Finite Group Databasebeta

← #27144435

Subgroups — #27144435

Order 12416.

696 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×#15690785C211yesmaximal, characteristic
2C2×C4×D388C211yesmaximal, characteristic, complemented
3C2×#15690772C211yesmaximal, characteristic, complemented
4#20260626C211yesmaximal, characteristic, complemented
5#20260624C211yesmaximal, characteristic, complemented
6#20260621C211yesmaximal, characteristic, complemented
7#20260613C211yesmaximal, characteristic, complemented
8#20260613C211yesmaximal, characteristic, complemented
9#20260613C211yesmaximal, characteristic, complemented
10#20260613C211yesmaximal, characteristic, complemented
11#20260609C211yesmaximal, characteristic, complemented
12#20260606C211yesmaximal, characteristic, complemented
13#20260594C211yesmaximal, characteristic, complemented
14#95×D97C211yesmaximal, characteristic, complemented
15#387×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic
24#15690785C418yescomplemented
25C4×D388—24no—
26#15690772—24no—
27C2×C4⋊Dic97C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×#13095490C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C2×#13095488C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C4⋊C4×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C23×D97C2311yescharacteristic
56C23×D97C2311yescharacteristic
57C2×D388—22no—
58C2×D388—22no—
59C2×D388—22no—
60C2×D388—22no—
61C2×D388—41no—
62C2×D388—41no—
63C2×D388C2×C414yescomplemented
64C2×D388C2×C414yescomplemented
65C2×D388C2×C414yescomplemented
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C2×C4×D97—22no—
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97—41no—
75C2×C4×D97—41no—
76C2×C4×D97—41no—
77C2×C4×D97—41no—
78C2×C4×D97—41no—
79C2×C4×D97—41no—
80C2×C4×D97—41no—
81C2×C4×D97—41no—
82C2×C4×D97—41no—
83C2×C4×D97—41no—
84C2×C4×D97C2×C414yescomplemented
85C2×C4×D97D414yescomplemented
86C4⋊Dic97—22no—
87#13095490—41no—
88#13095490—41no—
89#13095488—22no—
90#13095488—22no—
91#13095488—22no—
92#13095488—22no—
93#13095488—41no—
94#13095488—41no—
95#13095488—41no—
96#13095488—41no—
97#13095488C2×C414yescomplemented
98#13095488C2×C414yescomplemented
99C4×Dic97—22no—
100C22×C4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107C4⋊C4×C97—22no—
108C4⋊C4×C97C2×C414yescomplemented
109C42×C97—22no—
110C23×C97C2411yescharacteristic
111C22×D97—21no—
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—21no—
123C22×D97—21no—
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97C22×C412yes—
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×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97C22×C412yes—
150C2×C4×C97C22×C412yes—
151C2×C4×C97C22×C412yes—
152C2×C4×C97C2×D412yes—
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×Dic97C22×C412yes—
167C2×Dic97C2×D412yes—
168D388—28no—
169D388—28no—
170D388—44no—
171C4×D97—44no—
172C4×D97—44no—
173C4×D97—44no—
174C4×D97—44no—
175C4×D97—44no—
176C22×C97C23×C411yescharacteristic
177C22×C97C22×D411yescharacteristic
178C22×C97C2×#4111yescharacteristic
179C22×C97H(5,𝔽2)11yescharacteristic
180C22×C97H(5,𝔽2)11yescharacteristic
181C22×C97H(5,𝔽2)11yescharacteristic
182C22×C972−1+411yescharacteristic
183C2×D97—22no—
184C2×D97—22no—
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—22no—
195C2×D97—41no—
196C2×D97—41no—
197C2×D97—41no—
198C2×D97—41no—
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—22no—
208C4×C97—41no—
209C4×C97—41no—
210C4×C97—41no—
211Dic97—22no—
212Dic97—22no—
213Dic97—41no—
214Dic97—41no—
215Dic97—41no—
216C2×C97C2×C4×D411yescharacteristic
217C2×C97#51811yescharacteristic
218C2×C97#52011yescharacteristic
219C2×C97#53411yescharacteristic
220C2×C97#53711yescharacteristic
221C2×C97#53811yescharacteristic
222C2×C97#53911yescharacteristic
223D97—24no—
224D97—42no—
225D97—42no—
226#2325—971nomaximal, Sylow
227C97#232511yescharacteristic, complemented, Sylow
228C2×#121—971no—
229C2×C4×D4—971no—
230C2×C4×D4—971no—
231#388—971no—
232#388—971no—
233#387—971no—
234#387D9711yescharacteristic, complemented
235#386—971no—
236#386—971no—
237#386—971no—
238#386—971no—
239#385—971no—
240#385—971no—
241#380—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⋊C4—971no—
257C2×C4⋊C4C2×D9711yescharacteristic
258C2×C4⋊C4C2×D9711yescharacteristic
259C2×C22⋊C4—971no—
260C2×C22⋊C4—971no—
261C2×C22⋊C4—971no—
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×C42—971no—
270C2×C42C2×D9711yescharacteristic
271#95—971no—
272#95—971no—
273#95—971no—
274#95—971no—
275#95—971no—
276#95—971no—
277#95C2×D9711yescharacteristic
278#95C2×D9711yescharacteristic
279#95C2×D9711yescharacteristic
280#95C2×D9711yescharacteristic
281C24—971no—
282C24—971no—
283C24—971no—
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—974no—
296C22×C4—974no—
297C22×C4—971no—
298C22×C4—971no—
299C22×C4—971no—
300C22×C4—3881no—
301C22×C4—3881no—
302C22×C4—3881no—
303C22×C4—3881no—
304C22×C4—3881no—
305C22×C4—3881no—
306C22×C4—3881no—
307C22×C4—3881no—
308C22×C4—3881no—
309C22×C4—3881no—
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
319C22×C4C22×D9711yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—3881no—
322C4⋊C4—3881no—
323C4⋊C4—1942no—
324C4⋊C4C4×D9714yescomplemented
325C22⋊C4—974no—
326C22⋊C4—974no—
327C22⋊C4—3881no—
328C22⋊C4—3881no—
329C22⋊C4—3881no—
330C22⋊C4—3881no—
331C22⋊C4—1942no—
332C22⋊C4—1942no—
333C22⋊C4—1942no—
334C22⋊C4—1942no—
335C42—22no—
336C42—1942no—
337C23—972no—
338C23—972no—
339C23—972no—
340C23—972no—
341C23—972no—
342C23—1941no—
343C23—1941no—
344C23—1941no—
345C23—1941no—
346C23—1941no—
347C23—1941no—
348C23—1941no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23—1941no—
355C23—1941no—
356C23—1941no—
357C23—1941no—
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—21no—
378C2×C4—21no—
379C2×C4—972no—
380C2×C4—972no—
381C2×C4—3884no—
382C2×C4—3884no—
383C2×C4—3884no—
384C2×C4—3884no—
385C2×C4—3884no—
386C2×C4—1941no—
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×C4C2×C4×D9712yes—
400C2×C4C2×C4×D9712yes—
401C2×C4C2×C4×D9712yes—
402C2×C4D4×D9712yes—
403C22—974no—
404C22—3881no—
405C22—3881no—
406C22—3881no—
407C22—3881no—
408C22—3881no—
409C22—3881no—
410C22—3881no—
411C22—3881no—
412C22—1942no—
413C22—1942no—
414C22—1942no—
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#1569085611yescharacteristic
428C22#1569085611yescharacteristic
429C22#1569085611yescharacteristic
430C22#1569085811yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—3881no—
439C4—3881no—
440C4—3881no—
441C4—1942no—
442C4—1942no—
443C2—3882no—
444C2—3882no—
445C2—1944no—
446C2C4×D4×D9711yescharacteristic
447C2#2026120911yescharacteristic
448C2#2026121711yescharacteristic
449C2#2026129611yescharacteristic
450C2#2026131511yescharacteristic
451C2#2026132411yescharacteristic
452C2#2026133311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#387×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690785C2
hypercenter#387D97
hypocenterC97#2325
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×D388C2semidirect
3. C2×#15690772C2semidirect
4. #20260626C2semidirect
5. #20260624C2semidirect
6. #20260621C2semidirect
7. #20260613C2semidirect
8. #20260613C2semidirect
9. #20260613C2semidirect
10. #20260613C2semidirect
11. #20260609C2semidirect
12. #20260606C2semidirect
13. #20260594C2semidirect
14. #95×D97C2semidirect
15. #387×C97C2semidirect
24. #15690785C4semidirect
63. C2×D388C2×C4semidirect
64. C2×D388C2×C4semidirect
65. C2×D388C2×C4semidirect
84. C2×C4×D97C2×C4semidirect
85. C2×C4×D97D4semidirect
97. #13095488C2×C4semidirect
98. #13095488C2×C4semidirect
108. C4⋊C4×C97C2×C4semidirect
227. C97#2325semidirect
234. #387D97semidirect
324. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907852
C2×C4×D3882
C2×#156907722
#202606262
#202606242
#202606212
#202606132
#202606092
#202606062
#202605942
#95×D972
#387×C972
#232597