Finite Group Databasebeta

← #460504

Subgroups — #460504

Order 512.

1048 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 194 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#35154C211yesmaximal, characteristic
2#23746C212yesmaximal
3#13942C211yesmaximal, characteristic
4#11392C212yesmaximal
5#11296C212yesmaximal
6#11226C212yesmaximal
7#11162C214yesmaximal
8#10763C211yesmaximal, characteristic
9C2×#544C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C22×#101C2212yes—
12#2855—22no—
13#2627—22no—
14#2453—24no—
15#2451—24no—
16#2316C2211yescharacteristic
17#2310C2212yes—
18#2296C2212yes—
19C2×#391C2212yes—
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#2101—24no—
23#2101—24no—
24C2×#343C2212yes—
25#1736C2214yes—
26#1736C2214yes—
27C2×#339C2214yes—
28C2×#339C2214yes—
29#1728C2212yes—
30#1441—22no—
31C4⋊#95C2212yes—
32#1434—22no—
33C23×Q8C2311yescharacteristic
34C22×M16C2314yes—
35#544—24no—
36#544—42no—
37C2×C4⋊Q8—22no—
38C2×#125—22no—
39C2×C22⋊Q8—22no—
40C2×C4×Q8—24no—
41C2×C4×Q8—24no—
42C2×C4×Q8—42no—
43C2×#117C2312yes—
44C2×#117C2312yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C42D412yes—
50C22×C42C2312yes—
51C22×C42C2311yescharacteristic
52C2×#101—24no—
53C2×#101—42no—
54C2×#101—42no—
55#400—24no—
56#398—28no—
57#395—24no—
58#391—24no—
59#391—22no—
60#391—24no—
61#391—22no—
62#391—28no—
63#389—24no—
64#389—24no—
65#389D414yes—
66#389D414yes—
67#387—28no—
68#385—22no—
69#385—22no—
70#382—24no—
71#381—24no—
72#380—22no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—42no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95—21no—
80#343—24no—
81#340—24no—
82#340—24no—
83#340—28no—
84#339—44no—
85#339—44no—
86#336—24no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—21no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—21no—
95C22×Q8#4114yes—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4C2×D412yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2411yescharacteristic
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—24no—
109C22×C8—24no—
110C4⋊Q8—48no—
111#125—48no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C4×Q8—416no—
115#117—24no—
116#117—24no—
117#117—22no—
118#117—22no—
119#117—22no—
120#117—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C2×D412yes—
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4#4114yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—22no—
152C2×C42—42no—
153C2×C42C2×D414yes—
154#101—48no—
155#95—24no—
156#95—28no—
157#95—28no—
158#95—24no—
159#95—24no—
160#95—48no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—48no—
165C24C22×D411yescharacteristic
166C2×Q8—24no—
167C2×Q8—24no—
168C2×Q8—28no—
169C2×Q8—28no—
170C2×Q8—44no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—84no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4#12012yes—
209C22×C4#12012yes—
210C22×C4#12314yes—
211C22×C4#12414yes—
212C22×C4C22×D412yes—
213C22×C4C2×#4112yes—
214C22×C4C2×#4112yes—
215C22×C4H(5,𝔽2)11yescharacteristic
216C22×C4H(5,𝔽2)12yes—
217C22×C42−1+411yescharacteristic
218C22×C42−1+412yes—
219C22×C42−1+412yes—
220M16—44no—
221M16—44no—
222M16—44no—
223M16—44no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C4⋊C4—28no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—48no—
233C4⋊C4—48no—
234C4⋊C4—44no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—48no—
238C4⋊C4—84no—
239C4⋊C4—84no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C42—28no—
245C42—48no—
246C42—42no—
247C42—42no—
248C42—44no—
249C42—44no—
250C42—44no—
251C23—22no—
252C23—22no—
253C23C2×#12011yescharacteristic
254C23C2×#12311yescharacteristic
255C23C2×#12411yescharacteristic
256C23#53411yescharacteristic
257C23#53611yescharacteristic
258C23#53611yescharacteristic
259C23#53711yescharacteristic
260Q8—48no—
261Q8—48no—
262Q8—164no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#53612yes—
303C2×C4#53712yes—
304C2×C4#53812yes—
305C2×C4#53912yes—
306C2×C4#54112yes—
307C2×C4#54212yes—
308C2×C4#54614yes—
309C2×C4#54814yes—
310C2×C4#55214yes—
311C2×C4#56314yes—
312C8—84no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22#240511yescharacteristic
322C22#243611yescharacteristic
323C22#250311yescharacteristic
324C22#261611yescharacteristic
325C22#262511yescharacteristic
326C22#263911yescharacteristic
327C22#281711yescharacteristic
328C22#301812yes—
329C22#302012yes—
330C4—42no—
331C4—44no—
332C4—42no—
333C4—44no—
334C4—84no—
335C4—84no—
336C4—82no—
337C4—82no—
338C4—162no—
339C4#269014yes—
340C4#270214yes—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2#1636611yescharacteristic
346C2#2402111yescharacteristic
347C2#2402311yescharacteristic
348C2#2592112yes—
349C2#2970812yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#351542
#237462
#139422
#113922
#112962
#112262
#111622
#107632