Finite Group Databasebeta

← #8407

Subgroups — #8407

Order 256.

450 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 87 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×#539C211yesmaximal, characteristic
2#1874C211yesmaximal, characteristic
3#1872C211yesmaximal, characteristic, complemented
4C2×#328C211yesmaximal, characteristic, complemented
5C2×#327C211yesmaximal, characteristic
6C2×#324C211yesmaximal, characteristic, complemented
7#1441C211yesmaximal, characteristic, complemented
8#539—22no—
9#539—22no—
10#539—22no—
11#539C414yescomplemented
12C2×#126—21no—
13C2×#125—21no—
14C2×#123—21no—
15C2×#123—21no—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4—21no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#103—21no—
23C2×#102—21no—
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#95—21no—
27C2×#95—21no—
28C2×#95C2211yescharacteristic
29#340—21no—
30#340—21no—
31#328—22no—
32#327—22no—
33#324—22no—
34C22×Q8D411yescharacteristic
35C22×D4—21no—
36C22×D4D411yescharacteristic
37C23×C4—21no—
38C23×C4—21no—
39C23×C4—21no—
40C23×C4D411yescharacteristic
41C23×C4D411yescharacteristic
42C23×C4C2311yescharacteristic
43C22×C8—21no—
44C22×C8—21no—
45#126—24no—
46#126—24no—
47#125—24no—
48#125—24no—
49#123—24no—
50#123—24no—
51#123—24no—
52#123—24no—
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55C22⋊Q8—21no—
56C22⋊Q8—21no—
57C22⋊Q8C2×C412yes—
58C22⋊Q8D412yes—
59#121—21no—
60#121—21no—
61#121C2×C412yes—
62#121D412yes—
63C4×D4—24no—
64C4×D4—24no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4C2×C412yes—
77C2×C4⋊C4D412yes—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4D411yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C42—21no—
86#103—42no—
87#102—42no—
88C22⋊C8—21no—
89C22⋊C8—21no—
90C22⋊C8—21no—
91C22⋊C8—21no—
92#95—21no—
93#95—21no—
94#95—42no—
95#95—42no—
96C24—21no—
97C24C2×D411yescharacteristic
98C2×Q8—21no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8C22⋊C412yes—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—41no—
111C2×D4C22⋊C412yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4C22⋊C412yes—
145C22×C4C22⋊C412yes—
146C22×C4C22×C411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4C22⋊C412yes—
174C4⋊C4C22⋊C412yes—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C42—24no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23C2×C22⋊C411yescharacteristic
195C23#12011yescharacteristic
196C23#12011yescharacteristic
197C23#13511yescharacteristic
198C23#13511yescharacteristic
199C23AGL(1,ℤ/8)11yescharacteristic
200C23#13711yescharacteristic
201Q8—42no—
202Q8—81no—
203D4—42no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—42no—
208D4—81no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—84no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4C2×C22⋊C411yescharacteristic
254C2×C4C2×C22⋊C411yescharacteristic
255C2×C4#12011yescharacteristic
256C2×C4#12011yescharacteristic
257C8—81no—
258C8—81no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22#37911yescharacteristic
279C22#41611yescharacteristic
280C22#41711yescharacteristic
281C22#44911yescharacteristic
282C22#44911yescharacteristic
283C22#45211yescharacteristic
284C22#45211yescharacteristic
285C22#45411yescharacteristic
286C22#45611yescharacteristic
287C22B(4,𝔽2)11yescharacteristic
288C22#45811yescharacteristic
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—42no—
294C4—42no—
295C4—42no—
296C4—42no—
297C4—42no—
298C4—82no—
299C4—82no—
300C2—21no—
301C2—21no—
302C2—21no—
303C2—21no—
304C2—82no—
305C2#159711yescharacteristic
306C2#160311yescharacteristic
307C2#160911yescharacteristic
308C2#161511yescharacteristic
309C2#178811yescharacteristic
310C2#179011yescharacteristic
311C2#179311yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1872C2semidirect
4. C2×#328C2semidirect
6. C2×#324C2semidirect
7. #1441C2semidirect
11. #539C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5392
#18742
#18722
C2×#3282
C2×#3272
C2×#3242
#14412