Finite Group Databasebeta

← #9263

Subgroups — #9263

Order 256.

239 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 61 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#3190C211yesmaximal, characteristic
2#2977C211yesmaximal, characteristic
3#1504C211yesmaximal, characteristic
4C2×#328C211yesmaximal, characteristic, complemented
5#1472C211yesmaximal, characteristic, complemented
6#1298C211yesmaximal, characteristic, complemented
7#1280C211yesmaximal, characteristic, complemented
8#525C2211yescharacteristic
9C2×C22⋊Q8C2211yescharacteristic
10#485—21no—
11#483C411yescharacteristic, complemented
12#480C411yescharacteristic, complemented
13#467—21no—
14#465—21no—
15#452—21no—
16#451C411yescharacteristic, complemented
17#448C411yescharacteristic, complemented
18#435—21no—
19#435—21no—
20C8×D4—21no—
21C8×D4—21no—
22#433—21no—
23#408—21no—
24#407C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×#95C2211yescharacteristic
29#328—21no—
30#328—21no—
31#328—21no—
32#327—21no—
33#324—21no—
34#324—21no—
35C2×#41D411yescharacteristic
36C22×Q8C2×C411yescharacteristic
37C23×C4—21no—
38C23×C4C2311yescharacteristic
39C2×Q16—21no—
40C2×SD16—21no—
41C2×M16—21no—
42C2×M16—21no—
43C22×C8—21no—
44C22×C8—21no—
45C22×C8—21no—
46C22×C8—21no—
47#123—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8—21no—
52C22⋊Q8C2×C411yescharacteristic
53C22⋊Q8D411yescharacteristic
54#121C2×C411yescharacteristic
55#121D411yescharacteristic
56#120—21no—
57C4×D4—21no—
58C4×D4—21no—
59#117D411yescharacteristic
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2×C411yescharacteristic
62C2×C22⋊C4—21no—
63#107—21no—
64#106—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67#103—21no—
68#103—21no—
69#103—21no—
70#102—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8C2×C411yescharacteristic
79C22⋊C8C2×C411yescharacteristic
80C22⋊C8D411yescharacteristic
81C22⋊C8D411yescharacteristic
82C22⋊C8D411yescharacteristic
83C22⋊C8D411yescharacteristic
84#97—21no—
85C4×C8—21no—
86#95—21no—
87#95—21no—
88C24#4111yescharacteristic
89#41—81no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—81no—
93C2×Q8C22⋊C411yescharacteristic
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4C22⋊C411yescharacteristic
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—81no—
105C22×C4—81no—
106C22×C4—81no—
107C22×C4C22×C411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4#4111yescharacteristic
113Q16—81no—
114SD16—81no—
115M16—81no—
116M16—81no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—41no—
124C2×C8—81no—
125C2×C8—81no—
126C2×C8—81no—
127C2×C8—81no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4C22⋊C411yescharacteristic
135C4⋊C4C22⋊C411yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C42—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—41no—
150C23C4×D411yescharacteristic
151C23#12111yescharacteristic
152C23#12311yescharacteristic
153Q8—41no—
154Q8—41no—
155Q8—41no—
156Q8—81no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—81no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—82no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—82no—
184C2×C4C2×C22⋊C411yescharacteristic
185C2×C4C4×D411yescharacteristic
186C2×C4#12011yescharacteristic
187C2×C4#12111yescharacteristic
188C8—41no—
189C8—41no—
190C8—41no—
191C8—41no—
192C8—41no—
193C8—41no—
194C8—81no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22#38611yescharacteristic
210C22#44311yescharacteristic
211C22#44411yescharacteristic
212C22#47111yescharacteristic
213C22#47311yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—81no—
223C4—81no—
224C2—21no—
225C2—21no—
226C2—21no—
227C2—21no—
228C2—41no—
229C2—81no—
230C2#187611yescharacteristic
231C2#190811yescharacteristic
232C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#328C2semidirect
5. #1472C2semidirect
6. #1298C2semidirect
7. #1280C2semidirect
11. #483C4semidirect
12. #480C4semidirect
16. #451C4semidirect
17. #448C4semidirect

Maximal subgroups

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

subgroupindex
#31902
#29772
#15042
C2×#3282
#14722
#12982
#12802