Finite Group Databasebeta

← #27144580

Subgroups — #27144580

Order 12416.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690767C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D97C211yesmaximal, characteristic
3C2×C22⋊C4×D97C211yesmaximal, characteristic
4#20260629C211yesmaximal, characteristic, complemented
5#20260628C211yesmaximal, characteristic, complemented
6#20260623C211yesmaximal, characteristic, complemented
7#20260609C211yesmaximal, characteristic, complemented
8#20260606C211yesmaximal, characteristic, complemented
9#20260597C211yesmaximal, characteristic, complemented
10#20260592C211yesmaximal, characteristic, complemented
11#20260590C211yesmaximal, characteristic
12C4×#13095488C211yesmaximal, characteristic, complemented
13#20260582C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690767—24no—
22C2×C4⋊Dic97C2211yescharacteristic, complemented
23C2×#13095490C2211yescharacteristic, complemented
24C2×#13095490C2211yescharacteristic, complemented
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C22⋊C4×D97—24no—
37C22⋊C4×D97C418yescomplemented
38C2×C4×Dic97C2211yescharacteristic, complemented
39C2×C4×Dic97C2211yescharacteristic, complemented
40C2×C4×Dic97C2211yescharacteristic, complemented
41#15690717C2211yescharacteristic, complemented
42#15690717C2211yescharacteristic, complemented
43#15690717C2211yescharacteristic, complemented
44#15690717C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47C2×C4⋊C4×C97C2211yescharacteristic, complemented
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C42×C97C2211yescharacteristic, complemented
52#95×C97C2211yescharacteristic, complemented
53#95×C97C2211yescharacteristic, complemented
54C23×D97—22no—
55C23×D97—22no—
56C23×D97—22no—
57C23×D97—22no—
58C23×D97—41no—
59C23×D97—41no—
60C23×D97C2×C414yes—
61C23×D97C2311yescharacteristic
62C23×D97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C2×C4×D97—22no—
70C2×C4×D97—22no—
71C2×C4×D97—22no—
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97—41no—
76C2×C4×D97—41no—
77C2×C4×D97—41no—
78C2×C4×D97—41no—
79C2×C4×D97C2×C414yescomplemented
80C2×C4×D97C2×C414yescomplemented
81C4⋊Dic97—41no—
82#13095490—22no—
83#13095490—22no—
84C22⋊Dic97—22no—
85C22⋊Dic97—41no—
86C22⋊Dic97C2×C414yescomplemented
87#13095488—22no—
88#13095488—22no—
89#13095488—22no—
90#13095488—22no—
91#13095488—22no—
92#13095488C2×C414yes—
93#13095488C2×C414yes—
94#13095488D414yes—
95C4×Dic97—22no—
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—22no—
107C22⋊C4×C97—41no—
108C22⋊C4×C97C2×C414yescomplemented
109C42×C97—22no—
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—28no—
127C22×D97—28no—
128C22×D97—44no—
129C22×D97—44no—
130C22×D97—44no—
131C22×D97—44no—
132C22×D97—44no—
133C22×D97C22×C412yes—
134C22×D97C22×C412yes—
135C22×D97C2×D412yes—
136C22×D97#4112yes—
137C22×D97#4112yes—
138C22×D97#4112yes—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
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×C97C22×C412yes—
155C2×C4×C97C22×C412yes—
156C2×C4×C97C2×D412yes—
157C2×Dic97—21no—
158C2×Dic97—21no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—21no—
163C2×Dic97—21no—
164C2×Dic97—21no—
165C2×Dic97—21no—
166C2×Dic97—21no—
167C2×Dic97—21no—
168C2×Dic97—21no—
169C2×Dic97—21no—
170C2×Dic97—21no—
171C2×Dic97—21no—
172C2×Dic97C22×C412yes—
173C2×Dic97C22×C412yes—
174C2×Dic97C2×D412yes—
175C4×D97—28no—
176C4×D97—44no—
177C4×D97—44no—
178C4×D97—44no—
179C22×C97—22no—
180C22×C97—22no—
181C22×C97—22no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97C23×C411yescharacteristic
187C22×C97C22×D411yescharacteristic
188C22×C97C2×#4111yescharacteristic
189C22×C97C2×#4111yescharacteristic
190C22×C97C2×#4111yescharacteristic
191C22×C97H(5,𝔽2)11yescharacteristic
192C22×C97H(5,𝔽2)11yescharacteristic
193C2×D97—22no—
194C2×D97—22no—
195C2×D97—22no—
196C2×D97—22no—
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—416no—
206C2×D97#11714yescomplemented
207C2×D97C4×D414yes—
208C4×C97—22no—
209C4×C97—22no—
210C4×C97—22no—
211C4×C97—41no—
212C4×C97—41no—
213C4×C97—41no—
214Dic97—22no—
215Dic97—22no—
216Dic97—22no—
217Dic97—41no—
218Dic97—41no—
219Dic97—41no—
220C2×C97—42no—
221C2×C97C2×#11711yescharacteristic
222C2×C97C2×C4×D411yescharacteristic
223C2×C97#51811yescharacteristic
224C2×C97#54611yescharacteristic
225C2×C97#54611yescharacteristic
226C2×C97#55111yescharacteristic
227C2×C97#55111yescharacteristic
228D97—24no—
229D97—24no—
230#2342—971nomaximal, Sylow
231C97#234211yescharacteristic, complemented, Sylow
232C2×#117—971no—
233C22×C22⋊C4—971no—
234C22×C22⋊C4—971no—
235#388—971no—
236#388—971no—
237#388—971no—
238#388D9711yescharacteristic, complemented
239#385—971no—
240#385—971no—
241#381—971no—
242#380—971no—
243#379—971no—
244C4×C22⋊C4—971no—
245C4×C22⋊C4—971no—
246C4×C22⋊C4—971no—
247C25—971no—
248C23×C4—971no—
249C23×C4—971no—
250C23×C4—971no—
251C23×C4—971no—
252#117—1944no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4C2×D9711yescharacteristic, complemented
257C2×C22⋊C4—971no—
258C2×C22⋊C4—971no—
259C2×C22⋊C4—971no—
260C2×C22⋊C4—971no—
261C2×C22⋊C4—978no—
262C2×C22⋊C4—971no—
263C2×C22⋊C4—971no—
264C2×C22⋊C4—971no—
265C2×C22⋊C4—971no—
266C2×C22⋊C4—971no—
267C2×C22⋊C4—971no—
268C2×C22⋊C4—971no—
269C2×C22⋊C4—1944no—
270C2×C22⋊C4C2×D9711yescharacteristic
271C2×C22⋊C4C2×D9711yescharacteristic
272C2×C22⋊C4C2×D9711yescharacteristic
273C2×C42—971no—
274C2×C42—971no—
275C2×C42—971no—
276C2×C42C2×D9711yescharacteristic, complemented
277#95—971no—
278#95—971no—
279#95—971no—
280#95—971no—
281#95—971no—
282#95—971no—
283#95C2×D9711yescharacteristic, complemented
284#95C2×D9711yescharacteristic, complemented
285C24—971no—
286C24—971no—
287C24—974no—
288C24—3881no—
289C24—3881no—
290C24—1942no—
291C24—1942no—
292C24—1942no—
293C24—1942no—
294C24C22×D9711yescharacteristic
295C22×C4—971no—
296C22×C4—971no—
297C22×C4—971no—
298C22×C4—974no—
299C22×C4—974no—
300C22×C4—971no—
301C22×C4—971no—
302C22×C4—971no—
303C22×C4—3881no—
304C22×C4—3881no—
305C22×C4—3881no—
306C22×C4—3881no—
307C22×C4—1942no—
308C22×C4—1942no—
309C22×C4—1942no—
310C22×C4—1942no—
311C22×C4—1942no—
312C22×C4—1942no—
313C22×C4C22×D9711yescharacteristic
314C22×C4C22×D9711yescharacteristic
315C22×C4C22×D9711yescharacteristic
316C22×C4C22×D9711yescharacteristic
317C22×C4C22×D9711yescharacteristic
318C22×C4C22×D9711yescharacteristic
319C4⋊C4—41no—
320C4⋊C4—3881no—
321C4⋊C4—1942no—
322C4⋊C4—1942no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—974no—
326C22⋊C4—974no—
327C22⋊C4—974no—
328C22⋊C4—974no—
329C22⋊C4—3881no—
330C22⋊C4—1942no—
331C22⋊C4—1942no—
332C22⋊C4—1942no—
333C22⋊C4—1942no—
334C22⋊C4—1942no—
335C22⋊C4—1942no—
336C22⋊C4C4×D9714yescomplemented
337C42—22no—
338C42—3881no—
339C42—3881no—
340C42—1942no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—972no—
348C23—972no—
349C23—972no—
350C23—972no—
351C23—972no—
352C23—972no—
353C23—3884no—
354C23—3884no—
355C23—3884no—
356C23—3884no—
357C23—3884no—
358C23—1948no—
359C23—1948no—
360C23—1941no—
361C23—1941no—
362C23—1941no—
363C23—1941no—
364C23—1941no—
365C23—1941no—
366C23—1941no—
367C23—1941no—
368C23C2×C4×D9712yes—
369C23C23×D9711yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—972no—
386C2×C4—972no—
387C2×C4—972no—
388C2×C4—3884no—
389C2×C4—3884no—
390C2×C4—3884no—
391C2×C4—1948no—
392C2×C4—1941no—
393C2×C4—1941no—
394C2×C4—1941no—
395C2×C4—1941no—
396C2×C4—1941no—
397C2×C4—1941no—
398C2×C4—1941no—
399C2×C4—1941no—
400C2×C4—1941no—
401C2×C4—1941no—
402C2×C4—1941no—
403C2×C4—1941no—
404C2×C4—1941no—
405C2×C4—1941no—
406C2×C4—1941no—
407C2×C4C2×C4×D9712yes—
408C2×C4C2×C4×D9712yes—
409C2×C4D4×D9712yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—974no—
418C22—974no—
419C22—38816no—
420C22—1942no—
421C22—1942no—
422C22—1942no—
423C22—1942no—
424C22—1942no—
425C22—1942no—
426C22—1942no—
427C22—1942no—
428C22—1942no—
429C22—1942no—
430C22—1942no—
431C22—1942no—
432C22C22×C4×D9711yescharacteristic
433C22C2×D4×D9711yescharacteristic
434C22#41×D9711yescharacteristic
435C22#41×D9711yescharacteristic
436C22#41×D9711yescharacteristic
437C22#1569085511yescharacteristic
438C22#1569085611yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—3881no—
446C4—3881no—
447C4—3881no—
448C4—1942no—
449C4—1942no—
450C4—1942no—
451C2—42no—
452C2—1944no—
453C2—1944no—
454C2#117×D9711yescharacteristic
455C2C4×D4×D9711yescharacteristic
456C2#2026120711yescharacteristic
457C2#2026137711yescharacteristic
458C2#2026138211yescharacteristic
459C2#2026141011yescharacteristic
460C2#2026141511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenter#388D97
hypocenterC97#2342
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#15690767C2semidirect
4. #20260629C2semidirect
5. #20260628C2semidirect
6. #20260623C2semidirect
7. #20260609C2semidirect
8. #20260606C2semidirect
9. #20260597C2semidirect
10. #20260592C2semidirect
12. C4×#13095488C2semidirect
13. #20260582C2semidirect
14. #20260582C2semidirect
15. #388×C97C2semidirect
22. C2×C4⋊Dic97C22semidirect
23. C2×#13095490C22semidirect
24. C2×#13095490C22semidirect
37. C22⋊C4×D97C4semidirect
38. C2×C4×Dic97C22semidirect
39. C2×C4×Dic97C22semidirect
40. C2×C4×Dic97C22semidirect
41. #15690717C22semidirect
42. #15690717C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. C2×C4⋊C4×C97C22semidirect
51. C2×C42×C97C22semidirect
52. #95×C97C22semidirect
53. #95×C97C22semidirect
79. C2×C4×D97C2×C4semidirect
80. C2×C4×D97C2×C4semidirect
86. C22⋊Dic97C2×C4semidirect
108. C22⋊C4×C97C2×C4semidirect
206. C2×D97#117semidirect
231. C97#2342semidirect
238. #388D97semidirect
256. C2×C4⋊C4C2×D97semidirect
276. C2×C42C2×D97semidirect
283. #95C2×D97semidirect
284. #95C2×D97semidirect
336. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907672
C2×C22⋊C4×D972
#202606292
#202606282
#202606232
#202606092
#202606062
#202605972
#202605922
#202605902
C4×#130954882
#202605822
#388×C972
#234297