Finite Group Databasebeta

← #99722

Subgroups — #99722

Order 512.

452 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#23627C211yesmaximal, characteristic
2#21909C211yesmaximal, characteristic
3#8709C212yesmaximal, complemented
4#7563C211yesmaximal, characteristic
5#7029C212yesmaximal, complemented
6C2×#416—21no—
7#2297C2211yescharacteristic
8#2285C2211yescharacteristic
9#2074—21no—
10#2073—21no—
11#2072C412yes—
12#2071C412yes—
13#1924—22no—
14#1923—22no—
15#1797C2212yes—
16#1780—21no—
17#1750—21no—
18C4×C22⋊C8C2212yes—
19#1735—21no—
20C2×#336C2211yescharacteristic
21#1279—22no—
22#1279—22no—
23#1272—21no—
24#1270—21no—
25C22×#41D411yescharacteristic
26C23×C8—21no—
27C2×#117—21no—
28C2×#117D411yescharacteristic
29C22×C42C2311yescharacteristic
30C2×C4⋊C8D412yes—
31C2×C4⋊C8D412yes—
32#416—44no—
33C2×#103—22no—
34C2×#102—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—21no—
40C2×#97—21no—
41C2×C4×C8—22no—
42C2×C4×C8—21no—
43#391C2×C411yescharacteristic
44#391D411yescharacteristic
45#390C2×C411yescharacteristic
46#390D411yescharacteristic
47#386—22no—
48#384C2×C412yes—
49#382—21no—
50#380—22no—
51C4×C4⋊C4—22no—
52C4×C22⋊C4—22no—
53#340—22no—
54#340—22no—
55#336—22no—
56#336—22no—
57#336—21no—
58#336—22no—
59#336C2×C412yes—
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—41no—
63C2×#41—41no—
64C2×#41—42no—
65C22×Q8C22⋊C411yescharacteristic
66C22×D4—21no—
67C22×D4C22⋊C411yescharacteristic
68C23×C4—21no—
69C23×C4C2×D411yescharacteristic
70C23×C4#4112yes—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8—42no—
79#117—41no—
80#117—41no—
81#117—42no—
82#117—42no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4C22⋊C411yescharacteristic
87C2×C4⋊C4C22⋊C411yescharacteristic
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—41no—
92C2×C42—21no—
93C2×C42—42no—
94C2×C42—41no—
95C2×C42C22×C411yescharacteristic
96C2×C42C2×D412yes—
97C2×C42C2×D411yescharacteristic
98C4⋊C8—42no—
99C4⋊C8—42no—
100#103—44no—
101#102—44no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—81no—
107C22⋊C8—81no—
108C22⋊C8—82no—
109#97—81no—
110C4×C8—42no—
111C4×C8—42no—
112C4×C8—81no—
113#95—22no—
114#95—22no—
115#95—42no—
116C24—21no—
117C24#12311yescharacteristic
118#41—44no—
119#41—44no—
120#41—82no—
121#41—82no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—42no—
131C2×D4—81no—
132C2×D4—81no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D412yes—
150C22×C4#12011yescharacteristic
151C22×C4#12112yes—
152C22×C4#13512yes—
153C22×C4AGL(1,ℤ/8)12yes—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—82no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8—82no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—82no—
178C22⋊C4—81no—
179C22⋊C4—82no—
180C42—22no—
181C42—22no—
182C42—41no—
183C42—41no—
184C42—41no—
185C42—41no—
186C42—82no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23—41no—
196C23#38611yescharacteristic
197C23#48212yes—
198Q8—42no—
199Q8—81no—
200D4—42no—
201D4—81no—
202D4—84no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4#41611yescharacteristic
229C2×C4#41811yescharacteristic
230C2×C4#44312yes—
231C2×C4#44412yes—
232C2×C4#44912yes—
233C2×C4#45511yescharacteristic
234C2×C4#45511yescharacteristic
235C2×C4#45812yes—
236C8—82no—
237C8—84no—
238C8—82no—
239C8—82no—
240C22—24no—
241C22—22no—
242C22—41no—
243C22—42no—
244C22—41no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22—82no—
249C22—81no—
250C22#187511yescharacteristic
251C22#187612yes—
252C22#188112yes—
253C22#188511yescharacteristic
254C22#189011yescharacteristic
255C4—21no—
256C4—21no—
257C4—22no—
258C4—22no—
259C4—42no—
260C4—81no—
261C4—81no—
262C4—81no—
263C4—162no—
264C4#161212yes—
265C4#161212yes—
266C2—42no—
267C2—161no—
268C2#899612yes—
269C2#900511yescharacteristic
270C2#902811yescharacteristic
271C2#903812yes—
272C2#906711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8709C2semidirect
5. #7029C2semidirect

Maximal subgroups

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

subgroupindex
#236272
#219092
#87092
#75632
#70292