Finite Group Databasebeta

← #34336

Subgroups — #34336

Order 256.

1012 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#557C211yesmaximal, characteristic
2#2370C213yesmaximal
3#2370C213yesmaximal
4#2369C213yesmaximal
5#2355C213yesmaximal
6#2351C211yesmaximal, characteristic
7#2351C211yesmaximal, characteristic
8Q8×C4⋊C4C211yesmaximal, characteristic
9#2331C213yesmaximal
10#2330C213yesmaximal
11C4×#125C213yesmaximal
12#2295C213yesmaximal
13#2276C211yesmaximal, characteristic
14#2276C211yesmaximal, characteristic
15C42×Q8C211yesmaximal, characteristic
16#557C4116yescomplemented
17C2×C4⋊Q8C2213yes—
18C2×#125C2213yes—
19C2×#125C2213yes—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2213yes—
27C2×C4×Q8C2211yescharacteristic
28#391C2213yes—
29#391C2216yes—
30#391C2213yes—
31#389C2213yes—
32#389C2213yes—
33#389C2213yes—
34#389C2213yes—
35#389C2213yes—
36#389C2216yes—
37#389C2213yes—
38#389C2213yes—
39#389C2213yes—
40#387C2213yes—
41#387C2216yes—
42#387C2216yes—
43#387C2216yes—
44#387C2213yes—
45#387C2213yes—
46#387C2211yescharacteristic
47#384C2213yes—
48#382C2213yes—
49#382C2213yes—
50C4×C4⋊C4C2213yes—
51C4×C4⋊C4C2213yes—
52C4×C4⋊C4C2216yes—
53C4×C4⋊C4C2216yes—
54C4×C4⋊C4C2213yes—
55C4×C4⋊C4C2213yes—
56C4×C4⋊C4C2213yes—
57C4×C4⋊C4C2213yes—
58C4×C4⋊C4C2213yes—
59C4×C4⋊C4C2213yes—
60C4×C4⋊C4C2213yes—
61C4×C4⋊C4C2213yes—
62C4×C4⋊C4C2213yes—
63C4×C4⋊C4C2213yes—
64#376C2213yes—
65#376C2213yes—
66C43C2213yes—
67C22×Q8C2313yes—
68C22×Q8C2311yescharacteristic
69C4⋊Q8C2×C4124yes—
70#125C2×C4124yes—
71#125C2×C4124yes—
72C4×Q8—24no—
73C4×Q8—24no—
74C4×Q8—24no—
75C4×Q8C2×C418yes—
76C4×Q8C2×C418yes—
77C4×Q8C2×C4124yes—
78C4×Q8C2×C418yes—
79C4×Q8Q818yescomplemented
80C2×C4⋊C4C2313yes—
81C2×C4⋊C4C2313yes—
82C2×C4⋊C4C2313yes—
83C2×C4⋊C4C2313yes—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2313yes—
86C2×C4⋊C4C2316yes—
87C2×C4⋊C4C2313yes—
88C2×C4⋊C4C2313yes—
89C2×C4⋊C4C2313yes—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2313yes—
92C2×C4⋊C4C2313yes—
93C2×C4⋊C4C2313yes—
94C2×C4⋊C4C2313yes—
95C2×C4⋊C4C2313yes—
96C2×C4⋊C4C2316yes—
97C2×C4⋊C4C2313yes—
98C2×C42C2313yes—
99C2×C42C2313yes—
100C2×C42C2313yes—
101C2×C42C2311yescharacteristic
102C2×C42C2313yes—
103C2×C42C2313yes—
104C2×C42C2313yes—
105C2×C42C2316yes—
106C2×C42C2316yes—
107C2×C42C2313yes—
108C2×C42C2313yes—
109C2×C42C2313yes—
110C2×C42C2313yes—
111#95C2311yescharacteristic
112#95C2311yescharacteristic
113#95C2313yes—
114#95C2313yes—
115#95C2313yes—
116#95C2313yes—
117#95C2313yes—
118#95C2313yes—
119#95C2313yes—
120#95C2311yescharacteristic
121#95C2316yes—
122#95C2313yes—
123#95C2311yescharacteristic
124#95C2313yes—
125#95C2316yes—
126#95C2316yes—
127#95C2313yes—
128C2×Q8—43no—
129C2×Q8—43no—
130C2×Q8C22×C4112yes—
131C2×Q8C22×C414yes—
132C2×Q8C2×Q814yes—
133C2×Q8#4114yes—
134C22×C4C2411yescharacteristic
135C22×C4C2413yes—
136C22×C4C2411yescharacteristic
137C22×C4C2413yes—
138C22×C4C2413yes—
139C22×C4C2411yescharacteristic
140C22×C4C2411yescharacteristic
141C22×C4C2413yes—
142C22×C4C2411yescharacteristic
143C22×C4C2413yes—
144C22×C4C2413yes—
145C22×C4C2411yescharacteristic
146C22×C4C2413yes—
147C22×C4C2411yescharacteristic
148C22×C4C2413yes—
149C4⋊C4—26no—
150C4⋊C4—26no—
151C4⋊C4—26no—
152C4⋊C4—26no—
153C4⋊C4—22no—
154C4⋊C4—43no—
155C4⋊C4—43no—
156C4⋊C4—43no—
157C4⋊C4—43no—
158C4⋊C4—46no—
159C4⋊C4C22×C414yes—
160C4⋊C4C22×C4112yes—
161C4⋊C4C22×C4112yes—
162C4⋊C4C22×C4112yes—
163C4⋊C4C22×C4112yes—
164C4⋊C4C22×C4112yes—
165C4⋊C4C22×C4124yes—
166C4⋊C4C2×Q8112yes—
167C42—26no—
168C42—26no—
169C42—26no—
170C42—22no—
171C42—43no—
172C42—43no—
173C42—46no—
174C42—46no—
175C42—43no—
176C42C22×C4112yes—
177C42C22×C4112yes—
178C42C22×C4112yes—
179C42C2×Q8112yes—
180C23C2511yescharacteristic
181Q8—412no—
182Q8C4×Q8116yes—
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—21no—
193C2×C4—23no—
194C2×C4—23no—
195C2×C4—23no—
196C2×C4—23no—
197C2×C4—21no—
198C2×C4—23no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—23no—
204C2×C4—23no—
205C2×C4—23no—
206C2×C4—21no—
207C2×C4—23no—
208C2×C4—23no—
209C2×C4—23no—
210C2×C4—23no—
211C2×C4—23no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—23no—
215C2×C4—21no—
216C2×C4C23×C412yes—
217C2×C4C23×C412yes—
218C2×C4C23×C412yes—
219C2×C4C23×C416yes—
220C2×C4C23×C416yes—
221C2×C4C23×C416yes—
222C2×C4C23×C416yes—
223C2×C4C22×Q812yes—
224C2×C4C22×Q816yes—
225C2×C4C22×Q816yes—
226C2×C4C2×#4116yes—
227C2×C42−1+412yes—
228C22C24×C411yescharacteristic
229C22C23×Q811yescharacteristic
230C22C22×#4111yescharacteristic
231C22C2×2−1+411yescharacteristic
232C22#58511yescharacteristic
233C22#58511yescharacteristic
234C22#58511yescharacteristic
235C4—26no—
236C4—26no—
237C4—22no—
238C4—22no—
239C4—26no—
240C4—22no—
241C4—22no—
242C4—26no—
243C4—22no—
244C4—41no—
245C4—43no—
246C4—41no—
247C4—43no—
248C4—43no—
249C4C2×C4×Q8112yes—
250C2C22×C4×Q811yescharacteristic
251C2#342811yescharacteristic
252C2C4×2−1+411yescharacteristic
253C2#350211yescharacteristic
254C2#350411yescharacteristic
255C2#351611yescharacteristic
256C2#352411yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC23C25
commutatorC22C24×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #557C4semidirect
79. C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
#23702
#23692
#23552
#23512
Q8×C4⋊C42
#23312
#23302
C4×#1252
#22952
#22762
C42×Q82