Finite Group Databasebeta

← #27151864

Subgroups — #27151864

Order 12416.

380 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261289C211yesmaximal, characteristic
2C4×Q8×D97C211yesmaximal, characteristic
3#20261067C211yesmaximal, characteristic
4#20261064C211yesmaximal, characteristic
5#20261019C211yesmaximal, characteristic, complemented
6#20261018C211yesmaximal, characteristic, complemented
7#20260944C211yesmaximal, characteristic, complemented
8#20260943C211yesmaximal, characteristic, complemented
9#20260892C211yesmaximal, characteristic
10#20260889C211yesmaximal, characteristic
11#20260843C211yesmaximal, characteristic, complemented
12#20260840C211yesmaximal, characteristic, complemented
13#20260840C211yesmaximal, characteristic, complemented
14#20260790C211yesmaximal, characteristic
15#441×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×Dic388C2211yescharacteristic
18C2×#13095501C2211yescharacteristic
19C2×#13095501C2211yescharacteristic
20#15690809C2211yescharacteristic
21#15690808C2211yescharacteristic
22#15690800—21no—
23#15690790C2211yescharacteristic
24#15690788—21no—
25#15690786C2211yescharacteristic
26Q8×Dic97C2211yescharacteristic
27C4×Dic194—21no—
28C4×Dic194—21no—
29C4×Dic194C2211yescharacteristic
30C4×Dic194C2211yescharacteristic
31#15690775—21no—
32#15690775C2211yescharacteristic
33#15690767—21no—
34C4⋊C4×D97—21no—
35C4⋊C4×D97C2211yescharacteristic
36C42×D97—21no—
37C42×D97C2211yescharacteristic
38#15690742C2211yescharacteristic
39#15690741C2211yescharacteristic
40#15690741C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690733C2211yescharacteristic
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690721C2211yescharacteristic
52C2×Q16×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62C2×Dic194C2311yescharacteristic
63Q8×D97—22no—
64Q8×D97—22no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97D411yescharacteristic
68C2×C4×D97D411yescharacteristic
69C2×C4×D97C2311yescharacteristic
70Dic388—22no—
71#13095501—22no—
72#13095501—22no—
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C4⋊Dic97—21no—
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095488—21no—
88#13095488—21no—
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97—21no—
93C4×Dic97D411yescharacteristic
94C4×Dic97D411yescharacteristic
95C4×Dic97C2311yescharacteristic
96C2×Q8×C97C2311yescharacteristic
97C2×Q8×C97C2311yescharacteristic
98Q16×C97—22no—
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C4⋊C4×C97—21no—
102C4⋊C4×C97—21no—
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C42×C97—21no—
106C42×C97—21no—
107C42×C97C2311yescharacteristic
108C22×D97C2×D411yescharacteristic
109Q8×C97—21no—
110Q8×C97—21no—
111Q8×C97—21no—
112Q8×C97#4112yes—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97C2×D411yescharacteristic
118C2×C4×C97C2×D411yescharacteristic
119C2×C4×C97C2411yescharacteristic
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97C2×D411yescharacteristic
125C2×Dic97C2×D411yescharacteristic
126C2×Dic97C2×D411yescharacteristic
127C4×D97—24no—
128C4×D97—22no—
129C4×D97—21no—
130C4×D97—21no—
131C4×D97—22no—
132C4×D97—42no—
133Dic194—24no—
134Dic194—21no—
135Dic194—21no—
136Dic194—21no—
137Dic194—42no—
138Dic194#4112yes—
139C8×C97—21no—
140C8×C97—21no—
141#11715895—21no—
142#11715895—21no—
143C22×C97C22×D411yescharacteristic
144C2×D97—21no—
145C2×D97—21no—
146C2×D97#13712yes—
147C4×C97—22no—
148C4×C97—21no—
149C4×C97—21no—
150C4×C97—41no—
151C4×C97—41no—
152C4×C97—41no—
153C4×C97C2×#4111yescharacteristic
154C4×C97C2×#4111yescharacteristic
155Dic97—21no—
156Dic97—22no—
157Dic97—21no—
158Dic97—41no—
159Dic97—41no—
160Dic97—41no—
161Dic97#13712yes—
162C2×C97#52511yescharacteristic
163C2×C97C2×#13711yescharacteristic
164C2×C97C2×#13711yescharacteristic
165D97—22no—
166#3113—971nomaximal, Sylow
167C97#311311yescharacteristic, complemented, Sylow
168#533—971no—
169C2×C4×Q8—971no—
170#483—971no—
171#483—971no—
172#475—971no—
173#475—971no—
174#462—971no—
175#462—971no—
176#451—971no—
177#451—971no—
178#441—971no—
179#441—971no—
180#441—971no—
181#441D9711yescharacteristic, complemented
182#432—971no—
183C22×Q8—971no—
184C2×Q16—971no—
185C2×Q16—971no—
186C2×Q16—971no—
187C2×Q16C2×D9711yescharacteristic
188C4⋊Q8—971no—
189C4⋊Q8—971no—
190#125—1941no—
191C22⋊Q8—971no—
192C22⋊Q8—971no—
193C22⋊Q8—1941no—
194C4×Q8—971no—
195C4×Q8—971no—
196C4×Q8—971no—
197C4×Q8—971no—
198C4×Q8—1941no—
199C4×Q8—1941no—
200C4×Q8—1941no—
201C4×Q8C2×D9711yescharacteristic
202C4×Q8C2×D9711yescharacteristic
203#117—1941no—
204C2×C4⋊C4—971no—
205C2×C4⋊C4—1941no—
206C2×C42—971no—
207C2×C42—1941no—
208#106—971no—
209#106—971no—
210#106—971no—
211#106C2×D9711yescharacteristic
212C4⋊C8—971no—
213C4⋊C8—971no—
214#103—971no—
215#103—971no—
216#103—971no—
217#103—971no—
218#103—971no—
219#103—971no—
220#103C2×D9711yescharacteristic
221#103C2×D9711yescharacteristic
222C22⋊C8—971no—
223C22⋊C8—971no—
224#97—971no—
225#97C2×D9711yescharacteristic
226C2×Q8—971no—
227C2×Q8—971no—
228C2×Q8—1941no—
229C2×Q8—1941no—
230C2×Q8—1942no—
231C2×Q8—1942no—
232C2×Q8C22×D9711yescharacteristic
233C2×Q8C22×D9711yescharacteristic
234C22×C4—971no—
235C22×C4—971no—
236C22×C4—971no—
237C22×C4—1941no—
238C22×C4—1941no—
239Q16—22no—
240Q16—1942no—
241Q16—1942no—
242Q16—1942no—
243C2×C8—971no—
244C2×C8—971no—
245C2×C8C22×D9711yescharacteristic
246C2×C8C22×D9711yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—971no—
250C4⋊C4—971no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4—1941no—
259C4⋊C4—1941no—
260C4⋊C4—1941no—
261C4⋊C4C22×D9711yescharacteristic
262C4⋊C4C22×D9711yescharacteristic
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C42—21no—
266C42—21no—
267C42—971no—
268C42—971no—
269C42—971no—
270C42—1941no—
271C42—1941no—
272C42—1941no—
273C42—1941no—
274C42C22×D9711yescharacteristic
275C23—971no—
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—972no—
280Q8—3882no—
281Q8—1944no—
282Q8—1941no—
283Q8—1941no—
284Q8—1941no—
285Q8#1309551112yes—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—971no—
291C2×C4—971no—
292C2×C4—971no—
293C2×C4—3882no—
294C2×C4—1942no—
295C2×C4—1942no—
296C2×C4—1944no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4D4×D9711yescharacteristic
304C2×C4D4×D9711yescharacteristic
305C2×C4C23×D9711yescharacteristic
306C8—21no—
307C8—21no—
308C8—1941no—
309C8—1941no—
310C22—972no—
311C22—1941no—
312C22—1941no—
313C22C2×D4×D9711yescharacteristic
314C4—22no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—972no—
321C4—3881no—
322C4—3881no—
323C4—3881no—
324C4—1941no—
325C4—1942no—
326C4—1941no—
327C4#41×D9711yescharacteristic
328C4C2×#1309551111yescharacteristic
329C2—1942no—
330C2#2026124411yescharacteristic
331C2#137×D9711yescharacteristic
332C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#441×C97C2
socleC22×C97C22×D4
prime radicalC22×D97C2×D4
hypercenter#441D97
hypocenterC97#3113
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
5. #20261019C2semidirect
6. #20261018C2semidirect
7. #20260944C2semidirect
8. #20260943C2semidirect
11. #20260843C2semidirect
12. #20260840C2semidirect
13. #20260840C2semidirect
15. #441×C97C2semidirect
167. C97#3113semidirect
181. #441D97semidirect

Maximal subgroups

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

subgroupindex
#202612892
C4×Q8×D972
#202610672
#202610642
#202610192
#202610182
#202609442
#202609432
#202608922
#202608892
#202608432
#202608402
#202607902
#441×C972
#311397