Finite Group Databasebeta

← #194929

Subgroups — #194929

Order 512.

1352 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1352 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2633C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#21838C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9028C214yesmaximal, complemented
6#9024C214yesmaximal, complemented
7#8018C212yesmaximal, complemented
8C22×#125C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2633—24no—
12#2633C418yescomplemented
13C2×#399C2212yes—
14C2×#398—21no—
15C2×#397C2212yes—
16C2×#394—21no—
17#2357—24no—
18#2354—24no—
19#2312C2214yes—
20#2309—21no—
21C2×#385—22no—
22#2285C2212yes—
23C2×#383C2211yescharacteristic
24#2280—22no—
25#2274—21no—
26#1992—24no—
27#1989—24no—
28#1824C2214yes—
29#1823C2214yes—
30#1466—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×D4D411yescharacteristic
34C24×C4D412yes—
35C2×#125—22no—
36C2×#125—22no—
37C2×#125—22no—
38C2×#125—24no—
39C2×#125C2×C414yes—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—24no—
43C2×C4×D4C2×C414yes—
44C2×C4×D4D414yes—
45C2×#117—22no—
46C2×#117—24no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57#432—24no—
58#432—24no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62#399—24no—
63#399C2×C418yes—
64#398—28no—
65#397—24no—
66#397C2×C418yes—
67#394—28no—
68#390—28no—
69#388—216no—
70#385—216no—
71#384D414yes—
72#383—22no—
73#383C2×C414yes—
74#382—22no—
75#381—24no—
76#380—24no—
77#380—22no—
78#380C2314yes—
79#379—24no—
80C4×C4⋊C4—28no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4D414yes—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—24no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95C2312yes—
89#340—28no—
90#340—28no—
91#323—28no—
92C25C2×D412yes—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—44no—
96C22×D4C22⋊C414yes—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—84no—
104C23×C4C22⋊C418yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—24no—
110#125—28no—
111#125—28no—
112#125—44no—
113#125—44no—
114C4×D4—28no—
115C4×D4—28no—
116C4×D4—28no—
117C4×D4—28no—
118#117—48no—
119#117—88no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—84no—
139C2×C4⋊C4C22⋊C414yes—
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C2×D418yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—48no—
151C2×C22⋊C4—84no—
152C2×C22⋊C4C22×C414yes—
153C2×C22⋊C4C2×D414yes—
154C2×C42—24no—
155C2×C42—28no—
156C2×C42C22×C412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42#4114yes—
161C4⋊C8—44no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164#97—44no—
165#95—28no—
166#95—24no—
167#95—24no—
168#95—28no—
169#95—24no—
170#95—22no—
171#95—24no—
172#95C22×C414yes—
173C24—22no—
174C24—24no—
175C24—22no—
176C24—24no—
177C24—44no—
178C24—44no—
179C24C2×C22⋊C414yes—
180C24C22×D411yescharacteristic
181C2×D4—28no—
182C2×D4—48no—
183C2×D4—48no—
184C2×D4—44no—
185C2×D4—44no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—28no—
190C22×C4—28no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—28no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—48no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—84no—
210C22×C4—88no—
211C22×C4—88no—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C414yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4#12011yescharacteristic
216C22×C4#12012yes—
217C22×C4#12011yescharacteristic
218C22×C4#12312yes—
219C22×C4#12312yes—
220C22×C4C23×C411yescharacteristic
221C22×C4C22×D412yes—
222C22×C4C22×D411yescharacteristic
223C22×C4C2×#4112yes—
224C2×C8—48no—
225C2×C8—44no—
226C4⋊C4—28no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—48no—
230C4⋊C4—48no—
231C4⋊C4—48no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—88no—
238C22⋊C4—28no—
239C22⋊C4—28no—
240C22⋊C4—416no—
241C22⋊C4—48no—
242C22⋊C4—88no—
243C22⋊C4—88no—
244C22⋊C4—88no—
245C42—48no—
246C42—88no—
247C42C4×D414yes—
248C42C4×D414yes—
249C42#12114yes—
250C42#12114yes—
251C23—22no—
252C23—22no—
253C23—24no—
254C23—24no—
255C23—24no—
256C23—44no—
257C23—44no—
258C23—416no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—82no—
264C23—82no—
265C23—88no—
266C23C22×C22⋊C411yescharacteristic
267C23C2×#12011yescharacteristic
268C23C2×#12311yescharacteristic
269C23#57514yes—
270D4—416no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—416no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—88no—
299C2×C4—82no—
300C2×C4—88no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4#38614yes—
306C2×C4#38612yes—
307C2×C4#38612yes—
308C2×C4C2×C4×D412yes—
309C2×C4C2×#12112yes—
310C8—84no—
311C22—28no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—48no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—84no—
321C22—88no—
322C22—88no—
323C22C2×#38611yescharacteristic
324C22#289612yes—
325C22#300512yes—
326C22#301912yes—
327C22#302812yes—
328C22#309312yes—
329C4—24no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—44no—
335C4—84no—
336C4—82no—
337C4—88no—
338C4—82no—
339C2—22no—
340C2—22no—
341C2—84no—
342C2#1837514yes—
343C2#2091011yescharacteristic
344C2#2095311yescharacteristic
345C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21838C2semidirect
5. #9028C2semidirect
6. #9024C2semidirect
7. #8018C2semidirect
12. #2633C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26332
#341862
#218382
C2×#14662
#90282
#90242
#80182