Finite Group Databasebeta

← #31795

Subgroups — #31795

Order 256.

355 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2485C211yesmaximal, characteristic
2#2415C211yesmaximal, characteristic
3#2087C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2066C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic
10#1943C211yesmaximal, characteristic, complemented
11#1935C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1825C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35#397—21no—
36#397—21no—
37#394—21no—
38#394—21no—
39#391C2211yescharacteristic
40#389—21no—
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#388—21no—
46#387—21no—
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4D411yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22⋊Q8—22no—
68C22⋊Q8—41no—
69C22⋊Q8—41no—
70C22⋊Q8—41no—
71C4×D4—41no—
72C4×D4—41no—
73#117—22no—
74#117—41no—
75#117—41no—
76#117—41no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
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⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103#107—22no—
104#107—41no—
105#106—41no—
106#106—41no—
107C4⋊C8—41no—
108#103—22no—
109#103—41no—
110#103—41no—
111#103—41no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116#97—41no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24C2×D411yescharacteristic
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8#4112yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4#4112yes—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—42no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23C22×D411yescharacteristic
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227D4—44no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C2×C4H(5,𝔽2)11yescharacteristic
277C2×C42−1+411yescharacteristic
278C8—22no—
279C8—41no—
280C8—41no—
281C8—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22#53411yescharacteristic
297C22#54611yescharacteristic
298C22#54711yescharacteristic
299C22#57511yescharacteristic
300C22#57511yescharacteristic
301C22#57711yescharacteristic
302C22#57811yescharacteristic
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4#54011yescharacteristic
316C4#54211yescharacteristic
317C4#55311yescharacteristic
318C4#56211yescharacteristic
319C2—81no—
320C2—81no—
321C2#270111yescharacteristic
322C2#319711yescharacteristic
323C2#320111yescharacteristic
324C2#330711yescharacteristic
325C2#331311yescharacteristic
326C2#334811yescharacteristic
327C2#334911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2087C2semidirect
4. #2077C2semidirect
5. #2066C2semidirect
6. #2036C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
10. #1943C2semidirect
11. #1935C2semidirect
12. #1873C2semidirect
13. #1854C2semidirect
14. #1854C2semidirect
15. #1825C2semidirect

Maximal subgroups

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

subgroupindex
#24852
#24152
#20872
#20772
#20662
#20362
#20352
#19552
#19432
#19352
#18732
#18542
#18252