Finite Group Databasebeta

← #29242

Subgroups — #29242

Order 256.

385 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊Q8C211yesmaximal, characteristic
2#2296C211yesmaximal, characteristic
3#2088C211yesmaximal, characteristic, complemented
4#2086C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2066C211yesmaximal, characteristic, complemented
7#1921C211yesmaximal, characteristic, complemented
8#1919C211yesmaximal, characteristic, complemented
9#1873C211yesmaximal, characteristic
10#1873C211yesmaximal, characteristic
11#1865C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic
14#1854C211yesmaximal, characteristic
15#1797C211yesmaximal, characteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C42C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#391C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#387—21no—
40#385—21no—
41#385—21no—
42#385—21no—
43#380C2211yescharacteristic
44C4×C4⋊C4—21no—
45C4×C4⋊C4—21no—
46C4×C4⋊C4—21no—
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4—21no—
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22⋊Q8—22no—
68C22⋊Q8Q814yes—
69C4×Q8—41no—
70C4×Q8—41no—
71#117—22no—
72#117—22no—
73#117—41no—
74#117—41no—
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⋊C4—21no—
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×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42D411yescharacteristic
100C2×C42D411yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—22no—
104#107—41no—
105#106—22no—
106#106—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109#103—22no—
110#103—22no—
111#103—41no—
112#103—41no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124C24C2×D411yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C2×Q812yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×Q812yes—
155C22×C4#4112yes—
156C22×C4C2411yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C4⋊C4—24no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4C2×Q812yes—
188C4⋊C4#4112yes—
189C4⋊C4#4112yes—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C42—24no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—41no—
200C42—41no—
201C42—42no—
202C42—42no—
203C42—42no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23C22⋊Q812yes—
211C23C22×D411yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8—41no—
215Q8—41no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—44no—
237C2×C4—44no—
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—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4C22⋊Q812yes—
259C2×C4#13512yes—
260C2×C4#13712yes—
261C2×C4C22×Q811yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22C2×C22⋊Q811yescharacteristic
279C22#52511yescharacteristic
280C22#53411yescharacteristic
281C22C2×#13511yescharacteristic
282C22C2×#13711yescharacteristic
283C22#57511yescharacteristic
284C22#57511yescharacteristic
285C4—22no—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4#55311yescharacteristic
299C4#55311yescharacteristic
300C4#55411yescharacteristic
301C4#55411yescharacteristic
302C2—42no—
303C2#257911yescharacteristic
304C2#307611yescharacteristic
305C2#307811yescharacteristic
306C2#310711yescharacteristic
307C2#311111yescharacteristic
308C2#319111yescharacteristic
309C2#319611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2088C2semidirect
4. #2086C2semidirect
5. #2068C2semidirect
6. #2066C2semidirect
7. #1921C2semidirect
8. #1919C2semidirect
11. #1865C2semidirect
12. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#22962
#20882
#20862
#20682
#20662
#19212
#19192
#18732
#18652
#18632
#18542
#17972