Finite Group Databasebeta

← #24334

Subgroups — #24334

Order 256.

521 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×#557C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic
3C2×C4×SD16C211yesmaximal, characteristic
4#2580C211yesmaximal, characteristic
5#2059C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic
8#2023C211yesmaximal, characteristic
9#1991C211yesmaximal, characteristic, complemented
10#1984C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic
12#1880C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1863C211yesmaximal, characteristic, complemented
15#1805C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17#557—24no—
18#557—24no—
19C2×C4⋊Q8—21no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×#125—21no—
23C2×#125C2211yescharacteristic
24C2×#125C2211yescharacteristic
25C2×C4×Q8—21no—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31#478—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#399—21no—
45#393—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#385—21no—
51#385—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—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16Q814yescomplemented
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—28no—
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78#125—28no—
79#125—28no—
80#125—22no—
81#125—22no—
82#125—22no—
83#125D414yes—
84C4×Q8—28no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8D414yes—
93C4×D4—22no—
94C4×D4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
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×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124C2×C42C2311yescharacteristic
125#107—22no—
126#106—22no—
127#106—22no—
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135C4×C8—22no—
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—41no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8C2×D412yes—
151C2×Q8C2×Q812yes—
152C2×Q8#4112yes—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4C2×Q812yes—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2411yescharacteristic
176SD16—28no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
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×C8C2×Q812yes—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4C2×D412yes—
212C4⋊C4C2×D412yes—
213C4⋊C4C2×D412yes—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—81no—
218C22⋊C4—81no—
219C42—24no—
220C42—24no—
221C42—24no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42C2×D412yes—
227C42C2×D412yes—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23C22×D411yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—44no—
240Q8—48no—
241Q8C22⋊Q814yes—
242D4—22no—
243D4—22no—
244D4—41no—
245D4—41no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
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—84no—
283C2×C4#12012yes—
284C2×C4#12012yes—
285C2×C4#13512yes—
286C2×C4C22×D411yescharacteristic
287C2×C4C22×D411yescharacteristic
288C2×C4C22×Q811yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C42−1+411yescharacteristic
291C2×C42−1+411yescharacteristic
292C8—22no—
293C8—41no—
294C8—41no—
295C8—41no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22C2×#12011yescharacteristic
304C22#54811yescharacteristic
305C22D4×Q811yescharacteristic
306C22C2×#13511yescharacteristic
307C22#57511yescharacteristic
308C22#57711yescharacteristic
309C22#57711yescharacteristic
310C4—24no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—42no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4C2×C22⋊Q811yescharacteristic
324C4#53611yescharacteristic
325C4#54811yescharacteristic
326C4D4×Q811yescharacteristic
327C2—82no—
328C2#240911yescharacteristic
329C2#300211yescharacteristic
330C2#301011yescharacteristic
331C2#336511yescharacteristic
332C2#337111yescharacteristic
333C2#338111yescharacteristic
334C2#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
1. C2×#557C2semidirect
5. #2059C2semidirect
6. #2058C2semidirect
9. #1991C2semidirect
10. #1984C2semidirect
13. #1865C2semidirect
14. #1863C2semidirect
15. #1805C2semidirect
70. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4782
C2×C4×SD162
#25802
#20592
#20582
#20252
#20232
#19912
#19842
#18802
#18652
#18632
#18052