Finite Group Databasebeta

← #24350

Subgroups — #24350

Order 256.

731 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 137 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#2848C212yesmaximal, complemented
3C4×#120C211yesmaximal, characteristic
4#2062C212yesmaximal, complemented
5#2029C212yesmaximal, complemented
6#1996C212yesmaximal, complemented
7#1889C212yesmaximal
8#1867C212yesmaximal, complemented
9#1799C211yesmaximal, characteristic, complemented
10C24×C4C2211yescharacteristic
11#554—22no—
12#554—24no—
13D4×Q8—22no—
14#544—21no—
15#543—21no—
16#543C2212yes—
17#533—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—22no—
20C2×C22⋊Q8C2212yes—
21C2×#120C2211yescharacteristic
22#518C2212yes—
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×#99C2212yes—
26#386—21no—
27#386C2211yescharacteristic
28#385—21no—
29#385—24no—
30#385C2211yescharacteristic
31#385C2214yes—
32#380—22no—
33#380C2212yes—
34#379C2214yes—
35#379C2211yescharacteristic
36C4×C22⋊C4—21no—
37C4×C22⋊C4C2211yescharacteristic
38#342C2212yes—
39#342C2214yes—
40#342C2214yes—
41C25C2311yescharacteristic
42C22×Q8—21no—
43C22×Q8D412yescomplemented
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—21no—
50C23×C4—44no—
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—24no—
57#125—22no—
58#125—24no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69C22⋊Q8—24no—
70C22⋊Q8—24no—
71C22⋊Q8—48no—
72#120—22no—
73#120Q814yescomplemented
74C4×Q8—22no—
75C4×Q8—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—24no—
79C4×D4—24no—
80C4×D4—44no—
81#117—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D412yescomplemented
89C2×C4⋊C4D414yescomplemented
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4C2311yescharacteristic
99C2×C22⋊C4C2314yes—
100C2×C22⋊C4C2311yescharacteristic
101C2×C22⋊C4C2314yes—
102C2×C22⋊C4C2311yescharacteristic
103C2×C42—21no—
104C2×C42D411yescharacteristic
105#99—24no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—24no—
110#95—22no—
111C24—22no—
112C24—21no—
113C24—21no—
114C24—22no—
115C24—42no—
116C24—42no—
117C24C2×D411yescharacteristic
118C24C2×Q812yescomplemented
119C24C2411yescharacteristic
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—82no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—24no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4C2×Q812yes—
132C2×D4#4112yes—
133C22×C4—24no—
134C22×C4—28no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—24no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D414yes—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D414yes—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D412yes—
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—24no—
170C4⋊C4—22no—
171C4⋊C4—24no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—48no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—22no—
182C22⋊C4—24no—
183C22⋊C4—21no—
184C22⋊C4—22no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—48no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—48no—
194C22⋊C4C2×Q812yescomplemented
195C42—24no—
196C42—21no—
197C42—21no—
198C42—42no—
199C23—24no—
200C23—24no—
201C23—22no—
202C23—22no—
203C23—21no—
204C23—22no—
205C23—24no—
206C23—21no—
207C23—41no—
208C23—44no—
209C23—41no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—41no—
214C23—44no—
215C23—48no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23C22⋊Q812yescomplemented
222C23C22×D411yescharacteristic
223C23C22×D412yes—
224C23C22×Q811yescharacteristic
225C23C2×#4111yescharacteristic
226C232−1+412yes—
227Q8—44no—
228Q8—82no—
229Q8—82no—
230Q8—82no—
231D4—24no—
232D4—24no—
233D4—48no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—48no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4#12011yescharacteristic
270C2×C4#12012yes—
271C2×C4#12011yescharacteristic
272C2×C4C22⋊Q812yescomplemented
273C22—22no—
274C22—21no—
275C22—22no—
276C22—21no—
277C22—24no—
278C22—28no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—44no—
287C22—42no—
288C22—42no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—42no—
295C22—81no—
296C22C2×#12011yescharacteristic
297C22C2×C22⋊Q811yescharacteristic
298C22#53611yescharacteristic
299C22#54812yes—
300C22D4×Q812yes—
301C4—22no—
302C4—21no—
303C4—22no—
304C4—42no—
305C4—42no—
306C4—41no—
307C4—44no—
308C4—44no—
309C4—41no—
310C4—84no—
311C4—81no—
312C4B(4,𝔽2)12yes—
313C2—21no—
314C2—22no—
315C2—21no—
316C2—22no—
317C2—44no—
318C2—42no—
319C2#240911yescharacteristic
320C2C2×B(4,𝔽2)11yescharacteristic
321C2#302511yescharacteristic
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. #2848C2semidirect
4. #2062C2semidirect
5. #2029C2semidirect
6. #1996C2semidirect
8. #1867C2semidirect
9. #1799C2semidirect
43. C22×Q8D4semidirect
73. #120Q8semidirect
88. C2×C4⋊C4D4semidirect
89. C2×C4⋊C4D4semidirect
118. C24C2×Q8semidirect
194. C22⋊C4C2×Q8semidirect
221. C23C22⋊Q8semidirect
272. C2×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#35012
#28482
C4×#1202
#20622
#20292
#19962
#18892
#18672
#17992