Finite Group Databasebeta

← #192090

Subgroups — #192090

Order 512.

1296 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1296 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
1#34186C211yesmaximal, characteristic
2#21365C212yesmaximal, complemented
3C2×#1466C211yesmaximal, characteristic
4#13847C211yesmaximal, characteristic
5#8858C218yesmaximal, complemented
6#8022C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C2×#432C2212yes—
9#2357—24no—
10#2354—22no—
11#2354—22no—
12#2312C2214yes—
13#2309—21no—
14#2285C2212yes—
15#2280—22no—
16#2274—21no—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4—22no—
19C2×#376C2211yescharacteristic
20#1948—22no—
21#1947—22no—
22#1853C2214yes—
23#1852C2214yes—
24#1776—24no—
25#1466C418yescomplemented
26C2×#323C2214yes—
27#1445—21no—
28#1439C2214yes—
29#1438—21no—
30#1294C2218yes—
31C23×D4D411yescharacteristic
32C24×C4D412yes—
33C2×C4×D4—22no—
34C2×C4×D4—22no—
35C2×C4×D4—24no—
36C2×C4×D4C2×C414yes—
37C2×C4×D4Q814yes—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4C2312yes—
43C22×C22⋊C4—24no—
44C22×C22⋊C4C2312yes—
45C22×C42—21no—
46C22×C42—21no—
47C22×C42C2311yescharacteristic
48#432—24no—
49#432C2×C418yes—
50C2×#107—24no—
51C2×#106—24no—
52C2×C4⋊C8—22no—
53C2×C22⋊C8C2314yes—
54C2×#97—22no—
55#390—24no—
56#390—24no—
57#388—28no—
58#388—28no—
59#384D412yes—
60#384D412yes—
61#382—22no—
62#381—24no—
63#380—24no—
64#380—22no—
65#380C2314yes—
66#379—24no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4D414yes—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—24no—
75C4×C22⋊C4—48no—
76#376—24no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—24no—
80C2×#95C2312yes—
81#340—24no—
82#340—24no—
83#323C2×C4116yes—
84C25C2×D412yes—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—44no—
88C22×D4C4⋊C414yes—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4C4⋊C418yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2411yescharacteristic
101C22×C8—24no—
102C4×D4—28no—
103C4×D4—28no—
104C4×D4—28no—
105C4×D4—28no—
106#117—48no—
107#117—84no—
108#117—84no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4C4⋊C414yes—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—48no—
128C2×C22⋊C4—48no—
129C2×C22⋊C4—48no—
130C2×C22⋊C4C22×C414yes—
131C2×C22⋊C4C2×Q814yes—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42—24no—
135C2×C42—24no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—44no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C22×C412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C2×C42C2×Q812yes—
145C2×C42#4114yes—
146#107—48no—
147#106—48no—
148C4⋊C8—28no—
149C22⋊C8—24no—
150C22⋊C8C22×C418yes—
151#97—28no—
152#95—28no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—48no—
157#95—42no—
158#95—42no—
159C24—22no—
160C24—24no—
161C24—22no—
162C24—24no—
163C24—42no—
164C24—44no—
165C24—42no—
166C24C2×C4⋊C414yes—
167C24C22×D411yescharacteristic
168C2×D4—28no—
169C2×D4—48no—
170C2×D4—48no—
171C2×D4—44no—
172C2×D4—44no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—42no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—88no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4C2×C4⋊C412yes—
202C22×C4C2×C4⋊C412yes—
203C22×C4C2×C4⋊C414yes—
204C22×C4#12011yescharacteristic
205C22×C4#12012yes—
206C22×C4#12011yescharacteristic
207C22×C4#12312yes—
208C22×C4#12312yes—
209C22×C4C23×C411yescharacteristic
210C22×C4C22×D412yes—
211C22×C4C22×Q811yescharacteristic
212C22×C4C2×#4112yes—
213C2×C8—28no—
214C2×C8—44no—
215C2×C8—44no—
216C4⋊C4—28no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C22⋊C4—28no—
228C22⋊C4—28no—
229C22⋊C4—432no—
230C22⋊C4—48no—
231C22⋊C4—88no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C42—48no—
235C42—48no—
236C42—82no—
237C42—84no—
238C42—84no—
239C42—82no—
240C42C4×D414yes—
241C42C4×D414yes—
242C42C22⋊Q814yes—
243C42C22⋊Q814yes—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—44no—
251C23—44no—
252C23—48no—
253C23—48no—
254C23—44no—
255C23—44no—
256C23—42no—
257C23—42no—
258C23—44no—
259C23—82no—
260C23—82no—
261C23—84no—
262C23—84no—
263C23C22×C4⋊C411yescharacteristic
264C23C2×#12011yescharacteristic
265C23C2×#12311yescharacteristic
266C23#57514yes—
267D4—416no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—416no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—88no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4#38514yes—
305C2×C4#38512yes—
306C2×C4#38512yes—
307C2×C4#45312yes—
308C2×C4#45512yes—
309C2×C4C2×C4×D412yes—
310C2×C4C2×C22⋊Q812yes—
311C8—48no—
312C22—24no—
313C22—24no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—48no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—84no—
323C22—88no—
324C22—84no—
325C22—84no—
326C22C2×#38511yescharacteristic
327C22#291112yes—
328C22#300512yes—
329C22C2×#45311yescharacteristic
330C22C2×#45511yescharacteristic
331C22#302512yes—
332C22#309312yes—
333C4—24no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—44no—
338C4—84no—
339C4—84no—
340C4—84no—
341C4—84no—
342C4#185614yes—
343C2—22no—
344C2—22no—
345C2—84no—
346C2#1819012yes—
347C2#1819112yes—
348C2#2084511yescharacteristic
349C2C2×#185611yescharacteristic
350C2#2088811yescharacteristic
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
2. #21365C2semidirect
5. #8858C2semidirect
6. #8022C2semidirect
25. #1466C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#213652
C2×#14662
#138472
#88582
#80222