Finite Group Databasebeta

← #23614

Subgroups — #23614

Order 256.

607 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 175 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×#428C211yesmaximal, characteristic, complemented
2C2×#424C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic
4C22×#102C211yesmaximal, characteristic
5#2316C211yesmaximal, characteristic, complemented
6#2309C211yesmaximal, characteristic
7#2067C214yesmaximal, complemented
8#2065C214yesmaximal, complemented
9#1735C211yesmaximal, characteristic, complemented
10C23×D4C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C23×C8C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#428—24no—
18C2×#107C2212yescomplemented
19C2×#106C2212yescomplemented
20#424—24no—
21C2×C4⋊C8C2214yescomplemented
22#418—24no—
23C2×#102—24no—
24C2×#102C2214yes—
25C2×#102C2214yes—
26#390—22no—
27#390C2212yes—
28#390C2212yes—
29#389—22no—
30#389C2214yescomplemented
31#380—22no—
32#379—22no—
33#340C2212yescomplemented
34#340C2212yescomplemented
35C25—21no—
36C22×D4—22no—
37C22×D4—22no—
38C22×D4—21no—
39C22×D4—22no—
40C22×D4—22no—
41C22×D4—22no—
42C22×D4Q814yes—
43C22×D4C2312yes—
44C23×C4—21no—
45C23×C4C2311yescharacteristic
46C2×M16—22no—
47C2×M16—22no—
48C2×M16D414yes—
49C22×C8—22no—
50C22×C8—22no—
51C22×C8—22no—
52C22×C8C2312yes—
53C22×C8C2312yes—
54#117—22no—
55#117—22no—
56#117—22no—
57#117—22no—
58#117—22no—
59#117—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4D414yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4D412yes—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C2312yes—
77#107—24no—
78#106—24no—
79C4⋊C8—28no—
80#102—28no—
81#102—28no—
82#95—24no—
83C24—22no—
84C24—24no—
85C24—42no—
86C24—42no—
87C24—82no—
88C24—82no—
89C24C2×D411yescharacteristic
90C2×D4—28no—
91C2×D4—28no—
92C2×D4—24no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—44no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—44no—
102C2×D4C2×Q814yes—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—82no—
109C22×C4C2×D414yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D412yes—
112C22×C4C2×D412yes—
113C22×C4C2×Q812yes—
114C22×C4#4112yes—
115C22×C4#4112yes—
116C22×C4#4112yes—
117C22×C4C2411yescharacteristic
118M16—28no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—44no—
125C2×C8C2×D414yes—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—84no—
136C4⋊C4C2×D414yes—
137C22⋊C4—24no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—84no—
141C22⋊C4—84no—
142C42—22no—
143C42—22no—
144C42—42no—
145C42—42no—
146C23—22no—
147C23—22no—
148C23—24no—
149C23—42no—
150C23—42no—
151C23—44no—
152C23—44no—
153C23—48no—
154C23—88no—
155C23—84no—
156C23—84no—
157C23—88no—
158C23#12112yes—
159C23#12112yes—
160C23C22⋊Q812yes—
161C23C22×D411yescharacteristic
162D4—24no—
163D4—24no—
164D4—42no—
165D4—42no—
166D4—48no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—44no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—84no—
180C2×C4#12114yes—
181C2×C4#12112yes—
182C2×C4#12112yes—
183C2×C4C22⋊Q814yes—
184C2×C4C22⋊Q814yes—
185C2×C4C22⋊Q812yes—
186C2×C4#12614yes—
187C2×C4C22×D411yescharacteristic
188C2×C4C22×D411yescharacteristic
189C2×C4C22×Q811yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C8—24no—
194C8—42no—
195C22—24no—
196C22—22no—
197C22—22no—
198C22—24no—
199C22—48no—
200C22—44no—
201C22—816no—
202C22—82no—
203C22—82no—
204C22—84no—
205C22#39614yes—
206C22C2×#12111yescharacteristic
207C22C2×#12111yescharacteristic
208C22C2×C22⋊Q811yescharacteristic
209C22#57612yes—
210C22#57712yes—
211C4—44no—
212C4—44no—
213C4—81no—
214C4—81no—
215C4—82no—
216C4#39614yes—
217C4C2×#12111yescharacteristic
218C4C2×C22⋊Q811yescharacteristic
219C4C2×C22⋊Q811yescharacteristic
220C4C2×#12611yescharacteristic
221C2—24no—
222C2—84no—
223C2C2×#39611yescharacteristic
224C2#304111yescharacteristic
225C2#304311yescharacteristic
226C2#306212yes—
227C2#308112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#428C2semidirect
2. C2×#424C2semidirect
5. #2316C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #1735C2semidirect
18. C2×#107C22semidirect
19. C2×#106C22semidirect
21. C2×C4⋊C8C22semidirect
30. #389C22semidirect
33. #340C22semidirect
34. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
C2×#4242
C2×#4182
C22×#1022
#23162
#23092
#20672
#20652
#17352