Finite Group Databasebeta

← #12009181

Subgroups — #12009181

Order 1280.

352 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#713C211yesmaximal, characteristic
2#10600077C211yesmaximal, characteristic, complemented
3#10599845C211yesmaximal, characteristic, complemented
4C5×Q8⋊C16C211yesmaximal, characteristic, complemented
5Q8⋊#672C211yesmaximal, characteristic, complemented
6#10598047C211yesmaximal, characteristic, complemented
7#10597994C211yesmaximal, characteristic, complemented
8C4×#713C411yescharacteristic
9C5×C8×Q8C2211yescharacteristic
10Q8×#168C411yescharacteristic
11C2×#3930—21no—
12#65123—21no—
13#65123C411yescharacteristic
14#65122—21no—
15#65122C411yescharacteristic
16C8×D20—21no—
17C8×D20C2211yescharacteristic
18C4×C8×D5—21no—
19C4×C8×D5C2211yescharacteristic
20C5×C4⋊C16C2211yescharacteristic
21C4×C5×C16C2211yescharacteristic
22#64727—21no—
23#64721—21no—
24#64721—21no—
25C4⋊#672C2211yescharacteristic
26C4×#672C2211yescharacteristic
27#11623—51nomaximal, Sylow
28C2×#713C811yescharacteristic
29C4×C5×Q8C2×C411yescharacteristic
30Q8×Dic5C811yescharacteristic
31C2×C8×D5—21no—
32C2×C8×D5—21no—
33C2×C8×D5D411yescharacteristic
34C2×C8×D5D411yescharacteristic
35#4039—21no—
36#4039C811yescharacteristic
37#4038—21no—
38#4038C811yescharacteristic
39C4×D20—21no—
40C4×D20C2×C411yescharacteristic
41C42×D5—21no—
42C42×D5C2×C411yescharacteristic
43C2×C5×C16—21no—
44C2×C5×C16—21no—
45C5×C4⋊C8—21no—
46C5×C4⋊C8C2×C411yescharacteristic
47C4×C5×C8—21no—
48C4×C5×C8C2311yescharacteristic
49#3952—21no—
50#3952—21no—
51#3952—21no—
52C8×Dic5—21no—
53C8×Dic5D411yescharacteristic
54C8×Dic5D411yescharacteristic
55C2×#672—21no—
56C2×#672—21no—
57C4⋊#168—21no—
58C4⋊#168C2×C411yescharacteristic
59C4×#168—21no—
60C4×#168C2×C411yescharacteristic
61#3930—42no—
62C8×#41—51no—
63#2162—51no—
64#2150—51no—
65Q8⋊C16—51no—
66Q8⋊C16D511yescharacteristic, complemented
67D4⋊C16—51no—
68D4⋊C16—51no—
69C2×C5×Q8C2×C811yescharacteristic
70#713—22no—
71#713—22no—
72C2×D20—21no—
73C2×D20C2×C811yescharacteristic
74C2×C4×D5—21no—
75C2×C4×D5—21no—
76C2×C4×D5—21no—
77C2×C4×D5C22⋊C411yescharacteristic
78C2×C4×D5C2×C811yescharacteristic
79C2×C4×D5M1611yescharacteristic
80C2×C5×C8—21no—
81C2×C5×C8C2×D411yescharacteristic
82C2×C5×C8C2×D411yescharacteristic
83C5×C4⋊C4—21no—
84C5×C4⋊C4C2×C811yescharacteristic
85C42×C5—21no—
86C42×C5C22×C411yescharacteristic
87#685—21no—
88#685—21no—
89#685—21no—
90C4⋊Dic5—21no—
91C4⋊Dic5C2×C811yescharacteristic
92C4×Dic5—21no—
93C4×Dic5—21no—
94C4×Dic5C22⋊C411yescharacteristic
95C4×Dic5C2×C811yescharacteristic
96C4×Dic5M1611yescharacteristic
97C2×#168—21no—
98C2×#168C22⋊C411yescharacteristic
99C2×#168C22⋊C411yescharacteristic
100C8×D5—21no—
101C8×D5—21no—
102C8×D5—41no—
103C8×D5—42no—
104C8×D5—41no—
105C8×D5—41no—
106C5×C16—41no—
107C5×C16—41no—
108#672—41no—
109#672—41no—
110C4×#41—51no—
111C2×M32—101no—
112C8×Q8—51no—
113C8×Q8C2×D511yescharacteristic
114C8×D4—51no—
115C8×D4—51no—
116C8×D4—101no—
117C8×D4—101no—
118#431—51no—
119#431—101no—
120C2×C4×C8—51no—
121C2×C4×C8—101no—
122C4⋊C16—51no—
123C4⋊C16—101no—
124C4⋊C16—101no—
125C4⋊C16C2×D511yescharacteristic
126C22⋊C16—101no—
127C4×C16—51no—
128C4×C16C2×D511yescharacteristic
129C22×D5—21no—
130C22×D5C22⋊C811yescharacteristic
131C5×Q8—21no—
132C5×Q8M3212yescomplemented
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5C2×C22⋊C411yescharacteristic
136C2×C4×C5C22×C811yescharacteristic
137C2×C4×C5C2×M1611yescharacteristic
138C2×Dic5—21no—
139C2×Dic5C22⋊C811yescharacteristic
140C2×Dic5C22⋊C811yescharacteristic
141C2×Dic5C22⋊C811yescharacteristic
142D20—24no—
143D20—21no—
144D20M3212yescomplemented
145C4×D5—24no—
146C4×D5—21no—
147C4×D5—21no—
148C4×D5—21no—
149C4×D5—21no—
150C4×D5—22no—
151C4×D5—41no—
152C4×D5—41no—
153C4×D5—42no—
154C5×C8—21no—
155C5×C8—41no—
156C5×C8AGL(1,ℤ/8)11yescharacteristic
157C5×C8#13711yescharacteristic
158#168—21no—
159#168—21no—
160#168—41no—
161C2×#41—51no—
162C22×C8—51no—
163C22×C8—51no—
164C22×C8—101no—
165C22×C8—101no—
166C4×Q8—51no—
167C4×Q8C4×D511yescharacteristic
168C4×D4—51no—
169C4×D4—51no—
170C4×D4—101no—
171C4×D4—101no—
172#117—51no—
173#117—101no—
174C2×C42—51no—
175C2×C42—101no—
176M32—202no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—101no—
180C2×C16—101no—
181C4⋊C8—21no—
182C4⋊C8—51no—
183C4⋊C8—101no—
184C4⋊C8C4×D511yescharacteristic
185C22⋊C8—101no—
186C22⋊C8—101no—
187C22⋊C8—101no—
188C4×C8—21no—
189C4×C8—51no—
190C4×C8—51no—
191C4×C8—51no—
192C4×C8—101no—
193C4×C8—101no—
194C4×C8C22×D511yescharacteristic
195C22×C5C2×C22⋊C811yescharacteristic
196C2×D5—21no—
197C2×D5—22no—
198C2×D5—21no—
199C2×D5—21no—
200C2×D5—41no—
201C2×D5—41no—
202C4×C5—22no—
203C4×C5—21no—
204C4×C5—41no—
205C4×C5#41711yescharacteristic
206C4×C5#42111yescharacteristic
207C4×C5C2×M3211yescharacteristic
208C4×C5C2×M3211yescharacteristic
209Dic5—21no—
210Dic5—21no—
211Dic5—41no—
212Dic5#35012yes—
213#41—102no—
214#41—102no—
215C2×Q8C8×D511yescharacteristic
216C2×D4—51no—
217C2×D4—101no—
218C22×C4—51no—
219C22×C4—51no—
220C22×C4—51no—
221C22×C4—101no—
222C22×C4—101no—
223C22×C4—101no—
224C2×C8—21no—
225C2×C8—51no—
226C2×C8—51no—
227C2×C8—101no—
228C2×C8—101no—
229C2×C8—101no—
230C2×C8—201no—
231C2×C8—202no—
232C2×C8—201no—
233C2×C8—201no—
234C2×C8D4×D511yescharacteristic
235C2×C8D4×D511yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—51no—
238C4⋊C4—101no—
239C4⋊C4C8×D511yescharacteristic
240C22⋊C4—101no—
241C22⋊C4—101no—
242C22⋊C4—101no—
243C42—21no—
244C42—51no—
245C42—51no—
246C42—51no—
247C42—101no—
248C42—101no—
249C42C2×C4×D511yescharacteristic
250C16—41no—
251C16—41no—
252C16—201no—
253C16—201no—
254C2×C5#147711yescharacteristic
255C2×C5#211211yescharacteristic
256C2×C5C2×#35011yescharacteristic
257D5—41no—
258D5—42no—
259C23—51no—
260C23—101no—
261Q8—21no—
262Q8#393012yescomplemented
263D4—52no—
264D4—104no—
265D4—101no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—51no—
269C2×C4—51no—
270C2×C4—51no—
271C2×C4—102no—
272C2×C4—104no—
273C2×C4—101no—
274C2×C4—101no—
275C2×C4—101no—
276C2×C4—101no—
277C2×C4—101no—
278C2×C4—202no—
279C2×C4—201no—
280C2×C4—201no—
281C2×C4D5×C22⋊C411yescharacteristic
282C2×C4C2×C8×D511yescharacteristic
283C2×C4D5×M1611yescharacteristic
284C8—21no—
285C8—41no—
286C8—101no—
287C8—101no—
288C8—201no—
289C8#406011yescharacteristic
290C8#406411yescharacteristic
291C5#1162311yescharacteristic, complemented, Sylow
292C22—101no—
293C22—102no—
294C22—101no—
295C22—101no—
296C22—201no—
297C22—201no—
298C22D5×C22⋊C811yescharacteristic
299C4—22no—
300C4—21no—
301C4—41no—
302C4—52no—
303C4—101no—
304C4—101no—
305C4—201no—
306C4#6509211yescharacteristic
307C4#6511011yescharacteristic
308C4C2×#393011yescharacteristic
309C4D5×M3211yescharacteristic
310C2—201no—
311C2—202no—
312C2#1059902311yescharacteristic
313C2#1059989911yescharacteristic
314C2D5×#35011yescharacteristic
subgroupG/H
centerC2×C8D4×D5
commutatorC2×C4×C5C22×C8
FrattiniC4×C8C22×D5
FittingC5×Q8⋊C16C2
socleC22×C5C2×C22⋊C8
prime radicalC2×#713C8
cosocleC4×C5×C8C23
hypercenterQ8⋊C16D5
hypocenterC5#11623
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ Q8⋊C16

Complements

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

NH ≅ G/Ntype
2. #10600077C2semidirect
3. #10599845C2semidirect
4. C5×Q8⋊C16C2semidirect
5. Q8⋊#672C2semidirect
6. #10598047C2semidirect
7. #10597994C2semidirect
66. Q8⋊C16D5semidirect
132. C5×Q8M32semidirect
144. D20M32semidirect
262. Q8#3930semidirect
291. C5#11623semidirect

Maximal subgroups

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

subgroupindex
C8×#7132
#106000772
#105998452
C5×Q8⋊C162
Q8⋊#6722
#105980472
#105979942
#116235