Finite Group Databasebeta

← #121371

Subgroups — #121371

Order 512.

800 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 154 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#21940C211yesmaximal, characteristic
2#20846C212yesmaximal
3#10713C211yesmaximal, characteristic
4C2×#1293C211yesmaximal, characteristic
5#7915C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#428—22no—
8C2×#407C2212yes—
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C2×#380C2212yes—
12#1854—24no—
13#1853—22no—
14#1852—22no—
15#1852—22no—
16#1852—22no—
17#1808—22no—
18#1736—22no—
19C2×#336—22no—
20#1293C414yescomplemented
21#1280—24no—
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C23×C8—21no—
25C23×C8D412yes—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—44no—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D412yes—
32C22×C4⋊C4D411yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—22no—
35C22×C42—21no—
36#428—48no—
37C2×#107—24no—
38C2×#107—22no—
39C2×#107—22no—
40C2×#106—24no—
41#407—22no—
42#407—22no—
43#407C2×C414yes—
44#407Q814yes—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8C2×C412yes—
52C2×C22⋊C8D414yes—
53C2×C22⋊C8Q812yes—
54C2×C22⋊C8Q812yes—
55#380—22no—
56#380—22no—
57#380—22no—
58#380—22no—
59C2×#95—22no—
60#340—24no—
61#340—24no—
62#340—24no—
63#336—44no—
64C25#4111yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C23×C4C2×D412yes—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×Q811yescharacteristic
77C23×C4C2×Q812yes—
78C23×C4C2×Q811yescharacteristic
79C23×C4#4112yes—
80C23×C4#4111yescharacteristic
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—44no—
84C2×M16—44no—
85C22×C8—22no—
86C22×C8—24no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8C4⋊C414yes—
96#117—28no—
97#117—44no—
98#117—44no—
99#117—44no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—44no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—84no—
116C2×C42—24no—
117C2×C42—42no—
118C2×C42—41no—
119C2×C42—41no—
120#107—48no—
121#107—82no—
122#107—82no—
123#106—48no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—24no—
130C22⋊C8—42no—
131C22⋊C8C4⋊C418yes—
132#95—42no—
133#95—42no—
134C24—22no—
135C24—22no—
136C24—41no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24C4×Q811yescharacteristic
141C24#12112yes—
142C24#12111yescharacteristic
143C24#12512yes—
144C24#12511yescharacteristic
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4C2×C4⋊C411yescharacteristic
178C22×C4C2×C4⋊C412yes—
179C22×C4C4×D412yes—
180C22×C4C4×D411yescharacteristic
181C22×C4#12012yes—
182C22×C4#12011yescharacteristic
183C22×C4C22⋊Q812yes—
184C22×C4C22⋊Q812yes—
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q812yes—
187C22×C4C22⋊Q812yes—
188C22×C4#12311yescharacteristic
189C22×C4#12312yes—
190C22×C4#12711yescharacteristic
191C22×C4C4⋊Q811yescharacteristic
192C22×C4C4⋊Q812yes—
193C22×C4C4⋊Q812yes—
194C22×C4C4⋊Q811yescharacteristic
195M16—48no—
196M16—48no—
197C2×C8—24no—
198C2×C8—28no—
199C2×C8—44no—
200C2×C8—44no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—42no—
206C2×C8—82no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—82no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C22⋊C4—48no—
219C22⋊C4—84no—
220C22⋊C4—84no—
221C42—44no—
222C42—44no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—24no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23#38912yes—
247C23#38911yescharacteristic
248C23#39211yescharacteristic
249C23#39812yes—
250C23#39811yescharacteristic
251C23#47212yes—
252C23#47312yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—164no—
289C2×C4—164no—
290C2×C4#38411yescharacteristic
291C2×C4#38512yes—
292C2×C4#38511yescharacteristic
293C2×C4#39512yes—
294C2×C4#39511yescharacteristic
295C2×C4#39911yescharacteristic
296C2×C4B(4,𝔽2)12yes—
297C2×C4#45812yes—
298C8—48no—
299C8—44no—
300C8—82no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—84no—
320C22—82no—
321C22—82no—
322C22#144411yescharacteristic
323C22#195212yes—
324C22#201711yescharacteristic
325C22#201711yescharacteristic
326C22#201811yescharacteristic
327C22#202011yescharacteristic
328C22#202111yescharacteristic
329C22#202211yescharacteristic
330C22#208912yes—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—22no—
335C4—82no—
336C4—82no—
337C4—162no—
338C4—162no—
339C4#186712yes—
340C4#206212yes—
341C2—22no—
342C2—22no—
343C2—42no—
344C2—42no—
345C2#997411yescharacteristic
346C2#997611yescharacteristic
347C2#997711yescharacteristic
348C2#997911yescharacteristic
349C2#1133111yescharacteristic
350C2#1133711yescharacteristic
351C2#1134411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1293C4semidirect

Maximal subgroups

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

subgroupindex
#219402
#208462
#107132
C2×#12932
#79152