Finite Group Databasebeta

← #25965

Subgroups — #25965

Order 256.

473 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#479C211yesmaximal, characteristic, complemented
2C22×#103C211yesmaximal, characteristic
3#2442C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2026C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1863C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic
12#1850C211yesmaximal, characteristic
13#1808C211yesmaximal, characteristic
14#1805C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#479—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#400—21no—
40#398—21no—
41#391—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#385—21no—
48#382C2211yescharacteristic
49#380—21no—
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—22no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8Q814yes—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80#125—22no—
81#125—41no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×Q8—41no—
85C4×Q8—41no—
86C2×C4⋊C4—22no—
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⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—41no—
111#106—22no—
112#106—41no—
113C4⋊C8—22no—
114C4⋊C8—41no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—41no—
121#103—41no—
122#103D414yes—
123C22⋊C8—22no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95D411yescharacteristic
135#95D411yescharacteristic
136C24C2×D411yescharacteristic
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—28no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8C2×D412yes—
153C2×Q8C2×Q812yes—
154C2×Q8C2×Q812yes—
155C2×Q8#4112yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—81no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×Q812yes—
180C22×C4#4112yes—
181C22×C4C2411yescharacteristic
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—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—44no—
194C2×C8C2×D412yes—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4C2×D412yes—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—42no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23C22⋊Q812yes—
229C23#13512yes—
230C23#13712yes—
231C23C22×D411yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—48no—
238Q8—41no—
239Q8—41no—
240Q8C22⋊Q814yes—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—48no—
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—84no—
285C2×C4C22⋊Q812yes—
286C2×C4C22×D411yescharacteristic
287C2×C4C22×Q811yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C2×C4H(5,𝔽2)11yescharacteristic
292C8—22no—
293C8—41no—
294C8—41no—
295C8—41no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22#47914yescomplemented
303C22C2×C22⋊Q811yescharacteristic
304C22D4211yescharacteristic
305C22#54611yescharacteristic
306C22C2×#13511yescharacteristic
307C22C2×#13711yescharacteristic
308C22#57711yescharacteristic
309C22#57811yescharacteristic
310C4—22no—
311C4—24no—
312C4—42no—
313C4—41no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4C2×C22⋊Q811yescharacteristic
324C4#54311yescharacteristic
325C4#54611yescharacteristic
326C4#55111yescharacteristic
327C2—24no—
328C2#243711yescharacteristic
329C2C2×#47911yescharacteristic
330C2#308211yescharacteristic
331C2#326611yescharacteristic
332C2#328511yescharacteristic
333C2#330111yescharacteristic
334C2#331211yescharacteristic
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×#479C2semidirect
5. #2097C2semidirect
6. #2068C2semidirect
7. #2066C2semidirect
8. #2026C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect
14. #1805C2semidirect
302. C22#479semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C22×#1032
#24422
Q8×C22⋊C42
#20972
#20682
#20662
#20262
#18652
#18632
#18532
#18502
#18082
#18052
#17872