Finite Group Databasebeta

← #27153588

Subgroups — #27153588

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261431C211yesmaximal, characteristic, complemented
2#20261381C211yesmaximal, characteristic, complemented
3#20261090C211yesmaximal, characteristic, complemented
4#20261071C211yesmaximal, characteristic, complemented
5#20261060C211yesmaximal, characteristic, complemented
6#20261036C211yesmaximal, characteristic, complemented
7#20260971C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260951C211yesmaximal, characteristic, complemented
10#20260882C211yesmaximal, characteristic, complemented
11#20260871C211yesmaximal, characteristic, complemented
12C4×D776C211yesmaximal, characteristic, complemented
13#20260732C211yesmaximal, characteristic, complemented
14C2×#15690732C211yesmaximal, characteristic, complemented
15C8×D4×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C2×#13095500C2211yescharacteristic, complemented
19C2×D776C2211yescharacteristic
20C2×D776C2211yescharacteristic
21#15690805—21no—
22#15690802C2211yescharacteristic, complemented
23#15690798C2211yescharacteristic
24#15690795—21no—
25#15690792—21no—
26#15690789C2211yescharacteristic
27#15690784C2211yescharacteristic
28#15690784C2211yescharacteristic
29#15690784C2211yescharacteristic, complemented
30#15690783—21no—
31#15690782—21no—
32#15690778—21no—
33C4×D388C2211yescharacteristic
34C4×D388C2211yescharacteristic, complemented
35#15690770—21no—
36#15690769C2211yescharacteristic, complemented
37C2×C4⋊Dic97C2211yescharacteristic
38C22⋊C4×D97—21no—
39#15690744C2211yescharacteristic, complemented
40#15690744C2211yescharacteristic, complemented
41#15690742C2211yescharacteristic, complemented
42#15690735C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690732C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic, complemented
52C22×C8×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic, complemented
54C4×D4×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic, complemented
56C22⋊C8×C97C2211yescharacteristic
57C22⋊C8×C97C2211yescharacteristic, complemented
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095511—22no—
61#13095511—41no—
62C2×Dic194C2311yescharacteristic
63C2×D388—22no—
64C2×D388—21no—
65C2×D388—41no—
66C2×D388C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×D388C2311yescharacteristic
69C2×C22⋊D97—21no—
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C22⋊D97—21no—
73C22×Dic97—21no—
74C2×C4×D97—21no—
75C2×C4×D97—21no—
76C2×C4×D97—21no—
77#13095500—41no—
78D776—22no—
79D776—41no—
80C4⋊Dic97—21no—
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88C22⋊Dic97—21no—
89C22⋊Dic97—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488—21no—
97C4×Dic97—21no—
98C2×D4×C97D411yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C2×C8×C97—41no—
102C2×C8×C97—41no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C22⋊C4×C97D411yescharacteristic
110C42×C97C2311yescharacteristic
111C23×C97C2×D411yescharacteristic
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—21no—
116C22×D97—42no—
117C22×D97—42no—
118C22×D97—81no—
119D4×C97—22no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—42no—
130C22⋊D97—42no—
131C22⋊D97—81no—
132C22⋊D97—81no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—81no—
139D388—21no—
140D388—21no—
141D388—21no—
142D388—21no—
143D388—21no—
144D388—42no—
145D388#4112yes—
146C4×D97—42no—
147C4×D97—81no—
148C4×D97—81no—
149Dic194—21no—
150Dic194—21no—
151C8×C97—21no—
152C8×C97—21no—
153C8×C97—21no—
154C8×C97—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—21no—
158C22×C97—21no—
159C22×C97—21no—
160C22×C97#13512yes—
161C22×C97C22×D411yescharacteristic
162C2×D97—22no—
163C2×D97—41no—
164C2×D97—41no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97—41no—
169C2×D97—41no—
170C2×D97—41no—
171C2×D97—84no—
172C4×C97—21no—
173C4×C97—21no—
174C4×C97—21no—
175C4×C97—21no—
176C4×C97C2×#4111yescharacteristic
177C4×C97H(5,𝔽2)11yescharacteristic
178Dic97—41no—
179Dic97—41no—
180Dic97—41no—
181Dic97—41no—
182Dic97—41no—
183C2×C97—22no—
184C2×C97—41no—
185C2×C97#54611yescharacteristic
186C2×C97C2×#13511yescharacteristic
187C2×C97#57611yescharacteristic
188D97—42no—
189D97—81no—
190D97—81no—
191#3297—971nomaximal, Sylow
192C97#329711yescharacteristic, complemented, Sylow
193#553—971no—
194#546—971no—
195#488—971no—
196#484—971no—
197#482—971no—
198#478—971no—
199#466—971no—
200#465—971no—
201#463—971no—
202#449—971no—
203#447—971no—
204C4×D8—971no—
205C8×D4D9711yescharacteristic, complemented
206#416—971no—
207C2×#102—971no—
208C2×#41—971no—
209C22×D4—971no—
210C2×SD16—971no—
211C2×D8—971no—
212C2×D8—971no—
213C22×C8C2×D9711yescharacteristic
214C22×C8C2×D9711yescharacteristic, complemented
215#126—1941no—
216#125—971no—
217#124—971no—
218#123—1941no—
219#123—1941no—
220C22⋊Q8—971no—
221#121—971no—
222#121—971no—
223#121—971no—
224#121—1941no—
225#121—1941no—
226#120—1941no—
227C4×D4—971no—
228C4×D4—971no—
229C4×D4—1941no—
230C4×D4C2×D9711yescharacteristic
231#117—971no—
232C2×C4⋊C4—971no—
233C2×C22⋊C4—1941no—
234#107—971no—
235#107—971no—
236#106—971no—
237C4⋊C8C2×D9711yescharacteristic, complemented
238#103—971no—
239#103—971no—
240#103—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102—971no—
245#102—971no—
246#102—971no—
247#102—971no—
248C22⋊C8C2×D9711yescharacteristic
249C22⋊C8C2×D9711yescharacteristic, complemented
250C4×C8C2×D9711yescharacteristic, complemented
251C24—1941no—
252#41—3881no—
253#41—1942no—
254C2×Q8—971no—
255C2×D4—971no—
256C2×D4—971no—
257C2×D4—971no—
258C2×D4—3881no—
259C2×D4—1942no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4—1941no—
265C2×D4D38811yescharacteristic
266C22×C4—1941no—
267C22×C4—1941no—
268C22×C4—1941no—
269C22×C4—1941no—
270C22×C4C22×D9711yescharacteristic
271C22×C4C22×D9711yescharacteristic
272SD16—3881no—
273D8—3881no—
274D8—1942no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D9711yescharacteristic
278C2×C8C22×D9711yescharacteristic
279C2×C8C22×D9711yescharacteristic
280C2×C8C22×D9711yescharacteristic
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—971no—
285C4⋊C4—1941no—
286C4⋊C4—1941no—
287C4⋊C4—1941no—
288C4⋊C4—1941no—
289C4⋊C4D38811yescharacteristic
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C22⋊C4—1941no—
298C22⋊C4—1941no—
299C22⋊C4D38811yescharacteristic
300C22⋊C4D38811yescharacteristic
301C42—1941no—
302C42C22×D9711yescharacteristic
303C23—7761no—
304C23—3882no—
305C23—3882no—
306C23—1941no—
307C23—1941no—
308C23—1941no—
309C23C2×D38811yescharacteristic
310C23C2×D38811yescharacteristic
311Q8—1941no—
312Q8—1941no—
313D4—22no—
314D4—7761no—
315D4—7761no—
316D4—972no—
317D4—3882no—
318D4—3882no—
319D4—3882no—
320D4—1941no—
321D4—1941no—
322D4—1941no—
323D4—1941no—
324D4—1941no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—7761no—
330C2×C4—7761no—
331C2×C4—7761no—
332C2×C4—3882no—
333C2×C4—1941no—
334C2×C4—1941no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4C2×D38811yescharacteristic
339C2×C4C2×D38811yescharacteristic
340C2×C4C2×D38811yescharacteristic
341C2×C4C2×D38811yescharacteristic
342C2×C4C23×D9711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—7764no—
353C22—3881no—
354C22—3881no—
355C22—3881no—
356C22—3881no—
357C22—3881no—
358C22—3881no—
359C22—3881no—
360C22—3881no—
361C22—1942no—
362C22#1569083212yes—
363C22C22×D38811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—3881no—
369C4—3881no—
370C4—3881no—
371C4—3881no—
372C4—3881no—
373C4#41×D9711yescharacteristic
374C4#1569085511yescharacteristic
375C2—22no—
376C2—41no—
377C2—7761no—
378C2—7761no—
379C2—3882no—
380C2#2026138011yescharacteristic
381C2C2×#1569083211yescharacteristic
382C2#2026154011yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC8×D4×C97C2
socleC22×C97C22×D4
hypercenterC8×D4D97
hypocenterC97#3297
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20261431C2semidirect
2. #20261381C2semidirect
3. #20261090C2semidirect
4. #20261071C2semidirect
5. #20261060C2semidirect
6. #20261036C2semidirect
7. #20260971C2semidirect
8. #20260964C2semidirect
9. #20260951C2semidirect
10. #20260882C2semidirect
11. #20260871C2semidirect
12. C4×D776C2semidirect
13. #20260732C2semidirect
14. C2×#15690732C2semidirect
15. C8×D4×C97C2semidirect
18. C2×#13095500C22semidirect
22. #15690802C22semidirect
29. #15690784C22semidirect
34. C4×D388C22semidirect
36. #15690769C22semidirect
39. #15690744C22semidirect
40. #15690744C22semidirect
41. #15690742C22semidirect
42. #15690735C22semidirect
43. #15690735C22semidirect
44. #15690735C22semidirect
51. #15690732C22semidirect
53. C22×C8×C97C22semidirect
55. C4⋊C8×C97C22semidirect
57. C22⋊C8×C97C22semidirect
58. C4×C8×C97C22semidirect
192. C97#3297semidirect
205. C8×D4D97semidirect
214. C22×C8C2×D97semidirect
237. C4⋊C8C2×D97semidirect
249. C22⋊C8C2×D97semidirect
250. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614312
#202613812
#202610902
#202610712
#202610602
#202610362
#202609712
#202609642
#202609512
#202608822
#202608712
C4×D7762
#202607322
C2×#156907322
C8×D4×C972
#329797