Finite Group Databasebeta

← #140646

Subgroups — #140646

Order 512.

406 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 96 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#21090C211yesmaximal, characteristic
2#11454C211yesmaximal, characteristic
3#11450C211yesmaximal, characteristic
4#11434C211yesmaximal, characteristic
5#11428C211yesmaximal, characteristic
6#8602C211yesmaximal, characteristic
7#8156C211yesmaximal, characteristic
8C2×#419C411yescharacteristic
9C22×#103C411yescharacteristic
10C2×#405C2211yescharacteristic
11#2317C2211yescharacteristic
12#2310C2211yescharacteristic
13#1935—21no—
14#1935—21no—
15#1933—21no—
16#1933—21no—
17#1736C411yescharacteristic
18#1735C411yescharacteristic
19#1564—21no—
20#1458C2211yescharacteristic
21#1458C2211yescharacteristic
22#1458C2211yescharacteristic
23#1458C2211yescharacteristic
24#1450—21no—
25#1326—22no—
26#1325—22no—
27#1319—22no—
28#1317—22no—
29#1281—21no—
30#1275—21no—
31C23×Q8C2×C411yescharacteristic
32C22×M16C2×C411yescharacteristic
33C23×C8C2×C411yescharacteristic
34C2×#117C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117C2311yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#419—22no—
43#419C814yescomplemented
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×#103—41no—
49C2×#103—41no—
50#405—41no—
51#405—41no—
52C2×#97D411yescharacteristic
53C2×#97D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8D411yescharacteristic
56#391—21no—
57#391—21no—
58#391—21no—
59#391—21no—
60#389—21no—
61#389—21no—
62#385—21no—
63#385—21no—
64#384—21no—
65#382—21no—
66#381—21no—
67#381—21no—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#335—41no—
73#334—41no—
74#324—21no—
75#324—21no—
76#324—21no—
77#324D412yes—
78#323—21no—
79#323—21no—
80#323—21no—
81#323D412yes—
82#322—41no—
83C82—41no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8—21no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8C2×C812yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C22×C411yescharacteristic
97C2×M16—41no—
98C2×M16—41no—
99C2×M16C2×C812yes—
100C2×M16M1612yes—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—41no—
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109C22×C8C22⋊C411yescharacteristic
110#117—21no—
111#117—21no—
112#117—21no—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117#117C2×C812yes—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4C2×D411yescharacteristic
127C2×C4⋊C4#4111yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4C2×D411yescharacteristic
131C2×C22⋊C4#4111yescharacteristic
132C2×C42—21no—
133C2×C42—21no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140#103—24no—
141#103—24no—
142#103—42no—
143#103—41no—
144#103—41no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#97—41no—
150#97—41no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—41no—
158C4×C8—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165C24C4×D411yescharacteristic
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—81no—
179C2×Q8—81no—
180C2×Q8—81no—
181C2×Q8—81no—
182C2×Q8—81no—
183C2×Q8—81no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—161no—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12411yescharacteristic
203C22×C4#12711yescharacteristic
204C22×C4C22×C811yescharacteristic
205C22×C4C2×M1611yescharacteristic
206C22×C4#13111yescharacteristic
207C22×C4#13111yescharacteristic
208M16—42no—
209M16—42no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8C22⋊C812yes—
228C2×C8C22⋊C812yes—
229C2×C8#9911yescharacteristic
230C2×C8#9911yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—81no—
240C4⋊C4—161no—
241C4⋊C4—161no—
242C22⋊C4—22no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—81no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—22no—
252C42—41no—
253C42—81no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23#39011yescharacteristic
259C23C8×D411yescharacteristic
260C23#43611yescharacteristic
261Q8—42no—
262Q8—42no—
263Q8—82no—
264Q8—82no—
265Q8—81no—
266Q8—81no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—162no—
300C2×C4C2×C22⋊C811yescharacteristic
301C2×C4#40811yescharacteristic
302C2×C4C2×#9911yescharacteristic
303C2×C4#41011yescharacteristic
304C2×C4#41311yescharacteristic
305C2×C4#41311yescharacteristic
306C2×C4C8×D411yescharacteristic
307C2×C4#43611yescharacteristic
308C8—22no—
309C8—22no—
310C8—21no—
311C8—21no—
312C8—41no—
313C8—82no—
314C8—82no—
315C8—161no—
316C8—161no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22#195411yescharacteristic
327C22#196411yescharacteristic
328C22#196711yescharacteristic
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—82no—
336C4—82no—
337C4—161no—
338C4—161no—
339C4#145811yescharacteristic
340C4#145911yescharacteristic
341C4#146811yescharacteristic
342C4#147311yescharacteristic
343C2—21no—
344C2—21no—
345C2—41no—
346C2—41no—
347C2#965111yescharacteristic
348C2#1322911yescharacteristic
349C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
43. #419C8semidirect

Maximal subgroups

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

subgroupindex
#210902
#114542
#114502
#114342
#114282
#86022
#81562