Finite Group Databasebeta

← #105161

Subgroups — #105161

Order 512.

394 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 88 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#28516C211yesmaximal, characteristic
2#9900C211yesmaximal, characteristic, complemented
3#9605C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7663C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic, complemented
8C2×#478—21no—
9C2×#478—21no—
10#2859C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#2094—21no—
13#2093—21no—
14#1961—21no—
15#1847—21no—
16#1842C2211yescharacteristic
17#1831C2211yescharacteristic
18#1767C2211yescharacteristic
19C4×C4⋊C8C2211yescharacteristic
20C4×C4⋊C8C2211yescharacteristic
21C2×#334—21no—
22C2×#331—21no—
23#1296—21no—
24#1296—21no—
25#1296—21no—
26#1277—21no—
27#1271—21no—
28#1271—21no—
29#1270—21no—
30#1270—21no—
31C2×#127D411yescharacteristic
32C2×#125—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36#478—42no—
37#478—42no—
38#478—42no—
39#478—42no—
40C2×#107—21no—
41C2×#107—21no—
42C2×#106—21no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×#102—21no—
53C2×#102—21no—
54C2×#102—21no—
55C2×#102—21no—
56C2×#97—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59#390—21no—
60#390—21no—
61#390—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#384D411yescharacteristic
67#384D411yescharacteristic
68#384D411yescharacteristic
69#382—21no—
70C43C2311yescharacteristic
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336D411yescharacteristic
76#336D411yescharacteristic
77#334—42no—
78#331—42no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C23×C4—41no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90#127—21no—
91#127—21no—
92#127—21no—
93#127SD1612yes—
94#125—41no—
95#125—41no—
96#125—41no—
97#125—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—81no—
103C4×D4—81no—
104C4×D4—82no—
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⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#107—42no—
127#107—42no—
128#106—42no—
129#106—42no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—41no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8D812yes—
143C4⋊C8SD1612yes—
144#102—42no—
145#102—42no—
146#102—42no—
147#102—42no—
148#97—41no—
149#97—41no—
150C4×C8—41no—
151C4×C8—41no—
152C4×C8—81no—
153#95—41no—
154#95—41no—
155C24—41no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4#12011yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12411yescharacteristic
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C22⋊C4—82no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42C2×D811yescharacteristic
235C42C2×SD1611yescharacteristic
236C42C2×SD1611yescharacteristic
237C42#13511yescharacteristic
238C42AGL(1,ℤ/8)11yescharacteristic
239C42AGL(1,ℤ/8)11yescharacteristic
240C42#13711yescharacteristic
241C42#13711yescharacteristic
242C23—42no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23#39411yescharacteristic
250D4—42no—
251D4—42no—
252D4—82no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—81no—
257D4—81no—
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—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—162no—
281C2×C4—162no—
282C2×C4#44711yescharacteristic
283C2×C4#44811yescharacteristic
284C2×C4#45011yescharacteristic
285C2×C4#45211yescharacteristic
286C2×C4#45411yescharacteristic
287C2×C4#45611yescharacteristic
288C2×C4#45911yescharacteristic
289C2×C4#46011yescharacteristic
290C2×C4#46511yescharacteristic
291C2×C4#47011yescharacteristic
292C2×C4#47111yescharacteristic
293C2×C4#47211yescharacteristic
294C2×C4#48311yescharacteristic
295C2×C4#48511yescharacteristic
296C2×C4#48611yescharacteristic
297C2×C4#49011yescharacteristic
298C8—42no—
299C8—42no—
300C8—42no—
301C8—81no—
302C8—82no—
303C8—161no—
304C8—161no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—161no—
309C22—161no—
310C22—161no—
311C22—161no—
312C22#203511yescharacteristic
313C22#203811yescharacteristic
314C22#204111yescharacteristic
315C22#204811yescharacteristic
316C22#205011yescharacteristic
317C22#205111yescharacteristic
318C22#205311yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—161no—
331C4—161no—
332C4—161no—
333C4#162711yescharacteristic
334C4#163511yescharacteristic
335C4#164711yescharacteristic
336C4#165111yescharacteristic
337C2—162no—
338C2#1042011yescharacteristic
339C2#1046011yescharacteristic
340C2#1046811yescharacteristic
341C2#1049711yescharacteristic
342C2#1050911yescharacteristic
343C2#1054611yescharacteristic
344C2#1054911yescharacteristic
subgroupG/H
center, socleC23#394
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9900C2semidirect
3. #9605C2semidirect
5. #7663C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#285162
#99002
#96052
#85882
#76632
#75692
#70222