Finite Group Databasebeta

← #101814

Subgroups — #101814

Order 512.

386 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 84 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#27878C211yesmaximal, characteristic
2#22628C211yesmaximal, characteristic
3#9888C211yesmaximal, characteristic
4#8717C211yesmaximal, characteristic
5#7727C211yesmaximal, characteristic
6#7024C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9#2285C2211yescharacteristic
10#2076—21no—
11#2076—21no—
12#1990C411yescharacteristic, complemented
13#1990C411yescharacteristic, complemented
14#1940—21no—
15#1921—21no—
16#1921—21no—
17#1917—21no—
18#1916—21no—
19#1845C411yescharacteristic, complemented
20#1834C411yescharacteristic, complemented
21#1817C2211yescharacteristic
22#1798—21no—
23#1797C2211yescharacteristic
24#1771—21no—
25#1750—21no—
26#1749C2211yescharacteristic
27C4×C22⋊C8C2211yescharacteristic
28#1434C2211yescharacteristic
29#1294—21no—
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1272—21no—
34#1272—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C22⋊Q8—21no—
37C2×#117—21no—
38C22×C42C2311yescharacteristic
39C2×#107—21no—
40C2×#107—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8D411yescharacteristic
48C2×C22⋊C8D411yescharacteristic
49C2×#97—21no—
50C2×#97—21no—
51C2×C4×C8—21no—
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8C2×C411yescharacteristic
54#391—21no—
55#389—21no—
56#389—21no—
57#384C2×C411yescharacteristic
58#384C2×C411yescharacteristic
59#384C2×C411yescharacteristic
60#382—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380D411yescharacteristic
65#380D411yescharacteristic
66C4×C4⋊C4—21no—
67C4×C4⋊C4D411yescharacteristic
68C4×C4⋊C4D411yescharacteristic
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336D411yescharacteristic
87#336D411yescharacteristic
88C22×Q8—21no—
89C23×C4—41no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C22⋊Q8—82no—
103#117—81no—
104#117—81no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
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—41no—
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—41no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4111yescharacteristic
130#107—42no—
131#107—42no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—81no—
135C4⋊C8—81no—
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—42no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8Q1612yes—
145#97—41no—
146#97—41no—
147#97—81no—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8—81no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—22no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163C24#12111yescharacteristic
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—81no—
167C2×Q8—81no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4C2×Q1611yescharacteristic
198C22×C4#13511yescharacteristic
199C22×C4#13711yescharacteristic
200C22×C4#13711yescharacteristic
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—82no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23#38611yescharacteristic
259C23#46211yescharacteristic
260C23#46411yescharacteristic
261Q8—161no—
262Q8—161no—
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—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—161no—
307C2×C4—161no—
308C2×C4—161no—
309C2×C4—161no—
310C2×C4#41011yescharacteristic
311C2×C4#41011yescharacteristic
312C2×C4C4×Q1611yescharacteristic
313C2×C4#44111yescharacteristic
314C2×C4#44311yescharacteristic
315C2×C4#44411yescharacteristic
316C2×C4#45111yescharacteristic
317C2×C4#45211yescharacteristic
318C8—42no—
319C8—42no—
320C8—81no—
321C8—81no—
322C8—81no—
323C8—81no—
324C8—161no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22#187611yescharacteristic
336C22#187811yescharacteristic
337C22#188411yescharacteristic
338C22#189911yescharacteristic
339C22#190311yescharacteristic
340C22#190411yescharacteristic
341C22#190911yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—162no—
357C4—161no—
358C4—161no—
359C2—41no—
360C2—41no—
361C2#905611yescharacteristic
362C2#924211yescharacteristic
363C2#925711yescharacteristic
364C2#927911yescharacteristic
365C2#928811yescharacteristic
366C2#941711yescharacteristic
367C2#944411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1990C4semidirect
13. #1990C4semidirect
19. #1845C4semidirect
20. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#278782
#226282
#98882
#87172
#77272
#70242
#70212