Finite Group Databasebeta

← #27151967

Subgroups — #27151967

Order 12416.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261356C211yesmaximal, characteristic
2#20261212C211yesmaximal, characteristic, complemented
3#20261077C211yesmaximal, characteristic, complemented
4#20261060C211yesmaximal, characteristic
5#20261031C211yesmaximal, characteristic, complemented
6#20261021C211yesmaximal, characteristic, complemented
7#20260958C211yesmaximal, characteristic, complemented
8#20260958C211yesmaximal, characteristic, complemented
9#20260893C211yesmaximal, characteristic, complemented
10#20260876C211yesmaximal, characteristic
11#20260843C211yesmaximal, characteristic, complemented
12#20260837C211yesmaximal, characteristic, complemented
13C4×Dic388C211yesmaximal, characteristic, complemented
14C4×#13095500C211yesmaximal, characteristic, complemented
15#433×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic
17C2×Dic388C2211yescharacteristic
18C2×Dic388C2211yescharacteristic
19C2×#13095500C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21#15690810C2211yescharacteristic
22#15690805—21no—
23#15690798C2211yescharacteristic
24#15690796—21no—
25#15690792—21no—
26#15690789C2211yescharacteristic
27#15690788—21no—
28#15690784C2211yescharacteristic
29C4×Dic194C2211yescharacteristic
30C4×Dic194C2211yescharacteristic
31C4×Dic194C2211yescharacteristic
32C4×Dic194C2211yescharacteristic
33C4×Dic194C2211yescharacteristic
34#15690775—21no—
35#15690775—21no—
36C4×D388C2211yescharacteristic
37#15690769C2211yescharacteristic
38#15690766—21no—
39#15690766—21no—
40#15690744C2211yescharacteristic
41#15690744C2211yescharacteristic
42#15690742C2211yescharacteristic
43#15690742C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690735C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690735C2211yescharacteristic
49#15690735C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52#117×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic
54C4⋊C8×C97C2211yescharacteristic
55C22⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—22no—
61C2×Dic194—22no—
62C2×Dic194—21no—
63C2×Dic194C2311yescharacteristic
64C2×Dic194C2311yescharacteristic
65C2×Dic194C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C22×Dic97—21no—
69C2×C4×D97—21no—
70Dic388—22no—
71Dic388—22no—
72#13095500—22no—
73#13095500—22no—
74C4⋊Dic97—21no—
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91C4×Dic97—21no—
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C22×C4×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97D411yescharacteristic
103C4⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C22⋊C4×C97D411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97C2×D411yescharacteristic
109C22×D97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97C2×D411yescharacteristic
113C2×C4×C97C2×D411yescharacteristic
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2×D411yescharacteristic
117C2×C4×C97C2411yescharacteristic
118C22⋊D97—81no—
119C2×Dic97—24no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127D388—21no—
128D388—21no—
129C4×D97—81no—
130Dic194—24no—
131Dic194—21no—
132Dic194—21no—
133Dic194—21no—
134Dic194—21no—
135Dic194—81no—
136Dic194#4112yes—
137Dic194#4112yes—
138C8×C97—21no—
139C8×C97—21no—
140C8×C97—21no—
141C8×C97—21no—
142C22×C97—21no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97C22×D411yescharacteristic
146C2×D97—41no—
147C2×D97—41no—
148C2×D97—41no—
149C4×C97—21no—
150C4×C97—21no—
151C4×C97—21no—
152C4×C97—21no—
153C4×C97—21no—
154C4×C97C2×#4111yescharacteristic
155C4×C97C2×#4111yescharacteristic
156Dic97—22no—
157Dic97—22no—
158Dic97—41no—
159Dic97—41no—
160Dic97—41no—
161Dic97—41no—
162Dic97—41no—
163C2×C97—41no—
164C2×C97#52511yescharacteristic
165C2×C97#57711yescharacteristic
166C2×C97#57811yescharacteristic
167D97—81no—
168#3121—971nomaximal, Sylow
169C97#312111yescharacteristic, complemented, Sylow
170#542—971no—
171#519—971no—
172#485—971no—
173#482—971no—
174#477—971no—
175#475—971no—
176#464—971no—
177#464—971no—
178#451—971no—
179#448—971no—
180#441—971no—
181#440—971no—
182C4×Q16—971no—
183C4×SD16—971no—
184#433D9711yescharacteristic, complemented
185C22×Q8—971no—
186C2×Q16—971no—
187C2×Q16—971no—
188C2×SD16—971no—
189C2×SD16—971no—
190C4⋊Q8—971no—
191#126—1941no—
192#124—971no—
193#124—1941no—
194#123—1941no—
195C22⋊Q8—971no—
196C22⋊Q8—1941no—
197#121—971no—
198C4×Q8—971no—
199C4×Q8—971no—
200C4×Q8—971no—
201C4×Q8—971no—
202C4×Q8—971no—
203C4×Q8—1941no—
204C4×Q8—1941no—
205C4×D4—971no—
206#117—971no—
207#117—1941no—
208#117—1941no—
209#117C2×D9711yescharacteristic
210#107—971no—
211#107—971no—
212#106—971no—
213#106—971no—
214C4⋊C8C2×D9711yescharacteristic
215C4⋊C8C2×D9711yescharacteristic
216#103—971no—
217#103—971no—
218#103—971no—
219#103—971no—
220#103—971no—
221#103—971no—
222#102—971no—
223#102—971no—
224C22⋊C8C2×D9711yescharacteristic
225C22⋊C8C2×D9711yescharacteristic
226#97C2×D9711yescharacteristic
227C4×C8C2×D9711yescharacteristic
228C2×Q8—971no—
229C2×Q8—971no—
230C2×Q8—971no—
231C2×Q8—1942no—
232C2×Q8—1942no—
233C2×Q8—1941no—
234C2×Q8—1941no—
235C2×D4—971no—
236C2×D4—1941no—
237C22×C4—1941no—
238C22×C4—1941no—
239C22×C4C22×D9711yescharacteristic
240Q16—1942no—
241Q16—1942no—
242SD16—1942no—
243SD16—1942no—
244C2×C8C22×D9711yescharacteristic
245C2×C8C22×D9711yescharacteristic
246C2×C8C22×D9711yescharacteristic
247C2×C8C22×D9711yescharacteristic
248C4⋊C4—971no—
249C4⋊C4—971no—
250C4⋊C4—971no—
251C4⋊C4—971no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4—1941no—
259C4⋊C4D38811yescharacteristic
260C4⋊C4D38811yescharacteristic
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C22⋊C4—1941no—
266C22⋊C4—1941no—
267C22⋊C4D38811yescharacteristic
268C22⋊C4D38811yescharacteristic
269C42—1941no—
270C42—1941no—
271C42—1941no—
272C42—1941no—
273C42—1941no—
274C42—1941no—
275C42C22×D9711yescharacteristic
276C42C22×D9711yescharacteristic
277C23—1941no—
278C23C2×D38811yescharacteristic
279Q8—7761no—
280Q8—972no—
281Q8—972no—
282Q8—1944no—
283Q8—1941no—
284Q8—1941no—
285Q8—1941no—
286Q8—1941no—
287D4—7761no—
288D4—1941no—
289D4—1941no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—7761no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1944no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4C2×D38811yescharacteristic
302C2×C4C2×D38811yescharacteristic
303C2×C4C2×D38811yescharacteristic
304C2×C4C2×D38811yescharacteristic
305C2×C4C2×D38811yescharacteristic
306C2×C4C23×D9711yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—3881no—
315C22—3881no—
316C22—3881no—
317C22C22×D38811yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—3881no—
324C4—3881no—
325C4—3881no—
326C4—3881no—
327C4—3881no—
328C4—1942no—
329C4—1942no—
330C4#41×D9711yescharacteristic
331C4#41×D9711yescharacteristic
332C2—41no—
333C2—7761no—
334C2#2026124711yescharacteristic
335C2#2026154711yescharacteristic
336C2#2026155211yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#433×C97C2
socleC22×C97C22×D4
prime radical#15690784C22
hypercenter#433D97
hypocenterC97#3121
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #20261212C2semidirect
3. #20261077C2semidirect
5. #20261031C2semidirect
6. #20261021C2semidirect
7. #20260958C2semidirect
8. #20260958C2semidirect
9. #20260893C2semidirect
11. #20260843C2semidirect
12. #20260837C2semidirect
13. C4×Dic388C2semidirect
14. C4×#13095500C2semidirect
15. #433×C97C2semidirect
169. C97#3121semidirect
184. #433D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202612122
#202610772
#202610602
#202610312
#202610212
#202609582
#202608932
#202608762
#202608432
#202608372
C4×Dic3882
C4×#130955002
#433×C972
#312197