Finite Group Databasebeta

← #9928

Subgroups — #9928

Order 256.

389 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 67 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×#484C211yesmaximal, characteristic, complemented
2#2626C211yesmaximal, characteristic
3C2×#399C211yesmaximal, characteristic
4#1824C211yesmaximal, characteristic, complemented
5C2×#328C211yesmaximal, characteristic, complemented
6C2×#323C211yesmaximal, characteristic
7#1294C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C22×C4⋊C4—21no—
10C22×C4⋊C4C2211yescharacteristic
11C22×C22⋊C4—21no—
12C22×C22⋊C4C2211yescharacteristic
13#484—22no—
14#484—22no—
15C2×#107—21no—
16C2×#103—21no—
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#399—21no—
20#399—22no—
21#399—22no—
22#396—22no—
23#393—21no—
24#385—21no—
25#380C2211yescharacteristic
26#379—21no—
27C2×#95C2211yescharacteristic
28#340—22no—
29#328—22no—
30#323—22no—
31C25D411yescharacteristic
32C22×Q8D411yescharacteristic
33C23×C4—21no—
34C23×C4—21no—
35C23×C4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C22×C8—21no—
38C22×C8—21no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4D412yes—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D412yes—
55C2×C22⋊C4—24no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—44no—
67#107—24no—
68#103—24no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8D412yes—
73#95—22no—
74#95—21no—
75#95—21no—
76#95D412yes—
77C24—21no—
78C24—21no—
79C24—21no—
80C24—21no—
81C24—21no—
82C24—21no—
83C24—42no—
84C24—42no—
85C24Q1612yes—
86C24C2×D411yescharacteristic
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—42no—
90C22×C4—24no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4#4111yescharacteristic
114C2×C8—22no—
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—41no—
118C4⋊C4—22no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—42no—
123C4⋊C4—48no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—82no—
129C22⋊C4—48no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C23—22no—
141C23—22no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—44no—
146C23—42no—
147C23—44no—
148C23—42no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—42no—
153C23—42no—
154C23—41no—
155C23—82no—
156C23—82no—
157C23—81no—
158C23—81no—
159C23#12011yescharacteristic
160C23#12011yescharacteristic
161C23#12111yescharacteristic
162C23C2×Q1611yescharacteristic
163C23#13511yescharacteristic
164C23#13711yescharacteristic
165C23#13711yescharacteristic
166Q8—42no—
167Q8—81no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—81no—
195C2×C4#12011yescharacteristic
196C2×C4#12111yescharacteristic
197C2×C4#12111yescharacteristic
198C2×C4#12711yescharacteristic
199C8—42no—
200C8—81no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—44no—
211C22—42no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—81no—
217C22—81no—
218C22#39211yescharacteristic
219C22#45111yescharacteristic
220C22#45111yescharacteristic
221C22#45211yescharacteristic
222C22#45211yescharacteristic
223C22#45312yes—
224C22#45812yes—
225C22#46711yescharacteristic
226C22#47011yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—44no—
232C4—82no—
233C4—81no—
234C4—81no—
235C4—82no—
236C4—81no—
237C2—21no—
238C2—21no—
239C2—21no—
240C2—21no—
241C2—82no—
242C2#161412yes—
243C2#161812yes—
244C2#200111yescharacteristic
245C2#200211yescharacteristic
246C2#202211yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
4. #1824C2semidirect
5. C2×#328C2semidirect
7. #1294C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
#26262
C2×#3992
#18242
C2×#3282
C2×#3232
#12942