Finite Group Databasebeta

← #8439

Subgroups — #8439

Order 256.

380 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 91 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
1Q8×#41C211yesmaximal, characteristic
2#1806C211yesmaximal, characteristic
3C4×#103C211yesmaximal, characteristic, complemented
4#1503C211yesmaximal, characteristic, complemented
5#1493C211yesmaximal, characteristic, complemented
6#1487C211yesmaximal, characteristic, complemented
7#1476C211yesmaximal, characteristic
8Q82C411yescharacteristic, complemented
9#557—21no—
10#557C411yescharacteristic, complemented
11#554—22no—
12#554—21no—
13#554—21no—
14#554C411yescharacteristic, complemented
15D4×Q8—21no—
16D4×Q8C411yescharacteristic, complemented
17#533—22no—
18#533—21no—
19#533—21no—
20#533C2211yescharacteristic
21C4×#41—21no—
22C4×#41C2211yescharacteristic
23C2×C4×Q8—21no—
24C2×C4×Q8C2211yescharacteristic
25#431C2211yescharacteristic
26#423C2211yescharacteristic
27C2×#104—21no—
28C2×#103—21no—
29C2×C4×C8C2211yescharacteristic
30C4×C4⋊C4C2211yescharacteristic
31#340—21no—
32#339—21no—
33#328—22no—
34Q8⋊C8—21no—
35Q8⋊C8—21no—
36Q8⋊C8—21no—
37Q8⋊C8—22no—
38D4⋊C8—21no—
39C2×#41D411yescharacteristic
40C22×Q8—21no—
41C22×Q8D411yescharacteristic
42C2×M16—21no—
43C22×C8—21no—
44C4⋊Q8—22no—
45C4⋊Q8—21no—
46C4⋊Q8—22no—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49C4⋊Q8—21no—
50C4⋊Q8C2×C411yescharacteristic
51C4⋊Q8C2×C411yescharacteristic
52#125—22no—
53#125—22no—
54#125—22no—
55#125—21no—
56#125—21no—
57#125—21no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—21no—
64C22⋊Q8—22no—
65C22⋊Q8—21no—
66C22⋊Q8—21no—
67C22⋊Q8—22no—
68C22⋊Q8D412yes—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8C2×C411yescharacteristic
80C4×Q8C2×C411yescharacteristic
81C4×Q8C2×C411yescharacteristic
82C4×Q8D411yescharacteristic
83C4×Q8D411yescharacteristic
84C4×Q8D411yescharacteristic
85C4×Q8D412yes—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4C2×C411yescharacteristic
93C4×D4D411yescharacteristic
94#117—22no—
95#117—21no—
96#117—21no—
97#117D411yescharacteristic
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C42—22no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111#104—42no—
112#103—24no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117#95—21no—
118#41—21no—
119#41—21no—
120#41SD1612yes—
121#41Q1612yes—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—41no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8C22⋊C411yescharacteristic
131C2×Q8C22⋊C411yescharacteristic
132C2×Q8C22⋊C411yescharacteristic
133C2×Q8C22⋊C412yes—
134C2×D4—21no—
135C2×D4C22⋊C411yescharacteristic
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147M16—41no—
148M16—41no—
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4C22⋊C411yescharacteristic
177C4⋊C4C22⋊C411yescharacteristic
178C4⋊C4C22⋊C411yescharacteristic
179C4⋊C4C22⋊C411yescharacteristic
180C4⋊C4C22⋊C412yes—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C42—22no—
188C42—22no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—41no—
198C42—41no—
199C42C22×C411yescharacteristic
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C23—21no—
204C23#12011yescharacteristic
205Q8—21no—
206Q8—21no—
207Q8—21no—
208Q8—21no—
209Q8—42no—
210Q8—42no—
211Q8—44no—
212Q8—41no—
213Q8—41no—
214Q8—82no—
215Q8#10312yescomplemented
216Q8#10412yescomplemented
217D4—22no—
218D4—22no—
219D4—21no—
220D4#10312yescomplemented
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4C2×C22⋊C411yescharacteristic
246C2×C4C2×C22⋊C411yescharacteristic
247C2×C4C2×C22⋊C411yescharacteristic
248C2×C4#12011yescharacteristic
249C2×C4#12011yescharacteristic
250C2×C4#12011yescharacteristic
251C2×C4C2×SD1611yescharacteristic
252C2×C4C2×Q1611yescharacteristic
253C2×C4#13711yescharacteristic
254C2×C4#13711yescharacteristic
255C8—42no—
256C8—81no—
257C8—81no—
258C22—21no—
259C22—21no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22#37911yescharacteristic
264C22#44811yescharacteristic
265C22#45111yescharacteristic
266C4—21no—
267C4—21no—
268C4—22no—
269C4—21no—
270C4—22no—
271C4—41no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—42no—
277C4—41no—
278C4—82no—
279C4C2×#10311yescharacteristic
280C4#41911yescharacteristic
281C4C2×#10411yescharacteristic
282C4#42111yescharacteristic
283C4#44811yescharacteristic
284C4#45111yescharacteristic
285C2—21no—
286C2—21no—
287C2—42no—
288C2#178711yescharacteristic
289C2#178911yescharacteristic
290C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#103C2semidirect
4. #1503C2semidirect
5. #1493C2semidirect
6. #1487C2semidirect
8. Q82C4semidirect
10. #557C4semidirect
14. #554C4semidirect
16. D4×Q8C4semidirect
215. Q8#103semidirect
216. Q8#104semidirect
220. D4#103semidirect

Maximal subgroups

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

subgroupindex
Q8×#412
#18062
C4×#1032
#15032
#14932
#14872
#14762