Finite Group Databasebeta

← #27144876

Subgroups — #27144876

Order 12416.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690780C211yesmaximal, characteristic
2#20260637C211yesmaximal, characteristic, complemented
3#20260636C211yesmaximal, characteristic, complemented
4#20260635C211yesmaximal, characteristic, complemented
5#20260628C211yesmaximal, characteristic, complemented
6#20260627C211yesmaximal, characteristic, complemented
7#20260626C211yesmaximal, characteristic, complemented
8#20260625C211yesmaximal, characteristic, complemented
9#20260624C211yesmaximal, characteristic, complemented
10#20260613C211yesmaximal, characteristic, complemented
11#20260613C211yesmaximal, characteristic, complemented
12#20260611C211yesmaximal, characteristic, complemented
13#20260590C211yesmaximal, characteristic, complemented
14#20260590C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic, complemented
21#15690780C418yescomplemented
22C2×C4⋊Dic97C2211yescharacteristic, complemented
23C2×#13095490C2211yescharacteristic, complemented
24C2×C22⋊Dic97C2211yescharacteristic
25C2×C22⋊Dic97C2211yescharacteristic
26C2×#13095488C2211yescharacteristic
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic, complemented
37C2×#13095488C2211yescharacteristic, complemented
38C2×#13095488C2211yescharacteristic, complemented
39C2×C4×Dic97C2211yescharacteristic, complemented
40C2×C4×Dic97C2211yescharacteristic, complemented
41#15690717C2211yescharacteristic, complemented
42#15690717C2211yescharacteristic, complemented
43#15690717C2211yescharacteristic, complemented
44#15690717C2211yescharacteristic, complemented
45C2×C4⋊C4×C97C2211yescharacteristic, complemented
46C2×C22⋊C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic, complemented
50#95×C97C2211yescharacteristic, complemented
51#95×C97C2211yescharacteristic, complemented
52C23×D97—41no—
53C23×D97—41no—
54C23×D97—41no—
55C23×D97—41no—
56C23×D97—41no—
57C23×D97—41no—
58C23×D97C2×C414yescomplemented
59C23×D97C2311yescharacteristic
60C23×D97C2311yescharacteristic
61C23×D97C2311yescharacteristic
62C2×D388—41no—
63C2×D388—41no—
64C2×D388—41no—
65C2×D388—41no—
66C2×D388C2×C414yes—
67C2×D388C2×C414yes—
68C2×C22⋊D97—41no—
69C2×C22⋊D97—41no—
70C2×C22⋊D97C2×C414yes—
71C22×Dic97C2311yescharacteristic
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C2×C4×D97—41no—
77C2×C4×D97—41no—
78C2×C4×D97—41no—
79C4⋊Dic97—41no—
80#13095490—41no—
81C22⋊Dic97—41no—
82C22⋊Dic97—41no—
83#13095488—41no—
84#13095488—41no—
85#13095488—41no—
86#13095488—41no—
87#13095488—41no—
88#13095488—41no—
89#13095488—41no—
90#13095488—41no—
91#13095488—41no—
92#13095488—41no—
93#13095488—41no—
94#13095488C2×C414yescomplemented
95#13095488C2×C414yescomplemented
96C4×Dic97—41no—
97C4×Dic97—41no—
98C24×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105C4⋊C4×C97—41no—
106C22⋊C4×C97—41no—
107C22⋊C4×C97—41no—
108C22⋊C4×C97C2×C414yescomplemented
109C42×C97—41no—
110C23×C97—21no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—21no—
115C23×C97—21no—
116C23×C97C22×C412yes—
117C23×C97C2411yescharacteristic
118C22×D97—21no—
119C22×D97—21no—
120C22×D97—21no—
121C22×D97—21no—
122C22×D97—21no—
123C22×D97—21no—
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—21no—
131C22×D97—21no—
132C22×D97—21no—
133C22×D97—21no—
134C22×D97—21no—
135C22×D97—21no—
136C22×D97—44no—
137C22×D97—44no—
138C22×D97—44no—
139C22×D97—44no—
140C22×D97—44no—
141C22×D97—44no—
142C22×D97—44no—
143C22×D97C22×C412yes—
144C22×D97C22×C412yes—
145C22×D97C22×C412yes—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—21no—
152C2×C4×C97—21no—
153C2×C4×C97—21no—
154C2×C4×C97—21no—
155C2×C4×C97—21no—
156C2×C4×C97—21no—
157C2×C4×C97—21no—
158C2×C4×C97—21no—
159C2×C4×C97—21no—
160C2×C4×C97—21no—
161C2×C4×C97—21no—
162C2×C4×C97C22×C412yes—
163C2×C4×C97C22×C412yes—
164C22⋊D97—44no—
165C2×Dic97—21no—
166C2×Dic97—21no—
167C2×Dic97—21no—
168C2×Dic97—21no—
169C2×Dic97—21no—
170C2×Dic97—21no—
171C2×Dic97—21no—
172C2×Dic97—21no—
173C2×Dic97—21no—
174C2×Dic97—21no—
175C2×Dic97—21no—
176C2×Dic97—21no—
177C2×Dic97—21no—
178C2×Dic97—21no—
179C2×Dic97C22×C412yes—
180D388—44no—
181D388—44no—
182C4×D97—44no—
183C22×C97—22no—
184C22×C97—22no—
185C22×C97—22no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97—41no—
190C22×C97C23×C411yescharacteristic
191C22×C97H(5,𝔽2)11yescharacteristic
192C22×C97H(5,𝔽2)11yescharacteristic
193C22×C97H(5,𝔽2)11yescharacteristic
194C22×C97H(5,𝔽2)11yescharacteristic
195C22×C97H(5,𝔽2)11yescharacteristic
196C22×C97H(5,𝔽2)11yescharacteristic
197C2×D97—22no—
198C2×D97—22no—
199C2×D97—22no—
200C2×D97—22no—
201C2×D97—22no—
202C2×D97—22no—
203C2×D97—22no—
204C2×D97—22no—
205C2×D97—22no—
206C2×D97—416no—
207C2×D97—41no—
208C2×D97—41no—
209C2×D97—41no—
210C2×D97—41no—
211C2×D97—41no—
212C2×D97—41no—
213C2×D97—41no—
214C2×D97—41no—
215C2×D97—41no—
216C2×D97—41no—
217C2×D97—41no—
218C2×D97—41no—
219C4×C97—22no—
220C4×C97—22no—
221C4×C97—41no—
222C4×C97—41no—
223C4×C97—41no—
224C4×C97—41no—
225Dic97—22no—
226Dic97—41no—
227Dic97—41no—
228Dic97—41no—
229Dic97—41no—
230C2×C97—42no—
231C2×C97#51811yescharacteristic
232C2×C97#51811yescharacteristic
233C2×C97#51811yescharacteristic
234C2×C97#56011yescharacteristic
235C2×C97#56011yescharacteristic
236C2×C97#56011yescharacteristic
237C2×C97H(3,𝔽4)11yescharacteristic
238D97—42no—
239D97—42no—
240D97—42no—
241#2375—971nomaximal, Sylow
242C97#237511yescharacteristic, complemented, Sylow
243C2×#120—971no—
244#390—971no—
245#390—971no—
246#390—971no—
247#388—971no—
248#388—971no—
249#388—971no—
250#388—971no—
251#388—971no—
252#388D9711yescharacteristic, complemented
253#386—971no—
254#386—971no—
255#386—971no—
256#379—971no—
257#379—971no—
258C25—971no—
259C22×D4—971no—
260C22×D4—971no—
261C22×D4—971no—
262C23×C4—971no—
263#120—978no—
264C2×C4⋊C4—971no—
265C2×C4⋊C4—971no—
266C2×C4⋊C4C2×D9711yescharacteristic, complemented
267C2×C22⋊C4—971no—
268C2×C22⋊C4—971no—
269C2×C22⋊C4—971no—
270C2×C22⋊C4—971no—
271C2×C22⋊C4—971no—
272C2×C22⋊C4—971no—
273C2×C22⋊C4—971no—
274C2×C22⋊C4—971no—
275C2×C22⋊C4—971no—
276C2×C22⋊C4—971no—
277C2×C22⋊C4—971no—
278C2×C22⋊C4—971no—
279C2×C22⋊C4—971no—
280C2×C22⋊C4—971no—
281C2×C22⋊C4—971no—
282C2×C22⋊C4C2×D9711yescharacteristic
283C2×C22⋊C4C2×D9711yescharacteristic
284C2×C22⋊C4C2×D9711yescharacteristic
285C2×C42—971no—
286C2×C42—971no—
287C2×C42C2×D9711yescharacteristic, complemented
288#95—971no—
289#95—971no—
290#95—971no—
291#95—971no—
292#95C2×D9711yescharacteristic, complemented
293#95C2×D9711yescharacteristic, complemented
294C24—974no—
295C24—971no—
296C24—971no—
297C24—971no—
298C24—3881no—
299C24—3881no—
300C24—3881no—
301C24—3881no—
302C24—3881no—
303C24—3881no—
304C24C22×D9711yescharacteristic
305C2×D4—974no—
306C2×D4—974no—
307C2×D4—974no—
308C2×D4—3881no—
309C2×D4—3881no—
310C2×D4—3881no—
311C2×D4—3881no—
312C2×D4—3881no—
313C2×D4—3881no—
314C22×C4—971no—
315C22×C4—971no—
316C22×C4—971no—
317C22×C4—971no—
318C22×C4—971no—
319C22×C4—3881no—
320C22×C4—3881no—
321C22×C4—3881no—
322C22×C4C22×D9711yescharacteristic
323C22×C4C22×D9711yescharacteristic
324C22×C4C22×D9711yescharacteristic
325C22×C4C22×D9711yescharacteristic
326C22×C4C22×D9711yescharacteristic
327C22×C4C22×D9711yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—3881no—
330C4⋊C4—3881no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—974no—
334C22⋊C4—974no—
335C22⋊C4—3881no—
336C22⋊C4—3881no—
337C22⋊C4—3881no—
338C22⋊C4—3881no—
339C22⋊C4—3881no—
340C22⋊C4—3881no—
341C22⋊C4—3881no—
342C22⋊C4—3881no—
343C22⋊C4—3881no—
344C22⋊C4—3881no—
345C22⋊C4—3881no—
346C22⋊C4—3881no—
347C22⋊C4—3881no—
348C22⋊C4C4×D9714yescomplemented
349C42—41no—
350C42—3881no—
351C42—3881no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—972no—
359C23—972no—
360C23—972no—
361C23—3884no—
362C23—3884no—
363C23—3884no—
364C23—3884no—
365C23—3884no—
366C23—3884no—
367C23—3884no—
368C23—1941no—
369C23—1941no—
370C23—1941no—
371C23—1941no—
372C23—1941no—
373C23—1941no—
374C23—1941no—
375C23—1941no—
376C23—1941no—
377C23—1941no—
378C23—1941no—
379C23—1941no—
380C23—1941no—
381C23—1941no—
382C23—1941no—
383C23—1941no—
384C23—1941no—
385C23—1941no—
386C23C2×C4×D9712yes—
387C23C23×D9711yescharacteristic
388D4—3884no—
389D4—3884no—
390D4—3884no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—972no—
408C2×C4—3884no—
409C2×C4—1941no—
410C2×C4—1941no—
411C2×C4—1941no—
412C2×C4—1941no—
413C2×C4—1941no—
414C2×C4—1941no—
415C2×C4—1941no—
416C2×C4—1941no—
417C2×C4—1941no—
418C2×C4—1941no—
419C2×C4—1941no—
420C2×C4—1941no—
421C2×C4—1941no—
422C2×C4—1941no—
423C2×C4C2×C4×D9712yes—
424C2×C4C2×C4×D9712yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—38816no—
433C22—3881no—
434C22—3881no—
435C22—3881no—
436C22—3881no—
437C22—3881no—
438C22—3881no—
439C22—3881no—
440C22—3881no—
441C22—3881no—
442C22—3881no—
443C22—3881no—
444C22—3881no—
445C22—1942no—
446C22—1942no—
447C22—1942no—
448C22—1942no—
449C22—1942no—
450C22—1942no—
451C22—1942no—
452C22—1942no—
453C22—1942no—
454C22C22×C4×D9711yescharacteristic
455C22#1569085511yescharacteristic
456C22#1569085511yescharacteristic
457C22#1569085611yescharacteristic
458C22#1569085611yescharacteristic
459C22#1569085611yescharacteristic
460C22#1569085611yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—3881no—
468C4—3881no—
469C4—3881no—
470C4—3881no—
471C4—1942no—
472C2—42no—
473C2—3882no—
474C2—3882no—
475C2—3882no—
476C2#2026120711yescharacteristic
477C2#2026120711yescharacteristic
478C2#2026120911yescharacteristic
479C2#2026147311yescharacteristic
480C2#2026147511yescharacteristic
481C2#2026147611yescharacteristic
482C2#2026147711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690780C2
hypercenter#388D97
hypocenterC97#2375
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20260637C2semidirect
3. #20260636C2semidirect
4. #20260635C2semidirect
5. #20260628C2semidirect
6. #20260627C2semidirect
7. #20260626C2semidirect
8. #20260625C2semidirect
9. #20260624C2semidirect
10. #20260613C2semidirect
11. #20260613C2semidirect
12. #20260611C2semidirect
13. #20260590C2semidirect
14. #20260590C2semidirect
15. #388×C97C2semidirect
20. C22×C4×D97C22semidirect
21. #15690780C4semidirect
22. C2×C4⋊Dic97C22semidirect
23. C2×#13095490C22semidirect
36. C2×#13095488C22semidirect
37. C2×#13095488C22semidirect
38. C2×#13095488C22semidirect
39. C2×C4×Dic97C22semidirect
40. C2×C4×Dic97C22semidirect
41. #15690717C22semidirect
42. #15690717C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. C2×C4⋊C4×C97C22semidirect
49. C2×C42×C97C22semidirect
50. #95×C97C22semidirect
51. #95×C97C22semidirect
58. C23×D97C2×C4semidirect
94. #13095488C2×C4semidirect
95. #13095488C2×C4semidirect
108. C22⋊C4×C97C2×C4semidirect
242. C97#2375semidirect
252. #388D97semidirect
266. C2×C4⋊C4C2×D97semidirect
287. C2×C42C2×D97semidirect
292. #95C2×D97semidirect
293. #95C2×D97semidirect
348. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907802
#202606372
#202606362
#202606352
#202606282
#202606272
#202606262
#202606252
#202606242
#202606132
#202606112
#202605902
#388×C972
#237597