Finite Group Databasebeta

← #416701

Subgroups — #416701

Order 512.

1460 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 242 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#35181C211yesmaximal, characteristic, complemented
2#21203C212yesmaximal
3#14154C211yesmaximal, characteristic
4#11063C218yesmaximal, complemented
5#10851C212yesmaximal, complemented
6C4×#343C211yesmaximal, characteristic, complemented
7C22×#127C2211yescharacteristic
8C2×#418C2218yes—
9C2×C4×M16C2212yes—
10#2851—22no—
11#2646—22no—
12#2571—22no—
13#2570—24no—
14#2552—22no—
15#2317C2214yes—
16C2×#390—22no—
17C2×#390C2214yes—
18#2285C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1961—24no—
22#1957—22no—
23#1957—22no—
24C2×#343C2214yes—
25C2×#339C2218yes—
26#1438—22no—
27C23×D4—21no—
28C23×D4—21no—
29C23×D4C2312yes—
30C22×M16C2314yes—
31C2×#127—22no—
32C2×#127—22no—
33C2×#127—24no—
34C2×#127—22no—
35C2×#127Q814yes—
36C2×#125—22no—
37C2×#117C2314yes—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2312yes—
40C22×C22⋊C4—28no—
41C22×C42D412yes—
42C22×C42C2312yes—
43C22×C42C2311yescharacteristic
44#418—216no—
45#418—48no—
46C2×#102—28no—
47C2×#102—28no—
48C4×M16—24no—
49C4×M16—24no—
50C2×#97—22no—
51C2×C4×C8—22no—
52#400—28no—
53#398—28no—
54#390—48no—
55#390—48no—
56#389—24no—
57#389—24no—
58#389—24no—
59#387—28no—
60#385—28no—
61#384—24no—
62#384D412yes—
63#384D412yes—
64#382—22no—
65#382—24no—
66#382D414yes—
67#380—22no—
68#380—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—24no—
74C2×#95—21no—
75C2×#95D412yes—
76#343—28no—
77#339—48no—
78C25—22no—
79C22×D4—24no—
80C22×D4—24no—
81C22×D4—24no—
82C22×D4—24no—
83C22×D4—28no—
84C22×D4—22no—
85C22×D4—28no—
86C22×D4—44no—
87C22×D4—44no—
88C22×D4—41no—
89C22×D4—42no—
90C22×D4—41no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—84no—
94C22×D4C2×Q814yes—
95C23×C4—22no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D414yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2411yescharacteristic
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16#4118yes—
106C22×C8—24no—
107#127—28no—
108#127—28no—
109#127—44no—
110#127—44no—
111#125—84no—
112#117—28no—
113#117—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D414yes—
128C2×C4⋊C4C2×D414yes—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—416no—
131C2×C22⋊C4—416no—
132C2×C22⋊C4—88no—
133C2×C42—22no—
134C2×C42—24no—
135C2×C42—24no—
136C2×C42—28no—
137C2×C42—22no—
138C2×C42—44no—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2×Q812yes—
142C2×C42#4114yes—
143C2×C42#4112yes—
144#102—416no—
145#102—416no—
146#97—44no—
147C4×C8—44no—
148#95—28no—
149#95—28no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—42no—
154C24—24no—
155C24—44no—
156C24—48no—
157C24—44no—
158C24—42no—
159C24—42no—
160C24—44no—
161C24—44no—
162C24C22×D411yescharacteristic
163C2×D4—28no—
164C2×D4—216no—
165C2×D4—416no—
166C2×D4—44no—
167C2×D4—48no—
168C2×D4—48no—
169C2×D4—48no—
170C2×D4—48no—
171C2×D4—48no—
172C2×D4—44no—
173C2×D4—48no—
174C2×D4—44no—
175C2×D4—82no—
176C2×D4—82no—
177C2×D4—88no—
178C2×D4—84no—
179C2×D4—84no—
180C2×D4—88no—
181C2×D4—88no—
182C2×D4—82no—
183C2×D4—82no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4#12011yescharacteristic
200C22×C4#12012yes—
201C22×C4#12011yescharacteristic
202C22×C4C22⋊Q812yes—
203C22×C4C22⋊Q812yes—
204C22×C4#12412yes—
205C22×C4#12412yes—
206C22×C4C22×D412yes—
207C22×C4C22×Q811yescharacteristic
208C22×C4C2×#4112yes—
209C22×C4C2×#4111yescharacteristic
210C22×C4C2×#4114yes—
211C22×C4H(5,𝔽2)14yes—
212M16—28no—
213M16—28no—
214M16—44no—
215M16—44no—
216C2×C8—28no—
217C2×C8—44no—
218C2×C8—44no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C4⋊C4—48no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—88no—
229C4⋊C4—88no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—832no—
233C42—48no—
234C42—44no—
235C42—48no—
236C42—82no—
237C42C22⋊Q814yes—
238C42C22⋊Q814yes—
239C42#12414yes—
240C42#12414yes—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—48no—
245C23—48no—
246C23—48no—
247C23—44no—
248C23—88no—
249C23—84no—
250C23—84no—
251C23—88no—
252C23—88no—
253C23—88no—
254C23—82no—
255C23—82no—
256C23—84no—
257C23—82no—
258C23—82no—
259C23—84no—
260C23C2×#12011yescharacteristic
261C23C2×C22⋊Q811yescharacteristic
262C23C2×#12411yescharacteristic
263C23#53212yes—
264C23#53512yes—
265D4—416no—
266D4—88no—
267D4—816no—
268D4—88no—
269D4—88no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4#39312yes—
295C2×C4#39314yes—
296C2×C4#39312yes—
297C2×C4#45312yes—
298C2×C4#45412yes—
299C2×C4C2×C22⋊Q812yes—
300C2×C4C2×#12412yes—
301C2×C4#52914yes—
302C2×C4#54014yes—
303C2×C4#54614yes—
304C2×C4#55014yes—
305C2×C4#55314yes—
306C2×C4#55414yes—
307C8—48no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—816no—
317C22—84no—
318C22—84no—
319C22—84no—
320C22—164no—
321C22—164no—
322C22—164no—
323C22—164no—
324C22C2×#39311yescharacteristic
325C22#243112yes—
326C22#256612yes—
327C22#264111yescharacteristic
328C22#286111yescharacteristic
329C22C2×#45311yescharacteristic
330C22C2×#45411yescharacteristic
331C22#302112yes—
332C4—22no—
333C4—24no—
334C4—84no—
335C4—84no—
336C4—84no—
337C4—84no—
338C4#202714yes—
339C4#245414yes—
340C4#249014yes—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—164no—
345C2#1522911yescharacteristic
346C2C2×#202711yescharacteristic
347C2#2346311yescharacteristic
348C2#2562212yes—
349C2#2562412yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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
1. #35181C2semidirect
4. #11063C2semidirect
5. #10851C2semidirect
6. C4×#343C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#212032
#141542
#110632
#108512
C4×#3432