Finite Group Databasebeta

← #27155328

Subgroups — #27155328

Order 12416.

1112 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261470C211yesmaximal, characteristic, complemented
2#20261467C212yesmaximal, complemented
3#20261460C211yesmaximal, characteristic, complemented
4#20261457C212yesmaximal, complemented
5#20261436C214yesmaximal, complemented
6#20261429C212yesmaximal, complemented
7#20261406C211yesmaximal, characteristic, complemented
8#20261395C212yesmaximal, complemented
9#20261286C211yesmaximal, characteristic, complemented
10#20261284C212yesmaximal, complemented
11#20261271C212yesmaximal, complemented
12#20261268C214yesmaximal, complemented
13C4×#13095512C212yesmaximal, complemented
14#20261203C212yesmaximal, complemented
15C4×#13095510C211yesmaximal, characteristic, complemented
16C4×Q8×D97C211yesmaximal, characteristic, complemented
17#556×C97C211yesmaximal, characteristic, complemented
18C2×#13095512C2212yes—
19C2×#13095511C2212yes—
20C2×#13095510C2211yescharacteristic
21C2×Q8×D97C2211yescharacteristic
22#15690808C2212yes—
23#15690807C2211yescharacteristic
24#15690806C2212yes—
25#15690805C2214yes—
26#15690804C2214yes—
27#15690801C2212yes—
28#15690800C2214yes—
29#15690799C2212yes—
30#15690799C2212yes—
31#15690798C2212yes—
32#15690796C2214yes—
33#15690795C2214yes—
34#15690794C2214yes—
35#15690792C2214yes—
36#15690788C2214yes—
37#15690785C2212yes—
38#15690785C2212yes—
39#15690784C2212yes—
40#15690783C2214yes—
41#15690783C2214yes—
42#15690783C2214yes—
43Q8×Dic97C2211yescharacteristic
44Q8×Dic97C2212yes—
45C4×Dic194C2211yescharacteristic
46C4×Dic194C2212yes—
47C4×Dic194C2211yescharacteristic
48#15690775C2212yes—
49#15690775C2211yescharacteristic
50#15690775C2212yes—
51#15690775C2211yescharacteristic
52C4×D388C2214yes—
53C4×D388C2212yes—
54D4×Dic97C2211yescharacteristic
55#15690772C2214yes—
56#15690772C2212yes—
57#15690772C2212yes—
58#15690772C2212yes—
59#15690772C2212yes—
60C4×C22⋊D97C2212yes—
61#15690770C2212yes—
62#15690770C2214yes—
63#15690770C2214yes—
64#15690770C2214yes—
65#15690768C2211yescharacteristic
66#15690768C2214yes—
67#15690767C2214yes—
68#15690766C2212yes—
69C4⋊C4×D97C2211yescharacteristic
70C4⋊C4×D97C2212yes—
71C2×C4×Dic97C2212yes—
72C42×D97C2211yescharacteristic
73C42×D97C2212yes—
74C42×D97C2214yes—
75C4⋊Q8×C97C2211yescharacteristic
76#126×C97C2214yes—
77#124×C97C2212yes—
78C22⋊Q8×C97C2212yes—
79C4×Q8×C97C2211yescharacteristic
80C4×Q8×C97C2212yes—
81C4×D4×C97C2211yescharacteristic
82#117×C97C2212yes—
83#13095512—28no—
84#13095511—28no—
85#13095510—24no—
86C2×Dic194C2312yes—
87C2×Dic194C2311yescharacteristic
88C2×Dic194C2311yescharacteristic
89Q8×D97—24no—
90C2×D388C2312yes—
91C2×D388C2314yes—
92C2×D388C2312yes—
93C2×D388C2312yes—
94C2×C22⋊D97C2314yes—
95C2×C22⋊D97C2312yes—
96C2×C22⋊D97C2314yes—
97C22×Dic97C2312yes—
98C22×Dic97C2312yes—
99C2×C4×D97C2311yescharacteristic
100C2×C4×D97C2312yes—
101C2×C4×D97C2314yes—
102C2×C4×D97C2314yes—
103C2×C4×D97C2311yescharacteristic
104C2×C4×D97C2312yes—
105C2×C4×D97C2314yes—
106C2×C4×D97C2311yescharacteristic
107C4⋊Dic97C2311yescharacteristic
108C4⋊Dic97C2312yes—
109C4⋊Dic97C2311yescharacteristic
110C4⋊Dic97C2314yes—
111#13095490C2312yes—
112#13095490C2312yes—
113#13095490C2312yes—
114#13095490C2312yes—
115#13095490C2312yes—
116#13095490C2314yes—
117#13095490C2314yes—
118C22⋊Dic97C2312yes—
119#13095488C2314yes—
120#13095488C2314yes—
121#13095488C2312yes—
122#13095488C2314yes—
123#13095488C2314yes—
124#13095488C2314yes—
125#13095488C2314yes—
126C4×Dic97C2312yes—
127C4×Dic97C2312yes—
128C4×Dic97C2311yescharacteristic
129C4×Dic97C2311yescharacteristic
130C4×Dic97C2312yes—
131C4×Dic97C2314yes—
132C4×Dic97C2312yes—
133C4×Dic97C2311yescharacteristic
134C4×Dic97C2312yes—
135C4×Dic97C2312yes—
136C2×Q8×C97C2311yescharacteristic
137C2×Q8×C97C2312yes—
138C2×D4×C97C2311yescharacteristic
139C22×C4×C97C2312yes—
140C4⋊C4×C97C2311yescharacteristic
141C4⋊C4×C97C2312yes—
142C4⋊C4×C97C2311yescharacteristic
143C4⋊C4×C97C2314yes—
144C4⋊C4×C97C2312yes—
145C4⋊C4×C97C2312yes—
146C22⋊C4×C97C2312yes—
147C22⋊C4×C97C2314yes—
148C22⋊C4×C97C2314yes—
149C42×C97C2314yes—
150C42×C97C2312yes—
151C42×C97C2311yescharacteristic
152C23×C97C2412yes—
153C22×D97C2414yes—
154C22×D97C2411yescharacteristic
155Q8×C97—22no—
156Q8×C97—24no—
157D4×C97—22no—
158C2×C4×C97—24no—
159C2×C4×C97C2412yes—
160C2×C4×C97C2412yes—
161C2×C4×C97C2411yescharacteristic
162C2×C4×C97C2412yes—
163C2×C4×C97C2414yes—
164C2×C4×C97C2411yescharacteristic
165C2×C4×C97C2411yescharacteristic
166C22⋊D97—24no—
167C22⋊D97—28no—
168C22⋊D97—44no—
169C2×Dic97—24no—
170C2×Dic97—42no—
171C2×Dic97C2411yescharacteristic
172C2×Dic97C2412yes—
173C2×Dic97C2411yescharacteristic
174C2×Dic97C2412yes—
175C2×Dic97C2412yes—
176C2×Dic97C2411yescharacteristic
177C2×Dic97C2412yes—
178D388—28no—
179D388—24no—
180D388—24no—
181D388—42no—
182C4×D97—22no—
183C4×D97—24no—
184C4×D97—28no—
185C4×D97—28no—
186C4×D97—44no—
187C4×D97—41no—
188C4×D97—42no—
189C4×D97#4114yescomplemented
190Dic194—22no—
191Dic194—24no—
192Dic194#4114yescomplemented
193C22×C97—22no—
194C22×C97—22no—
195C22×C97—22no—
196C22×C97C2511yescharacteristic
197C2×D97—21no—
198C2×D97—21no—
199C2×D97—24no—
200C2×D97—24no—
201C2×D97—24no—
202C2×D97C2×#4112yes—
203C4×C97—22no—
204C4×C97—22no—
205C4×C97—24no—
206C4×C97—22no—
207C4×C97—21no—
208C4×C97C2×#4112yes—
209C4×C97C2×#4112yes—
210Dic97—21no—
211Dic97—22no—
212Dic97—22no—
213Dic97—21no—
214Dic97—22no—
215Dic97C2×#4114yes—
216Dic97C2×#4112yes—
217C2×C97—42no—
218C2×C97C22×#4111yescharacteristic
219C2×C97C22×#4111yescharacteristic
220C2×C97#58511yescharacteristic
221D97—22no—
222D97—44no—
223#3480—971nomaximal, Sylow
224C97#348011yescharacteristic, complemented, Sylow
225#559—972no—
226#559—971no—
227#557—971no—
228#556—972no—
229#556D9711yescharacteristic, complemented
230#553—974no—
231#553—972no—
232#550—971no—
233#548—972no—
234#532—971no—
235#532—972no—
236#529—974no—
237#529—972no—
238C4×#41—972no—
239C4×#41—971no—
240C4×#41—972no—
241C2×C4×Q8—971no—
242C2×#41—972no—
243C2×#41—971no—
244C2×#41—972no—
245C22×Q8—971no—
246C4⋊Q8—972no—
247C4⋊Q8C2×D9711yescharacteristic
248#127—972no—
249#127—971no—
250#126—974no—
251#126—974no—
252#126C2×D9714yes—
253#125—974no—
254#125—972no—
255#124—972no—
256#124—972no—
257#124—974no—
258#124—972no—
259#124C2×D9712yes—
260#123—974no—
261#123—974no—
262#123—974no—
263C22⋊Q8—974no—
264C22⋊Q8C2×D9712yes—
265#121—972no—
266#121—972no—
267#121—972no—
268#121—974no—
269#121—974no—
270#121—974no—
271C4×Q8—971no—
272C4×Q8—972no—
273C4×Q8—971no—
274C4×Q8—972no—
275C4×Q8—972no—
276C4×Q8—971no—
277C4×Q8—971no—
278C4×Q8—972no—
279C4×Q8—971no—
280C4×Q8C2×D9711yescharacteristic
281C4×Q8C2×D9712yes—
282C4×D4—971no—
283C4×D4—972no—
284C4×D4—972no—
285C4×D4—974no—
286C4×D4—972no—
287C4×D4—972no—
288C4×D4—972no—
289C4×D4—974no—
290C4×D4—972no—
291C4×D4—974no—
292C4×D4—974no—
293C4×D4—974no—
294C4×D4—972no—
295C4×D4C2×D9711yescharacteristic
296#117—971no—
297#117—974no—
298#117—972no—
299#117—974no—
300#117C2×D9712yes—
301C2×C4⋊C4—972no—
302C2×C4⋊C4—971no—
303C2×C42—972no—
304C2×C42—971no—
305C2×C42—972no—
306C2×C42—974no—
307#41—1948no—
308#41—1944no—
309#41—1948no—
310C2×Q8—971no—
311C2×Q8—972no—
312C2×Q8—971no—
313C2×Q8—1944no—
314C2×Q8C22×D9711yescharacteristic
315C2×Q8C22×D9712yes—
316C2×D4—974no—
317C2×D4—972no—
318C2×D4—974no—
319C2×D4—972no—
320C2×D4—974no—
321C2×D4—972no—
322C2×D4—972no—
323C2×D4C22×D9711yescharacteristic
324C22×C4—971no—
325C22×C4—972no—
326C22×C4—972no—
327C22×C4—971no—
328C22×C4—972no—
329C22×C4—974no—
330C22×C4—972no—
331C22×C4—971no—
332C22×C4—974no—
333C22×C4—974no—
334C22×C4C22×D9712yes—
335C4⋊C4—972no—
336C4⋊C4—972no—
337C4⋊C4—972no—
338C4⋊C4—974no—
339C4⋊C4—972no—
340C4⋊C4—972no—
341C4⋊C4—974no—
342C4⋊C4—971no—
343C4⋊C4—972no—
344C4⋊C4—971no—
345C4⋊C4—974no—
346C4⋊C4C22×D9711yescharacteristic
347C4⋊C4C22×D9712yes—
348C4⋊C4C22×D9711yescharacteristic
349C4⋊C4C22×D9714yes—
350C4⋊C4C22×D9712yes—
351C4⋊C4C22×D9712yes—
352C22⋊C4—972no—
353C22⋊C4—972no—
354C22⋊C4—974no—
355C22⋊C4—974no—
356C22⋊C4—974no—
357C22⋊C4—974no—
358C22⋊C4—974no—
359C22⋊C4—974no—
360C22⋊C4C22×D9712yes—
361C22⋊C4C22×D9714yes—
362C22⋊C4C22×D9714yes—
363C42—972no—
364C42—972no—
365C42—971no—
366C42—971no—
367C42—972no—
368C42—972no—
369C42—974no—
370C42—972no—
371C42—971no—
372C42—972no—
373C42C22×D9714yes—
374C42C22×D9712yes—
375C42C22×D9711yescharacteristic
376C23—971no—
377C23—974no—
378C23C23×D9712yes—
379Q8—22no—
380Q8—24no—
381Q8—974no—
382Q8—1944no—
383Q8—1942no—
384D4—22no—
385D4—3882no—
386D4—3884no—
387D4—1948no—
388D4—1948no—
389D4—1944no—
390D4—1944no—
391D4—1944no—
392C2×C4—24no—
393C2×C4—971no—
394C2×C4—972no—
395C2×C4—974no—
396C2×C4—972no—
397C2×C4—971no—
398C2×C4—971no—
399C2×C4—972no—
400C2×C4—972no—
401C2×C4—3884no—
402C2×C4—3882no—
403C2×C4—3881no—
404C2×C4—3882no—
405C2×C4—1944no—
406C2×C4—1948no—
407C2×C4—1948no—
408C2×C4—1944no—
409C2×C4—1942no—
410C2×C4C23×D9712yes—
411C2×C4C23×D9712yes—
412C2×C4C23×D9712yes—
413C2×C4C23×D9714yes—
414C2×C4C23×D9711yescharacteristic
415C2×C4C23×D9711yescharacteristic
416C2×C4C23×D9711yescharacteristic
417C22—22no—
418C22—22no—
419C22—22no—
420C22—972no—
421C22—1944no—
422C22—1944no—
423C22—1944no—
424C22—1941no—
425C22—1941no—
426C22C24×D9711yescharacteristic
427C4—22no—
428C4—24no—
429C4—22no—
430C4—22no—
431C4—21no—
432C4—972no—
433C4—974no—
434C4—1942no—
435C4—1941no—
436C4—1942no—
437C4—1941no—
438C4—1942no—
439C4#41×D9712yes—
440C4#41×D9712yes—
441C2—42no—
442C2—3884no—
443C2—1942no—
444C2C2×#41×D9711yescharacteristic
445C2C2×#41×D9711yescharacteristic
446C2#2026157211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#556×C97C2
hypercenter#556D97
hypocenterC97#3480
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20261470C2semidirect
2. #20261467C2semidirect
3. #20261460C2semidirect
4. #20261457C2semidirect
5. #20261436C2semidirect
6. #20261429C2semidirect
7. #20261406C2semidirect
8. #20261395C2semidirect
9. #20261286C2semidirect
10. #20261284C2semidirect
11. #20261271C2semidirect
12. #20261268C2semidirect
13. C4×#13095512C2semidirect
14. #20261203C2semidirect
15. C4×#13095510C2semidirect
16. C4×Q8×D97C2semidirect
17. #556×C97C2semidirect
189. C4×D97#41semidirect
192. Dic194#41semidirect
224. C97#3480semidirect
229. #556D97semidirect

Maximal subgroups

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

subgroupindex
#202614702
#202614672
#202614602
#202614572
#202614362
#202614292
#202614062
#202613952
#202612862
#202612842
#202612712
#202612682
C4×#130955122
#202612032
C4×#130955102
C4×Q8×D972
#556×C972
#348097