Finite Group Databasebeta

← #27294

Subgroups — #27294

Order 256.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2506C211yesmaximal, characteristic, complemented
2D4×C4⋊C4C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2086C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#1942C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1916C211yesmaximal, characteristic, complemented
10#1862C211yesmaximal, characteristic
11#1862C211yesmaximal, characteristic
12#1862C211yesmaximal, characteristic
13#1862C211yesmaximal, characteristic
14#1802C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic
16C2×#125—21no—
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4—21no—
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31#400—21no—
32#398—21no—
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#387—21no—
40#385—21no—
41#385—21no—
42#384C2211yescharacteristic
43#384C2211yescharacteristic
44#380—21no—
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×D4C2311yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#125—44no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—41no—
73C4×D4—41no—
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⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—22no—
103#107—41no—
104#107—41no—
105C4⋊C8—41no—
106#102—22no—
107#102—22no—
108#102—22no—
109#102Q814yes—
110C4×C8—41no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4C2×Q812yes—
125C2×D4#4112yes—
126C2×D4#4112yes—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2411yescharacteristic
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—21no—
159C2×C8—21no—
160C2×C8—21no—
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—41no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4D814yes—
186C4⋊C4C2×Q812yes—
187C4⋊C4#4112yes—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—42no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23C22×D411yescharacteristic
206D4—22no—
207D4—22no—
208D4—22no—
209D4#12514yescomplemented
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—48no—
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×C4—84no—
249C2×C4C2×D812yes—
250C2×C4C2×D812yes—
251C2×C4C2×D812yes—
252C2×C4C22×Q811yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4H(5,𝔽2)11yescharacteristic
257C2×C42−1+411yescharacteristic
258C8—22no—
259C8—41no—
260C8—41no—
261C8—41no—
262C22—24no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22#52511yescharacteristic
270C22#54611yescharacteristic
271C22D4×Q811yescharacteristic
272C22C22×D811yescharacteristic
273C22#57511yescharacteristic
274C22#57711yescharacteristic
275C22#57811yescharacteristic
276C4—22no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—41no—
281C4—42no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4C2×#12511yescharacteristic
290C4#53911yescharacteristic
291C4#55211yescharacteristic
292C4#55411yescharacteristic
293C2—44no—
294C2#247311yescharacteristic
295C2#310011yescharacteristic
296C2#312011yescharacteristic
297C2#329411yescharacteristic
298C2#331511yescharacteristic
299C2Q8×D811yescharacteristic
300C2#338811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2506C2semidirect
3. #2099C2semidirect
4. #2086C2semidirect
5. #2083C2semidirect
6. #2083C2semidirect
7. #1942C2semidirect
8. #1918C2semidirect
9. #1916C2semidirect
209. D4#125semidirect

Maximal subgroups

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

subgroupindex
#25062
D4×C4⋊C42
#20992
#20862
#20832
#19422
#19182
#19162
#18622
#18022
C4×#1022