Finite Group Databasebeta

← #141439

Subgroups — #141439

Order 512.

409 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 80 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#28095C211yesmaximal, characteristic
2#13293C211yesmaximal, characteristic
3#13292C211yesmaximal, characteristic
4#11899C211yesmaximal, characteristic
5#11898C211yesmaximal, characteristic
6#8609C211yesmaximal, characteristic
7C8×#100C211yesmaximal, characteristic
8C2×C8×D4C2211yescharacteristic
9C22×#103C411yescharacteristic, complemented
10#2487C2211yescharacteristic
11#2409C2211yescharacteristic
12#2068—21no—
13#2036—21no—
14#1933—21no—
15#1892—21no—
16#1824C411yescharacteristic, complemented
17#1808—21no—
18#1808C411yescharacteristic, complemented
19#1799C2211yescharacteristic
20#1799C2211yescharacteristic
21#1787C411yescharacteristic, complemented
22C4×#100C2211yescharacteristic
23#1460C2211yescharacteristic
24C8×M16—21no—
25#1390—22no—
26#1390—22no—
27#1352—22no—
28#1351—22no—
29#1289—21no—
30#1283—21no—
31C23×Q8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C22⋊Q8—21no—
35C2×C4×D4C2311yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C8×D4—41no—
38C8×D4—41no—
39#431—21no—
40C2×#107—21no—
41C2×#107—41no—
42C2×C4⋊C8D411yescharacteristic
43C2×#103—21no—
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#103—41no—
48C2×#103—41no—
49C2×#100D412yes—
50C2×#99—21no—
51C2×#99—21no—
52C2×C22⋊C8C2×C411yescharacteristic
53C2×C22⋊C8D411yescharacteristic
54C4×M16—21no—
55C2×C4×C8D411yescharacteristic
56#400—21no—
57#399—21no—
58#395—21no—
59#393—21no—
60#391—21no—
61#389—21no—
62#386—21no—
63#386—21no—
64#385—21no—
65#385—21no—
66#380—21no—
67#380—21no—
68#380—21no—
69#380C2×C411yescharacteristic
70#380C2×C411yescharacteristic
71#343—21no—
72#342—21no—
73#342—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—41no—
78#323—21no—
79#323D412yes—
80#322—41no—
81C82—41no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8—42no—
87C22×Q8—81no—
88C22×D4C2×D411yescharacteristic
89C23×C4—21no—
90C23×C4—21no—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×M16—22no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—81no—
99C22×C8—81no—
100C22×C8C22⋊C411yescharacteristic
101C22×C8C22⋊C411yescharacteristic
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22⋊Q8—42no—
105C22⋊Q8—81no—
106C4×D4—41no—
107C4×D4—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4#4111yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4C2×D411yescharacteristic
126C2×C22⋊C4#4111yescharacteristic
127C2×C42C2×D411yescharacteristic
128#107—81no—
129#107—82no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132#103—42no—
133#103—42no—
134#103—41no—
135#103—41no—
136#103—81no—
137#100—22no—
138#100Q1614yes—
139#99—42no—
140#99—42no—
141C22⋊C8—22no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144#97—41no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—41no—
148C4×C8—41no—
149#95—21no—
150#95—21no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157C24C4×D411yescharacteristic
158C24#12711yescharacteristic
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—81no—
168C2×Q8—82no—
169C2×Q8—84no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4C2×Q1612yes—
175C2×D4#13512yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—82no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—161no—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C4×D411yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12411yescharacteristic
199M16—24no—
200M16—42no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8#9911yescharacteristic
215C2×C8#9911yescharacteristic
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—81no—
224C4⋊C4—82no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—161no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—82no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—161no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#39011yescharacteristic
251C23C4×Q1612yes—
252C23#49411yescharacteristic
253C23#49511yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—81no—
257Q8—82no—
258Q8—82no—
259Q8—81no—
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—162no—
293C2×C4—161no—
294C2×C4—161no—
295C2×C4—161no—
296C2×C4—161no—
297C2×C4—161no—
298C2×C4—161no—
299C2×C4C2×#9911yescharacteristic
300C2×C4#41011yescharacteristic
301C2×C4#46712yes—
302C8—21no—
303C8—21no—
304C8—44no—
305C8—41no—
306C8—41no—
307C8—41no—
308C8—82no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22#196411yescharacteristic
324C22#196911yescharacteristic
325C22#197611yescharacteristic
326C4—22no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—161no—
333C4—161no—
334C4—161no—
335C4—321no—
336C4—321no—
337C4#212011yescharacteristic
338C4#212111yescharacteristic
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2#978211yescharacteristic
345C2#1321911yescharacteristic
346C2#1322911yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×#103C4semidirect
16. #1824C4semidirect
18. #1808C4semidirect
21. #1787C4semidirect

Maximal subgroups

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

subgroupindex
#280952
#132932
#132922
#118992
#118982
#86092
C8×#1002