Finite Group Databasebeta

← #417132

Subgroups — #417132

Order 512.

1480 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 266 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
1C2×#3139C212yesmaximal
2#35173C211yesmaximal, characteristic
3C2×#1986C212yesmaximal
4#14155C211yesmaximal, characteristic
5#11055C218yesmaximal
6C4×#343C211yesmaximal, characteristic
7C2×#533C2212yes—
8C22×C4⋊Q8C2211yescharacteristic
9#3139—28no—
10#3139—28no—
11C2×#491—24no—
12C2×#487—22no—
13C2×#487—22no—
14C2×#419C2218yes—
15C2×C4×M16C2212yes—
16#2857—22no—
17#2659—22no—
18#2584—24no—
19#2567—22no—
20#2546—22no—
21#2316C2214yes—
22C2×#391C2214yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1986—28no—
26C2×#343C2214yes—
27C2×#339C2218yes—
28#1445—22no—
29C4⋊#95—22no—
30C23×Q8C2312yes—
31C22×M16C2314yes—
32#533—24no—
33#533—24no—
34#533—24no—
35#533—24no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—24no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8D414yes—
42C2×C4⋊Q8D412yes—
43C2×C4⋊Q8D412yes—
44C2×#125D414yes—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×#117C2314yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2312yes—
54C22×C42D412yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57#491—232no—
58#487—216no—
59#487—216no—
60#419—216no—
61#419—216no—
62C2×#103—28no—
63C2×#103—28no—
64C4×M16—24no—
65C4×M16D418yes—
66C2×#97—22no—
67C2×C4×C8—22no—
68#398—28no—
69#395—28no—
70#391—24no—
71#391—216no—
72#389—28no—
73#389—28no—
74#389—28no—
75#384—22no—
76#380—28no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—24no—
80C2×#95—28no—
81C2×#95—21no—
82C2×#95D412yes—
83#343—28no—
84#339—216no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—24no—
89C22×Q8—24no—
90C22×Q8—28no—
91C22×Q8—28no—
92C22×Q8C2×D414yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D414yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2411yescharacteristic
99C2×M16—24no—
100C2×M16—28no—
101C2×M16—28no—
102C2×M16C2×D418yes—
103C22×C8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—28no—
107C4⋊Q8—28no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—44no—
111C4⋊Q8—44no—
112C4⋊Q8—84no—
113#125—28no—
114#125—28no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×Q8—44no—
122C4×Q8—44no—
123#117—24no—
124#117—24no—
125#117—24no—
126#117—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C2×D414yes—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4#4118yes—
148C2×C22⋊C4—24no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—28no—
153C2×C42—22no—
154C2×C42—44no—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D414yes—
159C2×C42#4112yes—
160#103—232no—
161#103—232no—
162#97—28no—
163C4×C8—28no—
164#95—216no—
165#95—24no—
166#95—416no—
167#95—42no—
168C24C22×D411yescharacteristic
169C2×Q8—28no—
170C2×Q8—216no—
171C2×Q8—44no—
172C2×Q8—48no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—48no—
178C2×Q8—48no—
179C2×Q8—416no—
180C2×Q8—84no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4#12012yes—
201C22×C4#12011yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12112yes—
204C22×C4#12112yes—
205C22×C4#12418yes—
206C22×C4#12712yes—
207C22×C4#12712yes—
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D412yes—
210C22×C4C22×D412yes—
211C22×C4C2×#4111yescharacteristic
212C22×C4C2×#4114yes—
213C22×C42−1+414yes—
214M16—28no—
215M16—28no—
216M16—48no—
217C2×C8—28no—
218C2×C8—48no—
219C4⋊C4—216no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—48no—
225C4⋊C4—44no—
226C4⋊C4—48no—
227C4⋊C4—44no—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—88no—
233C4⋊C4—88no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C42—44no—
237C42—44no—
238C42—44no—
239C42—48no—
240C42—82no—
241C42#12014yes—
242C42#12014yes—
243C42#12114yes—
244C42#12114yes—
245C23—21no—
246C23—21no—
247C23—22no—
248C23C2×#12011yescharacteristic
249C23C2×#12111yescharacteristic
250C23C2×#12412yes—
251C23C2×#12711yescharacteristic
252C23#53612yes—
253Q8—416no—
254Q8—88no—
255Q8—88no—
256Q8—88no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—48no—
268C2×C4—44no—
269C2×C4—48no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—48no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—84no—
284C2×C4#39212yes—
285C2×C4#39214yes—
286C2×C4#39212yes—
287C2×C4#45512yes—
288C2×C4#45612yes—
289C2×C4C2×#12012yes—
290C2×C4C2×#12112yes—
291C2×C4#53214yes—
292C2×C4#53614yes—
293C2×C4#54714yes—
294C2×C4#54818yes—
295C2×C4#55614yes—
296C8—48no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×#39211yescharacteristic
306C22#243612yes—
307C22#254712yes—
308C22#265811yescharacteristic
309C22#285811yescharacteristic
310C22C2×#45511yescharacteristic
311C22C2×#45611yescharacteristic
312C22#302212yes—
313C4—22no—
314C4—24no—
315C4—48no—
316C4—84no—
317C4—84no—
318C4—84no—
319C4—84no—
320C4#200714yes—
321C4#243214yes—
322C4#254514yes—
323C2—21no—
324C2—21no—
325C2—22no—
326C2#1524311yescharacteristic
327C2C2×#200711yescharacteristic
328C2#2340811yescharacteristic
329C2#2591712yes—
330C2#2591912yes—
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
C2×#31392
#351732
C2×#19862
#141552
#110552
C4×#3432