Finite Group Databasebeta

← #460475

Subgroups — #460475

Order 512.

1092 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 210 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#23659C212yesmaximal
3#13942C211yesmaximal, characteristic
4#11392C212yesmaximal
5#11342C212yesmaximal
6#11226C212yesmaximal
7#11136C214yesmaximal
8#10763C211yesmaximal, characteristic
9C2×#544C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#428C2214yes—
12C22×#101C2212yes—
13#2855—22no—
14#2627—22no—
15#2453—24no—
16#2451—24no—
17#2316C2211yescharacteristic
18#2310C2212yes—
19#2296C2212yes—
20C2×#391C2212yes—
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2085—24no—
24#2085—24no—
25C2×#343C2212yes—
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
52#428—44no—
53#428—44no—
54C2×#107—24no—
55C2×#106—24no—
56C2×#101—42no—
57C2×#101—42no—
58C2×#101Q818yescomplemented
59#400—24no—
60#398—28no—
61#395—24no—
62#391—24no—
63#391—22no—
64#391—24no—
65#391—22no—
66#391—44no—
67#389—24no—
68#389—24no—
69#389D414yes—
70#389D414yes—
71#387—28no—
72#385—22no—
73#385—22no—
74#382—24no—
75#381—24no—
76#380—22no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—42no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—21no—
84#343—24no—
85#340—24no—
86#340—24no—
87#339—44no—
88#339—44no—
89#336—24no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—21no—
98C22×Q8C2×Q814yes—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2411yescharacteristic
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—24no—
111C2×M16C2×Q818yes—
112C22×C8—24no—
113C4⋊Q8—48no—
114#125—48no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117C4×Q8—416no—
118#117—24no—
119#117—24no—
120#117—22no—
121#117—22no—
122#117—22no—
123#117—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
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—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D414yes—
143C2×C4⋊C4#4114yes—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—42no—
156C2×C42C2×D414yes—
157#107—48no—
158#106—48no—
159#101—48no—
160#95—24no—
161#95—28no—
162#95—28no—
163#95—24no—
164#95—24no—
165#95—48no—
166#95—42no—
167#95—48no—
168#95—42no—
169#95—42no—
170C24C22×D411yescharacteristic
171C2×Q8—24no—
172C2×Q8—24no—
173C2×Q8—28no—
174C2×Q8—28no—
175C2×Q8—44no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—84no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4#12012yes—
214C22×C4#12012yes—
215C22×C4#12314yes—
216C22×C4C4⋊Q814yes—
217C22×C4C22×D412yes—
218C22×C4C22×Q812yes—
219C22×C4C2×#4112yes—
220C22×C4H(5,𝔽2)11yescharacteristic
221C22×C4H(5,𝔽2)12yes—
222C22×C4H(5,𝔽2)12yes—
223C22×C42−1+411yescharacteristic
224C22×C42−1+412yes—
225M16—28no—
226M16—28no—
227M16—44no—
228M16—44no—
229C2×C8—28no—
230C2×C8—44no—
231C2×C8—44no—
232C4⋊C4—28no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—48no—
238C4⋊C4—48no—
239C4⋊C4—44no—
240C4⋊C4—48no—
241C4⋊C4—48no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—28no—
251C42—48no—
252C42—42no—
253C42—42no—
254C42—44no—
255C42—48no—
256C23—22no—
257C23—22no—
258C23C2×#12011yescharacteristic
259C23C2×#12311yescharacteristic
260C23C2×C4⋊Q811yescharacteristic
261C23#53411yescharacteristic
262C23#53511yescharacteristic
263C23#53611yescharacteristic
264C23#53711yescharacteristic
265Q8—48no—
266Q8—48no—
267Q8—164no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#53512yes—
308C2×C4#53712yes—
309C2×C4#54112yes—
310C2×C4#54212yes—
311C2×C4#54312yes—
312C2×C4#54412yes—
313C2×C4#54614yes—
314C2×C4D4×Q814yes—
315C2×C4#55414yes—
316C2×C4#56214yes—
317C8—48no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22#240511yescharacteristic
327C22#243411yescharacteristic
328C22#250211yescharacteristic
329C22#261511yescharacteristic
330C22#262511yescharacteristic
331C22#265911yescharacteristic
332C22#281611yescharacteristic
333C22#301812yes—
334C22#302012yes—
335C4—42no—
336C4—44no—
337C4—42no—
338C4—44no—
339C4—84no—
340C4—82no—
341C4—82no—
342C4—84no—
343C4—162no—
344C4#269214yes—
345C4#270414yes—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2#1636511yescharacteristic
351C2#2402111yescharacteristic
352C2#2402311yescharacteristic
353C2#2582712yes—
354C2#2968312yes—
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

Complements

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

NH ≅ G/Ntype
58. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
#351542
#236592
#139422
#113922
#113422
#112262
#111362
#107632