Finite Group Databasebeta

← #25847

Subgroups — #25847

Order 256.

498 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×Q82C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic
3C2×#479C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic
5#2058C212yesmaximal
6#2057C212yesmaximal
7#1988C211yesmaximal, characteristic
8#1985C211yesmaximal, characteristic
9#1865C212yesmaximal
10#1863C212yesmaximal
11#1763C211yesmaximal, characteristic
12Q82—24no—
13Q82—24no—
14C2×C4⋊Q8—22no—
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#501—24no—
24#479—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#97C2211yescharacteristic
34#398—21no—
35#395—21no—
36#389—22no—
37#389C2212yes—
38#389C2212yes—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—22no—
46C22×Q8—21no—
47C22×Q8D411yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C4⋊Q8—216no—
53C4⋊Q8—28no—
54C4⋊Q8—24no—
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8D414yes—
58#125—22no—
59#125—22no—
60C4×Q8—28no—
61C4×Q8—24no—
62C4×Q8—24no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×Q8D414yes—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4D412yes—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C42—22no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#107—22no—
88#107—24no—
89#106—22no—
90#106—24no—
91C4⋊C8—22no—
92#103—22no—
93#103—22no—
94#103Q818yescomplemented
95#97—22no—
96#95—21no—
97#95—22no—
98#95—22no—
99C2×Q8—28no—
100C2×Q8—22no—
101C2×Q8—24no—
102C2×Q8—24no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—81no—
106C2×Q8—81no—
107C2×Q8—82no—
108C2×Q8—82no—
109C2×Q8C2×D412yes—
110C2×Q8C2×Q814yes—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2411yescharacteristic
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8C2×Q814yes—
129C4⋊C4—28no—
130C4⋊C4—28no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—24no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—44no—
145C4⋊C4—82no—
146C4⋊C4C2×D414yes—
147C4⋊C4C2×D412yes—
148C4⋊C4C2×Q814yes—
149C42—28no—
150C42—24no—
151C42—22no—
152C42—22no—
153C42—81no—
154C42C2×D412yes—
155C42C2×D412yes—
156C23C22×D411yescharacteristic
157Q8—28no—
158Q8—22no—
159Q8—24no—
160Q8—84no—
161Q8—88no—
162Q8C4⋊Q814yes—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4#12012yes—
188C2×C4#12012yes—
189C2×C4#13714yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C22×D411yescharacteristic
192C2×C4C22×Q812yes—
193C2×C42−1+412yes—
194C8—24no—
195C8—41no—
196C8—41no—
197C22C2×#12011yescharacteristic
198C22D4×Q812yes—
199C22C2×#13712yes—
200C22#57611yescharacteristic
201C22#57711yescharacteristic
202C4—24no—
203C4—22no—
204C4—24no—
205C4—22no—
206C4—44no—
207C4—42no—
208C4—44no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4C2×C4⋊Q811yescharacteristic
213C4#53611yescharacteristic
214C4D4×Q812yes—
215C2#243411yescharacteristic
216C2#300411yescharacteristic
217C2#301211yescharacteristic
218C2#338512yes—
219C2#338712yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
94. #103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q822
C2×#5012
C2×#4792
#25842
#20582
#20572
#19882
#19852
#18652
#18632
#17632