Finite Group Databasebeta

← #27144330

Subgroups — #27144330

Order 12416.

672 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×D97C211yesmaximal, characteristic
2#20260629C211yesmaximal, characteristic, complemented
3#20260628C211yesmaximal, characteristic, complemented
4#20260627C211yesmaximal, characteristic, complemented
5#20260626C211yesmaximal, characteristic, complemented
6#20260625C211yesmaximal, characteristic, complemented
7#20260624C211yesmaximal, characteristic, complemented
8#20260623C211yesmaximal, characteristic, complemented
9#20260594C211yesmaximal, characteristic, complemented
10#20260592C211yesmaximal, characteristic, complemented
11#20260590C211yesmaximal, characteristic
12#20260590C211yesmaximal, characteristic
13C4×#13095488C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C2×C4⋊Dic97C2211yescharacteristic, complemented
20C2×#13095490C2211yescharacteristic, complemented
21C2×#13095490C2211yescharacteristic, complemented
22C2×C22⋊Dic97C2211yescharacteristic
23C2×C22⋊Dic97C2211yescharacteristic
24C2×C22⋊Dic97C2211yescharacteristic
25C2×#13095488C2211yescharacteristic
26C2×#13095488C2211yescharacteristic
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C22⋊C4×D97C418yescomplemented
36C2×C4×Dic97C2211yescharacteristic, complemented
37C2×C4×Dic97C2211yescharacteristic, complemented
38C2×C4×Dic97C2211yescharacteristic, complemented
39#15690717C2211yescharacteristic, complemented
40#15690717C2211yescharacteristic, 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—22no—
53C23×D97—22no—
54C23×D97—41no—
55C23×D97—41no—
56C23×D97—41no—
57C23×D97—41no—
58C23×D97C2×C414yes—
59C23×D97C2311yescharacteristic
60C23×D97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C2×C4×D97—41no—
68C2×C4×D97—41no—
69C2×C4×D97—41no—
70C2×C4×D97—41no—
71C2×C4×D97C2×C414yes—
72C2×C4×D97C2×C414yes—
73C4⋊Dic97—41no—
74#13095490—41no—
75#13095490—41no—
76C22⋊Dic97—41no—
77C22⋊Dic97—41no—
78C22⋊Dic97C2×C414yes—
79#13095488—22no—
80#13095488—22no—
81#13095488—22no—
82#13095488—22no—
83#13095488—41no—
84#13095488—41no—
85#13095488—41no—
86#13095488—41no—
87#13095488C2×C414yes—
88#13095488C2×C414yes—
89C4×Dic97—41no—
90C4×Dic97—41no—
91C4×Dic97—41no—
92C24×C97C2311yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C22×C4×C97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97—41no—
100C22⋊C4×C97—41no—
101C22⋊C4×C97—41no—
102C22⋊C4×C97C2×C414yes—
103C42×C97—41no—
104C23×C97—21no—
105C23×C97—21no—
106C23×C97—21no—
107C23×C97—21no—
108C23×C97—21no—
109C23×C97—21no—
110C23×C97C22×C412yes—
111C23×C97C2411yescharacteristic
112C22×D97—21no—
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—21no—
116C22×D97—21no—
117C22×D97—21no—
118C22×D97—21no—
119C22×D97—21no—
120C22×D97—21no—
121C22×D97—21no—
122C22×D97—28no—
123C22×D97—44no—
124C22×D97—44no—
125C22×D97—44no—
126C22×D97—44no—
127C22×D97—44no—
128C22×D97—44no—
129C22×D97C22×C412yes—
130C22×D97C22×C412yes—
131C22×D97#4112yes—
132C22×D97#4112yes—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
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×C97C22×C412yes—
150C2×C4×C97C22×C412yes—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
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×Dic97C22×C412yes—
168C2×Dic97C22×C412yes—
169C4×D97—44no—
170C4×D97—44no—
171C22×C97—22no—
172C22×C97—22no—
173C22×C97—22no—
174C22×C97—41no—
175C22×C97—41no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97C23×C411yescharacteristic
179C22×C97C2×#4111yescharacteristic
180C22×C97C2×#4111yescharacteristic
181C22×C97H(5,𝔽2)11yescharacteristic
182C22×C97H(5,𝔽2)11yescharacteristic
183C22×C97H(5,𝔽2)11yescharacteristic
184C22×C97H(5,𝔽2)11yescharacteristic
185C2×D97—22no—
186C2×D97—22no—
187C2×D97—22no—
188C2×D97—22no—
189C2×D97—22no—
190C2×D97—22no—
191C2×D97—22no—
192C2×D97—22no—
193C2×D97—22no—
194C2×D97—416no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
199C2×D97#11714yes—
200C4×C97—22no—
201C4×C97—22no—
202C4×C97—41no—
203C4×C97—41no—
204C4×C97—41no—
205C4×C97—41no—
206Dic97—22no—
207Dic97—22no—
208Dic97—41no—
209Dic97—41no—
210Dic97—41no—
211Dic97—41no—
212C2×C97—42no—
213C2×C97C2×#11711yescharacteristic
214C2×C97#51811yescharacteristic
215C2×C97#51811yescharacteristic
216C2×C97#53811yescharacteristic
217C2×C97#53811yescharacteristic
218C2×C97#53811yescharacteristic
219C2×C97#53811yescharacteristic
220D97—24no—
221D97—42no—
222#2312—971nomaximal, Sylow
223C97#231211yescharacteristic, complemented, Sylow
224C22×C22⋊C4—971no—
225#388—971no—
226#388—971no—
227#388—971no—
228#388—971no—
229#388—971no—
230#388—971no—
231#388—971no—
232#388D9711yescharacteristic, complemented
233#380—971no—
234#380—971no—
235#379—971no—
236#379—971no—
237C4×C22⋊C4—971no—
238C4×C22⋊C4—971no—
239C25—971no—
240C23×C4—971no—
241C23×C4—971no—
242C2×C4⋊C4—971no—
243C2×C4⋊C4—971no—
244C2×C4⋊C4—971no—
245C2×C4⋊C4C2×D9711yescharacteristic, complemented
246C2×C22⋊C4—971no—
247C2×C22⋊C4—971no—
248C2×C22⋊C4—971no—
249C2×C22⋊C4—978no—
250C2×C22⋊C4—971no—
251C2×C22⋊C4—971no—
252C2×C22⋊C4—971no—
253C2×C22⋊C4—971no—
254C2×C22⋊C4—971no—
255C2×C22⋊C4—971no—
256C2×C22⋊C4—971no—
257C2×C22⋊C4—971no—
258C2×C22⋊C4—971no—
259C2×C22⋊C4—971no—
260C2×C22⋊C4C2×D9711yescharacteristic
261C2×C22⋊C4C2×D9711yescharacteristic
262C2×C22⋊C4C2×D9711yescharacteristic
263C2×C42—971no—
264C2×C42—971no—
265C2×C42—971no—
266C2×C42C2×D9711yescharacteristic, complemented
267#95—971no—
268#95—971no—
269#95—971no—
270#95—971no—
271#95—971no—
272#95—971no—
273#95C2×D9711yescharacteristic, complemented
274#95C2×D9711yescharacteristic, complemented
275C24—971no—
276C24—974no—
277C24—971no—
278C24—3881no—
279C24—3881no—
280C24—3881no—
281C24—3881no—
282C24—1942no—
283C24—1942no—
284C24C22×D9711yescharacteristic
285C22×C4—971no—
286C22×C4—971no—
287C22×C4—974no—
288C22×C4—974no—
289C22×C4—971no—
290C22×C4—971no—
291C22×C4—971no—
292C22×C4—971no—
293C22×C4—3881no—
294C22×C4—3881no—
295C22×C4—3881no—
296C22×C4—3881no—
297C22×C4C22×D9711yescharacteristic
298C22×C4C22×D9711yescharacteristic
299C22×C4C22×D9711yescharacteristic
300C22×C4C22×D9711yescharacteristic
301C22×C4C22×D9711yescharacteristic
302C22×C4C22×D9711yescharacteristic
303C4⋊C4—41no—
304C4⋊C4—3881no—
305C4⋊C4—3881no—
306C4⋊C4—3881no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—974no—
310C22⋊C4—974no—
311C22⋊C4—974no—
312C22⋊C4—3881no—
313C22⋊C4—3881no—
314C22⋊C4—3881no—
315C22⋊C4—3881no—
316C22⋊C4—3881no—
317C22⋊C4—3881no—
318C22⋊C4—1942no—
319C22⋊C4—1942no—
320C22⋊C4—1942no—
321C22⋊C4—1942no—
322C22⋊C4C4×D9714yes—
323C42—41no—
324C42—3881no—
325C42—3881no—
326C42—3881no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—972no—
334C23—972no—
335C23—972no—
336C23—972no—
337C23—3884no—
338C23—3884no—
339C23—3884no—
340C23—3884no—
341C23—3884no—
342C23—3884no—
343C23—1948no—
344C23—1941no—
345C23—1941no—
346C23—1941no—
347C23—1941no—
348C23—1941no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23C2×C4×D9712yes—
355C23C23×D9711yescharacteristic
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—972no—
373C2×C4—972no—
374C2×C4—3884no—
375C2×C4—3884no—
376C2×C4—1941no—
377C2×C4—1941no—
378C2×C4—1941no—
379C2×C4—1941no—
380C2×C4—1941no—
381C2×C4—1941no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4—1941no—
385C2×C4—1941no—
386C2×C4—1941no—
387C2×C4—1941no—
388C2×C4—1941no—
389C2×C4—1941no—
390C2×C4—1941no—
391C2×C4—1941no—
392C2×C4C2×C4×D9712yes—
393C2×C4C2×C4×D9712yes—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—974no—
402C22—38816no—
403C22—3881no—
404C22—3881no—
405C22—3881no—
406C22—3881no—
407C22—1942no—
408C22—1942no—
409C22—1942no—
410C22—1942no—
411C22—1942no—
412C22—1942no—
413C22—1942no—
414C22—1942no—
415C22—1942no—
416C22C22×C4×D9711yescharacteristic
417C22#41×D9711yescharacteristic
418C22#41×D9711yescharacteristic
419C22#1569085511yescharacteristic
420C22#1569085511yescharacteristic
421C22#1569085611yescharacteristic
422C22#1569085611yescharacteristic
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—3881no—
430C4—3881no—
431C4—3881no—
432C4—3881no—
433C4—1942no—
434C4—1942no—
435C2—42no—
436C2—3882no—
437C2—1944no—
438C2#117×D9711yescharacteristic
439C2#2026120711yescharacteristic
440C2#2026120711yescharacteristic
441C2#2026131711yescharacteristic
442C2#2026131911yescharacteristic
443C2#2026132111yescharacteristic
444C2#2026132411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenter#388D97
hypocenterC97#2312
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20260629C2semidirect
3. #20260628C2semidirect
4. #20260627C2semidirect
5. #20260626C2semidirect
6. #20260625C2semidirect
7. #20260624C2semidirect
8. #20260623C2semidirect
9. #20260594C2semidirect
10. #20260592C2semidirect
13. C4×#13095488C2semidirect
14. #20260582C2semidirect
15. #388×C97C2semidirect
19. C2×C4⋊Dic97C22semidirect
20. C2×#13095490C22semidirect
21. C2×#13095490C22semidirect
35. C22⋊C4×D97C4semidirect
36. C2×C4×Dic97C22semidirect
37. C2×C4×Dic97C22semidirect
38. C2×C4×Dic97C22semidirect
39. #15690717C22semidirect
40. #15690717C22semidirect
41. #15690717C22semidirect
42. #15690717C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. C2×C4⋊C4×C97C22semidirect
49. C2×C42×C97C22semidirect
50. #95×C97C22semidirect
51. #95×C97C22semidirect
223. C97#2312semidirect
232. #388D97semidirect
245. C2×C4⋊C4C2×D97semidirect
266. C2×C42C2×D97semidirect
273. #95C2×D97semidirect
274. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D972
#202606292
#202606282
#202606272
#202606262
#202606252
#202606242
#202606232
#202605942
#202605922
#202605902
C4×#130954882
#202605822
#388×C972
#231297