Finite Group Databasebeta

← #60335

Subgroups — #60335

Order 256.

873 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3482C211yesmaximal, characteristic
2#2994C213yesmaximal, complemented
3#2993C213yesmaximal, complemented
4#2990C213yesmaximal, complemented
5#2988C211yesmaximal, characteristic, complemented
6#2985C213yesmaximal, complemented
7#2984C211yesmaximal, characteristic, complemented
8#2982C211yesmaximal, characteristic, complemented
9#2971C211yesmaximal, characteristic, complemented
10#2969C213yesmaximal, complemented
11#2966C213yesmaximal, complemented
12#2965C213yesmaximal, complemented
13Q8×M16C211yesmaximal, characteristic, complemented
14#2962C213yesmaximal, complemented
15C2×C8×Q8C211yesmaximal, characteristic, complemented
16#557C411yescharacteristic
17#556C413yes—
18#552C413yes—
19#552C413yes—
20D4×Q8C411yescharacteristic
21#548C411yescharacteristic
22#548C411yescharacteristic
23#547C413yes—
24#533C2213yes—
25#529C2213yes—
26#529C2213yes—
27C4×#41C2213yes—
28C4×#41C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31#446C2213yes—
32#446C2213yes—
33#446C2213yes—
34#446C2216yes—
35#446C2213yes—
36#446C2213yes—
37#446C2213yes—
38C8×Q8C2213yes—
39C8×Q8C2211yescharacteristic
40C8×Q8C2211yescharacteristic
41C8×Q8C2213yes—
42C8×Q8C2213yes—
43C8×Q8C2211yescharacteristic
44C8×Q8C2213yes—
45C8×Q8C2211yescharacteristic
46#436C2213yes—
47#436C2213yes—
48#436C2213yes—
49#436C2213yes—
50#435C2211yescharacteristic
51#435C2213yes—
52#435C2213yes—
53#435C2213yes—
54#435C2216yes—
55C8×D4C2213yes—
56C8×D4C2216yes—
57C8×D4C2213yes—
58#433C2216yes—
59#433C2216yes—
60#432C2213yes—
61#432C2213yes—
62#431C2213yes—
63#431C2213yes—
64#431C2213yes—
65#431C2213yes—
66#424C2216yes—
67#423C2213yes—
68C2×C4⋊C8C2213yes—
69C2×C4⋊C8C2213yes—
70#405C2216yes—
71C4×M16C2213yes—
72C4×M16C2213yes—
73C2×C4×C8C2213yes—
74C2×#41C2×C411yescharacteristic
75C2×#41C2×C413yes—
76C22×Q8C2×C411yescharacteristic
77C22×Q8C2×C411yescharacteristic
78C2×M16C2311yescharacteristic
79C2×M16C2313yes—
80C2×M16C2313yes—
81C2×M16C2313yes—
82C22×C8C2311yescharacteristic
83C22×C8C2313yes—
84C22×C8C2316yes—
85C4⋊Q8C2×C413yes—
86C4⋊Q8C2×C413yes—
87#126C2×C416yes—
88#126C2×C416yes—
89#125C2×C413yes—
90#125C2×C413yes—
91#125C2×C416yes—
92#124C2×C413yes—
93#124C2×C413yes—
94#123C2×C416yes—
95#123C2×C416yes—
96C22⋊Q8C2×C413yes—
97C22⋊Q8C2×C413yes—
98C22⋊Q8C2×C416yes—
99C22⋊Q8C2×C413yes—
100C22⋊Q8C2×C413yes—
101#121C2×C413yes—
102#121C2×C413yes—
103C4×Q8C2×C411yescharacteristic
104C4×Q8C2×C411yescharacteristic
105C4×Q8C2×C413yes—
106C4×Q8C2×C413yes—
107C4×Q8C2×C411yescharacteristic
108C4×Q8C2×C413yes—
109C4×Q8C2311yescharacteristic
110C4×Q8C2313yes—
111C4×Q8C2313yes—
112C4×Q8C2313yes—
113C4×D4C2×C413yes—
114C4×D4C2×C413yes—
115C4×D4C2×C413yes—
116C4×D4C2×C413yes—
117C4×D4C2311yescharacteristic
118C4×D4C2313yes—
119C4×D4C2313yes—
120C4×D4C2313yes—
121#117C2×C413yes—
122#117C2×C413yes—
123#117C2×C413yes—
124#117C2×C413yes—
125#117C2316yes—
126C2×C4⋊C4C2×C413yes—
127C2×C4⋊C4C2×C413yes—
128C2×C42C2313yes—
129C2×C42C2313yes—
130C2×C42C2313yes—
131C4⋊C8C2313yes—
132C4⋊C8C2313yes—
133C4⋊C8C2313yes—
134C4⋊C8C2313yes—
135C4⋊C8C2316yes—
136C4⋊C8C2316yes—
137C4⋊C8C2313yes—
138C4⋊C8C2313yes—
139C4⋊C8C2311yescharacteristic
140C4⋊C8C2313yes—
141C4⋊C8C2313yes—
142C4⋊C8C2313yes—
143C22⋊C8C2311yescharacteristic
144C22⋊C8C2313yes—
145C22⋊C8C2311yescharacteristic
146C22⋊C8C2313yes—
147C22⋊C8C2313yes—
148C22⋊C8C2313yes—
149C22⋊C8C2316yes—
150#97C2311yescharacteristic
151#97C2313yes—
152#97C2313yes—
153#97C2313yes—
154#97C2316yes—
155C4×C8C2313yes—
156C4×C8C2313yes—
157C4×C8C2313yes—
158C4×C8C2313yes—
159C4×C8C2313yes—
160C4×C8C2313yes—
161C4×C8C2316yes—
162#41—24no—
163#41—46no—
164C2×Q8—24no—
165C2×Q8—24no—
166C2×Q8C22×C413yes—
167C2×Q8C22×C413yes—
168C2×Q8C22×C413yes—
169C2×Q8C22×C411yescharacteristic
170C2×D4C22×C411yescharacteristic
171C2×D4C22×C413yes—
172C2×D4C22×C413yes—
173C2×D4C22×C413yes—
174C22×C4C22×C413yes—
175C22×C4C22×C413yes—
176C22×C4C22×C413yes—
177C22×C4C22×C413yes—
178C22×C4C22×C413yes—
179C22×C4C22×C413yes—
180C22×C4C2411yescharacteristic
181C22×C4C2411yescharacteristic
182C22×C4C2413yes—
183M16—22no—
184M16—26no—
185M16—43no—
186M16—43no—
187C2×C8—22no—
188C2×C8—46no—
189C2×C8—43no—
190C2×C8C2411yescharacteristic
191C2×C8C2413yes—
192C2×C8C2411yescharacteristic
193C2×C8C2413yes—
194C2×C8C2411yescharacteristic
195C2×C8C2413yes—
196C2×C8C2413yes—
197C2×C8C2411yescharacteristic
198C4⋊C4C22×C413yes—
199C4⋊C4C22×C413yes—
200C4⋊C4C22×C413yes—
201C4⋊C4C22×C416yes—
202C4⋊C4C22×C416yes—
203C4⋊C4C22×C411yescharacteristic
204C4⋊C4C22×C413yes—
205C4⋊C4C22×C413yes—
206C4⋊C4C22×C413yes—
207C4⋊C4C22×C413yes—
208C4⋊C4C22×C413yes—
209C4⋊C4C22×C413yes—
210C4⋊C4C22×C416yes—
211C4⋊C4C22×C416yes—
212C22⋊C4C22×C411yescharacteristic
213C22⋊C4C22×C411yescharacteristic
214C22⋊C4C22×C413yes—
215C22⋊C4C22×C413yes—
216C22⋊C4C22×C416yes—
217C22⋊C4C22×C413yes—
218C22⋊C4C22×C413yes—
219C22⋊C4C22×C416yes—
220C22⋊C4C22×C416yes—
221C42C22×C413yes—
222C42C22×C413yes—
223C42C22×C413yes—
224C42C22×C413yes—
225C42C22×C413yes—
226C42C22×C413yes—
227C42C2413yes—
228C42C2413yes—
229C42C2413yes—
230C42C2411yescharacteristic
231C23C23×C413yes—
232C23C23×C411yescharacteristic
233C23C23×C411yescharacteristic
234Q8—26no—
235Q8—26no—
236Q8—43no—
237Q8#13114yes—
238D4—26no—
239D4—22no—
240D4—26no—
241D4—43no—
242C2×C4—26no—
243C2×C4—26no—
244C2×C4—26no—
245C2×C4—26no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—23no—
249C2×C4—21no—
250C2×C4—23no—
251C2×C4—21no—
252C2×C4—43no—
253C2×C4—43no—
254C2×C4C23×C411yescharacteristic
255C2×C4C23×C411yescharacteristic
256C2×C4C23×C411yescharacteristic
257C2×C4C23×C413yes—
258C2×C4C23×C411yescharacteristic
259C2×C4C23×C413yes—
260C2×C4C23×C413yes—
261C2×C4C23×C413yes—
262C2×C4C23×C413yes—
263C2×C4C23×C413yes—
264C2×C4C23×C413yes—
265C2×C4C2511yescharacteristic
266C8—21no—
267C8—21no—
268C8—23no—
269C8—23no—
270C8—23no—
271C8—23no—
272C8—21no—
273C82−1+412yes—
274C22—21no—
275C22—21no—
276C22—23no—
277C22—21no—
278C22—21no—
279C22—23no—
280C22—21no—
281C22—23no—
282C22#56812yes—
283C22C24×C411yescharacteristic
284C4—21no—
285C4—23no—
286C4—21no—
287C4—21no—
288C4—23no—
289C4—23no—
290C4—23no—
291C4—23no—
292C4—21no—
293C4—23no—
294C4C2×#13116yes—
295C4C2×2−1+411yescharacteristic
296C4#58511yescharacteristic
297C2—22no—
298C2—41no—
299C2—43no—
300C2C4×2−1+411yescharacteristic
301C2C22×#13111yescharacteristic
302C2C2×#56811yescharacteristic
subgroupG/H
center, Frattini, cosocleC2×C4C25
commutator, socleC22C24×C4
prime radical#3482C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2994C2semidirect
3. #2993C2semidirect
4. #2990C2semidirect
5. #2988C2semidirect
6. #2985C2semidirect
7. #2984C2semidirect
8. #2982C2semidirect
9. #2971C2semidirect
10. #2969C2semidirect
11. #2966C2semidirect
12. #2965C2semidirect
13. Q8×M16C2semidirect
14. #2962C2semidirect
15. C2×C8×Q8C2semidirect

Maximal subgroups

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

subgroupindex
#34822
#29942
#29932
#29902
#29882
#29852
#29842
#29822
#29712
#29692
#29662
#29652
Q8×M162
#29622
C2×C8×Q82