Finite Group Databasebeta

← #116873

Subgroups — #116873

Order 512.

756 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 140 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#21948C211yesmaximal, characteristic
2#20702C212yesmaximal
3#20685C211yesmaximal, characteristic
4#8107C212yesmaximal
5C2×#1289C211yesmaximal, characteristic
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10C2×#380C2212yes—
11#1855—24no—
12#1825—24no—
13#1824—22no—
14#1823—22no—
15#1818C414yescomplemented
16#1817—22no—
17#1810—22no—
18#1736—24no—
19#1729—22no—
20#1305—24no—
21#1289—22no—
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16D414yes—
25C23×C8D412yes—
26C2×#117—21no—
27C2×#117—21no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117C2×C412yes—
33C2×#117Q812yes—
34C22×C4⋊C4—22no—
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4—22no—
39C22×C42—21no—
40C2×#108—24no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8C2×C412yes—
53C2×C22⋊C8C2×C412yes—
54C2×C22⋊C8D412yes—
55#380—22no—
56#380—22no—
57#380—22no—
58#380D414yes—
59C2×#95—22no—
60#341—24no—
61#340—28no—
62#340—28no—
63#336—28no—
64C25#4111yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—44no—
68C23×C4—42no—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D412yes—
75C23×C4C2×Q811yescharacteristic
76C23×C4#4112yes—
77C23×C4#4112yes—
78C23×C4#4111yescharacteristic
79C2×M16—24no—
80C2×M16—24no—
81C2×M16—48no—
82C2×M16—44no—
83C2×M16—41no—
84C2×M16—41no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—24no—
89C22×C8—44no—
90C22×C8C22⋊C414yes—
91#117—24no—
92#117—24no—
93#117—24no—
94#117—24no—
95#117—41no—
96#117—42no—
97#117—41no—
98#117—41no—
99#117—42no—
100#117—41no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—48no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C4⋊C412yes—
111C2×C4⋊C4C4⋊C412yes—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—82no—
119C2×C22⋊C4—82no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—42no—
123C2×C42—42no—
124#108—84no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#95—24no—
134#95—42no—
135C24—22no—
136C24—21no—
137C24—21no—
138C24—42no—
139C24—81no—
140C24#11711yescharacteristic
141C24#12111yescharacteristic
142C24C22⋊Q811yescharacteristic
143C24#12312yes—
144C24#12412yes—
145C22×C4—28no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—44no—
163C22×C4—82no—
164C22×C4—84no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×C4⋊C411yescharacteristic
169C22×C4C4×D412yes—
170C22×C4C4×D412yes—
171C22×C4#12012yes—
172C22×C4#12011yescharacteristic
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4#12111yescharacteristic
176C22×C4#12112yes—
177C22×C4C22⋊Q812yes—
178C22×C4C22⋊Q812yes—
179C22×C4C22⋊Q811yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4#12312yes—
182C22×C4#12411yescharacteristic
183C22×C4#12612yes—
184C22×C4#12711yescharacteristic
185M16—48no—
186M16—82no—
187M16—82no—
188M16—84no—
189C2×C8—24no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—48no—
194C2×C8—42no—
195C2×C8—44no—
196C2×C8—82no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—48no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—84no—
205C22⋊C4—44no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—84no—
211C42—42no—
212C42—42no—
213C42—44no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—41no—
231C23—42no—
232C23—84no—
233C23—82no—
234C23—82no—
235C23#38011yescharacteristic
236C23#38812yes—
237C23#39311yescharacteristic
238C23#39412yes—
239C23#39911yescharacteristic
240C23#47112yes—
241C23#47211yescharacteristic
242C23#47311yescharacteristic
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—168no—
272C2×C4#38512yes—
273C2×C4#38611yescharacteristic
274C2×C4#39011yescharacteristic
275C2×C4#39211yescharacteristic
276C2×C4#39612yes—
277C2×C4#39711yescharacteristic
278C2×C4#45814yes—
279C8—44no—
280C8—82no—
281C8—84no—
282C22—22no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—22no—
289C22—21no—
290C22—22no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22—81no—
304C22—81no—
305C22#143911yescharacteristic
306C22#181511yescharacteristic
307C22#181611yescharacteristic
308C22#203012yes—
309C22#205112yes—
310C22#205212yes—
311C4—21no—
312C4—22no—
313C4—22no—
314C4—21no—
315C4—82no—
316C4—82no—
317C4—164no—
318C4#189012yes—
319C4#202212yes—
320C2—22no—
321C2—22no—
322C2—22no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2—41no—
327C2#1107011yescharacteristic
328C2#1110511yescharacteristic
329C2#1111011yescharacteristic
subgroupG/H
center, socleC22#1439
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
15. #1818C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#207022
#206852
#81072
C2×#12892