Finite Group Databasebeta

← #25822

Subgroups — #25822

Order 256.

491 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#499C211yesmaximal, characteristic, complemented
4#2629C211yesmaximal, characteristic
5#2059C211yesmaximal, characteristic, complemented
6#2059C211yesmaximal, characteristic, complemented
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1992C211yesmaximal, characteristic, complemented
10#1990C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1854C211yesmaximal, characteristic
13#1853C211yesmaximal, characteristic
14#1852C211yesmaximal, characteristic
15#1750C211yesmaximal, characteristic
16#544—24no—
17#544—24no—
18C2×C4⋊Q8—21no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×#125C2211yescharacteristic
23C2×#125C2211yescharacteristic
24C2×#125C2211yescharacteristic
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8—21no—
27C2×#117C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30#501—24no—
31#499—24no—
32C2×#107C2211yescharacteristic
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#106C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#398—21no—
45#398—21no—
46#396—21no—
47#396—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—28no—
72C4⋊Q8—22no—
73C4⋊Q8—22no—
74C4⋊Q8—22no—
75C4⋊Q8D414yes—
76#125—28no—
77#125—22no—
78#125—22no—
79#125—22no—
80#125—22no—
81#125—22no—
82#125D414yes—
83C22⋊Q8—28no—
84C22⋊Q8—28no—
85#117—22no—
86#117—22no—
87#117—22no—
88#117—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121C2×C42C2311yescharacteristic
122#107—22no—
123#107—22no—
124#107—22no—
125#107—22no—
126#106—22no—
127#106—22no—
128#106—22no—
129#106—22no—
130C22⋊C8Q814yescomplemented
131C22⋊C8Q814yescomplemented
132#97—22no—
133C4×C8—22no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24C2×D411yescharacteristic
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—81no—
144C2×Q8—81no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×Q812yes—
166C22×C4C2×Q812yes—
167C22×C4C2411yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8C2×Q812yes—
177C2×C8C2×Q812yes—
178C2×C8C2×Q812yes—
179C2×C8C2×Q812yes—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4C2×D412yes—
208C4⋊C4C2×D412yes—
209C4⋊C4C2×D412yes—
210C4⋊C4C2×D412yes—
211C22⋊C4—24no—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C42—24no—
217C42—21no—
218C42—21no—
219C42C2×D412yes—
220C42C2×D412yes—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23C4⋊Q812yes—
228C23C22×D411yescharacteristic
229Q8—84no—
230Q8—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4#12012yes—
273C2×C4#12012yes—
274C2×C4C4⋊Q812yes—
275C2×C4C22×D411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C2×C4C22×Q811yescharacteristic
278C2×C4C22×Q811yescharacteristic
279C2×C42−1+411yescharacteristic
280C2×C42−1+411yescharacteristic
281C8—22no—
282C8—22no—
283C8—22no—
284C8—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22C2×#12011yescharacteristic
293C22C2×C4⋊Q811yescharacteristic
294C22#53611yescharacteristic
295C22#57611yescharacteristic
296C22#57711yescharacteristic
297C22#57711yescharacteristic
298C22#57811yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—41no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4D4×Q811yescharacteristic
313C4D4×Q811yescharacteristic
314C4D4×Q811yescharacteristic
315C4D4×Q811yescharacteristic
316C2—42no—
317C2#243411yescharacteristic
318C2#301211yescharacteristic
319C2#301311yescharacteristic
320C2#316311yescharacteristic
321C2#316411yescharacteristic
322C2#324111yescharacteristic
323C2#324211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#501C2semidirect
3. C2×#499C2semidirect
5. #2059C2semidirect
6. #2059C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
9. #1992C2semidirect
10. #1990C2semidirect
130. C22⋊C8Q8semidirect
131. C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#5012
C2×#4992
#26292
#20592
#20582
#20572
#19922
#19902
#18542
#18532
#18522
#17502