Finite Group Databasebeta

← #100744

Subgroups — #100744

Order 512.

316 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 72 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#29581C211yesmaximal, characteristic
2#22025C211yesmaximal, characteristic
3#9894C211yesmaximal, characteristic
4#7703C211yesmaximal, characteristic
5#7611C211yesmaximal, characteristic
6#7048C211yesmaximal, characteristic
7#7043C211yesmaximal, characteristic
8#2859C2211yescharacteristic
9#2295C2211yescharacteristic
10#2099C411yescharacteristic
11#2096C411yescharacteristic
12#2083C411yescharacteristic
13#2072C411yescharacteristic
14#1985—21no—
15#1943—21no—
16#1940—21no—
17#1935—21no—
18#1920—21no—
19#1917—22no—
20#1863—21no—
21#1840—21no—
22#1839C2211yescharacteristic
23#1832—21no—
24#1767C2211yescharacteristic
25#1745C2211yescharacteristic
26#1745C2211yescharacteristic
27#1745C2211yescharacteristic
28#1295—22no—
29#1272—21no—
30#1271—21no—
31#1271—21no—
32#1271—21no—
33C2×#125—21no—
34C2×C4×Q8D411yescharacteristic
35C2×#107—21no—
36C2×#106—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#97—21no—
42C2×#97—21no—
43C2×#97—21no—
44C2×#97D412yes—
45#391C2×C411yescharacteristic
46#391D411yescharacteristic
47#389—21no—
48#389—21no—
49#389—21no—
50#389—21no—
51#389C2×C411yescharacteristic
52#389D411yescharacteristic
53#387—21no—
54#387—21no—
55#384C2×C411yescharacteristic
56#384C2×C411yescharacteristic
57#382—21no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4D411yescharacteristic
60#376—21no—
61C43C2311yescharacteristic
62#340—21no—
63#340—21no—
64#340—21no—
65#340—21no—
66#336—22no—
67#336—22no—
68#336—22no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336C2×C411yescharacteristic
78#336C2×C411yescharacteristic
79#336D411yescharacteristic
80#336D411yescharacteristic
81C22×Q8C22⋊C411yescharacteristic
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8—41no—
89#125—82no—
90C4×Q8—41no—
91C4×Q8—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4C22⋊C411yescharacteristic
104C2×C4⋊C4C22⋊C411yescharacteristic
105C2×C4⋊C4C22⋊C411yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C22×C411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115#107—42no—
116#106—42no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—82no—
120#103—42no—
121#103—42no—
122#97—22no—
123#97—42no—
124#97—41no—
125#97—41no—
126#97—41no—
127#97—42no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—41no—
136#95—41no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—41no—
140C2×Q8—41no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C4×D411yescharacteristic
149C22×C4C4×D411yescharacteristic
150C22×C4#12011yescharacteristic
151C22×C4#12111yescharacteristic
152C22×C4#12111yescharacteristic
153C22×C4#12311yescharacteristic
154C2×C8—22no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—42no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—42no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—81no—
171C2×C8—81no—
172C2×C8—81no—
173C2×C8—81no—
174C2×C8—81no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—41no—
197C42—41no—
198C42—41no—
199C42—41no—
200C42—41no—
201C42—42no—
202C42—41no—
203C42—41no—
204C23#38611yescharacteristic
205Q8—42no—
206Q8—81no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4#42111yescharacteristic
245C2×C4#42111yescharacteristic
246C2×C4#45511yescharacteristic
247C2×C4#45611yescharacteristic
248C8—42no—
249C8—81no—
250C8—82no—
251C8—81no—
252C8—81no—
253C8—82no—
254C8—161no—
255C22#188111yescharacteristic
256C22#188211yescharacteristic
257C22#188411yescharacteristic
258C22#189711yescharacteristic
259C22#190311yescharacteristic
260C22#190511yescharacteristic
261C22#191511yescharacteristic
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4—161no—
276C4—161no—
277C2#899811yescharacteristic
278C2#931311yescharacteristic
279C2#932811yescharacteristic
280C2#933611yescharacteristic
281C2#933811yescharacteristic
282C2#939011yescharacteristic
283C2#941811yescharacteristic
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
#295812
#220252
#98942
#77032
#76112
#70482
#70432