Finite Group Databasebeta

← #24352

Subgroups — #24352

Order 256.

599 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3501C211yesmaximal, characteristic, complemented
2#2626C212yesmaximal, complemented
3#2321C211yesmaximal, characteristic
4#2062C212yesmaximal, complemented
5#2029C212yesmaximal, complemented
6#1996C212yesmaximal, complemented
7#1889C212yesmaximal
8#1867C212yesmaximal, complemented
9#1799C211yesmaximal, characteristic, complemented
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544—21no—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×#120C2211yescharacteristic
20#518C2212yes—
21C2×C4×D4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C2×#99C2212yes—
24#399—22no—
25#396—24no—
26#393—22no—
27#390—21no—
28#388—21no—
29#388—21no—
30#386—21no—
31#386C2211yescharacteristic
32#385C2211yescharacteristic
33#385C2214yes—
34#380C2212yes—
35#379C2211yescharacteristic
36#379C2214yes—
37C4×C22⋊C4C2211yescharacteristic
38#342C2212yes—
39#342C2214yes—
40#342C2214yes—
41C25C2311yescharacteristic
42C22×Q8—21no—
43C22×Q8D412yescomplemented
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51C4⋊Q8—24no—
52#125—22no—
53#125—24no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66#120—22no—
67#120Q814yescomplemented
68C4×Q8—22no—
69C4×Q8—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—24no—
73C4×D4—24no—
74#117—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D414yescomplemented
82C2×C4⋊C4D412yescomplemented
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4C2311yescharacteristic
96C2×C22⋊C4C2314yes—
97C2×C22⋊C4C2311yescharacteristic
98C2×C22⋊C4C2314yes—
99C2×C22⋊C4C2311yescharacteristic
100C2×C42—21no—
101C2×C42D411yescharacteristic
102#99—24no—
103#95—21no—
104#95—22no—
105#95—21no—
106#95—24no—
107#95—21no—
108C24—22no—
109C24—21no—
110C24—21no—
111C24—22no—
112C24—81no—
113C24—81no—
114C24C2×D411yescharacteristic
115C24C2×Q812yescomplemented
116C24C2411yescharacteristic
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—82no—
122C2×Q8—82no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—24no—
126C2×D4—81no—
127C2×D4—81no—
128C2×D4C2×Q812yes—
129C2×D4#4112yes—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D414yes—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D414yes—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D412yes—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D412yes—
151C4⋊C4—22no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—24no—
162C4⋊C4—82no—
163C4⋊C4—81no—
164C4⋊C4—84no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—24no—
169C22⋊C4—21no—
170C22⋊C4—22no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—24no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C22⋊C4—81no—
179C22⋊C4—84no—
180C22⋊C4—81no—
181C22⋊C4—84no—
182C22⋊C4—81no—
183C22⋊C4C2×Q812yescomplemented
184C42—24no—
185C42—21no—
186C42—21no—
187C42—81no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—24no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—44no—
197C23—41no—
198C23—44no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—42no—
205C23—82no—
206C23—82no—
207C23—84no—
208C23C22⋊Q812yes—
209C23C22×D411yescharacteristic
210C23C22×D412yes—
211C23C22×Q811yescharacteristic
212C23C2×#4111yescharacteristic
213C232−1+412yes—
214Q8—44no—
215Q8—82no—
216Q8—82no—
217Q8—82no—
218D4—24no—
219D4—24no—
220D4—84no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—44no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4#12011yescharacteristic
253C2×C4#12011yescharacteristic
254C2×C4#12012yes—
255C2×C4C22⋊Q812yes—
256C22—22no—
257C22—21no—
258C22—22no—
259C22—21no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—48no—
268C22—41no—
269C22—41no—
270C22—42no—
271C22—81no—
272C22—84no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22C2×#12011yescharacteristic
278C22C2×C22⋊Q811yescharacteristic
279C22#53611yescharacteristic
280C22#54812yes—
281C22D4×Q812yes—
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—41no—
288C4—44no—
289C4—44no—
290C4—41no—
291C4—84no—
292C4—81no—
293C4—81no—
294C2—21no—
295C2—22no—
296C2—21no—
297C2—22no—
298C2—42no—
299C2—82no—
300C2#240911yescharacteristic
301C2#302611yescharacteristic
302C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radicalC2×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3501C2semidirect
2. #2626C2semidirect
4. #2062C2semidirect
5. #2029C2semidirect
6. #1996C2semidirect
8. #1867C2semidirect
9. #1799C2semidirect
43. C22×Q8D4semidirect
67. #120Q8semidirect
81. C2×C4⋊C4D4semidirect
82. C2×C4⋊C4D4semidirect
115. C24C2×Q8semidirect
183. C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35012
#26262
#23212
#20622
#20292
#19962
#18892
#18672
#17992