Finite Group Databasebeta

← #27151986

Subgroups — #27151986

Order 12416.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261346C211yesmaximal, characteristic, complemented
2#20261207C211yesmaximal, characteristic
3#20261075C211yesmaximal, characteristic, complemented
4#20261074C211yesmaximal, characteristic, complemented
5#20261036C211yesmaximal, characteristic, complemented
6#20261035C211yesmaximal, characteristic, complemented
7#20260937C211yesmaximal, characteristic, complemented
8#20260884C211yesmaximal, characteristic
9#20260884C211yesmaximal, characteristic
10#20260838C211yesmaximal, characteristic, complemented
11#20260836C211yesmaximal, characteristic, complemented
12SD16×Dic97C211yesmaximal, characteristic, complemented
13#20260820C211yesmaximal, characteristic, complemented
14#20260795C211yesmaximal, characteristic, complemented
15#461×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19#15690805—21no—
20#15690805—21no—
21#15690802C2211yescharacteristic, complemented
22#15690800—21no—
23#15690800—21no—
24#15690786C2211yescharacteristic
25#15690786C2211yescharacteristic
26Q8×Dic97C2211yescharacteristic, complemented
27Q8×Dic97C2211yescharacteristic, complemented
28C4×D388C2211yescharacteristic
29D4×Dic97C2211yescharacteristic
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#15690734C2211yescharacteristic, complemented
43#15690734C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic
45#15690732C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690722C2211yescharacteristic
49#15690722C2211yescharacteristic
50#15690720C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C2×SD16×C97C2211yescharacteristic
53C2×SD16×C97C2211yescharacteristic
54C4⋊Q8×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58#103×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62D4×D97—22no—
63D4×D97—22no—
64C22×Dic97—21no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97C2311yescharacteristic
68#13095498—22no—
69#13095498—22no—
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C4⋊Dic97—21no—
73C4⋊Dic97—21no—
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490D411yescharacteristic
82#13095490D411yescharacteristic
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488D411yescharacteristic
90#13095488D411yescharacteristic
91C4×Dic97—21no—
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97C2311yescharacteristic
95C2×Q8×C97C2311yescharacteristic
96C2×Q8×C97C2311yescharacteristic
97C2×D4×C97C2311yescharacteristic
98C22×C4×C97—21no—
99SD16×C97—22no—
100SD16×C97—22no—
101C2×C8×C97C2311yescharacteristic
102C2×C8×C97C2311yescharacteristic
103C4⋊C4×C97—21no—
104C4⋊C4×C97—21no—
105C4⋊C4×C97C2311yescharacteristic
106C22⋊C4×C97—21no—
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C22×D97—21no—
110C22×D97—24no—
111C22×D97—81no—
112C22×D97—81no—
113C22×D97C2×D411yescharacteristic
114Q8×C97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—21no—
118D4×C97—21no—
119D4×C97#4112yes—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—81no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2411yescharacteristic
127C22⋊D97—24no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—81no—
132C2×Dic97C2×D411yescharacteristic
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135D388—21no—
136D388#4112yes—
137C4×D97—21no—
138C4×D97—21no—
139C4×D97—81no—
140C4×D97—81no—
141C8×C97—21no—
142C8×C97—21no—
143#11715895—21no—
144#11715895—21no—
145C22×C97—22no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97C22×D411yescharacteristic
149C2×D97—21no—
150C2×D97—22no—
151C2×D97—21no—
152C2×D97—21no—
153C2×D97—41no—
154C2×D97—41no—
155C2×D97—84no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—41no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97C2×#4111yescharacteristic
162C4×C97C2×#4111yescharacteristic
163Dic97—21no—
164Dic97—21no—
165Dic97—21no—
166Dic97—41no—
167Dic97—41no—
168Dic97—41no—
169C2×C97—42no—
170C2×C97#52511yescharacteristic
171C2×C97#57711yescharacteristic
172C2×C97#57711yescharacteristic
173D97—41no—
174D97—42no—
175#3123—971nomaximal, Sylow
176C97#312311yescharacteristic, complemented, Sylow
177#541—971no—
178#518—971no—
179#485—971no—
180#485—971no—
181#478—971no—
182#478—971no—
183#461—971no—
184#461D9711yescharacteristic, complemented
185#450—971no—
186#450—971no—
187#440—971no—
188#440—971no—
189C4×SD16—971no—
190C4×SD16—971no—
191#433—971no—
192C22×D4—971no—
193C2×SD16—971no—
194C2×SD16—971no—
195C2×SD16C2×D9711yescharacteristic
196C2×SD16C2×D9711yescharacteristic
197C4⋊Q8C2×D9711yescharacteristic, complemented
198#126—1941no—
199#126—1941no—
200#125—971no—
201#125—1941no—
202#125—1941no—
203C22⋊Q8—971no—
204C22⋊Q8—971no—
205C4×Q8—971no—
206C4×Q8—971no—
207C4×D4—971no—
208C4×D4—971no—
209C4×D4—971no—
210C4×D4—1941no—
211C4×D4—1941no—
212C4×D4C2×D9711yescharacteristic
213#117—971no—
214#117—971no—
215C2×C22⋊C4—1941no—
216C2×C22⋊C4—1941no—
217#107—971no—
218#107—971no—
219#106—971no—
220#106—971no—
221C4⋊C8—971no—
222C4⋊C8C2×D9711yescharacteristic, complemented
223#103—971no—
224#103—971no—
225#103C2×D9711yescharacteristic, complemented
226#103C2×D9711yescharacteristic, complemented
227#102—971no—
228#102—971no—
229#102—971no—
230#102—971no—
231C22⋊C8—971no—
232C22⋊C8—971no—
233#97—971no—
234C4×C8—971no—
235C24—1941no—
236C2×Q8C22×D9711yescharacteristic
237C2×Q8C22×D9711yescharacteristic
238C2×D4—971no—
239C2×D4—1941no—
240C2×D4—1942no—
241C2×D4—1942no—
242C2×D4C22×D9711yescharacteristic
243C22×C4—21no—
244C22×C4—971no—
245C22×C4—1941no—
246C22×C4—1941no—
247C22×C4—1941no—
248SD16—22no—
249SD16—22no—
250SD16—1942no—
251SD16—1942no—
252C2×C8—971no—
253C2×C8—971no—
254C2×C8C22×D9711yescharacteristic
255C2×C8C22×D9711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—971no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4—971no—
263C4⋊C4—1941no—
264C4⋊C4—1941no—
265C4⋊C4—1941no—
266C4⋊C4—1941no—
267C4⋊C4—1941no—
268C4⋊C4—1941no—
269C4⋊C4C22×D9711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C22⋊C4—1941no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C42—971no—
280C42—1941no—
281C42—1941no—
282C42—1941no—
283C42C22×D9711yescharacteristic
284C23—21no—
285C23—7761no—
286C23—7761no—
287C23—971no—
288C23—1944no—
289C23—1941no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294D4—21no—
295D4—972no—
296D4—1944no—
297D4—1941no—
298D4#1309551012yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—7761no—
304C2×C4—7761no—
305C2×C4—7761no—
306C2×C4—971no—
307C2×C4—971no—
308C2×C4—971no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4D4×D9711yescharacteristic
315C2×C4D4×D9711yescharacteristic
316C2×C4C23×D9711yescharacteristic
317C8—21no—
318C8—21no—
319C8—1941no—
320C8—1941no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—7764no—
325C22—3881no—
326C22—3881no—
327C22—1941no—
328C22—1942no—
329C22—1941no—
330C22—1941no—
331C22C2×D4×D9711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3881no—
338C4—3881no—
339C4—3881no—
340C4—1941no—
341C4—1941no—
342C4—1941no—
343C4#41×D9711yescharacteristic
344C4C2×#1309551011yescharacteristic
345C2—42no—
346C2—3881no—
347C2—3882no—
348C2#2026124511yescharacteristic
349C2#2026154111yescharacteristic
350C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#461×C97C2
socleC22×C97C22×D4
prime radicalC2×D4×D97C22
hypercenter#461D97
hypocenterC97#3123
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20261346C2semidirect
3. #20261075C2semidirect
4. #20261074C2semidirect
5. #20261036C2semidirect
6. #20261035C2semidirect
7. #20260937C2semidirect
10. #20260838C2semidirect
11. #20260836C2semidirect
12. SD16×Dic97C2semidirect
13. #20260820C2semidirect
14. #20260795C2semidirect
15. #461×C97C2semidirect
21. #15690802C22semidirect
26. Q8×Dic97C22semidirect
27. Q8×Dic97C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. C4⋊#11715895C22semidirect
42. #15690734C22semidirect
43. #15690734C22semidirect
50. #15690720C22semidirect
51. C8×Dic97C22semidirect
54. C4⋊Q8×C97C22semidirect
56. C4⋊C8×C97C22semidirect
57. #103×C97C22semidirect
58. #103×C97C22semidirect
176. C97#3123semidirect
184. #461D97semidirect
197. C4⋊Q8C2×D97semidirect
222. C4⋊C8C2×D97semidirect
225. #103C2×D97semidirect
226. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613462
#202612072
#202610752
#202610742
#202610362
#202610352
#202609372
#202608842
#202608382
#202608362
SD16×Dic972
#202608202
#202607952
#461×C972
#312397