Finite Group Databasebeta

← #27144780

Subgroups — #27144780

Order 12416.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690783C211yesmaximal, characteristic
2C2×#15690772C211yesmaximal, characteristic, complemented
3C2×#15690770C211yesmaximal, characteristic, complemented
4#20260624C211yesmaximal, characteristic, complemented
5#20260617C211yesmaximal, characteristic, complemented
6#20260616C211yesmaximal, characteristic, complemented
7#20260613C211yesmaximal, characteristic, complemented
8#20260612C211yesmaximal, characteristic, complemented
9#20260611C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
12C4×#13095488C211yesmaximal, characteristic, complemented
13C4×#13095488C211yesmaximal, characteristic, complemented
14#95×D97C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic
24#15690783C418yescomplemented
25#15690772—24no—
26#15690770—24no—
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38C2×C4×Dic97C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C42×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C23×D97C2311yescharacteristic
56C2×D388—22no—
57C2×D388—22no—
58C2×D388C2×C414yescomplemented
59C2×C22⋊D97—22no—
60C2×C22⋊D97—22no—
61C2×C22⋊D97—41no—
62C2×C22⋊D97—41no—
63C2×C22⋊D97C2×C414yescomplemented
64C2×C22⋊D97C2×C414yescomplemented
65C22×Dic97—22no—
66C22×Dic97—22no—
67C22×Dic97—41no—
68C22×Dic97C2311yescharacteristic
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C22×Dic97C2311yescharacteristic
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C2×C4×D97—22no—
75C2×C4×D97—22no—
76C2×C4×D97—22no—
77C2×C4×D97—22no—
78C2×C4×D97—22no—
79C2×C4×D97—22no—
80C2×C4×D97—41no—
81C2×C4×D97—41no—
82C2×C4×D97—41no—
83C2×C4×D97—41no—
84C2×C4×D97—41no—
85C2×C4×D97C2×C414yescomplemented
86#13095490—22no—
87#13095490C2×C414yescomplemented
88C22⋊Dic97—22no—
89#13095488—22no—
90#13095488—22no—
91#13095488—22no—
92#13095488—41no—
93#13095488C2×C414yescomplemented
94#13095488D414yescomplemented
95C4×Dic97—22no—
96C4×Dic97—22no—
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—22no—
106C22⋊C4×C97—22no—
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×D97C22×C412yes—
129C22×D97C22×C412yes—
130C22×D97C2×D412yes—
131C22×D97#4112yes—
132C2×C4×C97—21no—
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×C97C22×C412yes—
148C2×C4×C97C22×C412yes—
149C2×C4×C97C2×D412yes—
150C22⋊D97—28no—
151C22⋊D97—44no—
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—44no—
166C2×Dic97C22×C412yes—
167C2×Dic97C22×C412yes—
168C2×Dic97C2×D412yes—
169C2×Dic97#4112yes—
170C2×Dic97#4112yes—
171D388—28no—
172C4×D97—44no—
173C4×D97—44no—
174C4×D97—44no—
175C4×D97—44no—
176C22×C97—22no—
177C22×C97—22no—
178C22×C97—22no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97C23×C411yescharacteristic
184C22×C97C22×D411yescharacteristic
185C22×C97C2×#4111yescharacteristic
186C22×C97C2×#4111yescharacteristic
187C22×C97C2×#4111yescharacteristic
188C22×C97H(5,𝔽2)11yescharacteristic
189C22×C97H(5,𝔽2)11yescharacteristic
190C2×D97—22no—
191C2×D97—22no—
192C2×D97—22no—
193C2×D97—22no—
194C2×D97—22no—
195C2×D97—22no—
196C2×D97—22no—
197C2×D97—22no—
198C2×D97—22no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C2×D97C4×D414yes—
204C4×C97—22no—
205C4×C97—22no—
206C4×C97—22no—
207C4×C97—41no—
208C4×C97—41no—
209C4×C97—41no—
210Dic97—22no—
211Dic97—22no—
212Dic97—22no—
213Dic97—22no—
214Dic97—22no—
215Dic97—41no—
216C2×C97—42no—
217C2×C97C2×C4×D411yescharacteristic
218C2×C97C4×#4111yescharacteristic
219C2×C97#51811yescharacteristic
220C2×C97#54611yescharacteristic
221C2×C97#54611yescharacteristic
222C2×C97#55311yescharacteristic
223C2×C97#55311yescharacteristic
224D97—24no—
225D97—42no—
226#2364—971nomaximal, Sylow
227C97#236411yescharacteristic, complemented, Sylow
228C2×#121—971no—
229C2×C4×D4—971no—
230C2×C4×D4—971no—
231#388—971no—
232#388D9711yescharacteristic, complemented
233#387—971no—
234#387—971no—
235#386—971no—
236#386—971no—
237#386—971no—
238#386—971no—
239C4×C22⋊C4—971no—
240C4×C22⋊C4—971no—
241C4×C22⋊C4—971no—
242C2×#95—971no—
243C22×D4—971no—
244C22×D4—971no—
245C22×D4—971no—
246C23×C4—971no—
247C23×C4—971no—
248C23×C4—971no—
249C23×C4—971no—
250C23×C4—971no—
251#121—978no—
252C4×D4—1944no—
253C4×D4—1944no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4C2×D9711yescharacteristic
257C2×C22⋊C4—971no—
258C2×C22⋊C4—971no—
259C2×C22⋊C4—971no—
260C2×C22⋊C4—971no—
261C2×C22⋊C4—971no—
262C2×C22⋊C4—971no—
263C2×C22⋊C4—971no—
264C2×C22⋊C4C2×D9711yescharacteristic
265C2×C22⋊C4C2×D9711yescharacteristic
266C2×C22⋊C4C2×D9711yescharacteristic
267C2×C42—971no—
268C2×C42—971no—
269C2×C42—971no—
270C2×C42C2×D9711yescharacteristic
271#95—971no—
272#95—971no—
273#95—971no—
274#95—971no—
275#95—971no—
276#95—971no—
277#95—971no—
278#95—971no—
279#95C2×D9711yescharacteristic
280#95C2×D9711yescharacteristic
281C24—971no—
282C24—971no—
283C24C22×D9711yescharacteristic
284C2×D4—974no—
285C2×D4—974no—
286C2×D4—974no—
287C2×D4—3881no—
288C2×D4—3881no—
289C2×D4—1942no—
290C2×D4—1942no—
291C2×D4—1942no—
292C2×D4—1942no—
293C22×C4—971no—
294C22×C4—971no—
295C22×C4—971no—
296C22×C4—971no—
297C22×C4—971no—
298C22×C4—974no—
299C22×C4—971no—
300C22×C4—3881no—
301C22×C4—3881no—
302C22×C4—3881no—
303C22×C4—3881no—
304C22×C4—3881no—
305C22×C4—3881no—
306C22×C4—1942no—
307C22×C4—1942no—
308C22×C4—1942no—
309C22×C4—1942no—
310C22×C4—1942no—
311C22×C4—1942no—
312C22×C4—1942no—
313C22×C4—1942no—
314C22×C4C22×D9711yescharacteristic
315C22×C4C22×D9711yescharacteristic
316C22×C4C22×D9711yescharacteristic
317C22×C4C22×D9711yescharacteristic
318C22×C4C22×D9711yescharacteristic
319C22×C4C22×D9711yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—974no—
322C4⋊C4—1942no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—974no—
326C22⋊C4—974no—
327C22⋊C4—3881no—
328C22⋊C4—1942no—
329C22⋊C4—1942no—
330C22⋊C4—1942no—
331C22⋊C4—1942no—
332C22⋊C4C4×D9714yescomplemented
333C42—41no—
334C42—3881no—
335C42—1942no—
336C42—1942no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—972no—
344C23—972no—
345C23—972no—
346C23—972no—
347C23—1941no—
348C23—1941no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23—1941no—
355C23—1941no—
356C23—1941no—
357C23C2×C4×D9712yes—
358C23C23×D9711yescharacteristic
359D4—3884no—
360D4—1948no—
361D4—1948no—
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—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—972no—
378C2×C4—972no—
379C2×C4—972no—
380C2×C4—972no—
381C2×C4—972no—
382C2×C4—3884no—
383C2×C4—3884no—
384C2×C4—3884no—
385C2×C4—3884no—
386C2×C4—3884no—
387C2×C4—1941no—
388C2×C4—1941no—
389C2×C4—1941no—
390C2×C4—1941no—
391C2×C4—1941no—
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×C4C2×C4×D9712yes—
401C2×C4C2×C4×D9712yes—
402C2×C4D4×D9712yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—974no—
411C22—3881no—
412C22—3881no—
413C22—3881no—
414C22—3881no—
415C22—1942no—
416C22—1942no—
417C22—1942no—
418C22—1942no—
419C22—1942no—
420C22—1942no—
421C22—1942no—
422C22—1942no—
423C22—1942no—
424C22C22×C4×D9711yescharacteristic
425C22C2×D4×D9711yescharacteristic
426C22#41×D9711yescharacteristic
427C22#41×D9711yescharacteristic
428C22#41×D9711yescharacteristic
429C22#1569085511yescharacteristic
430C22#1569085611yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—3881no—
438C4—1942no—
439C4—1942no—
440C4—1942no—
441C4—1942no—
442C4—1942no—
443C2—42no—
444C2—3882no—
445C2—1944no—
446C2C4×D4×D9711yescharacteristic
447C2#2026120311yescharacteristic
448C2#2026120711yescharacteristic
449C2#2026137411yescharacteristic
450C2#2026137811yescharacteristic
451C2#2026143011yescharacteristic
452C2#2026143611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690783C2
hypercenter#388D97
hypocenterC97#2364
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#15690772C2semidirect
3. C2×#15690770C2semidirect
4. #20260624C2semidirect
5. #20260617C2semidirect
6. #20260616C2semidirect
7. #20260613C2semidirect
8. #20260612C2semidirect
9. #20260611C2semidirect
10. #20260611C2semidirect
11. C22⋊C4×Dic97C2semidirect
12. C4×#13095488C2semidirect
13. C4×#13095488C2semidirect
14. #95×D97C2semidirect
15. #388×C97C2semidirect
24. #15690783C4semidirect
58. C2×D388C2×C4semidirect
63. C2×C22⋊D97C2×C4semidirect
64. C2×C22⋊D97C2×C4semidirect
85. C2×C4×D97C2×C4semidirect
87. #13095490C2×C4semidirect
93. #13095488C2×C4semidirect
94. #13095488D4semidirect
108. C22⋊C4×C97C2×C4semidirect
227. C97#2364semidirect
232. #388D97semidirect
332. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907832
C2×#156907722
C2×#156907702
#202606242
#202606172
#202606162
#202606132
#202606122
#202606112
C22⋊C4×Dic972
C4×#130954882
#95×D972
#388×C972
#236497