Finite Group Databasebeta

← #101810

Subgroups — #101810

Order 512.

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

normal subgroups only

This group has 394 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#27892C211yesmaximal, characteristic
2#22628C211yesmaximal, characteristic
3#9889C211yesmaximal, characteristic
4#8717C211yesmaximal, characteristic
5#7725C211yesmaximal, characteristic
6#7024C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9#2285C2211yescharacteristic
10#2036—21no—
11#2036—21no—
12#1956C411yescharacteristic, complemented
13#1956C411yescharacteristic, complemented
14#1940—21no—
15#1928—21no—
16#1928—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#1271—21no—
34#1271—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×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8D411yescharacteristic
52C2×C22⋊C8D411yescharacteristic
53C2×#97—21no—
54C2×#97—21no—
55C2×C4×C8—21no—
56C2×C4×C8C2×C411yescharacteristic
57C2×C4×C8C2×C411yescharacteristic
58#391—21no—
59#389—21no—
60#389—21no—
61#384C2×C411yescharacteristic
62#384C2×C411yescharacteristic
63#384C2×C411yescharacteristic
64#382—21no—
65#380—21no—
66#380—21no—
67#380—21no—
68#380D411yescharacteristic
69#380D411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4D411yescharacteristic
72C4×C4⋊C4D411yescharacteristic
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—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—81no—
103C22⋊Q8—81no—
104#117—81no—
105#117—81no—
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—21no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—41no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131#107—81no—
132#107—81no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—81no—
136C4⋊C8—81no—
137#103—42no—
138#103—42no—
139#103—81no—
140#103—81no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C22⋊C8—21no—
144C22⋊C8—42no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8Q1612yes—
150#97—41no—
151#97—41no—
152#97—81no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—81no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—22no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95—41no—
167#95—41no—
168C24#12111yescharacteristic
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4C2×Q1611yescharacteristic
205C22×C4#13511yescharacteristic
206C22×C4#13711yescharacteristic
207C22×C4#13711yescharacteristic
208C2×C8—22no—
209C2×C8—22no—
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—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
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—81no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—82no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C22⋊C4—82no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23#38611yescharacteristic
266C23#46211yescharacteristic
267C23#46411yescharacteristic
268Q8—81no—
269Q8—81no—
270Q8—81no—
271Q8—81no—
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—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4—161no—
317C2×C4—161no—
318C2×C4—161no—
319C2×C4#41011yescharacteristic
320C2×C4#41011yescharacteristic
321C2×C4C4×Q1611yescharacteristic
322C2×C4#44111yescharacteristic
323C2×C4#44311yescharacteristic
324C2×C4#44411yescharacteristic
325C2×C4#45111yescharacteristic
326C2×C4#45211yescharacteristic
327C8—42no—
328C8—42no—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—81no—
333C8—161no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#187611yescharacteristic
345C22#187811yescharacteristic
346C22#188411yescharacteristic
347C22#189911yescharacteristic
348C22#190211yescharacteristic
349C22#190411yescharacteristic
350C22#191011yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—162no—
366C4—161no—
367C4—161no—
368C2—41no—
369C2—41no—
370C2#905611yescharacteristic
371C2#924011yescharacteristic
372C2#925711yescharacteristic
373C2#928111yescharacteristic
374C2#929011yescharacteristic
375C2#941511yescharacteristic
376C2#944611yescharacteristic
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. #1956C4semidirect
13. #1956C4semidirect
19. #1845C4semidirect
20. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#278922
#226282
#98892
#87172
#77252
#70242
#70212