Finite Group Databasebeta

← #31422

Subgroups — #31422

Order 256.

349 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2496C211yesmaximal, characteristic
2#2440C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2086C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic
7#2075C211yesmaximal, characteristic
8#2057C211yesmaximal, characteristic, complemented
9#1990C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1942C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic
13#1918C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic
15#1823C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—21no—
18C2×C22⋊Q8—21no—
19C22×C4⋊C4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—21no—
29#398—21no—
30#397—21no—
31#395—21no—
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#387—21no—
37#385—21no—
38#384C2211yescharacteristic
39#384C2211yescharacteristic
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C22⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
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
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C4⋊Q8—22no—
67C4⋊Q8—41no—
68#126—44no—
69C22⋊Q8—44no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
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—41no—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—22no—
99#107—22no—
100#107—41no—
101#107—41no—
102C4⋊C8—22no—
103C22⋊C8—22no—
104C22⋊C8—41no—
105C4×C8—41no—
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
116C24C2×D411yescharacteristic
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—81no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
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—42no—
135C22×C4—42no—
136C22×C4—81no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
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—
154C2×C8#4112yes—
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—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4#4112yes—
181C22⋊C4—22no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4D814yes—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—42no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23C2×D812yes—
198C23C22×D411yescharacteristic
199Q8—84no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
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—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—84no—
241C2×C4C2×D812yes—
242C2×C4C2×D812yes—
243C2×C4#13712yes—
244C2×C4C2×#4111yescharacteristic
245C2×C4C2×#4111yescharacteristic
246C2×C4H(5,𝔽2)11yescharacteristic
247C2×C42−1+411yescharacteristic
248C2×C42−1+411yescharacteristic
249C2×C42−1+411yescharacteristic
250C8—22no—
251C8—41no—
252C8—41no—
253C8—41no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22#53611yescharacteristic
262C22#54611yescharacteristic
263C22#54711yescharacteristic
264C22C22×D811yescharacteristic
265C22C2×#13711yescharacteristic
266C22#57711yescharacteristic
267C22#57811yescharacteristic
268C4—22no—
269C4—22no—
270C4—22no—
271C4—41no—
272C4—41no—
273C4—42no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4#53911yescharacteristic
282C4#54111yescharacteristic
283C4#55611yescharacteristic
284C4#56311yescharacteristic
285C2—42no—
286C2#268911yescharacteristic
287C2#322611yescharacteristic
288C2#324211yescharacteristic
289C2#329411yescharacteristic
290C2#331211yescharacteristic
291C2#333411yescharacteristic
292C2#335611yescharacteristic
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. #2099C2semidirect
4. #2099C2semidirect
5. #2086C2semidirect
8. #2057C2semidirect
9. #1990C2semidirect
10. #1947C2semidirect
11. #1942C2semidirect
13. #1918C2semidirect

Maximal subgroups

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

subgroupindex
#24962
#24402
#20992
#20862
#20752
#20572
#19902
#19472
#19422
#19232
#19182
#18522
#18232