Finite Group Databasebeta

← #27023

Subgroups — #27023

Order 256.

345 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#2454C211yesmaximal, characteristic
2#2370C211yesmaximal, characteristic, complemented
3#2065C211yesmaximal, characteristic
4#2059C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic
7#1989C211yesmaximal, characteristic, complemented
8#1943C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic
10#1925C211yesmaximal, characteristic
11#1919C211yesmaximal, characteristic, complemented
12#1918C211yesmaximal, characteristic, complemented
13#1917C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic
15#1762C211yesmaximal, characteristic
16C2×#125C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#97C2211yescharacteristic
29#399—21no—
30#396—21no—
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#389—21no—
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#387—21no—
39#387—21no—
40#385—21no—
41#380—21no—
42C4×C4⋊C4—21no—
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
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—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65#125—22no—
66#125—22no—
67#125—22no—
68#125—41no—
69C4×D4—22no—
70C4×D4—41no—
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⋊C4D411yescharacteristic
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—22no—
100#107—22no—
101#106—22no—
102#106—41no—
103C4⋊C8—41no—
104#102—22no—
105#102—22no—
106#102—41no—
107#102Q814yes—
108#97—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
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—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—42no—
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×C8C2×Q812yes—
154C4⋊C4—24no—
155C4⋊C4—24no—
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—41no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4C2×Q812yes—
175C4⋊C4#4112yes—
176C4⋊C4#4112yes—
177C4⋊C4#4112yes—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—81no—
191C42—81no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23C22×D411yescharacteristic
200D4—22no—
201D4—22no—
202D4—41no—
203D4—41no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
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—21no—
225C2×C4—21no—
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—84no—
244C2×C4C22×Q811yescharacteristic
245C2×C4C2×#4111yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C2×C4C2×#4111yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C2×C42−1+411yescharacteristic
250C8—22no—
251C8—41no—
252C8—41no—
253C8—41no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22#52511yescharacteristic
262C22#54611yescharacteristic
263C22D4×Q811yescharacteristic
264C22#57511yescharacteristic
265C22#57511yescharacteristic
266C22#57611yescharacteristic
267C22#57711yescharacteristic
268C4—41no—
269C4—42no—
270C4—41no—
271C4—42no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—42no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4#53311yescharacteristic
282C4#54211yescharacteristic
283C4#55411yescharacteristic
284C4#55611yescharacteristic
285C2—82no—
286C2#246511yescharacteristic
287C2#311111yescharacteristic
288C2#311911yescharacteristic
289C2#331711yescharacteristic
290C2#332211yescharacteristic
291C2#338811yescharacteristic
292C2#338911yescharacteristic
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
2. #2370C2semidirect
4. #2059C2semidirect
5. #2059C2semidirect
7. #1989C2semidirect
8. #1943C2semidirect
11. #1919C2semidirect
12. #1918C2semidirect
13. #1917C2semidirect

Maximal subgroups

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

subgroupindex
#24542
#23702
#20652
#20592
#20252
#19892
#19432
#19262
#19252
#19192
#19182
#19172
#18622
#17622