Finite Group Databasebeta

← #25973

Subgroups — #25973

Order 256.

460 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#479C211yesmaximal, characteristic, complemented
2C2×#419C211yesmaximal, characteristic
3#2441C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic
5#2096C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2026C211yesmaximal, characteristic, complemented
9#1865C211yesmaximal, characteristic, complemented
10#1863C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1851C211yesmaximal, characteristic
13#1809C211yesmaximal, characteristic
14#1803C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#479—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29#419—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#399—21no—
40#396—21no—
41#391—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#389—21no—
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#387—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—22no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8Q814yes—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79#125—22no—
80#125—41no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8—41no—
84C4×Q8—41no—
85#117—22no—
86#117—41no—
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—21no—
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⋊C4—21no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—22no—
112#107—41no—
113#106—22no—
114#106—41no—
115C4⋊C8—22no—
116C4⋊C8—41no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—41no—
123#103—41no—
124#103D414yes—
125C22⋊C8—22no—
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×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×Q812yes—
177C22×C4#4112yes—
178C22×C4C2411yescharacteristic
179M16—44no—
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—21no—
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—21no—
202C4⋊C4—21no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4C2×D412yes—
212C22⋊C4—22no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—81no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—42no—
222C42—42no—
223C42—42no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23C22⋊Q812yes—
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—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
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—21no—
262C2×C4—21no—
263C2×C4—48no—
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×C4C22⋊Q812yes—
284C2×C4#13712yes—
285C2×C4C22×D411yescharacteristic
286C2×C4C22×Q811yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4H(5,𝔽2)11yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C8—22no—
292C8—41no—
293C8—41no—
294C8—41no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22C2×C22⋊Q811yescharacteristic
303C22D4211yescharacteristic
304C22#54611yescharacteristic
305C22C2×#13711yescharacteristic
306C22#57511yescharacteristic
307C22#57711yescharacteristic
308C22#57811yescharacteristic
309C4—22no—
310C4—24no—
311C4—22no—
312C4—42no—
313C4—41no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4C2×C22⋊Q811yescharacteristic
323C4#54311yescharacteristic
324C4#54611yescharacteristic
325C4#55111yescharacteristic
326C2—42no—
327C2#243711yescharacteristic
328C2#307811yescharacteristic
329C2#308211yescharacteristic
330C2#326911yescharacteristic
331C2#327811yescharacteristic
332C2#331211yescharacteristic
333C2#332611yescharacteristic
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. #2096C2semidirect
6. #2068C2semidirect
7. #2068C2semidirect
8. #2026C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect
14. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C2×#4192
#24412
Q8×C22⋊C42
#20962
#20682
#20262
#18652
#18632
#18542
#18512
#18092
#18032
#17872