Finite Group Databasebeta

← #329713

Subgroups — #329713

Order 512.

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

normal subgroups only

This group has 727 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#32906C212yesmaximal
2#29878C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10479C212yesmaximal, complemented
6#10478C212yesmaximal, complemented
7#10265C212yesmaximal, complemented
8#9138C212yesmaximal
9#8600C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×#462C2212yes—
13C2×#462C2212yes—
14C2×#451—22no—
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2857C2212yes—
18#2087—22no—
19#2076—22no—
20#1990C2212yes—
21#1985—22no—
22#1984C2212yes—
23#1984C2212yes—
24#1878—22no—
25#1878—22no—
26#1872—22no—
27#1834C2211yescharacteristic
28#1817—22no—
29#1803C2212yes—
30#1803C2212yes—
31#1797C2212yes—
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#332C2212yes—
35C2×#332C2212yes—
36#1296C2212yes—
37#1296C2212yes—
38#1277C2212yes—
39C23×Q8D411yescharacteristic
40C22×Q16—22no—
41C22×Q16—22no—
42C2×C4⋊Q8C2312yes—
43C2×C22⋊Q8—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—41no—
47C2×C4×Q8—41no—
48C2×C4×Q8—41no—
49C2×C4×Q8—41no—
50C2×C4×Q8D412yes—
51C2×C4×Q8C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#500—41no—
57#500—41no—
58#462—42no—
59#462—42no—
60#462—42no—
61#462—42no—
62#451—48no—
63#432—41no—
64#432—41no—
65#431—41no—
66#431—41no—
67C2×#107—22no—
68C2×C4⋊C8D412yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
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—24no—
90#340—22no—
91#340—22no—
92#336—24no—
93#336—24no—
94#336—22no—
95#335—41no—
96#334—41no—
97#332—42no—
98#332—42no—
99C22×Q8—21no—
100C22×Q8—24no—
101C22×Q8—22no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—42no—
105C22×Q8C2×D412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C2×Q16—44no—
112C2×Q16—44no—
113C2×Q16—44no—
114C2×Q16—44no—
115C2×Q16—44no—
116C2×Q16—44no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8C2×D412yes—
121C22×C8C2×D412yes—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4C2×D412yes—
152C2×C22⋊C4—42no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42C2411yescharacteristic
165#107—44no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—42no—
173#103—44no—
174#103—44no—
175#103—44no—
176#103—44no—
177C22⋊C8—42no—
178C22⋊C8—42no—
179C22⋊C8—44no—
180#97—42no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—21no—
184C4×C8—21no—
185#95—42no—
186#95—42no—
187C24#12111yescharacteristic
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—44no—
194C2×Q8—42no—
195C2×Q8—44no—
196C2×Q8—48no—
197C2×Q8—44no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8AGL(1,ℤ/8)14yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4C2×#4111yescharacteristic
231Q16—88no—
232Q16—88no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—44no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—82no—
259C4⋊C4—84no—
260C22⋊C4—84no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—44no—
268C42—42no—
269C42—42no—
270C42H(5,𝔽2)11yescharacteristic
271C42H(5,𝔽2)11yescharacteristic
272C42H(5,𝔽2)11yescharacteristic
273C42H(5,𝔽2)11yescharacteristic
274C23—21no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23#47312yes—
279C23C2×#12111yescharacteristic
280Q8—24no—
281Q8—24no—
282Q8—48no—
283Q8—42no—
284Q8—42no—
285Q8—82no—
286Q8—82no—
287Q8—82no—
288Q8—82no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—48no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4#53411yescharacteristic
320C2×C4#53411yescharacteristic
321C2×C4#53411yescharacteristic
322C2×C4#53511yescharacteristic
323C2×C4#53811yescharacteristic
324C2×C4#53811yescharacteristic
325C2×C4C2×AGL(1,ℤ/8)12yes—
326C2×C4#57512yes—
327C2×C4#57814yes—
328C8—42no—
329C8—42no—
330C8—82no—
331C8—82no—
332C8—82no—
333C22—24no—
334C22—22no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22#261711yescharacteristic
339C22#304412yes—
340C22#305512yes—
341C22C2×#47311yescharacteristic
342C22#306711yescharacteristic
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—44no—
348C4—41no—
349C4—82no—
350C4—162no—
351C4—162no—
352C4#319211yescharacteristic
353C4#319412yes—
354C4#319711yescharacteristic
355C4#320212yes—
356C4#322312yes—
357C2—42no—
358C2#1947612yes—
359C2#1947912yes—
360C2#2972411yescharacteristic
361C2#2974211yescharacteristic
362C2#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
#329062
#298782
#218202
#106262
#104792
#104782
#102652
#91382
#86002