Finite Group Databasebeta

← #30600

Subgroups — #30600

Order 256.

412 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3#2429C211yesmaximal, characteristic
4#2423C211yesmaximal, characteristic
5#2095C211yesmaximal, characteristic, complemented
6#2087C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2043C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic
10#2001C211yesmaximal, characteristic
11#1892C211yesmaximal, characteristic
12#1878C211yesmaximal, characteristic, complemented
13#1863C211yesmaximal, characteristic, complemented
14#1850C211yesmaximal, characteristic
15#1803C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25#467—24no—
26#462—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40#399—21no—
41#398—21no—
42#398—21no—
43#396—21no—
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#389—21no—
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#388—21no—
50#388—21no—
51#387—21no—
52#384C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C2×Q16—22no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16D414yes—
71C22×C8—22no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C4⋊Q8—22no—
79C4⋊Q8—41no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—41no—
89C4×Q8—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#107—22no—
117#107—41no—
118C4⋊C8—22no—
119C4⋊C8—41no—
120#103—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103—41no—
127#103—41no—
128C22⋊C8—41no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24C2×D411yescharacteristic
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8C2×D412yes—
156C2×Q8C2×D412yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177Q16—44no—
178Q16—44no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—44no—
191C2×C8C2×D412yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4#4112yes—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—81no—
218C22⋊C4Q1614yescomplemented
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—42no—
224C42—42no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23C2×Q1612yes—
232C23C22×D411yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
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—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4C2×Q1612yes—
287C2×C4C2×Q1612yes—
288C2×C4C22×D411yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C2×C4H(5,𝔽2)11yescharacteristic
292C2×C4H(5,𝔽2)11yescharacteristic
293C2×C42−1+411yescharacteristic
294C8—22no—
295C8—41no—
296C8—41no—
297C8—41no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22#53711yescharacteristic
306C22D4211yescharacteristic
307C22#54611yescharacteristic
308C22C22×Q1611yescharacteristic
309C22#57511yescharacteristic
310C22#57711yescharacteristic
311C22#57811yescharacteristic
312C4—22no—
313C4—22no—
314C4—41no—
315C4—42no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4#53411yescharacteristic
326C4#54011yescharacteristic
327C4#54711yescharacteristic
328C4#56211yescharacteristic
329C2—42no—
330C2#267111yescharacteristic
331C2#324911yescharacteristic
332C2#325811yescharacteristic
333C2#327111yescharacteristic
334C2D4×Q1611yescharacteristic
335C2#329911yescharacteristic
336C2#331511yescharacteristic
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×#462C2semidirect
5. #2095C2semidirect
6. #2087C2semidirect
7. #2056C2semidirect
8. #2043C2semidirect
12. #1878C2semidirect
13. #1863C2semidirect
15. #1803C2semidirect
218. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4622
#24292
#24232
#20952
#20872
#20562
#20432
#20362
#20012
#18922
#18782
#18632
#18502
#18032