Finite Group Databasebeta

← #133880

Subgroups — #133880

Order 512.

1112 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2857C211yesmaximal, characteristic
2#12860C212yesmaximal, complemented
3#12700C212yesmaximal, complemented
4#8550C212yesmaximal
5C22×C4⋊Q8C2211yescharacteristic
6C22×C22⋊Q8C411yescharacteristic
7C22×C22⋊Q8C2211yescharacteristic
8C23×C42C411yescharacteristic
9#2857—24no—
10#2857—24no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#389—22no—
14C2×#384C411yescharacteristic
15C2×#380—22no—
16C2×#356—24no—
17#1784C2214yes—
18#1463—22no—
19#1462—22no—
20#1394—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8C2311yescharacteristic
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—44no—
27C2×C4⋊Q8D414yes—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—42no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8—42no—
35C2×C22⋊Q8—42no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4D412yes—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D412yes—
44C22×C42—21no—
45C22×C42—22no—
46C22×C42—22no—
47C22×C42—44no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×#101—24no—
51C2×#101—24no—
52#407—24no—
53#391—22no—
54#391—22no—
55#391—22no—
56#391D414yes—
57#389—216no—
58#384—22no—
59#384—22no—
60#380—216no—
61C2×#95—22no—
62C2×#95—22no—
63#356—48no—
64#324—44no—
65#323—44no—
66C25C22⋊C411yescharacteristic
67C22×Q8—21no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8—42no—
73C22×Q8—42no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C22⋊C412yes—
77C22×Q8C2×D412yes—
78C22×Q8C2×D412yes—
79C23×C4—21no—
80C23×C4—28no—
81C23×C4—24no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—44no—
90C23×C4—44no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D412yes—
97C2×M16—44no—
98C4⋊Q8—28no—
99C4⋊Q8—44no—
100C4⋊Q8—48no—
101C22⋊Q8—48no—
102C22⋊Q8—48no—
103C22⋊Q8—48no—
104C22⋊Q8—48no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—48no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C42—21no—
130C2×C42—24no—
131C2×C42—24no—
132C2×C42—21no—
133C2×C42—48no—
134C2×C42—44no—
135C2×C42—44no—
136C2×C42—44no—
137C2×C42—42no—
138C2×C42C22⋊C412yes—
139C2×C42C22⋊C412yes—
140#101—48no—
141#101—48no—
142C22⋊C8—44no—
143C22⋊C8—44no—
144#95—28no—
145#95—28no—
146C24—24no—
147C24—21no—
148C24—21no—
149C24—41no—
150C24—42no—
151C24—41no—
152C24#9912yes—
153C24C2×C22⋊C411yescharacteristic
154C2×Q8—24no—
155C2×Q8—24no—
156C2×Q8—24no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—84no—
167C2×Q8—84no—
168C2×Q8—84no—
169C2×Q8—84no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—216no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—432no—
198C22×C4—48no—
199C22×C4—48no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4#9912yes—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4#12014yes—
219M16—84no—
220M16—84no—
221C2×C8—84no—
222C4⋊C4—48no—
223C4⋊C4—416no—
224C4⋊C4—42no—
225C4⋊C4—416no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—48no—
230C22⋊C4—48no—
231C22⋊C4—48no—
232C42—24no—
233C42—22no—
234C42—42no—
235C42—48no—
236C42—42no—
237C42#9914yes—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—24no—
243C23—24no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—48no—
249C23—42no—
250C23—42no—
251C23—48no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23#35614yescomplemented
256C23#37911yescharacteristic
257C23C2×#9911yescharacteristic
258C23C2×#9911yescharacteristic
259Q8—84no—
260Q8—84no—
261Q8—84no—
262Q8—84no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—216no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—432no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—48no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4C2×#9912yes—
304C2×C4#41012yes—
305C8—164no—
306C22—24no—
307C22—24no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—48no—
319C22—44no—
320C22—42no—
321C22—42no—
322C22#178111yescharacteristic
323C22#178111yescharacteristic
324C22#178211yescharacteristic
325C22C2×#35612yes—
326C22#213212yes—
327C4—28no—
328C4—22no—
329C4—22no—
330C4—44no—
331C4—44no—
332C4—42no—
333C4—82no—
334C4—82no—
335C4—82no—
336C2—21no—
337C2—22no—
338C2—22no—
339C2—21no—
340C2—44no—
341C2#854211yescharacteristic
342C2#1270011yescharacteristic
343C2#1270111yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12860C2semidirect
3. #12700C2semidirect
255. C23#356semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#128602
#127002
#85502