Finite Group Databasebeta

← #105154

Subgroups — #105154

Order 512.

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

normal subgroups only

This group has 376 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#28722C211yesmaximal, characteristic
2#9900C211yesmaximal, characteristic, complemented
3#9605C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7660C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic, complemented
8C2×#490—21no—
9#2859C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#2071—21no—
12#2071—21no—
13#1992—21no—
14#1842C2211yescharacteristic
15#1831C2211yescharacteristic
16#1804—21no—
17#1804—21no—
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#490—42no—
37#490—42no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×#102—21no—
47C2×#102—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×#97—21no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53#390—21no—
54#390—21no—
55#390—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#389—21no—
60#384D411yescharacteristic
61#384D411yescharacteristic
62#384D411yescharacteristic
63#382—21no—
64C43C2311yescharacteristic
65#340—21no—
66#340—21no—
67#340—21no—
68#340—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336D411yescharacteristic
74#336D411yescharacteristic
75#334—42no—
76#331—42no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C23×C4—41no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—41no—
88#127—21no—
89#127—21no—
90#127—21no—
91#127SD1612yes—
92#125—41no—
93#125—41no—
94#125—41no—
95#125—41no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—81no—
101C4×D4—81no—
102C4×D4—82no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
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—41no—
113C2×C4⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C4⋊C8—22no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—41no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138#102—42no—
139#102—42no—
140#102—42no—
141#102—42no—
142#97—41no—
143#97—41no—
144C4×C8—41no—
145C4×C8—41no—
146C4×C8—81no—
147#95—41no—
148#95—41no—
149C24—41no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4—81no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4#12011yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4#12311yescharacteristic
177C22×C4#12411yescharacteristic
178C2×C8—22no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C22⋊C4—82no—
217C22⋊C4—81no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42C2×SD1611yescharacteristic
229C42#13511yescharacteristic
230C42#13511yescharacteristic
231C42#13511yescharacteristic
232C42AGL(1,ℤ/8)11yescharacteristic
233C42AGL(1,ℤ/8)11yescharacteristic
234C42AGL(1,ℤ/8)11yescharacteristic
235C42#13711yescharacteristic
236C23—42no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23#39411yescharacteristic
244D4—42no—
245D4—42no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—82no—
250D4—81no—
251D4—81no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
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—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—162no—
275C2×C4—162no—
276C2×C4#44911yescharacteristic
277C2×C4#44911yescharacteristic
278C2×C4#45011yescharacteristic
279C2×C4#45211yescharacteristic
280C2×C4#45411yescharacteristic
281C2×C4#45611yescharacteristic
282C2×C4#46311yescharacteristic
283C2×C4#46411yescharacteristic
284C2×C4#46511yescharacteristic
285C2×C4#46911yescharacteristic
286C2×C4#47111yescharacteristic
287C2×C4#47211yescharacteristic
288C2×C4#48111yescharacteristic
289C2×C4#48211yescharacteristic
290C2×C4#48811yescharacteristic
291C2×C4#48911yescharacteristic
292C8—42no—
293C8—81no—
294C8—82no—
295C8—81no—
296C8—81no—
297C8—161no—
298C8—161no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—161no—
303C22—161no—
304C22—161no—
305C22—161no—
306C22#203411yescharacteristic
307C22#203811yescharacteristic
308C22#204211yescharacteristic
309C22#204811yescharacteristic
310C22#205011yescharacteristic
311C22#205111yescharacteristic
312C22#205311yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—161no—
325C4—161no—
326C4—161no—
327C4#162811yescharacteristic
328C4#163511yescharacteristic
329C4#165011yescharacteristic
330C4#165411yescharacteristic
331C2—162no—
332C2#1041711yescharacteristic
333C2#1045711yescharacteristic
334C2#1046511yescharacteristic
335C2#1049711yescharacteristic
336C2#1050911yescharacteristic
337C2#1054711yescharacteristic
338C2#1055311yescharacteristic
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. #7660C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#287222
#99002
#96052
#85882
#76602
#75692
#70222