Finite Group Databasebeta

← #100050

Subgroups — #100050

Order 512.

429 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 98 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#22373C211yesmaximal, characteristic, complemented
2#22226C211yesmaximal, characteristic
3#9674C211yesmaximal, characteristic
4#7603C211yesmaximal, characteristic
5#7585C211yesmaximal, characteristic, complemented
6#7023C211yesmaximal, characteristic, complemented
7#7023C211yesmaximal, characteristic, complemented
8C2×#490C411yescharacteristic
9C4×#127C2211yescharacteristic
10C4×#125C2211yescharacteristic
11#1992C411yescharacteristic
12#1961C411yescharacteristic
13#1939—21no—
14#1926—21no—
15#1925—21no—
16#1916—21no—
17#1916—21no—
18#1916—21no—
19#1916—21no—
20#1864—21no—
21#1862—21no—
22#1847C411yescharacteristic
23#1839C2211yescharacteristic
24#1839C2211yescharacteristic
25C8×C4⋊C4—21no—
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1745C2211yescharacteristic
29C2×D4⋊C8—21no—
30C2×D4⋊C8—21no—
31#1271—21no—
32#1271—21no—
33#1270—21no—
34#1270—21no—
35C2×#127C2×C411yescharacteristic
36C2×#125C2×C411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#490—22no—
41#490C814yescomplemented
42C2×#107—21no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×#102—21no—
51C2×#102—21no—
52C2×#97C2×C411yescharacteristic
53C2×#97C2×C411yescharacteristic
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57C2×C4×C8D411yescharacteristic
58C2×C4×C8D411yescharacteristic
59#390—21no—
60#390—21no—
61#390—21no—
62#389D411yescharacteristic
63#389D411yescharacteristic
64#387—21no—
65#387—21no—
66#384C2×C411yescharacteristic
67#382C2×C411yescharacteristic
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4D411yescharacteristic
71C4×C4⋊C4D411yescharacteristic
72C43C2311yescharacteristic
73#340—21no—
74#340—21no—
75#336—21no—
76#336—21no—
77#336—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#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90D4⋊C8—42no—
91D4⋊C8—42no—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4—21no—
95C23×C4—41no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—41no—
103C22×C8—41no—
104#127—21no—
105#127—21no—
106#127—21no—
107#127C2×C812yes—
108#125—21no—
109#125—21no—
110#125C2×C812yes—
111#125M1612yes—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—41no—
115C4×D4—41no—
116C4×D4—81no—
117C4×D4—81no—
118C4×D4—81no—
119C4×D4—81no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143#107—81no—
144#106—81no—
145C4⋊C8—22no—
146C4⋊C8—41no—
147C4⋊C8—42no—
148C4⋊C8—42no—
149C4⋊C8—81no—
150#102—24no—
151#102—24no—
152#102—81no—
153#102—81no—
154#97—21no—
155#97—21no—
156#97—21no—
157#97C2×C812yes—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—42no—
161C4×C8—81no—
162C4×C8—81no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169C24—41no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—81no—
185C2×D4—81no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12311yescharacteristic
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—42no—
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—81no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8—81no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4C22⋊C812yes—
245C4⋊C4C22⋊C812yes—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42C22×C811yescharacteristic
263C42C2×M1611yescharacteristic
264C42#13111yescharacteristic
265C42#13111yescharacteristic
266C23—42no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23#38611yescharacteristic
274D4—42no—
275D4—42no—
276D4—42no—
277D4—42no—
278D4—82no—
279D4—81no—
280D4—81no—
281D4—81no—
282D4—81no—
283D4—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4—161no—
317C2×C4—161no—
318C2×C4—161no—
319C2×C4C2×C22⋊C811yescharacteristic
320C2×C4#40811yescharacteristic
321C2×C4C8×D411yescharacteristic
322C2×C4C8×D411yescharacteristic
323C2×C4#43511yescharacteristic
324C2×C4#43511yescharacteristic
325C2×C4#45311yescharacteristic
326C2×C4#45411yescharacteristic
327C2×C4#47111yescharacteristic
328C2×C4#47211yescharacteristic
329C8—42no—
330C8—42no—
331C8—81no—
332C8—81no—
333C8—81no—
334C8—81no—
335C8—82no—
336C8—161no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—161no—
341C22—161no—
342C22—161no—
343C22—161no—
344C22#186911yescharacteristic
345C22#188111yescharacteristic
346C22#188311yescharacteristic
347C22#190611yescharacteristic
348C22#190711yescharacteristic
349C22#191011yescharacteristic
350C22#191211yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—42no—
365C4—81no—
366C4—81no—
367C4—161no—
368C2—162no—
369C2#895411yescharacteristic
370C2#909511yescharacteristic
371C2#910211yescharacteristic
372C2#937011yescharacteristic
373C2#937211yescharacteristic
374C2#937811yescharacteristic
375C2#938211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22373C2semidirect
5. #7585C2semidirect
6. #7023C2semidirect
7. #7023C2semidirect
41. #490C8semidirect

Maximal subgroups

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

subgroupindex
#223732
#222262
#96742
#76032
#75852
#70232