Finite Group Databasebeta

← #121713

Subgroups — #121713

Order 512.

656 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 126 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#20660C211yesmaximal, characteristic
2#10729C211yesmaximal, characteristic
3C22⋊#338C211yesmaximal, characteristic
4#8115C212yesmaximal
5#7946C212yesmaximal
6C23×M16C2211yescharacteristic
7C2×#427C412yescomplemented
8C2×#407C2211yescharacteristic
9C2×#407C2212yes—
10#2280C2211yescharacteristic
11#2280C2212yes—
12#1809—22no—
13#1809—22no—
14C2×#344—22no—
15#1735C412yescomplemented
16C2×#338—22no—
17#1306—24no—
18#1284—24no—
19C24×C4C2311yescharacteristic
20C22×M16—22no—
21C22×M16—22no—
22C22×M16—22no—
23C22×M16—21no—
24C22×M16—44no—
25C22×M16D412yes—
26C22×M16D412yes—
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C2×#117—21no—
30C2×#117C2×C414yes—
31#427—44no—
32C2×#107—22no—
33C2×#106—22no—
34#407—22no—
35#407—22no—
36#407—22no—
37#407—22no—
38#407—22no—
39#407—21no—
40#407D412yes—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8Q814yes—
44#381—22no—
45#381D412yes—
46#380—22no—
47#380—22no—
48#380—22no—
49#379—21no—
50#379—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54#344—44no—
55#340—24no—
56#338—44no—
57C25#4111yescharacteristic
58C23×C4—42no—
59C23×C4—42no—
60C23×C4C22×C411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×Q812yes—
66C23×C4#4112yes—
67C23×C4#4112yes—
68C23×C4#4111yescharacteristic
69C23×C4#4112yes—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—24no—
73C2×M16—24no—
74C2×M16—24no—
75C2×M16—22no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—48no—
79C2×M16—48no—
80C2×M16—44no—
81C2×M16—44no—
82C2×M16—42no—
83C2×M16—41no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C22×C8—22no—
88C22×C8—24no—
89C22×C8—22no—
90C22×C8—24no—
91C22×C8—42no—
92C22×C8—41no—
93C22×C8C22⋊C412yes—
94C22×C8C22⋊C412yes—
95#117—42no—
96#117—42no—
97#117—44no—
98#117—44no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—41no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C42—24no—
107C2×C42—41no—
108#107—44no—
109#106—44no—
110C22⋊C8—24no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#95—24no—
117#95—42no—
118#95—42no—
119C24—42no—
120C24—82no—
121C24C4×D411yescharacteristic
122C24#12111yescharacteristic
123C24#12411yescharacteristic
124C24#12512yes—
125C24#12612yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—44no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—44no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—44no—
142C22×C4—42no—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4#11712yes—
145C22×C4C4×D411yescharacteristic
146C22×C4C4×Q812yes—
147C22×C4#12111yescharacteristic
148C22×C4C22⋊Q812yes—
149C22×C4C22⋊Q812yes—
150C22×C4C22⋊Q812yes—
151C22×C4#12311yescharacteristic
152C22×C4#12311yescharacteristic
153C22×C4#12312yes—
154C22×C4#12312yes—
155C22×C4#12312yes—
156C22×C4#12411yescharacteristic
157C22×C4#12512yes—
158C22×C4#12612yes—
159C22×C4#12711yescharacteristic
160C22×C4C4⋊Q811yescharacteristic
161M16—44no—
162M16—44no—
163M16—48no—
164M16—44no—
165M16—84no—
166M16—82no—
167M16—82no—
168M16—82no—
169C2×C8—28no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—44no—
176C2×C8—48no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8#9914yes—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—82no—
185C22⋊C4—44no—
186C22⋊C4—84no—
187C22⋊C4—84no—
188C22⋊C4—84no—
189C22⋊C4—84no—
190C42—84no—
191C42—81no—
192C42—81no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—88no—
206C23#38712yes—
207C23#39011yescharacteristic
208C23#39711yescharacteristic
209C23#39811yescharacteristic
210C23#40012yes—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—84no—
234C2×C4—84no—
235C2×C4—84no—
236C2×C4—84no—
237C2×C4#38111yescharacteristic
238C2×C4#38712yes—
239C2×C4#39111yescharacteristic
240C2×C4#39711yescharacteristic
241C2×C4#39811yescharacteristic
242C2×C4#39912yes—
243C2×C4C2×#9912yes—
244C2×C4#41012yes—
245C8—24no—
246C8—42no—
247C8—84no—
248C8—82no—
249C22—22no—
250C22—21no—
251C22—22no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—44no—
261C22—82no—
262C22—82no—
263C22—84no—
264C22—84no—
265C22#144511yescharacteristic
266C22#196412yes—
267C4—21no—
268C4—22no—
269C4—22no—
270C4—21no—
271C4—164no—
272C4—162no—
273C4#181911yescharacteristic
274C4#182011yescharacteristic
275C4#199612yes—
276C2—22no—
277C2—41no—
278C2—42no—
279C2—41no—
280C2—42no—
281C2#1135311yescharacteristic
282C2#1136012yes—
subgroupG/H
center, socleC22#1445
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#427C4semidirect
15. #1735C4semidirect

Maximal subgroups

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

subgroupindex
#206602
#107292
C22⋊#3382
#81152
#79462