Finite Group Databasebeta

← #100874

Subgroups — #100874

Order 512.

373 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 86 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#28755C211yesmaximal, characteristic
2#23371C211yesmaximal, characteristic
3#9903C211yesmaximal, characteristic
4#9903C211yesmaximal, characteristic
5#7636C211yesmaximal, characteristic
6#7042C211yesmaximal, characteristic
7#7042C211yesmaximal, characteristic
8C2×#491C412yes—
9#2868C2211yescharacteristic
10C4×#125C2211yescharacteristic
11#2043—21no—
12#2043—21no—
13#1981—21no—
14#1981—21no—
15#1960C412yes—
16#1940—21no—
17#1939—21no—
18#1917—21no—
19#1917—21no—
20#1916—21no—
21#1916—21no—
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24#1767C2211yescharacteristic
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27#1296—21no—
28#1296—21no—
29#1296—21no—
30#1296—21no—
31#1271—21no—
32#1271—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8C2×C412yes—
36C2×#125C2×C411yescharacteristic
37#491—44no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×#103—22no—
48C2×#103—22no—
49C2×#103—22no—
50C2×#103—22no—
51C2×#97—21no—
52C2×#97—21no—
53C2×#97—21no—
54C2×#97C2×C412yes—
55C2×C4×C8—21no—
56#389D411yescharacteristic
57#389D411yescharacteristic
58#387—21no—
59#387—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#382C2×C411yescharacteristic
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4D411yescharacteristic
68C4×C4⋊C4D411yescharacteristic
69C43C2311yescharacteristic
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81C22×Q8—22no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8—41no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—82no—
96#125—22no—
97#125—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C22⋊C411yescharacteristic
109C2×C4⋊C4C22⋊C411yescharacteristic
110C2×C4⋊C4C22⋊C411yescharacteristic
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—81no—
132C4⋊C8—81no—
133#103—44no—
134#103—44no—
135#103—82no—
136#103—82no—
137#97—42no—
138#97—81no—
139#97—81no—
140#97—81no—
141C4×C8—81no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4C4×D411yescharacteristic
161C22×C4C4×D411yescharacteristic
162C22×C4#12011yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12311yescharacteristic
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—81no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8—81no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4#10012yes—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—41no—
223C42—41no—
224C23#38611yescharacteristic
225Q8—82no—
226Q8—82no—
227Q8—82no—
228Q8—82no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4C2×#10011yescharacteristic
260C2×C4#41311yescharacteristic
261C2×C4#45411yescharacteristic
262C2×C4#45411yescharacteristic
263C2×C4#45511yescharacteristic
264C2×C4#45511yescharacteristic
265C2×C4#45611yescharacteristic
266C2×C4#45611yescharacteristic
267C2×C4#47111yescharacteristic
268C2×C4#47111yescharacteristic
269C2×C4#47311yescharacteristic
270C2×C4#47311yescharacteristic
271C8—82no—
272C8—82no—
273C8—82no—
274C8—81no—
275C8—81no—
276C8—81no—
277C8—81no—
278C22#188211yescharacteristic
279C22#188211yescharacteristic
280C22#188811yescharacteristic
281C22#190011yescharacteristic
282C22#190811yescharacteristic
283C22#190811yescharacteristic
284C22#191011yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—41no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—162no—
299C4#166211yescharacteristic
300C4#166311yescharacteristic
301C4#166411yescharacteristic
302C4#166411yescharacteristic
303C2#900411yescharacteristic
304C2#935311yescharacteristic
305C2#935411yescharacteristic
306C2#937611yescharacteristic
307C2#937611yescharacteristic
308C2#943611yescharacteristic
309C2#945811yescharacteristic
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
#287552
#233712
#99032
#76362
#70422