Finite Group Databasebeta

← #329710

Subgroups — #329710

Order 512.

699 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 144 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#32954C212yesmaximal
2#29808C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10487C212yesmaximal, complemented
6#10486C212yesmaximal, complemented
7#10276C212yesmaximal, complemented
8#9140C212yesmaximal
9#8600C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#475C2212yes—
13C2×#475C2212yes—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2056—22no—
18#2001—22no—
19#1985—22no—
20#1984C2212yes—
21#1984C2212yes—
22#1956C2212yes—
23#1865—22no—
24#1865—22no—
25#1853—22no—
26#1834C2211yescharacteristic
27#1817—22no—
28#1803C2212yes—
29#1803C2212yes—
30#1797C2212yes—
31#1787—22no—
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#332C2212yes—
35C2×#332C2212yes—
36#1296C2212yes—
37#1296C2212yes—
38#1277C2212yes—
39C23×Q8D411yescharacteristic
40C22×Q16—22no—
41C2×C4⋊Q8C2312yes—
42C2×C22⋊Q8—22no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—41no—
46C2×C4×Q8—41no—
47C2×C4×Q8—41no—
48C2×C4×Q8—41no—
49C2×C4×Q8D412yes—
50C2×C4×Q8C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#494—41no—
56#494—41no—
57#475—42no—
58#475—42no—
59#475—42no—
60#475—42no—
61#432—41no—
62#432—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—22no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8D412yes—
79C2×#97—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—24no—
100C22×Q8—22no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—42no—
104C22×Q8C2×D412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×Q16—44no—
111C2×Q16—84no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C22⋊Q8—84no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—42no—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—42no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42C2411yescharacteristic
159#106—44no—
160#106—44no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—42no—
168#103—44no—
169#103—44no—
170#103—44no—
171#103—44no—
172#103—84no—
173C22⋊C8—42no—
174C22⋊C8—44no—
175C22⋊C8—42no—
176#97—42no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—21no—
181#95—42no—
182#95—42no—
183C24#12111yescharacteristic
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—44no—
190C2×Q8—42no—
191C2×Q8—44no—
192C2×Q8—48no—
193C2×Q8—44no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8#13714yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225Q16—88no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—44no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C22⋊C4—84no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—44no—
261C42—42no—
262C42—42no—
263C42H(5,𝔽2)11yescharacteristic
264C42H(5,𝔽2)11yescharacteristic
265C42H(5,𝔽2)11yescharacteristic
266C42H(5,𝔽2)11yescharacteristic
267C23—21no—
268C23—22no—
269C23—22no—
270C23—21no—
271C23#47312yes—
272C23C2×#12111yescharacteristic
273Q8—24no—
274Q8—24no—
275Q8—48no—
276Q8—42no—
277Q8—42no—
278Q8—84no—
279Q8—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—48no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—84no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4#53411yescharacteristic
311C2×C4#53411yescharacteristic
312C2×C4#53411yescharacteristic
313C2×C4#53511yescharacteristic
314C2×C4#53811yescharacteristic
315C2×C4#53811yescharacteristic
316C2×C4C2×#13712yes—
317C2×C4#57512yes—
318C2×C4#57714yes—
319C8—42no—
320C8—42no—
321C8—82no—
322C8—82no—
323C8—82no—
324C22—24no—
325C22—22no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22#261711yescharacteristic
330C22#304312yes—
331C22#305612yes—
332C22C2×#47311yescharacteristic
333C22#306711yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—44no—
339C4—41no—
340C4—82no—
341C4—162no—
342C4—162no—
343C4#319511yescharacteristic
344C4#319612yes—
345C4#319711yescharacteristic
346C4#320012yes—
347C4#322112yes—
348C2—42no—
349C2#1947112yes—
350C2#1947212yes—
351C2#2972211yescharacteristic
352C2#2974311yescharacteristic
353C2#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
#329542
#298082
#218202
#106262
#104872
#104862
#102762
#91402
#86002