Finite Group Databasebeta

← #26358

Subgroups — #26358

Order 256.

410 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#428C211yesmaximal, characteristic
3#2439C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic
12#1873C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic
15#1851C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#477—24no—
23#428—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#396—21no—
38#393—21no—
39#391—21no—
40#391C2211yescharacteristic
41#389—21no—
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#387—21no—
46#387—21no—
47#385—21no—
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74C22⋊Q8—22no—
75C22⋊Q8Q814yescomplemented
76C4×Q8—22no—
77C4×Q8—41no—
78#117—22no—
79#117—41no—
80C2×C4⋊C4—21no—
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—41no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4D411yescharacteristic
100C2×C22⋊C4D411yescharacteristic
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#107—41no—
109#107D414yes—
110#106—22no—
111#106—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—41no—
118C22⋊C8—22no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95D411yescharacteristic
129#95D411yescharacteristic
130C24C2×D411yescharacteristic
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×Q8C2×Q812yes—
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—42no—
153C22×C4—42no—
154C22×C4—81no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×Q812yes—
161C22×C4#4112yes—
162C22×C4C2411yescharacteristic
163M16—28no—
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—21no—
174C2×C8C2×D412yes—
175C2×C8#4112yes—
176C4⋊C4—24no—
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—21no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4C2×D412yes—
195C4⋊C4C2×D412yes—
196C4⋊C4C2×Q812yes—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—42no—
208C42—81no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23C22⋊Q812yes—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—84no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
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—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—84no—
264C2×C4C22⋊Q812yes—
265C2×C4#13712yes—
266C2×C4C22×D411yescharacteristic
267C2×C4C22×Q811yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C2×C42−1+411yescharacteristic
272C8—22no—
273C8—22no—
274C8—41no—
275C8—41no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22C2×C22⋊Q811yescharacteristic
284C22D4211yescharacteristic
285C22#54711yescharacteristic
286C22C2×#13711yescharacteristic
287C22#57511yescharacteristic
288C22#57611yescharacteristic
289C22#57811yescharacteristic
290C4—22no—
291C4—22no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#52511yescharacteristic
304C4#53911yescharacteristic
305C4D4×Q811yescharacteristic
306C4#55411yescharacteristic
307C2—42no—
308C2#244911yescharacteristic
309C2#307811yescharacteristic
310C2#308311yescharacteristic
311C2#327011yescharacteristic
312C2#327311yescharacteristic
313C2#334911yescharacteristic
314C2#335711yescharacteristic
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
1. C2×#477C2semidirect
5. #2097C2semidirect
6. #2088C2semidirect
7. #2068C2semidirect
8. #2057C2semidirect
9. #1949C2semidirect
10. #1918C2semidirect
13. #1865C2semidirect
75. C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4282
#24392
#23602
#20972
#20882
#20682
#20572
#19492
#19182
#18942
#18732
#18652
#18522
#18512