Finite Group Databasebeta

← #24337

Subgroups — #24337

Order 256.

535 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic
2C2×#479C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic
4#2581C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#2026C211yesmaximal, characteristic, complemented
8#2024C211yesmaximal, characteristic, complemented
9#1991C211yesmaximal, characteristic, complemented
10#1984C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic
13#1864C211yesmaximal, characteristic
14#1862C211yesmaximal, characteristic
15#1804C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17#554—24no—
18#554—24no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8—21no—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C22×C4⋊C4—21no—
31#479—24no—
32C4×SD16—24no—
33C2×#107C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#400—21no—
45#395—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#387—21no—
51#387—21no—
52#382C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8D411yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16Q814yescomplemented
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79#125—22no—
80#125—22no—
81#125—22no—
82#125D414yes—
83C22⋊Q8—28no—
84C22⋊Q8—28no—
85C22⋊Q8—28no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4D414yes—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122#106—22no—
123#106—22no—
124C4⋊C8—22no—
125#103—22no—
126#103—22no—
127#103—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131C4×C8—22no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140C24—21no—
141C2×Q8—24no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8C2×Q812yes—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4C2×D412yes—
157C2×D4C2×Q812yes—
158C2×D4#4112yes—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2411yescharacteristic
182SD16—28no—
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—21no—
194C2×C8C2×Q812yes—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—21no—
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—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4C2×D412yes—
216C4⋊C4C2×D412yes—
217C4⋊C4C2×D412yes—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—42no—
226C42C2×D412yes—
227C42C2×D412yes—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23C22×D411yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—44no—
239Q8—41no—
240Q8—41no—
241Q8—84no—
242D4—22no—
243D4—22no—
244D4—22no—
245D4C22⋊Q814yes—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—48no—
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—84no—
284C2×C4—84no—
285C2×C4#12012yes—
286C2×C4#12012yes—
287C2×C4#13512yes—
288C2×C4C22×D411yescharacteristic
289C2×C4C22×D411yescharacteristic
290C2×C4C22×Q811yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C42−1+411yescharacteristic
293C2×C42−1+411yescharacteristic
294C8—22no—
295C8—41no—
296C8—41no—
297C8—41no—
298C22—24no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22C2×#12011yescharacteristic
306C22#54811yescharacteristic
307C22D4×Q811yescharacteristic
308C22C2×#13511yescharacteristic
309C22#57511yescharacteristic
310C22#57711yescharacteristic
311C22#57711yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—41no—
317C4—42no—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—42no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4C2×C22⋊Q811yescharacteristic
326C4#53611yescharacteristic
327C4#54811yescharacteristic
328C4D4×Q811yescharacteristic
329C2—44no—
330C2#240911yescharacteristic
331C2#300211yescharacteristic
332C2#300911yescharacteristic
333C2#336011yescharacteristic
334C2#337611yescharacteristic
335C2#338111yescharacteristic
336C2#338811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#479C2semidirect
4. #2581C2semidirect
5. #2059C2semidirect
6. #2058C2semidirect
7. #2026C2semidirect
8. #2024C2semidirect
9. #1991C2semidirect
10. #1984C2semidirect
72. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4792
C2×C4×SD162
#25812
#20592
#20582
#20262
#20242
#19912
#19842
#18772
#18642
#18622
#18042