Finite Group Databasebeta

← #36475

Subgroups — #36475

Order 256.

795 conjugacy classes of subgroups.

The subgroup lattice has 251 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2864C213yesmaximal
2#2570C213yesmaximal
3#2570C213yesmaximal
4#2509C213yesmaximal
5#2508C213yesmaximal
6#2507C213yesmaximal
7#2506C213yesmaximal
8#2351C211yesmaximal, characteristic
9#2351C211yesmaximal, characteristic
10#2351C211yesmaximal, characteristic
11#2351C211yesmaximal, characteristic
12Q8×C4⋊C4C211yesmaximal, characteristic
13Q8×C4⋊C4C211yesmaximal, characteristic
14C4×#125C213yesmaximal
15C42×Q8C211yesmaximal, characteristic
16C2×#125C2213yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24#400C2216yes—
25#400C2216yes—
26#398C2216yes—
27#398C2216yes—
28#389C2213yes—
29#389C2213yes—
30#389C2213yes—
31#389C2213yes—
32#389C2213yes—
33#389C2213yes—
34#389C2213yes—
35#389C2213yes—
36#389C2213yes—
37#389C2213yes—
38#389C2213yes—
39#389C2213yes—
40#389C2213yes—
41#389C2213yes—
42#387C2211yescharacteristic
43#387C2213yes—
44#387C2213yes—
45#387C2213yes—
46#387C2216yes—
47#387C2216yes—
48#387C2216yes—
49#384C2213yes—
50#384C2213yes—
51#382C2213yes—
52#382C2213yes—
53#382C2213yes—
54#382C2213yes—
55#382C2213yes—
56C4×C4⋊C4C2213yes—
57C4×C4⋊C4C2213yes—
58C4×C4⋊C4C2213yes—
59C4×C4⋊C4C2213yes—
60C4×C4⋊C4C2213yes—
61C4×C4⋊C4C2213yes—
62C4×C4⋊C4C2213yes—
63C4×C4⋊C4C2213yes—
64C4×C4⋊C4C2213yes—
65C43C2213yes—
66C22×Q8C2311yescharacteristic
67#125—212no—
68C4×Q8—24no—
69C4×Q8—24no—
70C4×Q8—24no—
71C4×Q8—24no—
72C4×Q8—24no—
73C4×Q8—24no—
74C4×Q8Q818yes—
75C2×C4⋊C4C2313yes—
76C2×C4⋊C4C2313yes—
77C2×C4⋊C4C2313yes—
78C2×C4⋊C4C2313yes—
79C2×C4⋊C4C2313yes—
80C2×C4⋊C4C2313yes—
81C2×C4⋊C4C2313yes—
82C2×C4⋊C4C2313yes—
83C2×C4⋊C4C2313yes—
84C2×C4⋊C4C2313yes—
85C2×C4⋊C4C2313yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2313yes—
88C2×C4⋊C4C2313yes—
89C2×C4⋊C4C2313yes—
90C2×C4⋊C4C2313yes—
91C2×C4⋊C4C2313yes—
92C2×C4⋊C4C2316yes—
93C2×C4⋊C4C2316yes—
94C2×C4⋊C4C2313yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C42C2313yes—
97C2×C42C2313yes—
98C2×C42C2313yes—
99C2×C42C2313yes—
100C2×C42C2313yes—
101C2×C42C2313yes—
102C2×C42C2313yes—
103C2×C42C2311yescharacteristic
104C2×C42C2313yes—
105C2×C42C2313yes—
106C2×C42C2313yes—
107#95C2311yescharacteristic
108#95C2311yescharacteristic
109#95C2311yescharacteristic
110#95C2311yescharacteristic
111#95C2316yes—
112#95C2313yes—
113#95C2313yes—
114#95C2313yes—
115#95C2313yes—
116#95C2313yes—
117#95C2313yes—
118#95C2313yes—
119#95C2313yes—
120#95C2316yes—
121#95C2316yes—
122#95C2316yes—
123#95C2316yes—
124C2×Q8C2×Q814yes—
125C2×Q8#4114yes—
126C2×Q8#4114yes—
127C22×C4C2411yescharacteristic
128C22×C4C2413yes—
129C22×C4C2413yes—
130C22×C4C2411yescharacteristic
131C22×C4C2413yes—
132C22×C4C2413yes—
133C22×C4C2413yes—
134C22×C4C2413yes—
135C22×C4C2411yescharacteristic
136C22×C4C2413yes—
137C22×C4C2411yescharacteristic
138C22×C4C2411yescharacteristic
139C22×C4C2411yescharacteristic
140C22×C4C2411yescharacteristic
141C22×C4C2413yes—
142C4⋊C4—26no—
143C4⋊C4—26no—
144C4⋊C4—26no—
145C4⋊C4—26no—
146C4⋊C4—26no—
147C4⋊C4—26no—
148C4⋊C4—26no—
149C4⋊C4—26no—
150C4⋊C4—26no—
151C4⋊C4—26no—
152C4⋊C4—26no—
153C4⋊C4—22no—
154C4⋊C4—26no—
155C4⋊C4—43no—
156C4⋊C4—43no—
157C4⋊C4—43no—
158C4⋊C4—43no—
159C4⋊C4—46no—
160C4⋊C4—46no—
161C4⋊C4C2×Q8112yes—
162C4⋊C4#4114yes—
163C42—26no—
164C42—26no—
165C42—26no—
166C42—26no—
167C42—26no—
168C42—26no—
169C42—26no—
170C42—22no—
171C42—43no—
172C42—43no—
173C42C2×Q8112yes—
174C23C2511yescharacteristic
175Q8#125116yescomplemented
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—23no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—23no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—23no—
191C2×C4—23no—
192C2×C4—23no—
193C2×C4—23no—
194C2×C4—23no—
195C2×C4—23no—
196C2×C4—23no—
197C2×C4—23no—
198C2×C4—23no—
199C2×C4—23no—
200C2×C4—21no—
201C2×C4—23no—
202C2×C4—23no—
203C2×C4—23no—
204C2×C4—23no—
205C2×C4—23no—
206C2×C4—21no—
207C2×C4—23no—
208C2×C4—21no—
209C2×C4—23no—
210C2×C4—21no—
211C2×C4—23no—
212C2×C4C22×Q812yes—
213C2×C4C22×Q816yes—
214C2×C4C22×Q816yes—
215C2×C4C2×#4112yes—
216C2×C4C2×#4112yes—
217C2×C4C2×#4112yes—
218C2×C4C2×#4116yes—
219C2×C4C2×#4116yes—
220C2×C42−1+412yes—
221C22C23×Q811yescharacteristic
222C22C22×#4111yescharacteristic
223C22C22×#4111yescharacteristic
224C22C22×#4111yescharacteristic
225C22C2×2−1+411yescharacteristic
226C22#58511yescharacteristic
227C22#58511yescharacteristic
228C4—26no—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—43no—
235C4—43no—
236C4—43no—
237C4—41no—
238C4—43no—
239C4—41no—
240C4—43no—
241C4—43no—
242C4C2×#125112yes—
243C2C22×#12511yescharacteristic
244C2Q8×#4111yescharacteristic
245C2#348011yescharacteristic
246C2#348211yescharacteristic
247C2#350211yescharacteristic
248C2#351611yescharacteristic
249C2#352411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
175. Q8#125semidirect

Maximal subgroups

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

subgroupindex
#28642
#25702
#25092
#25082
#25072
#25062
#23512
Q8×C4⋊C42
C4×#1252
C42×Q82