Finite Group Databasebeta

← #27241

Subgroups — #27241

Order 256.

465 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#475C211yesmaximal, characteristic, complemented
2C2×#451C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic
6#2056C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic
8#2024C211yesmaximal, characteristic, complemented
9#1971C211yesmaximal, characteristic, complemented
10#1927C211yesmaximal, characteristic, complemented
11#1924C211yesmaximal, characteristic
12#1878C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1787C211yesmaximal, characteristic
15#1763C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#475—24no—
24#451—24no—
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39#400—21no—
40#393—21no—
41#391—21no—
42#391—21no—
43#391—21no—
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#388—21no—
49#380—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—21no—
64C22×Q8—21no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—41no—
86C22⋊Q8—22no—
87C22⋊Q8—41no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—41no—
91C4×Q8—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
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#106—22no—
112#106—22no—
113C4⋊C8—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—41no—
119#103—41no—
120#103—41no—
121#103D414yes—
122C22⋊C8—22no—
123C22⋊C8—41no—
124#97—41no—
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#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24C2×D411yescharacteristic
139C2×Q8—24no—
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—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—42no—
162C2×Q8—81no—
163C2×Q8C2×D412yes—
164C2×Q8#4112yes—
165C2×Q8#4112yes—
166C2×Q8#4112yes—
167C22×C4—21no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2411yescharacteristic
190Q16—44no—
191Q16—44no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8C2×D412yes—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4C2×D412yes—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C42—42no—
229C42—81no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#13712yes—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—22no—
242Q8—22no—
243Q8—48no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—84no—
251Q8#12414yes—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—48no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4#13712yes—
296C2×C4C22×D411yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4H(5,𝔽2)11yescharacteristic
301C2×C4H(5,𝔽2)11yescharacteristic
302C8—22no—
303C8—41no—
304C8—41no—
305C8—41no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22#52511yescharacteristic
314C22D4211yescharacteristic
315C22#54611yescharacteristic
316C22C2×#13711yescharacteristic
317C22C2×#13711yescharacteristic
318C22#57711yescharacteristic
319C22#57811yescharacteristic
320C4—24no—
321C4—22no—
322C4—41no—
323C4—41no—
324C4—42no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4C2×#12411yescharacteristic
334C4#53811yescharacteristic
335C4#54611yescharacteristic
336C4#55111yescharacteristic
337C2—42no—
338C2#247211yescharacteristic
339C2#311311yescharacteristic
340C2#312111yescharacteristic
341C2#326611yescharacteristic
342C2#327011yescharacteristic
343C2#331611yescharacteristic
344C2#332811yescharacteristic
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
6. #2056C2semidirect
8. #2024C2semidirect
9. #1971C2semidirect
10. #1927C2semidirect
12. #1878C2semidirect
13. #1878C2semidirect
15. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4512
#24922
Q8×C22⋊C42
#20772
#20562
#20352
#20242
#19712
#19272
#19242
#18782
#17872
#17632