Finite Group Databasebeta

← #232304

Subgroups — #232304

Order 512.

1588 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1588 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
1C22×#483C212yesmaximal
2#35011C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4#13904C211yesmaximal, characteristic
5#9973C218yesmaximal, complemented
6#7907C211yesmaximal, characteristic
7C22×C22⋊Q8C2212yes—
8C23×C4⋊C4C2211yescharacteristic
9C2×#483—28no—
10C2×#483—24no—
11C2×#483—28no—
12C2×#483C2218yes—
13C22×#106—22no—
14C22×#103—22no—
15C22×C22⋊C8C2212yes—
16#2857—22no—
17#2659C2214yes—
18#2583—24no—
19#2502—22no—
20#2434—22no—
21#2315—22no—
22C2×#381—22no—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1520—28no—
26C2×#328C2218yes—
27#1445C2214yes—
28C4⋊#95—22no—
29#1280C2214yes—
30C23×Q8D412yes—
31C24×C4—21no—
32C24×C4C2311yescharacteristic
33C23×C8—22no—
34C2×C4⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8D414yes—
42C2×C22⋊Q8C2314yes—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4D412yes—
56C22×C42D412yes—
57#483—232no—
58#483—216no—
59#483—216no—
60C2×#106—216no—
61C2×#106—28no—
62C2×#103—216no—
63C2×#103—28no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8D418yes—
67C2×C22⋊C8C2314yes—
68#398—28no—
69#395—28no—
70#391—24no—
71#389—28no—
72#385—28no—
73#385—28no—
74#384—22no—
75#381—48no—
76#380—28no—
77#380—24no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C2×#95—28no—
81C2×#95C2312yes—
82C2×#95C2312yes—
83#328—216no—
84C25C2×D411yescharacteristic
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—24no—
88C22×Q8—48no—
89C22×Q8C2×D414yes—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—44no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—44no—
99C23×C4—44no—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D414yes—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D414yes—
106C23×C4#4112yes—
107C23×C4C2411yescharacteristic
108C22×C8—28no—
109C22×C8—24no—
110C22×C8—48no—
111C4⋊Q8—84no—
112C22⋊Q8—28no—
113C22⋊Q8—28no—
114C22⋊Q8—24no—
115C22⋊Q8—24no—
116C22⋊Q8—24no—
117C22⋊Q8—24no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4C2×D414yes—
145C2×C4⋊C4C2×D414yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4SD1618yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—44no—
158#106—232no—
159#103—232no—
160C22⋊C8—28no—
161C22⋊C8—24no—
162C22⋊C8C2×D418yes—
163#95—216no—
164#95—24no—
165#95—24no—
166#95—416no—
167C24—42no—
168C24—42no—
169C24#12012yes—
170C24#12112yes—
171C24#12712yes—
172C24C2×SD1614yes—
173C24#13714yes—
174C24C22×D411yescharacteristic
175C2×Q8—24no—
176C2×Q8—24no—
177C2×Q8—48no—
178C2×Q8—416no—
179C2×Q8—44no—
180C2×Q8—48no—
181C2×Q8—88no—
182C22×C4—24no—
183C22×C4—28no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—48no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4—416no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4—88no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4#12014yes—
214C22×C4#12011yescharacteristic
215C22×C4#12014yes—
216C22×C4#12011yescharacteristic
217C22×C4#12114yes—
218C22×C4#12112yes—
219C22×C4#12114yes—
220C22×C4#12712yes—
221C22×C4C2×SD1618yes—
222C22×C4C22×D412yes—
223C22×C4C22×D411yescharacteristic
224C22×C4C22×D412yes—
225C22×C4C2×#4111yescharacteristic
226C2×C8—28no—
227C2×C8—48no—
228C2×C8—416no—
229C4⋊C4—28no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—84no—
240C4⋊C4—88no—
241C4⋊C4—88no—
242C22⋊C4—216no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—48no—
246C22⋊C4—84no—
247C42—44no—
248C42—82no—
249C23—22no—
250C23—24no—
251C23—22no—
252C23—24no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—48no—
258C23—48no—
259C23#39214yes—
260C23#44818yes—
261C23#46118yes—
262C23#49214yes—
263C23#49714yes—
264C23C2×#12011yescharacteristic
265C23C2×#12111yescharacteristic
266C23C2×#12711yescharacteristic
267C23C22×SD1612yes—
268C23C2×#13712yes—
269Q8—48no—
270Q8—84no—
271Q8—88no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—28no—
276C2×C4—24no—
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—84no—
294C2×C4—88no—
295C2×C4—88no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—816no—
300C2×C4—82no—
301C2×C4—88no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4#39212yes—
305C2×C4#39212yes—
306C2×C4#45814yes—
307C2×C4C2×#12012yes—
308C2×C4C2×#12112yes—
309C8—48no—
310C22—24no—
311C22—24no—
312C22—216no—
313C22—21no—
314C22—24no—
315C22—21no—
316C22—21no—
317C22—24no—
318C22—21no—
319C22—48no—
320C22—44no—
321C22—44no—
322C22#201318yes—
323C22C2×#39211yescharacteristic
324C22C2×#44812yes—
325C22#302212yes—
326C22C2×#45812yes—
327C22C2×#46112yes—
328C22C2×#49211yescharacteristic
329C22C2×#49711yescharacteristic
330C4—22no—
331C4—22no—
332C4—22no—
333C4—48no—
334C4—84no—
335C4—84no—
336C4—84no—
337C4—162no—
338C4#202214yes—
339C2—21no—
340C2—21no—
341C2—22no—
342C2—28no—
343C2#1820414yes—
344C2#2340811yescharacteristic
345C2C2×#201311yescharacteristic
346C2C2×#202211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9973C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
#350112
C2×#15202
#139042
#99732
#79072