Finite Group Databasebeta

← #25955

Subgroups — #25955

Order 256.

495 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#475C211yesmaximal, characteristic, complemented
2C22×#103C211yesmaximal, characteristic
3#2440C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic
5#2096C211yesmaximal, characteristic, complemented
6#2066C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2024C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1865C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic
12#1850C211yesmaximal, characteristic
13#1807C211yesmaximal, characteristic
14#1803C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#475—24no—
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103—24no—
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40#398—21no—
41#395—21no—
42#391—21no—
43#391—21no—
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#385—21no—
49#384C2211yescharacteristic
50#380—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8Q814yescomplemented
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4⋊Q8—22no—
82C4⋊Q8—41no—
83C22⋊Q8—44no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—41no—
87C4×Q8—41no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
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—41no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#106—22no—
113#106—22no—
114#106—41no—
115#106—41no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—22no—
123#103—41no—
124#103—41no—
125#103D414yes—
126C22⋊C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95D411yescharacteristic
136#95D411yescharacteristic
137C24C2×D411yescharacteristic
138C2×Q8—24no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—28no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—81no—
156C2×Q8C2×D412yes—
157C2×Q8C2×Q812yes—
158C2×Q8C2×Q812yes—
159C2×Q8#4112yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—81no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×Q812yes—
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—44no—
198C2×C8C2×D412yes—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4C2×D412yes—
219C22⋊C4—22no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4SD1614yes—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C42—42no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23C22⋊Q812yes—
234C23C2×SD1612yes—
235C23#13712yes—
236C23C22×D411yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—22no—
242Q8—48no—
243Q8—41no—
244Q8—41no—
245Q8—84no—
246Q8C22⋊Q814yescomplemented
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—48no—
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×C4—41no—
287C2×C4—41no—
288C2×C4—84no—
289C2×C4C22⋊Q812yes—
290C2×C4C2×SD1612yes—
291C2×C4C2×SD1612yes—
292C2×C4C22×D411yescharacteristic
293C2×C4C22×Q811yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C8—22no—
299C8—41no—
300C8—41no—
301C8—41no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22#47514yescomplemented
309C22C2×C22⋊Q811yescharacteristic
310C22D4211yescharacteristic
311C22#54611yescharacteristic
312C22C22×SD1611yescharacteristic
313C22C2×#13711yescharacteristic
314C22#57711yescharacteristic
315C22#57811yescharacteristic
316C4—22no—
317C4—24no—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—41no—
322C4—42no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4C2×C22⋊Q811yescharacteristic
330C4#54311yescharacteristic
331C4#54611yescharacteristic
332C4#55111yescharacteristic
333C2—24no—
334C2#243711yescharacteristic
335C2C2×#47511yescharacteristic
336C2#308211yescharacteristic
337C2#327011yescharacteristic
338C2D4×SD1611yescharacteristic
339C2#329811yescharacteristic
340C2#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
1. C2×#475C2semidirect
5. #2096C2semidirect
6. #2066C2semidirect
7. #2066C2semidirect
8. #2024C2semidirect
9. #1865C2semidirect
10. #1865C2semidirect
14. #1803C2semidirect
68. C22×Q8Q8semidirect
246. Q8C22⋊Q8semidirect
308. C22#475semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C22×#1032
#24402
Q8×C22⋊C42
#20962
#20662
#20242
#18652
#18532
#18502
#18072
#18032
#17872