Finite Group Databasebeta

← #141584

Subgroups — #141584

Order 512.

400 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 70 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#28095C211yesmaximal, characteristic
2#11786C211yesmaximal, characteristic
3#11785C211yesmaximal, characteristic
4#11576C211yesmaximal, characteristic
5#11575C211yesmaximal, characteristic
6#8622C211yesmaximal, characteristic
7C8×#99C211yesmaximal, characteristic
8C2×C8×D4C2211yescharacteristic
9C22×#103C411yescharacteristic, complemented
10#2487C2211yescharacteristic
11#2409C2211yescharacteristic
12#2068—21no—
13#2036—21no—
14#1933—21no—
15#1892—21no—
16#1824C411yescharacteristic, complemented
17#1808C411yescharacteristic, complemented
18#1787C411yescharacteristic, complemented
19C4×#99C2211yescharacteristic
20C8×C22⋊C4—21no—
21#1568—21no—
22#1466C2211yescharacteristic
23#1466C2211yescharacteristic
24#1459C2211yescharacteristic
25#1352—22no—
26#1351—22no—
27#1326—22no—
28#1326—22no—
29#1290—21no—
30#1282—21no—
31C23×Q8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C22⋊Q8—21no—
35C2×C4×D4C2311yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C8×D4—41no—
38C8×D4—41no—
39#432—21no—
40#432—21no—
41C2×#107—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×#103—21no—
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#103—41no—
48C2×#103—41no—
49C2×#99—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8C2×C411yescharacteristic
52C2×C22⋊C8D411yescharacteristic
53C2×C4×C8—41no—
54C2×C4×C8D411yescharacteristic
55#400—21no—
56#399—21no—
57#395—21no—
58#393—21no—
59#391—21no—
60#389—21no—
61#386—21no—
62#386—21no—
63#385—21no—
64#385—21no—
65#380—21no—
66#380C2×C411yescharacteristic
67#380C2×C411yescharacteristic
68C4×C22⋊C4—21no—
69#342D412yes—
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×Q8—21no—
83C22×Q8—21no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8—42no—
87C22×Q8—81no—
88C22×D4#4111yescharacteristic
89C23×C4—21no—
90C23×C4—21no—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8—81no—
99C22×C8—81no—
100C22×C8C22⋊C411yescharacteristic
101C22×C8C22⋊C411yescharacteristic
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22⋊Q8—42no—
105C22⋊Q8—81no—
106C4×D4—41no—
107C4×D4—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4#4111yescharacteristic
117C2×C22⋊C4—22no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4C2×D411yescharacteristic
123C2×C22⋊C4C2×D411yescharacteristic
124C2×C42—41no—
125C2×C42C2×D411yescharacteristic
126#107—81no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130#103—42no—
131#103—42no—
132#103—41no—
133#103—41no—
134#103—81no—
135#99—24no—
136C22⋊C8—22no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142#97—41no—
143#97—41no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—82no—
147#95—21no—
148#95—21no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154C24C4×D411yescharacteristic
155C24#12111yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—81no—
165C2×Q8—82no—
166C2×Q8—84no—
167C2×Q8—81no—
168C2×Q8—81no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—82no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—161no—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12411yescharacteristic
193C22×C4#12711yescharacteristic
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8#9911yescharacteristic
211C2×C8#9911yescharacteristic
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—81no—
220C4⋊C4—82no—
221C4⋊C4—161no—
222C22⋊C4—24no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—82no—
229C22⋊C4—161no—
230C42—21no—
231C42—21no—
232C42—81no—
233C42—81no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23#39011yescharacteristic
243C23#44312yes—
244C23#47312yes—
245Q8—42no—
246Q8—42no—
247Q8—81no—
248Q8—82no—
249Q8—82no—
250Q8—81no—
251D4—44no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—162no—
287C2×C4C2×#9911yescharacteristic
288C2×C4#41011yescharacteristic
289C8—21no—
290C8—21no—
291C8—41no—
292C8—41no—
293C8—41no—
294C8—82no—
295C8—161no—
296C8—161no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22#196411yescharacteristic
310C22#197411yescharacteristic
311C22#197911yescharacteristic
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—84no—
318C4—82no—
319C4—161no—
320C4—161no—
321C4—161no—
322C4#212011yescharacteristic
323C4#212111yescharacteristic
324C2—21no—
325C2—21no—
326C2—42no—
327C2—42no—
328C2#981411yescharacteristic
329C2#1322211yescharacteristic
330C2#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×#103C4semidirect
16. #1824C4semidirect
17. #1808C4semidirect
18. #1787C4semidirect

Maximal subgroups

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

subgroupindex
#280952
#117862
#117852
#115762
#115752
#86222
C8×#992