Finite Group Databasebeta

← #27148

Subgroups — #27148

Order 256.

326 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2513C211yesmaximal, characteristic
2#2369C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic
5#2057C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic
7#1949C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic
9#1926C211yesmaximal, characteristic
10#1925C211yesmaximal, characteristic
11#1925C211yesmaximal, characteristic
12#1921C211yesmaximal, characteristic, complemented
13#1918C211yesmaximal, characteristic, complemented
14#1917C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#397—21no—
28#397—21no—
29#391—21no—
30#391—21no—
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#382C2211yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2211yescharacteristic
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×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⋊C4D411yescharacteristic
72C2×C4⋊C4D411yescharacteristic
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×C42—21no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92C2×C42C2311yescharacteristic
93#107—22no—
94#107—22no—
95#107—22no—
96#106—22no—
97C4⋊C8—22no—
98#102—22no—
99#102—22no—
100#102—22no—
101#102—22no—
102#97—22no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
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#4112yes—
150C4⋊C4—24no—
151C4⋊C4—22no—
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—41no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4#4112yes—
168C4⋊C4#4112yes—
169C4⋊C4#4112yes—
170C22⋊C4—42no—
171C22⋊C4—42no—
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—42no—
183C42—81no—
184C42—81no—
185C42—81no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23C22×D411yescharacteristic
194Q8—84no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—41no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
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—21no—
218C2×C4—21no—
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#13712yes—
238C2×C4C2×#4111yescharacteristic
239C2×C4C2×#4111yescharacteristic
240C2×C4C2×#4111yescharacteristic
241C2×C4C2×#4111yescharacteristic
242C2×C42−1+411yescharacteristic
243C2×C42−1+411yescharacteristic
244C8—22no—
245C8—41no—
246C8—41no—
247C8—41no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22#52511yescharacteristic
256C22#54711yescharacteristic
257C22#54811yescharacteristic
258C22C2×#13711yescharacteristic
259C22#57511yescharacteristic
260C22#57611yescharacteristic
261C22#57611yescharacteristic
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—42no—
270C4—42no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4#52911yescharacteristic
276C4#54111yescharacteristic
277C4#55611yescharacteristic
278C4#55611yescharacteristic
279C2—161no—
280C2#246911yescharacteristic
281C2#311711yescharacteristic
282C2#312211yescharacteristic
283C2#334711yescharacteristic
284C2#335711yescharacteristic
285C2#337011yescharacteristic
286C2#337511yescharacteristic
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. #2369C2semidirect
3. #2101C2semidirect
5. #2057C2semidirect
7. #1949C2semidirect
12. #1921C2semidirect
13. #1918C2semidirect
14. #1917C2semidirect
15. #1776C2semidirect

Maximal subgroups

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

subgroupindex
#25132
#23692
#21012
#20672
#20572
#20232
#19492
#19362
#19262
#19252
#19212
#19182
#19172
#17762