×C191#121semidirect
277. C191#2424semidirect
282. C2×#120D191semidirect
405. C22⋊C4C22⋊D191semidirect
525. C22#20259293semidirect

Maximal subgroups

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

subgroupindex
C2×#202592942
C2×#202592932
C2×#202592912
C2×(C22⋊D191)⋊C42
C22×C22⋊Dic1912
#271325862
#271325792
#271325772
#271325712
#271325662
#271325512
#271325432
#271325392
#271325382
C2×#120×C1912
#2424191