Finite Group Databasebeta

← #329749

Subgroups — #329749

Order 512.

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

normal subgroups only

This group has 642 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#29878C211yesmaximal, characteristic
2#29664C212yesmaximal
3#21819C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10479C212yesmaximal, complemented
6#10478C212yesmaximal, complemented
7#10265C212yesmaximal, complemented
8#9130C212yesmaximal
9#8600C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×#462C2212yes—
13C2×#462C2212yes—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2087—22no—
18#2076—22no—
19#2024—22no—
20#2016—22no—
21#2001—22no—
22#2001—22no—
23#1990C2212yes—
24#1984—22no—
25#1984C2212yes—
26#1984C2212yes—
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—
40C22×Q16—22no—
41#533—41no—
42#533—41no—
43#533—41no—
44#533—41no—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8D412yes—
52C2×C4×Q8—21no—
53C2×C4×Q8C2312yes—
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#500—41no—
57#500—41no—
58#462—42no—
59#462—42no—
60#462—42no—
61#462—42no—
62#431—41no—
63#431—41no—
64#431—41no—
65#431—41no—
66C2×#107—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×C22⋊C8—21no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8D412yes—
78C2×C4×C8—21no—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391—22no—
82#389—22no—
83#389—22no—
84#384C2312yes—
85#380—22no—
86#380—22no—
87#340—22no—
88#340—24no—
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—44no—
106C2×Q16—82no—
107C2×Q16—82no—
108C2×Q16—82no—
109C2×Q16—82no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123#125—42no—
124#125—42no—
125C22⋊Q8—42no—
126C22⋊Q8—42no—
127C22⋊Q8—42no—
128C22⋊Q8—42no—
129C22⋊Q8—82no—
130C22⋊Q8—82no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—42no—
136C4×Q8—42no—
137#117—42no—
138#117—42no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C2×D412yes—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—42no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42C2411yescharacteristic
162#107—44no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—42no—
170#103—44no—
171#103—44no—
172#103—82no—
173#103—82no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—82no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—42no—
182#95—42no—
183#95—42no—
184C24#12111yescharacteristic
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C2×#4111yescharacteristic
221Q16—88no—
222Q16—88no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—44no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—42no—
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—44no—
250C4⋊C4—82no—
251C4⋊C4—84no—
252C22⋊C4—44no—
253C22⋊C4—84no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—42no—
261C42—42no—
262C42H(5,𝔽2)11yescharacteristic
263C42H(5,𝔽2)11yescharacteristic
264C42H(5,𝔽2)11yescharacteristic
265C42H(5,𝔽2)11yescharacteristic
266C23—21no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23C2×#12111yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—42no—
275Q8—42no—
276Q8—84no—
277Q8—82no—
278Q8—82no—
279Q8—82no—
280Q8—82no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4#47312yes—
309C2×C4#53411yescharacteristic
310C2×C4#53411yescharacteristic
311C2×C4#53411yescharacteristic
312C2×C4#53511yescharacteristic
313C2×C4#53811yescharacteristic
314C2×C4#53811yescharacteristic
315C2×C4#57612yes—
316C2×C4#57712yes—
317C2×C4#57814yes—
318C8—42no—
319C8—42no—
320C8—82no—
321C8—82no—
322C8—82no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22#261711yescharacteristic
329C22#304412yes—
330C22#306212yes—
331C22C2×#47311yescharacteristic
332C22#306711yescharacteristic
333C4—21no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—82no—
339C4—82no—
340C4—162no—
341C4—162no—
342C4#319812yes—
343C4#319911yescharacteristic
344C4#320212yes—
345C4#320411yescharacteristic
346C4#322312yes—
347C2—81no—
348C2#1957212yes—
349C2#1957512yes—
350C2#2972411yescharacteristic
351C2#2974511yescharacteristic
352C2#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. #10479C2semidirect
6. #10478C2semidirect
7. #10265C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#296642
#218192
#106262
#104792
#104782
#102652
#91302
#86002