Finite Group Databasebeta

← #510071

Subgroups — #510071

Order 512.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#28705C211yesmaximal, characteristic
2#24268C211yesmaximal, characteristic
3#13494C212yesmaximal, complemented
4#13382C212yesmaximal, complemented
5#13364C212yesmaximal, complemented
6#13132C212yesmaximal, complemented
7#13062C212yesmaximal, complemented
8#12977C211yesmaximal, characteristic
9#12732C212yesmaximal, complemented
10C23×Q16C2211yescharacteristic
11C22×M32C2211yescharacteristic
12C2×#500C2211yescharacteristic
13C2×#467C2212yes—
14C2×C8×D4C2211yescharacteristic
15#2563—21no—
16#2409—21no—
17C2×#367C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×C22⋊C16C2211yescharacteristic
23C2×C22⋊C16C2212yes—
24#2076—21no—
25#2076—21no—
26#2056—21no—
27#2001—21no—
28#1987—21no—
29#1969C2212yes—
30#1808C2212yes—
31#1808C2212yes—
32#1808C2211yescharacteristic
33#1787—21no—
34C4×#107C2211yescharacteristic
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1374C2212yes—
39C23×Q8—21no—
40C22×Q16—24no—
41C22×Q16—21no—
42C22×Q16—42no—
43C22×Q16—41no—
44C22×Q16—41no—
45C22×Q16C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C4×D4D411yescharacteristic
51C2×M32—42no—
52C2×M32—41no—
53C22×C16C2312yes—
54C22×C16C2312yes—
55#500—41no—
56#500—41no—
57#467—24no—
58#467—42no—
59C8×D4—41no—
60C8×D4—41no—
61C2×#107—21no—
62C2×#107C2312yes—
63C2×#107C2312yes—
64C2×#107C2312yes—
65C2×C4⋊C8D411yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×C22⋊C8D411yescharacteristic
69C2×C22⋊C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#395—41no—
72#393—41no—
73#393—41no—
74#393—41no—
75#391—22no—
76#390—41no—
77#386—41no—
78#385—41no—
79#382—41no—
80#380—22no—
81#380—22no—
82#380—21no—
83C4×C4⋊C4—21no—
84#367—42no—
85#358—24no—
86#358—42no—
87#358—42no—
88#358—42no—
89C22⋊C16—41no—
90C22⋊C16—42no—
91#340—21no—
92#340—22no—
93#340—22no—
94#340—21no—
95C22×Q8—22no—
96C22×Q8—44no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—82no—
102C22×D4D811yescharacteristic
103C23×C4—41no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×Q16—22no—
107C2×Q16—22no—
108C2×Q16—22no—
109C2×Q16—22no—
110C2×Q16—22no—
111C2×Q16—44no—
112C2×Q16—42no—
113C2×Q16—44no—
114C2×Q16—48no—
115C2×Q16—44no—
116C2×Q16#4114yes—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2411yescharacteristic
129C4⋊Q8—42no—
130C4⋊Q8—42no—
131C22⋊Q8—44no—
132C22⋊Q8—82no—
133C4×D4—41no—
134C4×D4—41no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4D811yescharacteristic
141C2×C22⋊C4—42no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4D811yescharacteristic
146C2×C22⋊C4D811yescharacteristic
147C2×C42—41no—
148C2×C42C2×D411yescharacteristic
149M32—44no—
150C2×C16—22no—
151C2×C16—22no—
152C2×C16—22no—
153C2×C16—22no—
154C2×C16—22no—
155C2×C16—22no—
156#107—22no—
157#107—22no—
158#107—22no—
159#107—22no—
160#107—22no—
161#107—22no—
162#107—42no—
163C4⋊C8—41no—
164#103—44no—
165#103—82no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C4×C8—21no—
169C4×C8—21no—
170#95—41no—
171#95—42no—
172#95—42no—
173#95—41no—
174#95—42no—
175#95—42no—
176C24C2×D811yescharacteristic
177C24C2×D811yescharacteristic
178C2×Q8—24no—
179C2×Q8—42no—
180C2×Q8—48no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—84no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×Q8—88no—
189C2×Q8—84no—
190C2×Q8—162no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—84no—
207C22×C4—82no—
208C22×C4C2×D811yescharacteristic
209C22×C4C2×D811yescharacteristic
210C22×C4C2×D811yescharacteristic
211C22×C4C2×D811yescharacteristic
212C22×C4C22×D411yescharacteristic
213Q16—24no—
214Q16—24no—
215Q16—48no—
216Q16—42no—
217Q16—42no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8C2×#4112yes—
231C2×C8H(5,𝔽2)11yescharacteristic
232C2×C8H(5,𝔽2)12yes—
233C2×C8H(5,𝔽2)11yescharacteristic
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C4⋊C4—82no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—84no—
246C22⋊C4—82no—
247C22⋊C4—162no—
248C22⋊C4—162no—
249C42—21no—
250C42—21no—
251C42—82no—
252C16—42no—
253C16—42no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—21no—
262C23—22no—
263C23#51012yes—
264C23C22×D811yescharacteristic
265Q8—44no—
266Q8—44no—
267Q8—88no—
268Q8—82no—
269Q8—82no—
270Q8—164no—
271D4—44no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—88no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#51012yes—
300C2×C4#53411yescharacteristic
301C2×C4#54612yes—
302C2×C4#57511yescharacteristic
303C2×C4#57712yes—
304C8—41no—
305C8—41no—
306C8—41no—
307C8#53812yes—
308C8#55111yescharacteristic
309C8H(3,𝔽4)11yescharacteristic
310C22—24no—
311C22—22no—
312C22—41no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22#318611yescharacteristic
321C22#329412yes—
322C22C2×#51011yescharacteristic
323C22C2×#51011yescharacteristic
324C22#341612yes—
325C4—22no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—84no—
330C4—162no—
331C4—162no—
332C4—162no—
333C4#268411yescharacteristic
334C4#320011yescharacteristic
335C4#331112yes—
336C2—42no—
337C2—81no—
338C2#3126011yescharacteristic
339C2#3376111yescharacteristic
340C2#3376611yescharacteristic
341C2#3388312yes—
342C2#3389612yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13494C2semidirect
4. #13382C2semidirect
5. #13364C2semidirect
6. #13132C2semidirect
7. #13062C2semidirect
9. #12732C2semidirect

Maximal subgroups

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

subgroupindex
#287052
#242682
#134942
#133822
#133642
#131322
#130622
#129772
#127322