Finite Group Databasebeta

← #101333

Subgroups — #101333

Order 512.

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

normal subgroups only

This group has 370 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#23780C211yesmaximal, characteristic
2#22838C211yesmaximal, characteristic
3#7652C211yesmaximal, characteristic, complemented
4#7624C211yesmaximal, characteristic, complemented
5#7594C211yesmaximal, characteristic
6#7049C211yesmaximal, characteristic, complemented
7#7041C211yesmaximal, characteristic, complemented
8C4×C4⋊Q8C2211yescharacteristic
9#2293C2211yescharacteristic
10#2100C411yescharacteristic
11#2094C411yescharacteristic
12#2088—21no—
13#2067—21no—
14#2057C411yescharacteristic
15#2023C411yescharacteristic
16#1955—21no—
17#1924—21no—
18#1923—21no—
19#1839C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1818—21no—
23#1776—21no—
24#1771—21no—
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27#1762—21no—
28#1750—21no—
29#1272—21no—
30#1272—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×D4D411yescharacteristic
38C2×#117—21no—
39C2×#107—21no—
40C2×#106—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#102—21no—
46C2×#102—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×#97—21no—
52C2×#97—21no—
53C2×#97—21no—
54C2×C4×C8—21no—
55#391—21no—
56#390C2×C411yescharacteristic
57#390D411yescharacteristic
58#389—21no—
59#389—21no—
60#389C2×C411yescharacteristic
61#389D411yescharacteristic
62#388—21no—
63#388—21no—
64#384C2×C411yescharacteristic
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4D411yescharacteristic
68C4×C22⋊C4—21no—
69C43C2311yescharacteristic
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336C2×C411yescharacteristic
83#336C2×C411yescharacteristic
84#336D411yescharacteristic
85#336D411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88C22×Q8—21no—
89C22×D4C22⋊C411yescharacteristic
90C23×C4—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8—41no—
99C4⋊Q8—42no—
100C4×Q8—81no—
101C4×Q8—81no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106#117—81no—
107#117—81no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42—41no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133#107—42no—
134#106—42no—
135C4⋊C8—42no—
136C4⋊C8—42no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—81no—
140#102—42no—
141#102—42no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144C22⋊C8—81no—
145C22⋊C8—81no—
146#97—41no—
147#97—41no—
148#97—41no—
149#97—41no—
150#97—81no—
151C4×C8—81no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—41no—
156#95—41no—
157C24—21no—
158C2×Q8—22no—
159C2×Q8—41no—
160C2×Q8—81no—
161C2×Q8—81no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4—41no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12311yescharacteristic
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—21no—
232C42—21no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—82no—
241C42#13511yescharacteristic
242C42#13511yescharacteristic
243C42#13711yescharacteristic
244C42#13711yescharacteristic
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23#38611yescharacteristic
253Q8—82no—
254Q8—82no—
255D4—42no—
256D4—81no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
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—81no—
293C2×C4#41611yescharacteristic
294C2×C4#41911yescharacteristic
295C2×C4#45211yescharacteristic
296C2×C4#45211yescharacteristic
297C2×C4#47111yescharacteristic
298C2×C4#47211yescharacteristic
299C2×C4#48511yescharacteristic
300C2×C4#48511yescharacteristic
301C8—42no—
302C8—42no—
303C8—81no—
304C8—81no—
305C8—81no—
306C8—81no—
307C8—161no—
308C8—161no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22#187311yescharacteristic
317C22#188411yescharacteristic
318C22#189111yescharacteristic
319C22#190211yescharacteristic
320C22#190411yescharacteristic
321C22#190611yescharacteristic
322C22#190711yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—82no—
336C4—81no—
337C4—81no—
338C4—161no—
339C2—161no—
340C2#903411yescharacteristic
341C2#916811yescharacteristic
342C2#917711yescharacteristic
343C2#941511yescharacteristic
344C2#943211yescharacteristic
345C2#952911yescharacteristic
346C2#953111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7652C2semidirect
4. #7624C2semidirect
6. #7049C2semidirect
7. #7041C2semidirect

Maximal subgroups

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

subgroupindex
#237802
#228382
#76522
#76242
#75942
#70492
#70412