Finite Group Databasebeta

← #100796

Subgroups — #100796

Order 512.

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

normal subgroups only

This group has 353 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
1#28692C211yesmaximal, characteristic
2#23363C211yesmaximal, characteristic
3#8593C211yesmaximal, characteristic
4#7723C211yesmaximal, characteristic
5#7723C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic
7#7041C211yesmaximal, characteristic
8C2×#501—21no—
9#2859C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#2084C412yes—
12#2083C412yes—
13#1992—21no—
14#1984—22no—
15#1949—22no—
16#1916—22no—
17#1916—22no—
18#1839C2211yescharacteristic
19#1839C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1767C2211yescharacteristic
23C2×#332—21no—
24#1296—21no—
25#1271—22no—
26#1271—22no—
27C2×C4⋊Q8D411yescharacteristic
28C2×#125—21no—
29C2×C4×Q8—21no—
30#501—42no—
31#501—42no—
32C2×#107—22no—
33C2×#106—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38C2×#97—21no—
39C2×C4×C8—22no—
40#391—21no—
41#389—21no—
42#389—21no—
43#389—21no—
44#389—21no—
45#389—21no—
46#389C2×C412yes—
47#384C2×C411yescharacteristic
48#384C2×C411yescharacteristic
49#384D411yescharacteristic
50#382—21no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4D412yes—
53C43C2311yescharacteristic
54#340—22no—
55#340—22no—
56#336—24no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336C2×C412yes—
66#336D412yes—
67#336D412yes—
68#332—42no—
69C22×Q8—21no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8—41no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8D812yes—
78C4⋊Q8SD1612yes—
79#125—41no—
80#125—41no—
81#125—41no—
82#125—41no—
83C4×Q8—81no—
84C4×Q8—81no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4C22⋊C412yes—
98C2×C4⋊C4C22⋊C412yes—
99C2×C42—22no—
100C2×C42C22×C411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42#4111yescharacteristic
106C2×C42#4111yescharacteristic
107#107—44no—
108#106—44no—
109C4⋊C8—24no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—82no—
115#97—41no—
116#97—41no—
117C4×C8—82no—
118#95—22no—
119#95—22no—
120#95—41no—
121#95—41no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—81no—
125C2×Q8—81no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C4×D411yescharacteristic
132C22×C4C4×D411yescharacteristic
133C22×C4#12011yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4#12311yescharacteristic
137C2×C8—24no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—81no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8—81no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—82no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4#10214yes—
172C42—22no—
173C42—41no—
174C42—41no—
175C42—41no—
176C42—41no—
177C42—41no—
178C42—42no—
179C42—42no—
180C42C2×D811yescharacteristic
181C42C2×SD1611yescharacteristic
182C42#13511yescharacteristic
183C42#13511yescharacteristic
184C23#38611yescharacteristic
185Q8—82no—
186Q8—82no—
187C2×C4—21no—
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—24no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4C2×#10211yescharacteristic
217C2×C4#41611yescharacteristic
218C2×C4#45612yes—
219C2×C4#46511yescharacteristic
220C2×C4#46511yescharacteristic
221C2×C4#46611yescharacteristic
222C2×C4#46611yescharacteristic
223C8—44no—
224C8—82no—
225C8—82no—
226C8—161no—
227C8—161no—
228C22#188111yescharacteristic
229C22#188111yescharacteristic
230C22#188611yescharacteristic
231C22#189311yescharacteristic
232C22#190411yescharacteristic
233C22#190511yescharacteristic
234C22#191511yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—21no—
241C4—22no—
242C4—41no—
243C4—44no—
244C4—41no—
245C4—41no—
246C4—82no—
247C4—161no—
248C4—161no—
249C2#900611yescharacteristic
250C2#930511yescharacteristic
251C2#930611yescharacteristic
252C2#933511yescharacteristic
253C2#933611yescharacteristic
254C2#939411yescharacteristic
255C2#942711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#286922
#233632
#85932
#77232
#70412