Finite Group Databasebeta

← #124691

Subgroups — #124691

Order 512.

407 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 88 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×#1944C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic
3#7861C211yesmaximal, characteristic
4#7127C214yesmaximal, complemented
5#2920—21no—
6C22×#107C411yescharacteristic
7C22×#106C411yescharacteristic
8C22×#97C2211yescharacteristic
9C2×#389—21no—
10#2112—21no—
11#1944—21no—
12#1944—21no—
13#1944C412yes—
14#1944C2212yes—
15C2×#340—21no—
16#1377—21no—
17#1377C2212yes—
18#1275—21no—
19#1275C2212yes—
20C23×C8C2×C411yescharacteristic
21C23×C8D411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—41no—
25C22×C42D411yescharacteristic
26C2×M32—22no—
27#432—41no—
28#432—41no—
29#431—22no—
30#431—22no—
31C2×#107—21no—
32C2×#107—41no—
33C2×#107—41no—
34C2×#107C2×C412yes—
35C2×#106—21no—
36C2×#106—24no—
37C2×#106C2×C412yes—
38#423—41no—
39#407—41no—
40C4×M16—22no—
41C2×#97—41no—
42C2×#97—41no—
43C2×#97C2×C411yescharacteristic
44C2×#97D411yescharacteristic
45C2×#97D411yescharacteristic
46C2×#97C2311yescharacteristic
47#389—22no—
48#389—22no—
49#389—41no—
50#389—41no—
51C2×#95—41no—
52C22⋊C16—41no—
53C22⋊C16—41no—
54#340—22no—
55#340—22no—
56C23×C4—41no—
57C23×C4—41no—
58C23×C4C22⋊C411yescharacteristic
59C23×C4D811yescharacteristic
60C23×C4SD1611yescharacteristic
61C2×M16—42no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—22no—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—84no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42D811yescharacteristic
93C2×C42SD1611yescharacteristic
94C2×C42C2×D411yescharacteristic
95M32—42no—
96M32—42no—
97C2×C16—42no—
98#107—42no—
99#107—42no—
100#106—24no—
101#106—22no—
102#106M1614yescomplemented
103C4⋊C8—42no—
104C4⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—41no—
108#97—41no—
109#97—41no—
110#97—41no—
111#97—42no—
112C4×C8—24no—
113#95—42no—
114#95—42no—
115C24#10211yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—82no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4#9911yescharacteristic
136C22×C4#9911yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4#10411yescharacteristic
141C22×C4#10411yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C2×D811yescharacteristic
144C22×C4C2×SD1611yescharacteristic
145M16—84no—
146C2×C8—24no—
147C2×C8—24no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8#10112yes—
161C2×C8C2×M1612yes—
162C2×C8#13712yes—
163C4⋊C4—24no—
164C4⋊C4—44no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—84no—
169C4⋊C4—84no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C16—82no—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—21no—
185C23#32711yescharacteristic
186C23#32711yescharacteristic
187C23C2×#10211yescharacteristic
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—88no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—162no—
218C2×C4—162no—
219C2×C4#32711yescharacteristic
220C2×C4#32711yescharacteristic
221C2×C4#40711yescharacteristic
222C2×C4C2×#9911yescharacteristic
223C2×C4C2×#10111yescharacteristic
224C2×C4#41911yescharacteristic
225C2×C4C2×#10411yescharacteristic
226C2×C4#42111yescharacteristic
227C8—22no—
228C8—22no—
229C8—22no—
230C8—41no—
231C8—84no—
232C22—24no—
233C22—24no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22#133912yes—
241C22#134212yes—
242C22#148611yescharacteristic
243C22C2×#32711yescharacteristic
244C22#154611yescharacteristic
245C4—21no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—84no—
252C4—162no—
253C4#146311yescharacteristic
254C4#149111yescharacteristic
255C4#151111yescharacteristic
256C4#152811yescharacteristic
257C2—21no—
258C2—21no—
259C2—22no—
260C2—22no—
261C2#761011yescharacteristic
262C2C2×#133911yescharacteristic
263C2C2×#134211yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7127C2semidirect
102. #106M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19442
#128462
#78612
#71272