Finite Group Databasebeta

← #17668

Subgroups — #17668

Order 256.

492 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
1C2×#529C211yesmaximal, characteristic
2#2920C211yesmaximal, characteristic, complemented
3C2×#432C211yesmaximal, characteristic, complemented
4#1535C214yesmaximal, complemented
5#1532C214yesmaximal, complemented
6#1528C212yesmaximal, complemented
7#1527C212yesmaximal, complemented
8#529—21no—
9#529—21no—
10#529—21no—
11#529—21no—
12#529C412yes—
13#529C412yes—
14#529C2212yes—
15#529C2212yes—
16C2×#126—21no—
17C2×#125C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×#123—21no—
20C2×C22⋊Q8C411yescharacteristic
21C2×#121C411yescharacteristic
22C2×C4×Q8C411yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C411yescharacteristic
25C2×#117—21no—
26C22×C42C2211yescharacteristic
27#432—21no—
28#432—21no—
29#432C2212yes—
30#432C2212yescomplemented
31#432C2212yescomplemented
32#432C2212yescomplemented
33#431—21no—
34#431—21no—
35#423—21no—
36C2×C4⋊C8—21no—
37#407—21no—
38C2×C22⋊C8—21no—
39C4×M16C2212yes—
40C2×#97C2212yescomplemented
41#330C2218yescomplemented
42#329C2218yescomplemented
43C22×Q8C2×C411yescharacteristic
44C22×D4—21no—
45C22×D4C2×C411yescharacteristic
46C23×C4—21no—
47C23×C4C2×C411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C2×M16—22no—
51C22×C8—22no—
52#126—22no—
53#126—22no—
54#126—22no—
55#126—22no—
56#125—21no—
57#125—21no—
58#125C2×C412yes—
59#125C2312yes—
60#124—21no—
61#124—21no—
62#124C2×C412yes—
63#124C2312yes—
64#123—22no—
65#123—22no—
66#123—22no—
67#123—22no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C22⋊Q8C2×C412yes—
71C22⋊Q8C2×C412yes—
72#121—21no—
73#121—21no—
74#121C2×C412yes—
75#121C2×C412yes—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8C2×C412yes—
79C4×Q8C2×C412yes—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4C2×C412yes—
87C4×D4C2×C412yes—
88#117—22no—
89#117—22no—
90#117—22no—
91#117—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×C411yescharacteristic
97C2×C4⋊C4C2×C411yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C42—21no—
104C2×C42C2×C411yescharacteristic
105C2×C42C2×C411yescharacteristic
106C2×C42C2×C411yescharacteristic
107C2×C42D411yescharacteristic
108C2×C42D411yescharacteristic
109C2×C42D411yescharacteristic
110C2×C42D411yescharacteristic
111C2×C42D411yescharacteristic
112C2×C42D411yescharacteristic
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124#97—22no—
125#97C2314yes—
126#97C2314yes—
127C4×C8—22no—
128C24—21no—
129C24C22⋊C411yescharacteristic
130C2×Q8—21no—
131C2×Q8—42no—
132C2×Q8C22×C412yes—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—21no—
136C2×D4—42no—
137C2×D4—44no—
138C2×D4C22×C412yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4C22⋊C411yescharacteristic
149C22×C4C22⋊C411yescharacteristic
150C22×C4C22⋊C411yescharacteristic
151C22×C4C22⋊C411yescharacteristic
152C22×C4C22⋊C411yescharacteristic
153C22×C4C22⋊C411yescharacteristic
154C22×C4C22⋊C411yescharacteristic
155C22×C4C22⋊C411yescharacteristic
156C22×C4C22⋊C411yescharacteristic
157C22×C4C22×C411yescharacteristic
158C22×C4C22×C411yescharacteristic
159C22×C4C22×C411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166M16—44no—
167C2×C8—24no—
168C2×C8—24no—
169C2×C8—44no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4C22×C412yes—
181C4⋊C4C22×C412yes—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C42—22no—
193C42—22no—
194C42C22⋊C411yescharacteristic
195C42C22⋊C411yescharacteristic
196C42C22⋊C411yescharacteristic
197C42C22⋊C411yescharacteristic
198C42C22⋊C411yescharacteristic
199C42C22⋊C411yescharacteristic
200C42C22×C411yescharacteristic
201C42C22×C411yescharacteristic
202C42C22×C411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2×D411yescharacteristic
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2×D411yescharacteristic
209C42C2411yescharacteristic
210C23—22no—
211C23—22no—
212C23—22no—
213C23—41no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23C2×C22⋊C411yescharacteristic
218C23C2×C22⋊C411yescharacteristic
219C23C2×C22⋊C411yescharacteristic
220Q8—42no—
221Q8—42no—
222D4—42no—
223D4—44no—
224D4—44no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4C2×C22⋊C411yescharacteristic
253C2×C4C2×C22⋊C411yescharacteristic
254C2×C4C2×C22⋊C411yescharacteristic
255C2×C4C2×C22⋊C411yescharacteristic
256C2×C4C2×C22⋊C411yescharacteristic
257C2×C4C2×C22⋊C411yescharacteristic
258C2×C4C2×C22⋊C411yescharacteristic
259C2×C4C2×C22⋊C411yescharacteristic
260C2×C4C2×C22⋊C411yescharacteristic
261C2×C4C23×C411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4C22×D411yescharacteristic
264C8—44no—
265C8—44no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—44no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—84no—
278C22#42112yes—
279C22#42112yes—
280C22C22×C22⋊C411yescharacteristic
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—42no—
294C4—42no—
295C4#41312yes—
296C2—22no—
297C2—22no—
298C2—41no—
299C2—82no—
300C2C2×#41311yescharacteristic
301C2C2×#42111yescharacteristic
302C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radicalC2×#121C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2920C2semidirect
3. C2×#432C2semidirect
4. #1535C2semidirect
5. #1532C2semidirect
6. #1528C2semidirect
7. #1527C2semidirect
30. #432C22semidirect
31. #432C22semidirect
32. #432C22semidirect
40. C2×#97C22semidirect
41. #330C22semidirect
42. #329C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
#29202
C2×#4322
#15352
#15322
#15282
#15272