Finite Group Databasebeta

← #8419

Subgroups — #8419

Order 256.

389 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 75 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#3522C211yesmaximal, characteristic
2#1980C211yesmaximal, characteristic, complemented
3#1763C211yesmaximal, characteristic, complemented
4#1530C211yesmaximal, characteristic, complemented
5#1524C211yesmaximal, characteristic, complemented
6#1509C211yesmaximal, characteristic, complemented
7#1491C211yesmaximal, characteristic, complemented
8#559—21no—
9#559C411yescharacteristic, complemented
10Q82C411yescharacteristic, complemented
11#554—21no—
12#554C411yescharacteristic, complemented
13#550C411yescharacteristic, complemented
14#548—21no—
15#547—22no—
16#547—21no—
17#542—22no—
18#542—21no—
19#532—21no—
20#532C2211yescharacteristic
21#529—21no—
22#529C2211yescharacteristic
23#520—21no—
24C2×C4×Q8C2211yescharacteristic
25#432C2211yescharacteristic
26C2×#104—22no—
27C2×#103—21no—
28C4×M16C2211yescharacteristic
29C2×#97C2211yescharacteristic
30C4×C4⋊C4C2211yescharacteristic
31#340—21no—
32#330—21no—
33#329—21no—
34#329—22no—
35#328—21no—
36#327—21no—
37Q8⋊C8—21no—
38Q8⋊C8—21no—
39C2×#41—22no—
40C2×#41—21no—
41C2×#41D412yes—
42C22×Q8D411yescharacteristic
43C2×M16—21no—
44C22×C8—21no—
45C4⋊Q8—22no—
46C4⋊Q8—21no—
47C4⋊Q8—21no—
48C4⋊Q8C2×C411yescharacteristic
49#127—21no—
50#127C2×C411yescharacteristic
51#126—22no—
52#126—21no—
53#125—21no—
54#125D411yescharacteristic
55#124—22no—
56#124—22no—
57#124—21no—
58#124—21no—
59#124D411yescharacteristic
60#124D412yes—
61#123—22no—
62#123—22no—
63#123—22no—
64#123—22no—
65#123—21no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C22⋊Q8—21no—
71C22⋊Q8D411yescharacteristic
72#121—22no—
73#121—22no—
74#121—21no—
75#121—21no—
76#121—21no—
77#121D411yescharacteristic
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8—21no—
81C4×Q8—21no—
82C4×Q8C2×C411yescharacteristic
83C4×Q8C2×C411yescharacteristic
84C4×Q8C2×C411yescharacteristic
85C4×Q8D411yescharacteristic
86C4×Q8D411yescharacteristic
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4C2×C411yescharacteristic
98#117—22no—
99#117—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109#104—44no—
110#103—24no—
111C22⋊C8—21no—
112#97—21no—
113#97—21no—
114#97—21no—
115#97—21no—
116C4×C8—21no—
117#95—21no—
118#41—42no—
119#41—42no—
120#41—42no—
121#41—42no—
122#41—82no—
123#41—82no—
124C2×Q8—22no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—41no—
128C2×Q8C22⋊C411yescharacteristic
129C2×Q8C22⋊C411yescharacteristic
130C2×Q8C22⋊C411yescharacteristic
131C2×Q8C22⋊C412yes—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4C22⋊C411yescharacteristic
139C2×D4C22⋊C412yes—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151M16—22no—
152M16—41no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
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—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—41no—
175C4⋊C4C22⋊C411yescharacteristic
176C4⋊C4C22⋊C411yescharacteristic
177C4⋊C4C22⋊C411yescharacteristic
178C4⋊C4C22⋊C411yescharacteristic
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C42—22no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42C22×C411yescharacteristic
199C42C2×D411yescharacteristic
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C23—22no—
203C23—21no—
204C23#12011yescharacteristic
205Q8—21no—
206Q8—21no—
207Q8—21no—
208Q8—42no—
209Q8—41no—
210Q8—42no—
211Q8—82no—
212Q8#10412yescomplemented
213D4—41no—
214D4—42no—
215D4—42no—
216D4—44no—
217D4—42no—
218D4—41no—
219D4—42no—
220D4—41no—
221D4—41no—
222D4—41no—
223D4—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4C2×C22⋊C411yescharacteristic
250C2×C4C2×C22⋊C411yescharacteristic
251C2×C4C2×C22⋊C411yescharacteristic
252C2×C4#12011yescharacteristic
253C2×C4#12011yescharacteristic
254C2×C4#12011yescharacteristic
255C8—42no—
256C8—42no—
257C8—81no—
258C22—21no—
259C22—21no—
260C22—42no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—42no—
265C22—41no—
266C22#37911yescharacteristic
267C22#45311yescharacteristic
268C22#45511yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—22no—
275C4—41no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4C2×#10411yescharacteristic
284C4#42111yescharacteristic
285C2—21no—
286C2—21no—
287C2—82no—
288C2—81no—
289C2#178911yescharacteristic
290C2#179211yescharacteristic
291C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3522C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1980C2semidirect
3. #1763C2semidirect
4. #1530C2semidirect
5. #1524C2semidirect
6. #1509C2semidirect
7. #1491C2semidirect
9. #559C4semidirect
10. Q82C4semidirect
12. #554C4semidirect
13. #550C4semidirect
212. Q8#104semidirect

Maximal subgroups

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

subgroupindex
#35222
#19802
#17632
#15302
#15242
#15092
#14912