Finite Group Databasebeta

← #25959

Subgroups — #25959

Order 256.

483 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#416C211yesmaximal, characteristic
3#2439C211yesmaximal, characteristic, complemented
4#2341C211yesmaximal, characteristic
5#2092C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1863C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1850C211yesmaximal, characteristic, complemented
13#1808C211yesmaximal, characteristic, complemented
14#1805C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#477—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29#416—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#396—21no—
40#393—21no—
41#391—21no—
42#390—21no—
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#387—21no—
49#382C2211yescharacteristic
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
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—41no—
65C2×#41—41no—
66C2×#41Q814yescomplemented
67C22×Q8—21no—
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×Q8—41no—
85C4×Q8—41no—
86#117—22no—
87#117—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#107—22no—
114#107—22no—
115#107—41no—
116#106—41no—
117C4⋊C8—22no—
118C4⋊C8—41no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103D414yes—
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127C22⋊C8—22no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95D411yescharacteristic
135#95D411yescharacteristic
136C24—21no—
137C24C2×D411yescharacteristic
138#41—28no—
139#41—44no—
140#41—44no—
141#41—44no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—81no—
148C2×Q8C2×D412yes—
149C2×Q8C2×Q812yes—
150C2×Q8#4112yes—
151C2×D4—24no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4C2×Q812yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—24no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
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—81no—
211C4⋊C4C2×D412yes—
212C22⋊C4—22no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—81no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—42no—
226C42—42no—
227C42—42no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23C22⋊Q812yes—
242C23C22×D411yescharacteristic
243Q8—22no—
244Q8—22no—
245Q8—22no—
246Q8—84no—
247Q8C22⋊Q814yes—
248D4—22no—
249D4—22no—
250D4—48no—
251D4—41no—
252D4—41no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—48no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4C22⋊Q812yes—
293C2×C4#13512yes—
294C2×C4C22×D411yescharacteristic
295C2×C4C22×Q811yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C8—22no—
301C8—41no—
302C8—41no—
303C8—41no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22C2×C22⋊Q811yescharacteristic
319C22D4211yescharacteristic
320C22#54611yescharacteristic
321C22C2×#13511yescharacteristic
322C22#57511yescharacteristic
323C22#57611yescharacteristic
324C22#57811yescharacteristic
325C4—22no—
326C4—24no—
327C4—22no—
328C4—41no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4C2×C22⋊Q811yescharacteristic
338C4#54311yescharacteristic
339C4#54611yescharacteristic
340C4#55111yescharacteristic
341C2—42no—
342C2—82no—
343C2#243711yescharacteristic
344C2#307611yescharacteristic
345C2#308311yescharacteristic
346C2#326811yescharacteristic
347C2#328511yescharacteristic
348C2#330511yescharacteristic
349C2#331311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
3. #2439C2semidirect
5. #2092C2semidirect
6. #2067C2semidirect
7. #2067C2semidirect
8. #2023C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect
11. #1854C2semidirect
12. #1850C2semidirect
13. #1808C2semidirect
14. #1805C2semidirect
66. C2×#41Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4162
#24392
#23412
#20922
#20672
#20232
#18652
#18632
#18542
#18502
#18082
#18052
#17882