Finite Group Databasebeta

← #27870

Subgroups — #27870

Order 256.

340 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#2625C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2097C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#1928C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1920C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic
13#1809C211yesmaximal, characteristic
14#1808C211yesmaximal, characteristic
15#1729C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28#400—21no—
29#398—21no—
30#397—21no—
31#397—21no—
32#391C2211yescharacteristic
33#391C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#388—21no—
38#387—21no—
39#387—21no—
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C2×M16—22no—
62C2×M16—22no—
63C2×M16—41no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73#117—22no—
74#117—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D411yescharacteristic
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×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—22no—
100#107—41no—
101#107—41no—
102#106—22no—
103#103—22no—
104#103—22no—
105#103—41no—
106#103—41no—
107#95—21no—
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#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—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—21no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4#4112yes—
145C22×C4#4112yes—
146C22×C4C2411yescharacteristic
147M16—44no—
148C2×C8—21no—
149C2×C8—21no—
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×C8—44no—
159C2×C8#4112yes—
160C2×C8#4112yes—
161C4⋊C4—22no—
162C4⋊C4—22no—
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—21no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—42no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—81no—
193C42—81no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23#12312yes—
201C23C22×D411yescharacteristic
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—41no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
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×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#12312yes—
249C2×C4C2×#4111yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C4C2×#4111yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C2×C42−1+411yescharacteristic
255C8—22no—
256C8—22no—
257C8—41no—
258C8—41no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22C2×#12311yescharacteristic
267C22#53211yescharacteristic
268C22#53711yescharacteristic
269C22#57611yescharacteristic
270C22#57711yescharacteristic
271C22#57811yescharacteristic
272C22#57811yescharacteristic
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4#55111yescharacteristic
287C4#55111yescharacteristic
288C4#55211yescharacteristic
289C4#55211yescharacteristic
290C2—42no—
291C2#249811yescharacteristic
292C2#309711yescharacteristic
293C2#309811yescharacteristic
294C2#317511yescharacteristic
295C2#317811yescharacteristic
296C2#325711yescharacteristic
297C2#326111yescharacteristic
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
3. #2101C2semidirect
4. #2097C2semidirect
5. #2083C2semidirect
6. #2072C2semidirect
7. #1928C2semidirect
8. #1928C2semidirect
9. #1920C2semidirect
10. #1918C2semidirect

Maximal subgroups

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

subgroupindex
#26252
#23142
#21012
#20972
#20832
#20722
#19282
#19202
#19182
#18942
#18922
#18092
#18082
#17292