Finite Group Databasebeta

← #27153262

Subgroups — #27153262

Order 12416.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690835C211yesmaximal, characteristic, complemented
2C2×#15690830C211yesmaximal, characteristic, complemented
3#20261381C211yesmaximal, characteristic, complemented
4#20261127C211yesmaximal, characteristic, complemented
5#20260991C211yesmaximal, characteristic, complemented
6#20260966C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260937C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260895C211yesmaximal, characteristic, complemented
11#20260885C211yesmaximal, characteristic, complemented
12#20260880C211yesmaximal, characteristic, complemented
13#20260845C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15#547×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18#15690835—24no—
19#15690830—24no—
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095498C2211yescharacteristic
24C2×#13095498C2211yescharacteristic
25C2×#13095498C2211yescharacteristic
26C2×#13095498C2211yescharacteristic
27C2×#13095496C2211yescharacteristic
28C2×#13095496C2211yescharacteristic
29C2×#13095495C2211yescharacteristic
30C22×#11715895C2211yescharacteristic, complemented
31#15690798C2211yescharacteristic
32#15690795—21no—
33#15690789C2211yescharacteristic
34#15690784C2211yescharacteristic
35#15690783—21no—
36#15690778—21no—
37C4×D388C2211yescharacteristic
38C22⋊C4×D97—21no—
39#15690741C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690734C2211yescharacteristic, complemented
42#15690734C2211yescharacteristic, complemented
43#15690733C2211yescharacteristic, complemented
44#15690731C2211yescharacteristic, complemented
45#15690730C2211yescharacteristic
46#15690730C2211yescharacteristic
47C22⋊#11715895C2211yescharacteristic
48C22⋊#11715895C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50C2×#41×C97C2211yescharacteristic
51C2×#41×C97C2211yescharacteristic, complemented
52C4⋊Q8×C97C2211yescharacteristic, complemented
53#123×C97—21no—
54#123×C97—21no—
55C22⋊Q8×C97C2211yescharacteristic
56C22⋊Q8×C97C2211yescharacteristic, complemented
57#121×C97—21no—
58C4×D4×C97C2211yescharacteristic
59C4×D4×C97C2211yescharacteristic, complemented
60C2×C4⋊C4×C97—21no—
61C23×D97—21no—
62#13095511—22no—
63#13095511—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388—22no—
66C2×D388—21no—
67C2×D388—41no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74#13095501—22no—
75#13095501—41no—
76#13095499—22no—
77#13095498—22no—
78#13095498—22no—
79#13095498—22no—
80#13095498—41no—
81#13095496—22no—
82#13095496D414yescomplemented
83#13095495—22no—
84C2×#11715895—22no—
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#41×C97—22no—
97#41×C97—22no—
98#41×C97—41no—
99#41×C97—41no—
100C2×Q8×C97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C2×D4×C97—21no—
103C2×D4×C97—21no—
104C2×D4×C97D411yescharacteristic
105C2×D4×C97C2311yescharacteristic
106C22×C4×C97—21no—
107C22×C4×C97—21no—
108C22×C4×C97—21no—
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C4⋊C4×C97—21no—
112C4⋊C4×C97—21no—
113C4⋊C4×C97—21no—
114C4⋊C4×C97—21no—
115C4⋊C4×C97D411yescharacteristic
116C4⋊C4×C97C2311yescharacteristic
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C22⋊C4×C97—21no—
120C22⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C42×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97C2×D411yescharacteristic
125C23×C97C2×D411yescharacteristic
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—42no—
129C22×D97—42no—
130C22×D97—81no—
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97—21no—
134Q8×C97—21no—
135D4×C97—21no—
136D4×C97—41no—
137D4×C97—42no—
138D4×C97—42no—
139D4×C97C2×D412yes—
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—42no—
148C2×C4×C97—42no—
149C2×C4×C97—42no—
150C2×C4×C97C2×D411yescharacteristic
151C2×C4×C97C2×D411yescharacteristic
152C2×C4×C97C2×D411yescharacteristic
153C2×C4×C97C2×D411yescharacteristic
154C2×C4×C97C2411yescharacteristic
155C22⋊D97—42no—
156C22⋊D97—81no—
157C2×Dic97—21no—
158C2×Dic97—21no—
159D388—21no—
160D388—21no—
161D388—21no—
162D388—42no—
163D388C2×D412yes—
164C4×D97—42no—
165C4×D97—81no—
166Dic194—21no—
167Dic194—21no—
168#11715895—21no—
169#11715895—21no—
170#11715895—21no—
171#11715895C2×D412yes—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—22no—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C97—21no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97C22×D411yescharacteristic
182C2×D97—22no—
183C2×D97—41no—
184C2×D97—41no—
185C2×D97—41no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—84no—
189C4×C97—21no—
190C4×C97—21no—
191C4×C97—21no—
192C4×C97—21no—
193C4×C97—41no—
194C4×C97—41no—
195C4×C97—41no—
196C4×C97C22×D411yescharacteristic
197C4×C97H(5,𝔽2)11yescharacteristic
198Dic97—41no—
199Dic97—41no—
200C2×C97—41no—
201C2×C97—41no—
202C2×C97—42no—
203C2×C97D4211yescharacteristic
204C2×C97#57511yescharacteristic
205C2×C97#57711yescharacteristic
206D97—42no—
207D97—81no—
208#3267—971nomaximal, Sylow
209C97#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—971no—
211C2×#135—971no—
212#547D9711yescharacteristic, complemented
213#546—971no—
214#497—971no—
215#470—971no—
216#465—971no—
217#463—971no—
218#461—971no—
219#452—971no—
220#452—971no—
221#450—971no—
222#449—971no—
223#442—971no—
224#435—971no—
225C2×#41—971no—
226C2×#41C2×D9711yescharacteristic
227C2×#41C2×D9711yescharacteristic, complemented
228C22×D4—971no—
229AGL(1,ℤ/8)—1944no—
230#135—1944no—
231C2×Q16—971no—
232C2×Q16—971no—
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—971no—
236C2×SD16—971no—
237C2×SD16—971no—
238C2×D8—971no—
239C2×D8—971no—
240C2×M16—971no—
241C22×C8—971no—
242C4⋊Q8C2×D9711yescharacteristic, complemented
243#124—971no—
244#123—21no—
245#123—21no—
246#123—1941no—
247C22⋊Q8—971no—
248C22⋊Q8C2×D9711yescharacteristic
249C22⋊Q8C2×D9711yescharacteristic, complemented
250#121—21no—
251#121—971no—
252#121—1941no—
253#120—1941no—
254C4×D4—971no—
255C4×D4C2×D9711yescharacteristic
256C4×D4C2×D9711yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—1941no—
259#106—971no—
260C4⋊C8—971no—
261#103—971no—
262#103—971no—
263#103—971no—
264#102—971no—
265#102—971no—
266#102—971no—
267C22⋊C8—971no—
268C22⋊C8—971no—
269#97—971no—
270C24—1941no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—3881no—
276#41—1942no—
277C2×Q8—971no—
278C2×Q8C22×D9711yescharacteristic
279C2×Q8C22×D9711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—971no—
283C2×D4—971no—
284C2×D4—3881no—
285C2×D4—1942no—
286C2×D4—1941no—
287C2×D4—1941no—
288C2×D4—1941no—
289C2×D4C22⋊D9711yescharacteristic
290C2×D4C22×D9711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1941no—
295C22×C4—1941no—
296C22×C4C22×D9711yescharacteristic
297C22×C4C22×D9711yescharacteristic
298Q16—3881no—
299Q16—1942no—
300SD16—3881no—
301SD16—1942no—
302SD16—1942no—
303SD16—1942no—
304SD16—1942no—
305D8—974no—
306D8—1942no—
307M16—1942no—
308C2×C8—971no—
309C2×C8—971no—
310C2×C8—971no—
311C2×C8—971no—
312C2×C8—1942no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—971no—
318C4⋊C4—1941no—
319C4⋊C4C22⋊D9711yescharacteristic
320C4⋊C4C22×D9711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C22⋊C4—1941no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C22⋊C4C22⋊D9711yescharacteristic
330C22⋊C4C22⋊D9711yescharacteristic
331C42C22×D9711yescharacteristic
332C23—21no—
333C23—7761no—
334C23—3882no—
335C23—3882no—
336C23—1941no—
337C23—1941no—
338C23C2×C22⋊D9711yescharacteristic
339C23C2×C22⋊D9711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—1941no—
345Q8—1941no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—7761no—
351D4—972no—
352D4—3882no—
353D4—3882no—
354D4—1941no—
355D4—1941no—
356D4—1941no—
357D4D4×D9712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—7761no—
369C2×C4—3882no—
370C2×C4—1941no—
371C2×C4—1941no—
372C2×C4C2×C22⋊D9711yescharacteristic
373C2×C4C2×C22⋊D9711yescharacteristic
374C2×C4C2×C22⋊D9711yescharacteristic
375C2×C4C2×C22⋊D9711yescharacteristic
376C2×C4C23×D9711yescharacteristic
377C8—972no—
378C8—1941no—
379C8—1941no—
380C8—1941no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—7764no—
391C22—3881no—
392C22—3881no—
393C22—3881no—
394C22—3881no—
395C22—3881no—
396C22—1942no—
397C22C22×C22⋊D9711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—3881no—
406C4—3881no—
407C4C2×D4×D9711yescharacteristic
408C4#1569085511yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—7761no—
413C2—3882no—
414C2D4×C22⋊D9711yescharacteristic
415C2#2026153311yescharacteristic
416C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#547×C97C2
socleC22×C97C22×D4
hypercenter#547D97
hypocenterC97#3267
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690830C2semidirect
3. #20261381C2semidirect
4. #20261127C2semidirect
5. #20260991C2semidirect
6. #20260966C2semidirect
7. #20260950C2semidirect
8. #20260937C2semidirect
9. #20260897C2semidirect
10. #20260895C2semidirect
11. #20260885C2semidirect
12. #20260880C2semidirect
13. #20260845C2semidirect
14. #20260807C2semidirect
15. #547×C97C2semidirect
20. C2×#13095501C22semidirect
21. C2×#13095501C22semidirect
22. C2×#13095499C22semidirect
30. C22×#11715895C22semidirect
39. #15690741C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690734C22semidirect
42. #15690734C22semidirect
43. #15690733C22semidirect
44. #15690731C22semidirect
48. C22⋊#11715895C22semidirect
49. #15690721C22semidirect
51. C2×#41×C97C22semidirect
52. C4⋊Q8×C97C22semidirect
56. C22⋊Q8×C97C22semidirect
59. C4×D4×C97C22semidirect
82. #13095496D4semidirect
209. C97#3267semidirect
212. #547D97semidirect
227. C2×#41C2×D97semidirect
242. C4⋊Q8C2×D97semidirect
249. C22⋊Q8C2×D97semidirect
256. C4×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156908302
#202613812
#202611272
#202609912
#202609662
#202609502
#202609372
#202608972
#202608952
#202608852
#202608802
#202608452
#202608072
#547×C972
#326797