Finite Group Databasebeta

← #27151977

Subgroups — #27151977

Order 12416.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261339C211yesmaximal, characteristic, complemented
2#20261207C211yesmaximal, characteristic, complemented
3#20261058C211yesmaximal, characteristic, complemented
4#20261057C211yesmaximal, characteristic, complemented
5#20261036C211yesmaximal, characteristic, complemented
6#20261035C211yesmaximal, characteristic, complemented
7#20260927C211yesmaximal, characteristic, complemented
8#20260867C211yesmaximal, characteristic, complemented
9#20260867C211yesmaximal, characteristic, complemented
10#20260847C211yesmaximal, characteristic, complemented
11#20260844C211yesmaximal, characteristic, complemented
12D8×Dic97C211yesmaximal, characteristic, complemented
13#20260815C211yesmaximal, characteristic, complemented
14#20260795C211yesmaximal, characteristic, complemented
15#459×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C2×#13095496C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19#15690802C2211yescharacteristic, complemented
20#15690791—21no—
21#15690791—21no—
22#15690782—21no—
23#15690782—21no—
24#15690781C2211yescharacteristic
25#15690781C2211yescharacteristic
26C4×D388C2211yescharacteristic
27D4×Dic97C2211yescharacteristic
28D4×Dic97C2211yescharacteristic, complemented
29D4×Dic97C2211yescharacteristic, complemented
30#15690772C2211yescharacteristic
31C4×C22⋊D97—21no—
32#15690770—21no—
33#15690768C2211yescharacteristic
34#15690767C2211yescharacteristic
35C2×#13095488—21no—
36C22⋊C4×D97—21no—
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41C4⋊#11715895C2211yescharacteristic, complemented
42#15690732C2211yescharacteristic
43#15690732C2211yescharacteristic
44#15690731C2211yescharacteristic
45#15690731C2211yescharacteristic
46#15690731C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic, complemented
48#15690722C2211yescharacteristic
49#15690722C2211yescharacteristic
50#15690720C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C2×D8×C97C2211yescharacteristic
53C2×D8×C97C2211yescharacteristic
54#127×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64D4×D97—22no—
65D4×D97—22no—
66C22×Dic97—21no—
67C22×Dic97—21no—
68C22×Dic97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97C2311yescharacteristic
72#13095496—22no—
73#13095496—22no—
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490D411yescharacteristic
82#13095490D411yescharacteristic
83C22⋊Dic97—21no—
84C22⋊Dic97—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488D411yescharacteristic
92#13095488D411yescharacteristic
93C4×Dic97—21no—
94C4×Dic97C2311yescharacteristic
95C2×D4×C97—21no—
96C2×D4×C97—21no—
97C2×D4×C97C2311yescharacteristic
98C2×D4×C97C2311yescharacteristic
99C2×D4×C97C2311yescharacteristic
100C22×C4×C97—21no—
101D8×C97—22no—
102D8×C97—22no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97C2311yescharacteristic
106C22⋊C4×C97—21no—
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97—21no—
110C23×C97—21no—
111C22×D97—21no—
112C22×D97—24no—
113C22×D97—81no—
114C22×D97—81no—
115C22×D97C2×D411yescharacteristic
116D4×C97—21no—
117D4×C97—21no—
118D4×C97—21no—
119D4×C97—21no—
120D4×C97—21no—
121D4×C97—81no—
122D4×C97—81no—
123D4×C97#4112yes—
124C2×C4×C97—21no—
125C2×C4×C97—81no—
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—24no—
130C22⋊D97—81no—
131C22⋊D97—81no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—81no—
136C2×Dic97—81no—
137C2×Dic97—81no—
138C2×Dic97C2×D411yescharacteristic
139C2×Dic97C2×D411yescharacteristic
140C2×Dic97C2×D411yescharacteristic
141D388—21no—
142D388#4112yes—
143C4×D97—21no—
144C4×D97—21no—
145C4×D97—81no—
146C4×D97—81no—
147C8×C97—21no—
148C8×C97—21no—
149#11715895—21no—
150#11715895—21no—
151C22×C97—22no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97—41no—
156C22×C97—41no—
157C22×C97—41no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97C22×D411yescharacteristic
161C2×D97—21no—
162C2×D97—22no—
163C2×D97—21no—
164C2×D97—21no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—84no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97—41no—
171C4×C97C2×#4111yescharacteristic
172C4×C97C2×#4111yescharacteristic
173Dic97—21no—
174Dic97—21no—
175Dic97—21no—
176Dic97—41no—
177Dic97—41no—
178Dic97—41no—
179C2×C97—42no—
180C2×C97—81no—
181C2×C97—81no—
182C2×C97#52511yescharacteristic
183C2×C97#57611yescharacteristic
184C2×C97#57611yescharacteristic
185D97—41no—
186D97—42no—
187#3122—971nomaximal, Sylow
188C97#312211yescharacteristic, complemented, Sylow
189#540—971no—
190#518—971no—
191#482—971no—
192#482—971no—
193#478—971no—
194#478—971no—
195#459—971no—
196#459D9711yescharacteristic, complemented
197#447—971no—
198#447—971no—
199#442—971no—
200#442—971no—
201C4×D8—971no—
202C4×D8—971no—
203#433—971no—
204C22×D4—971no—
205C2×D8—971no—
206C2×D8—971no—
207C2×D8C2×D9711yescharacteristic
208C2×D8C2×D9711yescharacteristic
209#127C2×D9711yescharacteristic, complemented
210#125—971no—
211#123—1941no—
212#123—1941no—
213#121—971no—
214#121—971no—
215#121—1941no—
216#121—1941no—
217C4×D4—971no—
218C4×D4—971no—
219C4×D4—971no—
220C4×D4—971no—
221C4×D4—971no—
222C4×D4—1941no—
223C4×D4—1941no—
224C4×D4C2×D9711yescharacteristic
225#117—971no—
226#117—971no—
227C2×C22⋊C4—1941no—
228C2×C22⋊C4—1941no—
229#107—971no—
230#107—971no—
231#106—971no—
232#106—971no—
233C4⋊C8—971no—
234C4⋊C8C2×D9711yescharacteristic, complemented
235#102—971no—
236#102—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102—971no—
241#102C2×D9711yescharacteristic, complemented
242#102C2×D9711yescharacteristic, complemented
243C22⋊C8—971no—
244C22⋊C8—971no—
245#97—971no—
246C4×C8—971no—
247C24—1941no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—971no—
251C2×D4—1941no—
252C2×D4—1942no—
253C2×D4—1941no—
254C2×D4—1941no—
255C2×D4—1942no—
256C2×D4C22×D9711yescharacteristic
257C2×D4C22×D9711yescharacteristic
258C2×D4C22×D9711yescharacteristic
259C22×C4—21no—
260C22×C4—971no—
261C22×C4—1941no—
262C22×C4—1941no—
263C22×C4—1941no—
264C22×C4—1941no—
265C22×C4—1941no—
266D8—22no—
267D8—22no—
268D8—1942no—
269D8—1942no—
270C2×C8—971no—
271C2×C8—971no—
272C2×C8C22×D9711yescharacteristic
273C2×C8C22×D9711yescharacteristic
274C4⋊C4—971no—
275C4⋊C4—971no—
276C4⋊C4—971no—
277C4⋊C4—971no—
278C4⋊C4—971no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4C22×D9711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—971no—
284C22⋊C4—971no—
285C22⋊C4—1941no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C42—971no—
294C42—1941no—
295C42C22×D9711yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—7761no—
300C23—7761no—
301C23—971no—
302C23—1944no—
303C23—1941no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—81no—
311D4—7761no—
312D4—7761no—
313D4—972no—
314D4—1944no—
315D4—1941no—
316D4#1309551012yes—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—7761no—
320C2×C4—7761no—
321C2×C4—7761no—
322C2×C4—7761no—
323C2×C4—7761no—
324C2×C4—971no—
325C2×C4—971no—
326C2×C4—971no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4D4×D9711yescharacteristic
333C2×C4D4×D9711yescharacteristic
334C2×C4C23×D9711yescharacteristic
335C8—21no—
336C8—21no—
337C8—1941no—
338C8—1941no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—7764no—
349C22—3881no—
350C22—3881no—
351C22—1941no—
352C22—1942no—
353C22—1941no—
354C22—1941no—
355C22C2×D4×D9711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—3881no—
360C4—3881no—
361C4—3881no—
362C4—1941no—
363C4—1941no—
364C4—1941no—
365C4#41×D9711yescharacteristic
366C4C2×#1309551011yescharacteristic
367C2—42no—
368C2—81no—
369C2—81no—
370C2—3881no—
371C2—3882no—
372C2#2026124511yescharacteristic
373C2#2026153611yescharacteristic
374C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#459×C97C2
socleC22×C97C22×D4
hypercenter#459D97
hypocenterC97#3122
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20261339C2semidirect
2. #20261207C2semidirect
3. #20261058C2semidirect
4. #20261057C2semidirect
5. #20261036C2semidirect
6. #20261035C2semidirect
7. #20260927C2semidirect
8. #20260867C2semidirect
9. #20260867C2semidirect
10. #20260847C2semidirect
11. #20260844C2semidirect
12. D8×Dic97C2semidirect
13. #20260815C2semidirect
14. #20260795C2semidirect
15. #459×C97C2semidirect
19. #15690802C22semidirect
28. D4×Dic97C22semidirect
29. D4×Dic97C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. C4⋊#11715895C22semidirect
46. #15690731C22semidirect
47. #15690731C22semidirect
50. #15690720C22semidirect
51. C8×Dic97C22semidirect
54. #127×C97C22semidirect
56. C4⋊C8×C97C22semidirect
57. #102×C97C22semidirect
58. #102×C97C22semidirect
188. C97#3122semidirect
196. #459D97semidirect
209. #127C2×D97semidirect
234. C4⋊C8C2×D97semidirect
241. #102C2×D97semidirect
242. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613392
#202612072
#202610582
#202610572
#202610362
#202610352
#202609272
#202608672
#202608472
#202608442
D8×Dic972
#202608152
#202607952
#459×C972
#312297