Finite Group Databasebeta

← #27151965

Subgroups — #27151965

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#20261076C211yesmaximal, characteristic, complemented
3#20261059C211yesmaximal, characteristic
4#20261030C211yesmaximal, characteristic, complemented
5#20261020C211yesmaximal, characteristic, complemented
6#20260957C211yesmaximal, characteristic, complemented
7#20260957C211yesmaximal, characteristic, complemented
8#20260891C211yesmaximal, characteristic, complemented
9#20260874C211yesmaximal, characteristic
10#20260840C211yesmaximal, characteristic, complemented
11#20260833C211yesmaximal, characteristic, complemented
12C4×#13095501C211yesmaximal, characteristic, complemented
13C4×#13095498C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#519×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20#15690810C2211yescharacteristic
21#15690805—21no—
22#15690798C2211yescharacteristic
23#15690796—21no—
24#15690792—21no—
25#15690789C2211yescharacteristic
26#15690788—21no—
27#15690784C2211yescharacteristic
28C4×Dic194C2211yescharacteristic
29C4×D388C2211yescharacteristic
30#15690743C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690734C2211yescharacteristic
39#15690734C2211yescharacteristic
40#15690733C2211yescharacteristic
41#15690733C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48C22×Q8×C97C2211yescharacteristic
49C4×Q8×C97—21no—
50C4×Q8×C97—21no—
51C4×Q8×C97C2211yescharacteristic
52C4×Q8×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55#117×C97—21no—
56#117×C97—21no—
57#117×C97C2211yescharacteristic
58#117×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194C2311yescharacteristic
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C4×D97—21no—
64#13095501—22no—
65#13095501—22no—
66#13095498—22no—
67#13095498—22no—
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C4⋊Dic97—21no—
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77C22⋊Dic97—21no—
78#13095488—21no—
79#13095488—21no—
80#13095488—21no—
81#13095488—21no—
82C4×Dic97—21no—
83C2×Q8×C97—22no—
84C2×Q8×C97—22no—
85C2×Q8×C97—21no—
86C2×Q8×C97C2311yescharacteristic
87C2×Q8×C97C2311yescharacteristic
88C22×C4×C97—21no—
89C22×C4×C97C2311yescharacteristic
90C4⋊C4×C97—21no—
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97D411yescharacteristic
95C4⋊C4×C97D411yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97D411yescharacteristic
100C22⋊C4×C97D411yescharacteristic
101C42×C97—21no—
102C42×C97—21no—
103C42×C97—21no—
104C42×C97—21no—
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97C2×D411yescharacteristic
109C22×D97—21no—
110Q8×C97—24no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97#4112yes—
114Q8×C97#4112yes—
115C2×C4×C97—24no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97C2×D411yescharacteristic
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2411yescharacteristic
128C22⋊D97—81no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132D388—21no—
133D388—21no—
134C4×D97—81no—
135Dic194—21no—
136Dic194—21no—
137Dic194—81no—
138#11715895—21no—
139#11715895—21no—
140#11715895—21no—
141#11715895—21no—
142C22×C97—21no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97C22×D411yescharacteristic
146C2×D97—41no—
147C2×D97—41no—
148C2×D97—41no—
149C4×C97—22no—
150C4×C97—22no—
151C4×C97—21no—
152C4×C97—21no—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—41no—
157C4×C97—41no—
158C4×C97C2×#4111yescharacteristic
159C4×C97C2×#4111yescharacteristic
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#519D9711yescharacteristic, complemented
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#433—971no—
185C22×Q8C2×D9711yescharacteristic
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—21no—
199C4×Q8—21no—
200C4×Q8—971no—
201C4×Q8C2×D9711yescharacteristic
202C4×Q8C2×D9711yescharacteristic
203C4×Q8C2×D9711yescharacteristic
204C4×Q8C2×D9711yescharacteristic
205C4×D4—971no—
206#117—21no—
207#117—21no—
208#117C2×D9711yescharacteristic
209#117C2×D9711yescharacteristic
210#107—971no—
211#107—971no—
212#106—971no—
213#106—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
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⋊C8—971no—
225C22⋊C8—971no—
226#97—971no—
227C4×C8—971no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—971no—
232C2×Q8—1941no—
233C2×Q8C22×D9711yescharacteristic
234C2×Q8C22×D9711yescharacteristic
235C2×D4—971no—
236C2×D4—1941no—
237C22×C4—21no—
238C22×C4—1941no—
239C22×C4C22×D9711yescharacteristic
240Q16—1942no—
241Q16—1942no—
242SD16—1942no—
243SD16—1942no—
244C2×C8—971no—
245C2×C8—971no—
246C2×C8—971no—
247C2×C8—971no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—971no—
253C4⋊C4—971no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4C22⋊D9711yescharacteristic
258C4⋊C4C22⋊D9711yescharacteristic
259C4⋊C4C22×D9711yescharacteristic
260C4⋊C4C22×D9711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—1941no—
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C22⋊C4—1941no—
266C22⋊C4—1941no—
267C22⋊C4C22⋊D9711yescharacteristic
268C22⋊C4C22⋊D9711yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—1941no—
275C42C22×D9711yescharacteristic
276C42C22×D9711yescharacteristic
277C23—1941no—
278C23C2×C22⋊D9711yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—7761no—
283Q8—1941no—
284Q8—1941no—
285Q8#1309551112yes—
286Q8#1309551112yes—
287D4—7761no—
288D4—1941no—
289D4—1941no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—7761no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4C2×C22⋊D9711yescharacteristic
302C2×C4C2×C22⋊D9711yescharacteristic
303C2×C4C2×C22⋊D9711yescharacteristic
304C2×C4C2×C22⋊D9711yescharacteristic
305C2×C4C2×C22⋊D9711yescharacteristic
306C2×C4C23×D9711yescharacteristic
307C8—1941no—
308C8—1941no—
309C8—1941no—
310C8—1941no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—3881no—
315C22—3881no—
316C22—3881no—
317C22C22×C22⋊D9711yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—3881no—
328C4—3881no—
329C4—3881no—
330C4C2×#1309551111yescharacteristic
331C4C2×#1309551111yescharacteristic
332C2—41no—
333C2—7761no—
334C2#2026124611yescharacteristic
335C2#2026154311yescharacteristic
336C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#519×C97C2
socleC22×C97C22×D4
prime radical#15690784C22
hypercenter#519D97
hypocenterC97#3121
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #20261076C2semidirect
4. #20261030C2semidirect
5. #20261020C2semidirect
6. #20260957C2semidirect
7. #20260957C2semidirect
8. #20260891C2semidirect
10. #20260840C2semidirect
11. #20260833C2semidirect
12. C4×#13095501C2semidirect
13. C4×#13095498C2semidirect
14. #20260796C2semidirect
15. #519×C97C2semidirect
169. C97#3121semidirect
171. #519D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202610762
#202610592
#202610302
#202610202
#202609572
#202608912
#202608742
#202608402
#202608332
C4×#130955012
C4×#130954982
#202607962
#519×C972
#312197