Finite Group Databasebeta

← #122183

Subgroups — #122183

Order 512.

728 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 162 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#35286C211yesmaximal, characteristic
2#10857C212yesmaximal
3#8041C211yesmaximal, characteristic
4#8028C211yesmaximal, characteristic
5#7903C212yesmaximal
6C2×#423C2212yes—
7C2×C4×M16C2212yes—
8#2859—22no—
9#2647—22no—
10#2571—22no—
11#2552—22no—
12#2442—24no—
13#2317C412yes—
14#2316C412yes—
15#2285C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C4×#117C2211yescharacteristic
18C2×#343—24no—
19#1296—22no—
20#1295—22no—
21#1279—22no—
22#1279—22no—
23C22×M16—22no—
24C22×M16—22no—
25C2×#125—24no—
26C2×#117C2×C411yescharacteristic
27C2×#117C2×C412yes—
28C2×#117C2×C411yescharacteristic
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4D414yes—
31C22×C42C2311yescharacteristic
32#423—22no—
33#423—22no—
34#423—22no—
35#423Q814yes—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C4×M16—22no—
39C4×M16—22no—
40C4×M16—22no—
41C4×M16Q814yes—
42C2×#97—22no—
43C2×C4×C8—22no—
44#400—28no—
45#398—28no—
46#389—24no—
47#389—24no—
48#389—24no—
49#389—24no—
50#389—24no—
51#385—24no—
52#384—22no—
53#384—22no—
54#384—21no—
55#384—21no—
56#384D412yes—
57#382—24no—
58#382—22no—
59#382D412yes—
60#380—24no—
61#380—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4D412yes—
64C4×C4⋊C4D412yes—
65C4×C22⋊C4—22no—
66#376—21no—
67C43—21no—
68#343—48no—
69C23×C4C22⋊C414yes—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D412yes—
72C2×M16—24no—
73C2×M16—24no—
74C2×M16—24no—
75C2×M16—24no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—44no—
81C2×M16—44no—
82C22×C8—42no—
83C22×C8—42no—
84#125—84no—
85#125—84no—
86#117—42no—
87#117—42no—
88#117—44no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4C22⋊C411yescharacteristic
101C2×C4⋊C4C22⋊C411yescharacteristic
102C2×C4⋊C4C22⋊C412yes—
103C2×C4⋊C4C22⋊C414yes—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—22no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×Q812yes—
116C2×C42C2×Q812yes—
117C2×C42#4112yes—
118C2×C42#4112yes—
119C4⋊C8—24no—
120C4⋊C8—24no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123#97—24no—
124#97—42no—
125C4×C8—24no—
126C4×C8—42no—
127#95—28no—
128#95—28no—
129#95—22no—
130#95—22no—
131#95—24no—
132C24C2×C22⋊C411yescharacteristic
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—48no—
145C22×C4—44no—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C4×Q812yes—
149C22×C4C4×Q812yes—
150C22×C4#12012yes—
151C22×C4#12012yes—
152C22×C4C22⋊Q812yes—
153C22×C4C22⋊Q812yes—
154C22×C4#12412yes—
155C22×C4C4⋊Q812yes—
156M16—28no—
157M16—44no—
158M16—44no—
159M16—44no—
160M16—48no—
161M16—84no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—82no—
167C2×C8—82no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—84no—
181C4⋊C4—88no—
182C4⋊C4—88no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—44no—
186C42—42no—
187C42—42no—
188C42—44no—
189C42—42no—
190C42—42no—
191C42—42no—
192C42—42no—
193C42C22⋊Q812yes—
194C42C22⋊Q812yes—
195C42C22⋊Q812yes—
196C42C22⋊Q812yes—
197C42#12412yes—
198C42#12411yescharacteristic
199C42#12411yescharacteristic
200C42C4⋊Q811yescharacteristic
201C42C4⋊Q811yescharacteristic
202C42C4⋊Q812yes—
203C23—22no—
204C23—22no—
205C23—21no—
206C23—21no—
207C23#37911yescharacteristic
208C23#39112yes—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—48no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—164no—
236C2×C4#39111yescharacteristic
237C2×C4#39111yescharacteristic
238C2×C4#39112yes—
239C2×C4#39312yes—
240C2×C4#39312yes—
241C2×C4#39512yes—
242C2×C4#39512yes—
243C2×C4#41312yes—
244C2×C4#41312yes—
245C2×C4#45312yes—
246C2×C4#45412yes—
247C2×C4#45512yes—
248C2×C4#45612yes—
249C8—48no—
250C8—84no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22#144611yescharacteristic
258C22#178511yescharacteristic
259C22#178511yescharacteristic
260C22#179212yes—
261C22#179312yes—
262C4—21no—
263C4—21no—
264C4—22no—
265C4—22no—
266C4—82no—
267C4—84no—
268C4—81no—
269C4—84no—
270C4—81no—
271C4#199814yes—
272C4#202712yes—
273C4#202812yes—
274C4#206012yes—
275C4#206112yes—
276C2—42no—
277C2#847012yes—
278C2#847211yescharacteristic
279C2#847411yescharacteristic
280C2#1138311yescharacteristic
281C2#1139212yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#352862
#108572
#80412
#80282
#79032