Finite Group Databasebeta

← #8331

Subgroups — #8331

Order 256.

648 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#525C211yesmaximal, characteristic
2#1779C211yesmaximal, characteristic, complemented
3C2×#328C212yesmaximal, complemented
4C2×#327C212yesmaximal, complemented
5C22⋊#95C211yesmaximal, characteristic, complemented
6C22×#41C411yescharacteristic, complemented
7C24×C4C2211yescharacteristic
8#525—24no—
9#525—24no—
10C2×#123—21no—
11C2×C22⋊Q8C411yescharacteristic, complemented
12C2×C22⋊Q8C2211yescharacteristic
13C2×#121C411yescharacteristic, complemented
14C2×#121C2211yescharacteristic
15C2×#120—21no—
16C2×C4×D4—21no—
17C2×C4×D4—21no—
18C2×#117C411yescharacteristic, complemented
19C22×C22⋊C4—21no—
20C2×C22⋊C8—21no—
21C2×C22⋊C8—21no—
22C2×C22⋊C8C2212yes—
23C2×#95—21no—
24C2×#95C2212yes—
25#328—24no—
26#327—24no—
27C25D411yescharacteristic
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—42no—
31C2×#41—42no—
32C22×Q8C2×C411yescharacteristic
33C22×D4—21no—
34C22×D4—21no—
35C22×D4—21no—
36C22×D4C2×C411yescharacteristic
37C23×C4—21no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—42no—
42C23×C4—42no—
43C23×C4C2×C411yescharacteristic
44C23×C4C2×C411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8—22no—
50#123—28no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8D414yes—
55#121—22no—
56#121—22no—
57#121—22no—
58#121D414yes—
59#120—28no—
60C4×D4—28no—
61C4×D4—28no—
62#117—22no—
63#117—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2×C411yescharacteristic
67C2×C4⋊C4C2×C411yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C42—21no—
77C22⋊C8—22no—
78C22⋊C8—22no—
79C22⋊C8—42no—
80C22⋊C8—42no—
81#95—22no—
82#95—22no—
83#95—42no—
84C24—22no—
85C24—22no—
86C24—21no—
87C24—21no—
88C24—21no—
89C24—22no—
90C24—22no—
91C24—41no—
92C24—41no—
93C24C22⋊C411yescharacteristic
94C24C22⋊C411yescharacteristic
95C24C2×D411yescharacteristic
96#41—48no—
97#41—48no—
98C2×Q8—21no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8C22⋊C412yescomplemented
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—24no—
105C2×D4—21no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4C22⋊C412yescomplemented
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—48no—
141C22×C4—48no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4C22⋊C411yescharacteristic
151C22×C4C22⋊C411yescharacteristic
152C22×C4C22×C411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D412yes—
155C22×C4C2×D412yes—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C4⋊C4—24no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—24no—
163C4⋊C4C22⋊C412yescomplemented
164C4⋊C4C22⋊C412yescomplemented
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—48no—
171C42—24no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—24no—
177C23—24no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—44no—
192C23—44no—
193C23—44no—
194C23—44no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23#9912yes—
206C23#10412yes—
207C23C2×C22⋊C411yescharacteristic
208C23#12012yes—
209C23#13512yes—
210C23AGL(1,ℤ/8)11yescharacteristic
211C23#13711yescharacteristic
212Q8—42no—
213Q8—42no—
214D4—42no—
215D4—42no—
216D4—48no—
217D4—48no—
218D4—44no—
219D4—44no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—48no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4#9912yes—
256C2×C4C2×C22⋊C411yescharacteristic
257C2×C4C2×C22⋊C411yescharacteristic
258C2×C4#12012yes—
259C8—82no—
260C22—22no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—48no—
275C22—48no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22#37911yescharacteristic
292C22C2×#9911yescharacteristic
293C22C2×#9911yescharacteristic
294C22#41611yescharacteristic
295C22#41711yescharacteristic
296C22C2×#10411yescharacteristic
297C22#42111yescharacteristic
298C22#44912yes—
299C22#45212yes—
300C4—22no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—84no—
309C2—22no—
310C2—22no—
311C2—42no—
312C2—42no—
313C2—82no—
314C2#150111yescharacteristic
315C2#150411yescharacteristic
316C2#151311yescharacteristic
317C2#151611yescharacteristic
318C2#178111yescharacteristic
319C2#178811yescharacteristic
320C2#178911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#525C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1779C2semidirect
3. C2×#328C2semidirect
4. C2×#327C2semidirect
5. C22⋊#95C2semidirect
6. C22×#41C4semidirect
11. C2×C22⋊Q8C4semidirect
13. C2×#121C4semidirect
18. C2×#117C4semidirect
101. C2×Q8C22⋊C4semidirect
115. C2×D4C22⋊C4semidirect
163. C4⋊C4C22⋊C4semidirect
164. C4⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
#17792
C2×#3282
C2×#3272
C22⋊#952