Finite Group Databasebeta

← #27155355

Subgroups — #27155355

Order 12416.

1094 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261463C211yesmaximal, characteristic, complemented
2#20261453C212yesmaximal, complemented
3#20261426C212yesmaximal, complemented
4#20261421C214yesmaximal, complemented
5#20261400C211yesmaximal, characteristic, complemented
6#20261393C212yesmaximal, complemented
7#20261390C211yesmaximal, characteristic, complemented
8#20261387C212yesmaximal, complemented
9#20261290C211yesmaximal, characteristic, complemented
10#20261287C212yesmaximal, complemented
11#20261268C214yesmaximal, complemented
12#20261264C212yesmaximal, complemented
13C4×#13095512C211yesmaximal, characteristic, complemented
14#20261203C212yesmaximal, complemented
15C4×#13095510C211yesmaximal, characteristic, complemented
16C4×Q8×D97C212yesmaximal, complemented
17#556×C97C211yesmaximal, characteristic, complemented
18C2×#13095512C2211yescharacteristic
19C2×#13095511C2212yes—
20C2×#13095510C2211yescharacteristic
21C2×Q8×D97C2212yes—
22#15690810C2211yescharacteristic
23#15690809C2212yes—
24#15690808C2212yes—
25#15690804C2214yes—
26#15690803C2214yes—
27#15690802C2212yes—
28#15690801C2212yes—
29#15690800C2214yes—
30#15690800C2214yes—
31#15690796C2214yes—
32#15690795C2214yes—
33#15690792C2214yes—
34#15690792C2214yes—
35#15690790C2212yes—
36#15690789C2212yes—
37#15690788C2214yes—
38#15690787C2214yes—
39#15690786C2212yes—
40#15690786C2212yes—
41#15690783C2214yes—
42#15690781C2212yes—
43Q8×Dic97C2212yes—
44Q8×Dic97C2211yescharacteristic
45C4×Dic194C2212yes—
46C4×Dic194C2211yescharacteristic
47C4×Dic194C2214yes—
48#15690775C2212yes—
49#15690775C2211yescharacteristic
50#15690775C2214yes—
51#15690775C2212yes—
52C4×D388C2211yescharacteristic
53C4×D388C2212yes—
54D4×Dic97C2211yescharacteristic
55#15690772C2212yes—
56#15690772C2211yescharacteristic
57#15690772C2212yes—
58C4×C22⋊D97C2212yes—
59#15690770C2214yes—
60#15690770C2212yes—
61#15690770C2214yes—
62#15690768C2212yes—
63#15690768C2211yescharacteristic
64#15690768C2211yescharacteristic
65#15690768C2212yes—
66#15690767C2214yes—
67#15690766C2212yes—
68#15690766C2214yes—
69C4⋊C4×D97C2214yes—
70C4⋊C4×D97C2212yes—
71C2×C4×Dic97C2212yes—
72C42×D97C2214yes—
73C42×D97C2212yes—
74C42×D97C2211yescharacteristic
75C4⋊Q8×C97C2211yescharacteristic
76#126×C97C2214yes—
77#124×C97C2212yes—
78C22⋊Q8×C97C2212yes—
79C4×Q8×C97C2212yes—
80C4×Q8×C97C2211yescharacteristic
81C4×D4×C97C2211yescharacteristic
82#117×C97C2212yes—
83#13095512—24no—
84#13095511—28no—
85#13095510—24no—
86C2×Dic194C2312yes—
87C2×Dic194C2314yes—
88C2×Dic194C2311yescharacteristic
89Q8×D97—28no—
90C2×D388C2312yes—
91C2×D388C2311yescharacteristic
92C2×C22⋊D97C2314yes—
93C2×C22⋊D97C2312yes—
94C22×Dic97C2312yes—
95C22×Dic97C2312yes—
96C2×C4×D97C2312yes—
97C2×C4×D97C2314yes—
98C2×C4×D97C2312yes—
99C2×C4×D97C2314yes—
100C2×C4×D97C2312yes—
101C2×C4×D97C2311yescharacteristic
102C2×C4×D97C2311yescharacteristic
103C2×C4×D97C2311yescharacteristic
104C4⋊Dic97C2312yes—
105C4⋊Dic97C2314yes—
106C4⋊Dic97C2311yescharacteristic
107C4⋊Dic97C2312yes—
108C4⋊Dic97C2311yescharacteristic
109C4⋊Dic97C2312yes—
110#13095490C2312yes—
111#13095490C2314yes—
112#13095490C2314yes—
113#13095490C2312yes—
114#13095490C2314yes—
115#13095490C2314yes—
116#13095490C2314yes—
117#13095490C2314yes—
118C22⋊Dic97C2312yes—
119C22⋊Dic97C2314yes—
120#13095488C2312yes—
121#13095488C2314yes—
122#13095488C2312yes—
123#13095488C2312yes—
124#13095488C2312yes—
125#13095488C2314yes—
126C4×Dic97C2311yescharacteristic
127C4×Dic97C2312yes—
128C4×Dic97C2311yescharacteristic
129C4×Dic97C2311yescharacteristic
130C4×Dic97C2312yes—
131C4×Dic97C2314yes—
132C4×Dic97C2312yes—
133C4×Dic97C2314yes—
134C4×Dic97C2312yes—
135C4×Dic97C2312yes—
136C2×Q8×C97C2312yes—
137C2×Q8×C97C2311yescharacteristic
138C2×D4×C97C2311yescharacteristic
139C22×C4×C97C2312yes—
140C4⋊C4×C97C2311yescharacteristic
141C4⋊C4×C97C2314yes—
142C4⋊C4×C97C2311yescharacteristic
143C4⋊C4×C97C2312yes—
144C4⋊C4×C97C2312yes—
145C4⋊C4×C97C2312yes—
146C22⋊C4×C97C2312yes—
147C22⋊C4×C97C2314yes—
148C22⋊C4×C97C2314yes—
149C42×C97C2314yes—
150C42×C97C2312yes—
151C42×C97C2311yescharacteristic
152C23×C97C2412yes—
153C22×D97C2411yescharacteristic
154C22×D97C2412yes—
155Q8×C97—24no—
156Q8×C97—22no—
157D4×C97—22no—
158C2×C4×C97—24no—
159C2×C4×C97C2412yes—
160C2×C4×C97C2412yes—
161C2×C4×C97C2411yescharacteristic
162C2×C4×C97C2414yes—
163C2×C4×C97C2412yes—
164C2×C4×C97C2411yescharacteristic
165C2×C4×C97C2411yescharacteristic
166C22⋊D97—28no—
167C22⋊D97—24no—
168C2×Dic97—24no—
169C2×Dic97—24no—
170C2×Dic97C2411yescharacteristic
171C2×Dic97C2411yescharacteristic
172C2×Dic97C2412yes—
173C2×Dic97C2412yes—
174C2×Dic97C2414yes—
175C2×Dic97C2412yes—
176C2×Dic97C2411yescharacteristic
177D388—24no—
178D388#4114yescomplemented
179C4×D97—28no—
180C4×D97—24no—
181C4×D97—24no—
182C4×D97—22no—
183C4×D97—22no—
184C4×D97—28no—
185C4×D97—24no—
186C4×D97—22no—
187Dic194—24no—
188Dic194—28no—
189Dic194#4114yescomplemented
190C22×C97—22no—
191C22×C97—22no—
192C22×C97—22no—
193C22×C97C2511yescharacteristic
194C2×D97—22no—
195C2×D97—22no—
196C2×D97—21no—
197C2×D97—21no—
198C2×D97—21no—
199C2×D97C2×#4114yes—
200C4×C97—24no—
201C4×C97—22no—
202C4×C97—22no—
203C4×C97—22no—
204C4×C97—21no—
205C4×C97C2×#4112yes—
206C4×C97C2×#4112yes—
207Dic97—22no—
208Dic97—24no—
209Dic97—21no—
210Dic97—21no—
211Dic97—22no—
212Dic97C2×#4114yes—
213Dic972−1+412yes—
214C2×C97—42no—
215C2×C97C22×#4111yescharacteristic
216C2×C97C22×#4111yescharacteristic
217C2×C97C2×2−1+411yescharacteristic
218D97—24no—
219D97—41no—
220#3482—971nomaximal, Sylow
221C97#348211yescharacteristic, complemented, Sylow
222#557—971no—
223#556—972no—
224#556D9711yescharacteristic, complemented
225#552—974no—
226#552—972no—
227D4×Q8—971no—
228#548—972no—
229#547—972no—
230#547—971no—
231#533—971no—
232#533—972no—
233#529—974no—
234#529—972no—
235C4×#41—971no—
236C4×#41—971no—
237C4×#41—972no—
238C2×C4×Q8—972no—
239C2×#41—971no—
240C2×#41—971no—
241C2×#41—972no—
242C22×Q8—972no—
243C4⋊Q8—972no—
244C4⋊Q8—972no—
245C4⋊Q8—971no—
246C4⋊Q8C2×D9711yescharacteristic
247#126—974no—
248#126—974no—
249#126C2×D9714yes—
250#125—974no—
251#125—974no—
252#125—972no—
253#125—972no—
254#124—974no—
255#124C2×D9712yes—
256#123—974no—
257#123—974no—
258#123—974no—
259C22⋊Q8—972no—
260C22⋊Q8—972no—
261C22⋊Q8—972no—
262C22⋊Q8—974no—
263C22⋊Q8—972no—
264C22⋊Q8—974no—
265C22⋊Q8C2×D9712yes—
266#121—972no—
267#121—974no—
268C4×Q8—972no—
269C4×Q8—971no—
270C4×Q8—971no—
271C4×Q8—974no—
272C4×Q8—972no—
273C4×Q8—972no—
274C4×Q8—972no—
275C4×Q8—974no—
276C4×Q8—971no—
277C4×Q8C2×D9712yes—
278C4×Q8C2×D9711yescharacteristic
279C4×D4—971no—
280C4×D4—972no—
281C4×D4—971no—
282C4×D4—972no—
283C4×D4—972no—
284C4×D4—972no—
285C4×D4—974no—
286C4×D4—974no—
287C4×D4—971no—
288C4×D4—972no—
289C4×D4C2×D9711yescharacteristic
290#117—971no—
291#117—972no—
292#117—971no—
293#117—972no—
294#117—972no—
295#117—974no—
296#117—974no—
297#117C2×D9712yes—
298C2×C4⋊C4—974no—
299C2×C4⋊C4—972no—
300C2×C42—972no—
301C2×C42—972no—
302C2×C42—974no—
303C2×C42—971no—
304#41—1948no—
305#41—1944no—
306#41—1944no—
307C2×Q8—971no—
308C2×Q8—974no—
309C2×Q8—972no—
310C2×Q8—1948no—
311C2×Q8C22×D9712yes—
312C2×Q8C22×D9711yescharacteristic
313C2×D4—974no—
314C2×D4—972no—
315C2×D4—971no—
316C2×D4—972no—
317C2×D4C22×D9711yescharacteristic
318C22×C4—972no—
319C22×C4—972no—
320C22×C4—972no—
321C22×C4—972no—
322C22×C4—974no—
323C22×C4—974no—
324C22×C4—972no—
325C22×C4—971no—
326C22×C4—971no—
327C22×C4—971no—
328C22×C4C22×D9712yes—
329C4⋊C4—972no—
330C4⋊C4—974no—
331C4⋊C4—974no—
332C4⋊C4—972no—
333C4⋊C4—974no—
334C4⋊C4—974no—
335C4⋊C4—974no—
336C4⋊C4—974no—
337C4⋊C4—972no—
338C4⋊C4—974no—
339C4⋊C4—972no—
340C4⋊C4—971no—
341C4⋊C4—971no—
342C4⋊C4—972no—
343C4⋊C4C22×D9711yescharacteristic
344C4⋊C4C22×D9714yes—
345C4⋊C4C22×D9711yescharacteristic
346C4⋊C4C22×D9712yes—
347C4⋊C4C22×D9712yes—
348C4⋊C4C22×D9712yes—
349C22⋊C4—972no—
350C22⋊C4—974no—
351C22⋊C4—972no—
352C22⋊C4—974no—
353C22⋊C4—972no—
354C22⋊C4—972no—
355C22⋊C4—972no—
356C22⋊C4—974no—
357C22⋊C4C22×D9712yes—
358C22⋊C4C22×D9714yes—
359C22⋊C4C22×D9714yes—
360C42—972no—
361C42—971no—
362C42—971no—
363C42—971no—
364C42—972no—
365C42—972no—
366C42—974no—
367C42—972no—
368C42—972no—
369C42—974no—
370C42C22×D9714yes—
371C42C22×D9711yescharacteristic
372C42C22×D9712yes—
373C23—972no—
374C23—971no—
375C23C23×D9712yes—
376Q8—24no—
377Q8—22no—
378Q8—974no—
379Q8—1948no—
380Q8—1944no—
381D4—22no—
382D4—974no—
383D4—1944no—
384D4—1944no—
385D4—1948no—
386C2×C4—24no—
387C2×C4—971no—
388C2×C4—972no—
389C2×C4—971no—
390C2×C4—971no—
391C2×C4—972no—
392C2×C4—972no—
393C2×C4—974no—
394C2×C4—1948no—
395C2×C4—1942no—
396C2×C4—1944no—
397C2×C4—1948no—
398C2×C4—1944no—
399C2×C4—1944no—
400C2×C4—1944no—
401C2×C4—1944no—
402C2×C4—1942no—
403C2×C4—1942no—
404C2×C4C23×D9712yes—
405C2×C4C23×D9712yes—
406C2×C4C23×D9711yescharacteristic
407C2×C4C23×D9714yes—
408C2×C4C23×D9712yes—
409C2×C4C23×D9711yescharacteristic
410C2×C4C23×D9711yescharacteristic
411C22—22no—
412C22—22no—
413C22—22no—
414C22—974no—
415C22—1941no—
416C22—1941no—
417C22—1941no—
418C22—1942no—
419C22—1942no—
420C22C24×D9711yescharacteristic
421C4—22no—
422C4—22no—
423C4—24no—
424C4—22no—
425C4—21no—
426C4—974no—
427C4—972no—
428C4—1941no—
429C4—1941no—
430C4—1944no—
431C4—1942no—
432C4—1942no—
433C4#41×D9712yes—
434C4#41×D9712yes—
435C2—42no—
436C2—3881no—
437C2—1944no—
438C2C2×#41×D9711yescharacteristic
439C2C2×#41×D9711yescharacteristic
440C22−1+4×D9711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#556×C97C2
hypercenter#556D97
hypocenterC97#3482
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20261463C2semidirect
2. #20261453C2semidirect
3. #20261426C2semidirect
4. #20261421C2semidirect
5. #20261400C2semidirect
6. #20261393C2semidirect
7. #20261390C2semidirect
8. #20261387C2semidirect
9. #20261290C2semidirect
10. #20261287C2semidirect
11. #20261268C2semidirect
12. #20261264C2semidirect
13. C4×#13095512C2semidirect
14. #20261203C2semidirect
15. C4×#13095510C2semidirect
16. C4×Q8×D97C2semidirect
17. #556×C97C2semidirect
178. D388#41semidirect
189. Dic194#41semidirect
221. C97#3482semidirect
224. #556D97semidirect

Maximal subgroups

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

subgroupindex
#202614632
#202614532
#202614262
#202614212
#202614002
#202613932
#202613902
#202613872
#202612902
#202612872
#202612682
#202612642
C4×#130955122
#202612032
C4×#130955102
C4×Q8×D972
#556×C972
#348297