Finite Group Databasebeta

← #28785

Subgroups — #28785

Order 256.

359 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2442C211yesmaximal, characteristic
2#2439C211yesmaximal, characteristic
3#2085C211yesmaximal, characteristic, complemented
4#2085C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1991C211yesmaximal, characteristic, complemented
9#1944C211yesmaximal, characteristic, complemented
10#1941C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1853C211yesmaximal, characteristic
13#1809C211yesmaximal, characteristic
14#1753C211yesmaximal, characteristic
15#1736C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×#97C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30#400—21no—
31#398—21no—
32#396—21no—
33#393—21no—
34#391—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#387—21no—
40#385—21no—
41#382C2211yescharacteristic
42#382C2211yescharacteristic
43#380—21no—
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46C4×C22⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C2×M16—41no—
63C2×M16—41no—
64C2×M16Q814yescomplemented
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71#125—22no—
72#125—41no—
73#117—22no—
74#117—41no—
75C2×C4⋊C4—22no—
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⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—41no—
102#107—41no—
103#106—22no—
104#106—22no—
105#106—41no—
106#106—41no—
107C22⋊C8Q814yescomplemented
108#97—41no—
109C4×C8—41no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95D411yescharacteristic
121#95D411yescharacteristic
122C24C2×D411yescharacteristic
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—81no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—81no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×Q812yes—
147C22×C4C2×Q812yes—
148C22×C4C2411yescharacteristic
149M16—44no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8C2×Q812yes—
159C2×C8C2×Q812yes—
160C2×C8C2×Q812yes—
161C2×C8C2×Q812yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4#4112yes—
185C4⋊C4#4112yes—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—81no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23C4⋊Q812yes—
203C23C22×D411yescharacteristic
204Q8—84no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
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×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—84no—
248C2×C4C4⋊Q812yes—
249C2×C4C22×Q811yescharacteristic
250C2×C4C22×Q811yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C4C2×#4111yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C2×C4H(5,𝔽2)11yescharacteristic
255C8—22no—
256C8—22no—
257C8—22no—
258C8—22no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22C2×C4⋊Q811yescharacteristic
267C22#54611yescharacteristic
268C22#54611yescharacteristic
269C22#57511yescharacteristic
270C22#57511yescharacteristic
271C22#57711yescharacteristic
272C22#57711yescharacteristic
273C4—22no—
274C4—41no—
275C4—42no—
276C4—41no—
277C4—41no—
278C4—42no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4#53311yescharacteristic
287C4#54311yescharacteristic
288C4#55411yescharacteristic
289C4#55411yescharacteristic
290C2—42no—
291C2#254911yescharacteristic
292C2#316311yescharacteristic
293C2#316611yescharacteristic
294C2#331111yescharacteristic
295C2#332111yescharacteristic
296C2#332311yescharacteristic
297C2#332611yescharacteristic
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. #2085C2semidirect
4. #2085C2semidirect
5. #2059C2semidirect
6. #2058C2semidirect
7. #1993C2semidirect
8. #1991C2semidirect
9. #1944C2semidirect
10. #1941C2semidirect
64. C2×M16Q8semidirect
107. C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#24422
#24392
#20852
#20592
#20582
#19932
#19912
#19442
#19412
#18542
#18532
#18092
#17532
#17362