Finite Group Databasebeta

← #26003

Subgroups — #26003

Order 256.

347 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2441C211yesmaximal, characteristic
2#2346C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2087C211yesmaximal, characteristic, complemented
5#2086C211yesmaximal, characteristic, complemented
6#2059C211yesmaximal, characteristic, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1947C211yesmaximal, characteristic, complemented
9#1921C211yesmaximal, characteristic, complemented
10#1920C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1850C211yesmaximal, characteristic
13#1825C211yesmaximal, characteristic
14#1808C211yesmaximal, characteristic
15#1735C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27#399—21no—
28#396—21no—
29#389—21no—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#388—21no—
35#388—21no—
36#387—21no—
37#384C2211yescharacteristic
38#382—21no—
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41C4×C4⋊C4C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C22⋊C4C2211yescharacteristic
44#376—21no—
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C23×C4C2311yescharacteristic
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—41no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#125—22no—
68#125—41no—
69#117—22no—
70#117—22no—
71#117—41no—
72#117Q814yes—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—41no—
103#107—41no—
104#106—22no—
105C4⋊C8—22no—
106C4⋊C8—41no—
107C22⋊C8—22no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95D411yescharacteristic
117#95D411yescharacteristic
118C24C2×D411yescharacteristic
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×Q812yes—
137C22×C4#4112yes—
138C22×C4C2411yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—44no—
151C2×C8#4112yes—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4C2×Q812yes—
174C4⋊C4C2×Q812yes—
175C4⋊C4#4112yes—
176C22⋊C4—24no—
177C22⋊C4—22no—
178C22⋊C4—41no—
179C22⋊C4—42no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C42—24no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—42no—
188C42—42no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23C22⋊Q812yes—
197C23C22×D411yescharacteristic
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4C22⋊Q812yes—
240C2×C4#13512yes—
241C2×C4C22×Q811yescharacteristic
242C2×C4C2×#4111yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C2×C42−1+411yescharacteristic
247C8—22no—
248C8—41no—
249C8—41no—
250C8—41no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22C2×C22⋊Q811yescharacteristic
259C22#54611yescharacteristic
260C22#54711yescharacteristic
261C22C2×#13511yescharacteristic
262C22#57511yescharacteristic
263C22#57611yescharacteristic
264C22#57811yescharacteristic
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4#53311yescharacteristic
279C4#54411yescharacteristic
280C4#55111yescharacteristic
281C4#55211yescharacteristic
282C2—42no—
283C2#243811yescharacteristic
284C2#307611yescharacteristic
285C2#308311yescharacteristic
286C2#330511yescharacteristic
287C2#331511yescharacteristic
288C2#333611yescharacteristic
289C2#334711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2099C2semidirect
4. #2087C2semidirect
5. #2086C2semidirect
6. #2059C2semidirect
7. #1989C2semidirect
8. #1947C2semidirect
9. #1921C2semidirect
10. #1920C2semidirect

Maximal subgroups

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

subgroupindex
#24412
#23462
#20992
#20872
#20862
#20592
#19892
#19472
#19212
#19202
#18542
#18502
#18252
#18082
#17352