Finite Group Databasebeta

← #30002

Subgroups — #30002

Order 256.

322 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2647C211yesmaximal, characteristic, complemented
2#2319C211yesmaximal, characteristic
3#2088C211yesmaximal, characteristic, complemented
4#2087C211yesmaximal, characteristic, complemented
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2067C211yesmaximal, characteristic
9#2067C211yesmaximal, characteristic
10#2065C211yesmaximal, characteristic
11#1949C211yesmaximal, characteristic, complemented
12#1947C211yesmaximal, characteristic, complemented
13#1843C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic
15#1802C211yesmaximal, characteristic
16C2×C4×D4C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31#400—21no—
32#400—21no—
33#398—21no—
34#398—21no—
35#390C2211yescharacteristic
36#390C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#384C2211yescharacteristic
46#382C2211yescharacteristic
47#381—21no—
48#380—21no—
49C4×C4⋊C4C2211yescharacteristic
50#376C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59C22×D4C2311yescharacteristic
60C23×C4—21no—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C4×D4—22no—
66C4×D4Q814yes—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C42—21no—
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#107—41no—
92#107—41no—
93#107—41no—
94#106—41no—
95C4⋊C8—22no—
96C4⋊C8—22no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101#102—41no—
102#102—41no—
103#102—41no—
104#102—41no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95D411yescharacteristic
115#95D411yescharacteristic
116#95D411yescharacteristic
117#95D411yescharacteristic
118C24—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4C2×Q812yes—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2411yescharacteristic
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—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—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—21no—
166C4⋊C4—21no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4C2×Q812yes—
176C22⋊C4—24no—
177C22⋊C4—81no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—81no—
186C42C2×Q812yes—
187C42#4112yes—
188C23—22no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23C22×D411yescharacteristic
196D4—22no—
197D4—22no—
198D4—41no—
199D4—41no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
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—84no—
238C2×C4C22⋊Q812yes—
239C2×C4C22⋊Q812yes—
240C2×C4C22×Q811yescharacteristic
241C2×C4C2×#4111yescharacteristic
242C2×C4H(5,𝔽2)11yescharacteristic
243C2×C4H(5,𝔽2)11yescharacteristic
244C2×C42−1+411yescharacteristic
245C2×C42−1+411yescharacteristic
246C8—41no—
247C8—41no—
248C8—41no—
249C8—41no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22C2×C22⋊Q811yescharacteristic
258C22#53711yescharacteristic
259C22#53711yescharacteristic
260C22#57611yescharacteristic
261C22#57611yescharacteristic
262C22#57711yescharacteristic
263C22#57811yescharacteristic
264C4—22no—
265C4—22no—
266C4—42no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4#54011yescharacteristic
278C4#54111yescharacteristic
279C4#54411yescharacteristic
280C4#54411yescharacteristic
281C2—82no—
282C2#262911yescharacteristic
283C2#308111yescharacteristic
284C2#308311yescharacteristic
285C2#325711yescharacteristic
286C2#325811yescharacteristic
287C2#325911yescharacteristic
288C2#326211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2647C2semidirect
3. #2088C2semidirect
4. #2087C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
11. #1949C2semidirect
12. #1947C2semidirect
13. #1843C2semidirect

Maximal subgroups

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

subgroupindex
#26472
#23192
#20882
#20872
#20862
#20672
#20652
#19492
#19472
#18432
#18042
#18022