Finite Group Databasebeta

← #30157

Subgroups — #30157

Order 256.

307 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2644C211yesmaximal, characteristic
2#2331C211yesmaximal, characteristic, complemented
3#2088C211yesmaximal, characteristic, complemented
4#2086C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic
7#2065C211yesmaximal, characteristic
8#2065C211yesmaximal, characteristic
9#2058C211yesmaximal, characteristic, complemented
10#2023C211yesmaximal, characteristic
11#1926C211yesmaximal, characteristic
12#1925C211yesmaximal, characteristic
13#1920C211yesmaximal, characteristic, complemented
14#1919C211yesmaximal, characteristic, complemented
15#1832C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#106C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27#397—21no—
28#397—21no—
29#397—21no—
30#396—21no—
31#391—21no—
32#391—21no—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#387—21no—
42#387—21no—
43#382C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46#376C2211yescharacteristic
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×Q8—21no—
60C22×D4C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C4⋊Q8—22no—
66C4⋊Q8—22no—
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⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#107—41no—
92#106—22no—
93#106—22no—
94#106—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97#102—22no—
98#102—22no—
99#102—41no—
100#102—41no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95D411yescharacteristic
111#95D411yescharacteristic
112#95D411yescharacteristic
113#95D411yescharacteristic
114C24—21no—
115C2×Q8—24no—
116C2×Q8—81no—
117C2×Q8—81no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C4⋊C4—24no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—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—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4#4112yes—
169C4⋊C4#4112yes—
170C22⋊C4—81no—
171C22⋊C4—81no—
172C22⋊C4—81no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—81no—
183C42—81no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23C22×D411yescharacteristic
192Q8—84no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—41no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
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×C4C2×#4111yescharacteristic
237C2×C4C2×#4111yescharacteristic
238C2×C4H(5,𝔽2)11yescharacteristic
239C2×C42−1+411yescharacteristic
240C2×C42−1+411yescharacteristic
241C2×C42−1+411yescharacteristic
242C8—41no—
243C8—41no—
244C8—41no—
245C8—41no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22#52511yescharacteristic
254C22#53611yescharacteristic
255C22#53711yescharacteristic
256C22#57611yescharacteristic
257C22#57711yescharacteristic
258C22#57711yescharacteristic
259C22#57711yescharacteristic
260C4—41no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4#54111yescharacteristic
274C4#54111yescharacteristic
275C4#54211yescharacteristic
276C4#54211yescharacteristic
277C2—161no—
278C2#263711yescharacteristic
279C2#311811yescharacteristic
280C2#312611yescharacteristic
281C2#324111yescharacteristic
282C2#324411yescharacteristic
283C2#325711yescharacteristic
284C2#326311yescharacteristic
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
2. #2331C2semidirect
3. #2088C2semidirect
4. #2086C2semidirect
5. #2085C2semidirect
9. #2058C2semidirect
13. #1920C2semidirect
14. #1919C2semidirect
15. #1832C2semidirect

Maximal subgroups

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

subgroupindex
#26442
#23312
#20882
#20862
#20852
#20732
#20652
#20582
#20232
#19262
#19252
#19202
#19192
#18322