Finite Group Databasebeta

← #31120

Subgroups — #31120

Order 256.

409 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
1C2×#483C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic, complemented
3#2440C211yesmaximal, characteristic
4#2429C211yesmaximal, characteristic
5#2084C211yesmaximal, characteristic, complemented
6#2066C211yesmaximal, characteristic, complemented
7#2043C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic
9#2035C211yesmaximal, characteristic
10#2024C211yesmaximal, characteristic, complemented
11#1956C211yesmaximal, characteristic, complemented
12#1941C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic
14#1872C211yesmaximal, characteristic
15#1865C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#483—24no—
24#475—24no—
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#398—21no—
42#398—21no—
43#396—21no—
44#395—21no—
45#391C2211yescharacteristic
46#389—21no—
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#385—21no—
52#384C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—41no—
77C22⋊Q8—41no—
78C22⋊Q8—44no—
79C4×Q8—22no—
80C4×Q8—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#106—22no—
110#106—22no—
111#106—41no—
112#106D414yes—
113C4⋊C8—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—41no—
120#103—41no—
121#103—41no—
122C22⋊C8—22no—
123C22⋊C8—41no—
124C4×C8—41no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—81no—
148C2×Q8#4112yes—
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×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—81no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8C2×D412yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4C2×D412yes—
207C4⋊C4C2×D412yes—
208C22⋊C4—22no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4SD1614yes—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—42no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23C2×SD1612yes—
225C23C22×D411yescharacteristic
226Q8—22no—
227Q8—22no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—84no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
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×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—84no—
276C2×C4C2×SD1612yes—
277C2×C4C2×SD1612yes—
278C2×C4#13712yes—
279C2×C4C22×D411yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C4H(5,𝔽2)11yescharacteristic
282C2×C4H(5,𝔽2)11yescharacteristic
283C2×C42−1+411yescharacteristic
284C2×C42−1+411yescharacteristic
285C8—22no—
286C8—41no—
287C8—41no—
288C8—41no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22#53611yescharacteristic
297C22D4211yescharacteristic
298C22#54611yescharacteristic
299C22C22×SD1611yescharacteristic
300C22C2×#13711yescharacteristic
301C22#57711yescharacteristic
302C22#57811yescharacteristic
303C4—22no—
304C4—22no—
305C4—22no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—42no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4#53711yescharacteristic
317C4#53911yescharacteristic
318C4#54811yescharacteristic
319C4#56211yescharacteristic
320C2—42no—
321C2#268111yescharacteristic
322C2#322911yescharacteristic
323C2#324211yescharacteristic
324C2#327611yescharacteristic
325C2D4×SD1611yescharacteristic
326C2#329811yescharacteristic
327C2#331611yescharacteristic
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
2. C2×#475C2semidirect
5. #2084C2semidirect
6. #2066C2semidirect
7. #2043C2semidirect
10. #2024C2semidirect
11. #1956C2semidirect
12. #1941C2semidirect
15. #1865C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4752
#24402
#24292
#20842
#20662
#20432
#20352
#20242
#19562
#19412
#19232
#18722
#18652