Finite Group Databasebeta

← #141561

Subgroups — #141561

Order 512.

369 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 72 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#27658C211yesmaximal, characteristic
2#11786C211yesmaximal, characteristic
3#11773C211yesmaximal, characteristic
4#11576C211yesmaximal, characteristic
5#11558C211yesmaximal, characteristic
6#8620C211yesmaximal, characteristic
7C8×#99C211yesmaximal, characteristic
8C2×C8×D4C2211yescharacteristic
9C22×#106C411yescharacteristic, complemented
10#2487C2211yescharacteristic
11#2486C2211yescharacteristic
12#2088—21no—
13#2077—21no—
14#1941—21no—
15#1824C411yescharacteristic, complemented
16#1823C411yescharacteristic, complemented
17#1807—21no—
18#1807C411yescharacteristic, complemented
19C4×#99C2211yescharacteristic
20C8×C22⋊C4—21no—
21#1568—21no—
22#1466C2211yescharacteristic
23#1466C2211yescharacteristic
24#1459C2211yescharacteristic
25#1352—22no—
26#1344—22no—
27#1326—22no—
28#1317—22no—
29#1290—21no—
30#1282—21no—
31C23×C8C2×C411yescharacteristic
32C23×C8D411yescharacteristic
33C2×#123—21no—
34C2×C4×D4C2311yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C8×D4—41no—
38C8×D4—41no—
39#432—21no—
40#432—21no—
41C2×#106—21no—
42C2×#106—21no—
43C2×#106—21no—
44C2×#106—41no—
45C2×#106—41no—
46C2×C4⋊C8D411yescharacteristic
47C2×#99—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×C4×C8—41no—
52C2×C4×C8D411yescharacteristic
53#400—21no—
54#399—21no—
55#398—21no—
56#397—21no—
57#389—21no—
58#389—21no—
59#386—21no—
60#385—21no—
61#381—21no—
62#380—21no—
63#380—21no—
64#380C2×C411yescharacteristic
65#380C2×C411yescharacteristic
66C4×C22⋊C4—21no—
67#342D412yes—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#336—41no—
74#335—41no—
75#334—41no—
76#324—21no—
77#323—21no—
78#323—21no—
79#323—21no—
80#323—21no—
81#323D412yes—
82C22×D4#4111yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—41no—
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98#123—82no—
99C4×D4—41no—
100C4×D4—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4#4111yescharacteristic
114C2×C22⋊C4—22no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4C2×D411yescharacteristic
120C2×C22⋊C4C2×D411yescharacteristic
121C2×C42—41no—
122C2×C42C2×D411yescharacteristic
123#106—42no—
124#106—42no—
125#106—41no—
126#106—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130#99—24no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#97—41no—
138#97—41no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—82no—
142#95—21no—
143#95—21no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149C24C4×D411yescharacteristic
150C24#12111yescharacteristic
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—41no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—161no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12411yescharacteristic
176C22×C4#12711yescharacteristic
177C2×C8—22no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—81no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8#9911yescharacteristic
194C2×C8#9911yescharacteristic
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4—82no—
207C4⋊C4—161no—
208C22⋊C4—24no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—82no—
214C22⋊C4—81no—
215C22⋊C4—161no—
216C42—21no—
217C42—21no—
218C42—81no—
219C42—81no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23#39011yescharacteristic
229C23#44312yes—
230C23#47112yes—
231D4—44no—
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—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—44no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—162no—
265C2×C4—162no—
266C2×C4C2×#9911yescharacteristic
267C2×C4#41011yescharacteristic
268C8—21no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8—82no—
273C8—161no—
274C8—161no—
275C8#35212yes—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—81no—
288C22#196411yescharacteristic
289C22#197411yescharacteristic
290C22#197711yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—84no—
297C4—161no—
298C4—161no—
299C4—161no—
300C4—161no—
301C4C2×#35211yescharacteristic
302C4#212311yescharacteristic
303C2—21no—
304C2—21no—
305C2—42no—
306C2—42no—
307C2#981311yescharacteristic
308C2#1322011yescharacteristic
309C2#1322911yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×#106C4semidirect
15. #1824C4semidirect
16. #1823C4semidirect
18. #1807C4semidirect

Maximal subgroups

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

subgroupindex
#276582
#117862
#117732
#115762
#115582
#86202
C8×#992