Finite Group Databasebeta

← #329746

Subgroups — #329746

Order 512.

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

normal subgroups only

This group has 634 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#29808C211yesmaximal, characteristic
2#29688C212yesmaximal
3#21819C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10487C212yesmaximal, complemented
6#10486C212yesmaximal, complemented
7#10276C212yesmaximal, complemented
8#9132C212yesmaximal
9#8600C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#475C2212yes—
13C2×#475C2212yes—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2058—22no—
18#2056—22no—
19#2043—22no—
20#2035—22no—
21#2035—22no—
22#2001—22no—
23#1984—22no—
24#1984C2212yes—
25#1984C2212yes—
26#1956C2212yes—
27#1834C2211yescharacteristic
28#1818—22no—
29#1803C2212yes—
30#1803C2212yes—
31#1797C2212yes—
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#332C2212yes—
35C2×#332C2212yes—
36#1296C2212yes—
37#1296C2212yes—
38#1277C2212yes—
39C22×Q16—22no—
40#533—41no—
41#533—41no—
42#533—41no—
43#533—41no—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8D412yes—
51C2×C4×Q8—21no—
52C2×C4×Q8C2312yes—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#494—41no—
56#494—41no—
57#475—42no—
58#475—42no—
59#475—42no—
60#475—42no—
61#431—41no—
62#431—41no—
63#431—41no—
64#431—41no—
65C2×#106—22no—
66C2×#106—22no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—24no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8D412yes—
79C2×C4×C8—21no—
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82#391—22no—
83#389—22no—
84#389—22no—
85#384C2312yes—
86#380—22no—
87#380—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—24no—
92#336—24no—
93#336—22no—
94#335—41no—
95#334—41no—
96#332—42no—
97#332—42no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×Q16—44no—
105C2×Q16—84no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119#125—42no—
120#125—42no—
121C22⋊Q8—42no—
122C22⋊Q8—42no—
123C22⋊Q8—42no—
124C22⋊Q8—42no—
125C22⋊Q8—84no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—42no—
131C4×Q8—42no—
132#117—42no—
133#117—42no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—42no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—22no—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#106—44no—
158#106—44no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—42no—
166#103—44no—
167#103—44no—
168#103—84no—
169#103—82no—
170#103—82no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173C22⋊C8—82no—
174C4×C8—21no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—42no—
179#95—42no—
180#95—42no—
181C24#12111yescharacteristic
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—44no—
194C2×Q8—44no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C2×#4111yescharacteristic
216Q16—88no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—44no—
228C2×C8—42no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—82no—
245C4⋊C4—84no—
246C22⋊C4—44no—
247C22⋊C4—84no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—42no—
255C42—42no—
256C42H(5,𝔽2)11yescharacteristic
257C42H(5,𝔽2)11yescharacteristic
258C42H(5,𝔽2)11yescharacteristic
259C42H(5,𝔽2)11yescharacteristic
260C23—21no—
261C23—22no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23C2×#12111yescharacteristic
266Q8—42no—
267Q8—42no—
268Q8—42no—
269Q8—42no—
270Q8—84no—
271Q8—84no—
272Q8—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#47312yes—
301C2×C4#53411yescharacteristic
302C2×C4#53411yescharacteristic
303C2×C4#53411yescharacteristic
304C2×C4#53511yescharacteristic
305C2×C4#53811yescharacteristic
306C2×C4#53811yescharacteristic
307C2×C4#57714yes—
308C2×C4#57712yes—
309C2×C4#57812yes—
310C8—42no—
311C8—42no—
312C8—82no—
313C8—82no—
314C8—82no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22—42no—
320C22#261711yescharacteristic
321C22#304312yes—
322C22#306312yes—
323C22C2×#47311yescharacteristic
324C22#306711yescharacteristic
325C4—21no—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—82no—
331C4—82no—
332C4—162no—
333C4—162no—
334C4#320012yes—
335C4#320112yes—
336C4#320211yescharacteristic
337C4#320411yescharacteristic
338C4#322112yes—
339C2—81no—
340C2#1956712yes—
341C2#1956812yes—
342C2#2972211yescharacteristic
343C2#2974611yescharacteristic
344C2#2974911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10626C2semidirect
5. #10487C2semidirect
6. #10486C2semidirect
7. #10276C2semidirect

Maximal subgroups

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

subgroupindex
#298082
#296882
#218192
#106262
#104872
#104862
#102762
#91322
#86002