Finite Group Databasebeta

← #44510107

Subgroups — #44510107

Order 30848.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#21440612C211yesmaximal, characteristic
2C2×C4×D964C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D241C211yesmaximal, characteristic, complemented
4#29507618C211yesmaximal, characteristic, complemented
5#29507617C211yesmaximal, characteristic, complemented
6#29507610C211yesmaximal, characteristic, complemented
7#29507609C211yesmaximal, characteristic, complemented
8#29507608C211yesmaximal, characteristic, complemented
9#29507595C211yesmaximal, characteristic, complemented
10#29507595C211yesmaximal, characteristic, complemented
11#29507593C211yesmaximal, characteristic, complemented
12#29507588C211yesmaximal, characteristic, complemented
13#29507572C211yesmaximal, characteristic, complemented
14#29507564C211yesmaximal, characteristic, complemented
15#388×C241C211yesmaximal, characteristic, complemented
16C24×D241C2211yescharacteristic
17C22×D964C2211yescharacteristic
18C22×D964C2211yescharacteristic
19C22×C22⋊D241C2211yescharacteristic
20C22×C4×D241C2211yescharacteristic
21C22×C4×D241C2211yescharacteristic
22C22×C4×D241C2211yescharacteristic, complemented
23#21440612C418yescomplemented
24C4×D964—24no—
25C2×C4⋊Dic241C2211yescharacteristic, complemented
26C2×#16838749C2211yescharacteristic, complemented
27C2×C22⋊Dic241C2211yescharacteristic
28C2×C22⋊Dic241C2211yescharacteristic
29C2×#16838747C2211yescharacteristic
30C2×#16838747C2211yescharacteristic
31C2×#16838747C2211yescharacteristic
32C2×#16838747C2211yescharacteristic
33C2×#16838747C2211yescharacteristic
34C2×#16838747C2211yescharacteristic
35C2×#16838747C2211yescharacteristic
36C2×#16838747C2211yescharacteristic
37C2×#16838747C2211yescharacteristic, complemented
38C2×#16838747C2211yescharacteristic, complemented
39C2×#16838747C2211yescharacteristic, complemented
40C22⋊C4×D241—24no—
41C2×C4×Dic241C2211yescharacteristic, complemented
42C2×C4×Dic241C2211yescharacteristic, complemented
43#21440549C2211yescharacteristic, complemented
44#21440549C2211yescharacteristic, complemented
45#21440549C2211yescharacteristic, complemented
46#21440549C2211yescharacteristic, complemented
47C2×C4⋊C4×C241C2211yescharacteristic, complemented
48C2×C22⋊C4×C241C2211yescharacteristic
49C2×C22⋊C4×C241C2211yescharacteristic
50C2×C22⋊C4×C241C2211yescharacteristic
51C2×C42×C241C2211yescharacteristic, complemented
52#95×C241C2211yescharacteristic, complemented
53#95×C241C2211yescharacteristic, complemented
54C23×D241—22no—
55C23×D241—22no—
56C23×D241—41no—
57C23×D241—41no—
58C23×D241—41no—
59C23×D241—41no—
60C23×D241C2×C414yescomplemented
61C23×D241C2311yescharacteristic
62C23×D241C2311yescharacteristic
63C23×D241C2311yescharacteristic
64C2×D964—22no—
65C2×D964—22no—
66C2×D964—41no—
67C2×D964—41no—
68C2×D964C2×C414yes—
69C2×D964C2×C414yescomplemented
70C2×C22⋊D241—41no—
71C2×C22⋊D241—41no—
72C2×C22⋊D241C2×C414yescomplemented
73C22×Dic241C2311yescharacteristic
74C22×Dic241C2311yescharacteristic
75C22×Dic241C2311yescharacteristic
76C22×Dic241C2311yescharacteristic
77C22×Dic241C2311yescharacteristic
78C2×C4×D241—22no—
79C2×C4×D241—22no—
80C2×C4×D241—41no—
81C2×C4×D241—41no—
82C2×C4×D241—41no—
83C2×C4×D241—41no—
84C2×C4×D241—41no—
85C2×C4×D241—41no—
86C2×C4×D241D414yescomplemented
87C4⋊Dic241—22no—
88#16838749—41no—
89C22⋊Dic241—22no—
90C22⋊Dic241—41no—
91#16838747—22no—
92#16838747—22no—
93#16838747—22no—
94#16838747—22no—
95#16838747—22no—
96#16838747—41no—
97#16838747—41no—
98#16838747—41no—
99#16838747—41no—
100#16838747C2×C414yescomplemented
101#16838747C2×C414yescomplemented
102C4×Dic241—41no—
103C4×Dic241—41no—
104C24×C241C2311yescharacteristic
105C22×C4×C241C2311yescharacteristic
106C22×C4×C241C2311yescharacteristic
107C22×C4×C241C2311yescharacteristic
108C22×C4×C241C2311yescharacteristic
109C22×C4×C241C2311yescharacteristic
110C22×C4×C241C2311yescharacteristic
111C4⋊C4×C241—41no—
112C22⋊C4×C241—22no—
113C22⋊C4×C241—41no—
114C22⋊C4×C241C2×C414yescomplemented
115C42×C241—22no—
116C22×D241—21no—
117C22×D241—21no—
118C22×D241—21no—
119C22×D241—21no—
120C22×D241—21no—
121C22×D241—21no—
122C22×D241—21no—
123C22×D241—21no—
124C22×D241—21no—
125C22×D241—21no—
126C22×D241—21no—
127C22×D241—21no—
128C22×D241—21no—
129C22×D241—21no—
130C22×D241—21no—
131C22×D241—21no—
132C22×D241—28no—
133C22×D241—44no—
134C22×D241—44no—
135C22×D241—44no—
136C22×D241—44no—
137C22×D241—44no—
138C22×D241—44no—
139C22×D241C22×C412yes—
140C22×D241C22×C412yes—
141C22×D241C22×C412yes—
142C22×D241C2×D412yes—
143C22×D241#4112yes—
144D964—28no—
145D964—44no—
146C22⋊D241—44no—
147C2×Dic241—21no—
148C2×Dic241—21no—
149C2×Dic241—21no—
150C2×Dic241—21no—
151C2×Dic241—21no—
152C2×Dic241—21no—
153C2×Dic241—21no—
154C2×Dic241—21no—
155C2×Dic241—21no—
156C2×Dic241—21no—
157C2×Dic241—21no—
158C2×Dic241—21no—
159C2×Dic241—21no—
160C2×Dic241C22×C412yes—
161C2×Dic241C2×D412yes—
162C4×D241—44no—
163C4×D241—44no—
164C4×D241—44no—
165C23×C241—21no—
166C23×C241—21no—
167C23×C241—21no—
168C23×C241—21no—
169C23×C241—21no—
170C23×C241—21no—
171C23×C241C22×C412yes—
172C23×C241C2411yescharacteristic
173C2×C4×C241—21no—
174C2×C4×C241—21no—
175C2×C4×C241—21no—
176C2×C4×C241—21no—
177C2×C4×C241—21no—
178C2×C4×C241—21no—
179C2×C4×C241—21no—
180C2×C4×C241—21no—
181C2×C4×C241—21no—
182C2×C4×C241—21no—
183C2×C4×C241—21no—
184C2×C4×C241—21no—
185C2×C4×C241—21no—
186C2×C4×C241—21no—
187C2×C4×C241—21no—
188C2×C4×C241C22×C412yes—
189C2×C4×C241C22×C412yes—
190C2×C4×C241C2×D412yes—
191C22×C241—22no—
192C22×C241—22no—
193C22×C241—22no—
194C22×C241—41no—
195C22×C241—41no—
196C22×C241—41no—
197C22×C241—41no—
198C22×C241C23×C411yescharacteristic
199C22×C241C22×D411yescharacteristic
200C22×C241C2×#4111yescharacteristic
201C22×C241H(5,𝔽2)11yescharacteristic
202C22×C241H(5,𝔽2)11yescharacteristic
203C22×C241H(5,𝔽2)11yescharacteristic
204C22×C241H(5,𝔽2)11yescharacteristic
205C2×D241—22no—
206C2×D241—22no—
207C2×D241—22no—
208C2×D241—22no—
209C2×D241—22no—
210C2×D241—22no—
211C2×D241—22no—
212C2×D241—22no—
213C2×D241—22no—
214C2×D241—22no—
215C2×D241—22no—
216C2×D241—22no—
217C2×D241—416no—
218C2×D241—41no—
219C2×D241—41no—
220C2×D241—41no—
221C2×D241—41no—
222C2×D241—41no—
223C2×D241—41no—
224C2×D241—41no—
225C2×D241—41no—
226C2×D241C4×D414yes—
227C4×C241—22no—
228C4×C241—22no—
229C4×C241—22no—
230C4×C241—41no—
231C4×C241—41no—
232C4×C241—41no—
233Dic241—22no—
234Dic241—22no—
235Dic241—41no—
236Dic241—41no—
237Dic241—41no—
238C2×C241—42no—
239C2×C241C2×C4×D411yescharacteristic
240C2×C241#51811yescharacteristic
241C2×C241#51811yescharacteristic
242C2×C241#53411yescharacteristic
243C2×C241#53511yescharacteristic
244C2×C241#53811yescharacteristic
245C2×C241#53811yescharacteristic
246D241—24no—
247D241—42no—
248D241—42no—
249C241#232111yescharacteristic, complemented, Sylow
250#2321—2411nomaximal, Sylow
251C2×#120—2411no—
252C2×C4×D4—2411no—
253C22×C22⋊C4—2411no—
254#390—2411no—
255#390—2411no—
256#388—2411no—
257#388—2411no—
258#388—2411no—
259#388D24111yescharacteristic, complemented
260#386—2411no—
261#386—2411no—
262#386—2411no—
263#385—2411no—
264#379—2411no—
265C4×C22⋊C4—2411no—
266C25—2411no—
267C22×D4—2411no—
268C22×D4—2411no—
269C22×D4—2411no—
270C23×C4—2411no—
271C23×C4—2411no—
272C23×C4—2411no—
273#120—2418no—
274C4×D4—4824no—
275C2×C4⋊C4—2411no—
276C2×C4⋊C4—2411no—
277C2×C4⋊C4C2×D24111yescharacteristic, complemented
278C2×C22⋊C4—4824no—
279C2×C22⋊C4—2411no—
280C2×C22⋊C4—2411no—
281C2×C22⋊C4—2411no—
282C2×C22⋊C4—2411no—
283C2×C22⋊C4—2411no—
284C2×C22⋊C4—2411no—
285C2×C22⋊C4—2411no—
286C2×C22⋊C4—2411no—
287C2×C22⋊C4—2411no—
288C2×C22⋊C4—2411no—
289C2×C22⋊C4—2411no—
290C2×C22⋊C4—2411no—
291C2×C22⋊C4—2411no—
292C2×C22⋊C4C2×D24111yescharacteristic
293C2×C22⋊C4C2×D24111yescharacteristic
294C2×C22⋊C4C2×D24111yescharacteristic
295C2×C42—2411no—
296C2×C42—2411no—
297C2×C42C2×D24111yescharacteristic, complemented
298#95—2411no—
299#95—2411no—
300#95—2411no—
301#95—2411no—
302#95C2×D24111yescharacteristic, complemented
303#95C2×D24111yescharacteristic, complemented
304C24—9641no—
305C24—9641no—
306C24—9641no—
307C24—9641no—
308C24—4822no—
309C24—4822no—
310C24—2414no—
311C24—2411no—
312C24—2411no—
313C24—2411no—
314C24C22×D24111yescharacteristic
315C2×D4—9641no—
316C2×D4—9641no—
317C2×D4—9641no—
318C2×D4—9641no—
319C2×D4—4822no—
320C2×D4—4822no—
321C2×D4—2414no—
322C2×D4—2414no—
323C2×D4—2414no—
324C22×C4—9641no—
325C22×C4—9641no—
326C22×C4—9641no—
327C22×C4—9641no—
328C22×C4—9641no—
329C22×C4—9641no—
330C22×C4—4822no—
331C22×C4—4822no—
332C22×C4—2414no—
333C22×C4—2411no—
334C22×C4—2411no—
335C22×C4—2411no—
336C22×C4—2411no—
337C22×C4—2411no—
338C22×C4C22×D24111yescharacteristic
339C22×C4C22×D24111yescharacteristic
340C22×C4C22×D24111yescharacteristic
341C22×C4C22×D24111yescharacteristic
342C22×C4C22×D24111yescharacteristic
343C22×C4C22×D24111yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—9641no—
346C4⋊C4—4822no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—9641no—
350C22⋊C4—9641no—
351C22⋊C4—9641no—
352C22⋊C4—9641no—
353C22⋊C4—9641no—
354C22⋊C4—4822no—
355C22⋊C4—4822no—
356C22⋊C4—4822no—
357C22⋊C4—4822no—
358C22⋊C4—4822no—
359C22⋊C4—4822no—
360C22⋊C4—2414no—
361C22⋊C4—2414no—
362C22⋊C4C4×D24114yescomplemented
363C42—22no—
364C42—9641no—
365C42—9641no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—9644no—
373C23—9644no—
374C23—9644no—
375C23—9644no—
376C23—9644no—
377C23—9644no—
378C23—4828no—
379C23—4821no—
380C23—4821no—
381C23—4821no—
382C23—4821no—
383C23—4821no—
384C23—4821no—
385C23—4821no—
386C23—4821no—
387C23—4821no—
388C23—4821no—
389C23—4821no—
390C23—4821no—
391C23—4821no—
392C23—4821no—
393C23—4821no—
394C23—4821no—
395C23—2412no—
396C23—2412no—
397C23—2412no—
398C23—2412no—
399C23—2412no—
400C23C2×C4×D24112yes—
401C23C23×D24111yescharacteristic
402D4—9644no—
403D4—9644no—
404D4—4828no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—9644no—
421C2×C4—9644no—
422C2×C4—9644no—
423C2×C4—4821no—
424C2×C4—4821no—
425C2×C4—4821no—
426C2×C4—4821no—
427C2×C4—4821no—
428C2×C4—4821no—
429C2×C4—4821no—
430C2×C4—4821no—
431C2×C4—4821no—
432C2×C4—4821no—
433C2×C4—4821no—
434C2×C4—4821no—
435C2×C4—4821no—
436C2×C4—2412no—
437C2×C4—2412no—
438C2×C4C2×C4×D24112yes—
439C2×C4C2×C4×D24112yes—
440C2×C4D4×D24112yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—96416no—
449C22—9641no—
450C22—9641no—
451C22—9641no—
452C22—9641no—
453C22—9641no—
454C22—9641no—
455C22—9641no—
456C22—9641no—
457C22—4822no—
458C22—4822no—
459C22—4822no—
460C22—4822no—
461C22—4822no—
462C22—4822no—
463C22—4822no—
464C22—4822no—
465C22—4822no—
466C22—4822no—
467C22—4822no—
468C22—4822no—
469C22—2414no—
470C22C22×C4×D24111yescharacteristic
471C22C2×D4×D24111yescharacteristic
472C22#41×D24111yescharacteristic
473C22#2144068711yescharacteristic
474C22#2144068811yescharacteristic
475C22#2144068811yescharacteristic
476C22#2144068811yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—9641no—
484C4—9641no—
485C4—9641no—
486C4—4822no—
487C4—4822no—
488C2—42no—
489C2—9642no—
490C2—9642no—
491C2—4824no—
492C2C4×D4×D24111yescharacteristic
493C2#2950818911yescharacteristic
494C2#2950819111yescharacteristic
495C2#2950827811yescharacteristic
496C2#2950828311yescharacteristic
497C2#2950830111yescharacteristic
498C2#2950830611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D241
commutatorC22×C241C23×C4
Fitting#388×C241C2
socle, cosocleC23×C241C24
prime radicalC2×#21440612C2
hypercenter#388D241
hypocenterC241#2321
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D964C2semidirect
3. C2×C22⋊C4×D241C2semidirect
4. #29507618C2semidirect
5. #29507617C2semidirect
6. #29507610C2semidirect
7. #29507609C2semidirect
8. #29507608C2semidirect
9. #29507595C2semidirect
10. #29507595C2semidirect
11. #29507593C2semidirect
12. #29507588C2semidirect
13. #29507572C2semidirect
14. #29507564C2semidirect
15. #388×C241C2semidirect
22. C22×C4×D241C22semidirect
23. #21440612C4semidirect
25. C2×C4⋊Dic241C22semidirect
26. C2×#16838749C22semidirect
37. C2×#16838747C22semidirect
38. C2×#16838747C22semidirect
39. C2×#16838747C22semidirect
41. C2×C4×Dic241C22semidirect
42. C2×C4×Dic241C22semidirect
43. #21440549C22semidirect
44. #21440549C22semidirect
45. #21440549C22semidirect
46. #21440549C22semidirect
47. C2×C4⋊C4×C241C22semidirect
51. C2×C42×C241C22semidirect
52. #95×C241C22semidirect
53. #95×C241C22semidirect
60. C23×D241C2×C4semidirect
69. C2×D964C2×C4semidirect
72. C2×C22⋊D241C2×C4semidirect
86. C2×C4×D241D4semidirect
100. #16838747C2×C4semidirect
101. #16838747C2×C4semidirect
114. C22⋊C4×C241C2×C4semidirect
249. C241#2321semidirect
259. #388D241semidirect
277. C2×C4⋊C4C2×D241semidirect
297. C2×C42C2×D241semidirect
302. #95C2×D241semidirect
303. #95C2×D241semidirect
362. C22⋊C4C4×D241semidirect

Maximal subgroups

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

subgroupindex
C2×#214406122
C2×C4×D9642
C2×C22⋊C4×D2412
#295076182
#295076172
#295076102
#295076092
#295076082
#295075952
#295075932
#295075882
#295075722
#295075642
#388×C2412
#2321241