Finite Group Databasebeta

← #103764

Subgroups — #103764

Order 512.

463 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 114 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#29287C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#28689C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic
5#9673C211yesmaximal, characteristic
6#7571C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#500—21no—
9#2868C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12#2088—22no—
13#2083—22no—
14#2058—22no—
15#1990—21no—
16#1990—21no—
17#1988—21no—
18#1987—21no—
19#1948—22no—
20#1948—21no—
21#1948—21no—
22#1947—22no—
23#1844—21no—
24#1842C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×Q8⋊C8—22no—
29#1296—22no—
30#1272—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38#500—42no—
39#500—42no—
40C2×#107—24no—
41C2×#106—24no—
42C2×#106—22no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D412yes—
48C2×C4⋊C8D412yes—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52#391—21no—
53#391—21no—
54#389—21no—
55#389—21no—
56#389—21no—
57#389—21no—
58#384—21no—
59#384—21no—
60#384—21no—
61#384D411yescharacteristic
62#384D411yescharacteristic
63#384D412yes—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C43C2311yescharacteristic
69#340—24no—
70#340—24no—
71#340—22no—
72#340—22no—
73#340—22no—
74#336—21no—
75#336—21no—
76#336D412yes—
77Q8⋊C8—44no—
78C22×Q8—21no—
79C22×Q8—21no—
80C22×Q8—41no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8—21no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—82no—
96C4⋊Q8—82no—
97C4⋊Q8SD1612yes—
98C4⋊Q8Q1612yes—
99C4×Q8—42no—
100C4×Q8—42no—
101C4×Q8—82no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
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—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129#107—48no—
130#106—84no—
131#106—82no—
132C4⋊C8—24no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—42no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8SD1614yes—
142C4×C8—41no—
143C4×C8—42no—
144C4×C8—41no—
145C4×C8—41no—
146C4×C8—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—82no—
158C2×Q8—82no—
159C2×Q8—81no—
160C2×Q8—81no—
161C2×Q8—81no—
162C2×Q8—81no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4#12011yescharacteristic
169C22×C4#12011yescharacteristic
170C22×C4#12011yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12711yescharacteristic
175C2×C8—24no—
176C2×C8—24no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—81no—
186C2×C8—81no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C42—21no—
215C42—22no—
216C42—21no—
217C42—22no—
218C42—41no—
219C42—82no—
220C42—82no—
221C42C2×SD1611yescharacteristic
222C42C2×SD1612yes—
223C42C2×Q1611yescharacteristic
224C42#13511yescharacteristic
225C42#13511yescharacteristic
226C42AGL(1,ℤ/8)11yescharacteristic
227C42#13711yescharacteristic
228C42#13712yes—
229C42#13711yescharacteristic
230C42#13711yescharacteristic
231C23#39211yescharacteristic
232Q8—44no—
233Q8—82no—
234Q8—82no—
235Q8—82no—
236Q8—162no—
237Q8—162no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4#44811yescharacteristic
263C2×C4#44812yes—
264C2×C4#44911yescharacteristic
265C2×C4#45011yescharacteristic
266C2×C4#45111yescharacteristic
267C2×C4#45111yescharacteristic
268C2×C4#45211yescharacteristic
269C2×C4#45211yescharacteristic
270C2×C4#45211yescharacteristic
271C2×C4#45511yescharacteristic
272C2×C4#45611yescharacteristic
273C2×C4#45612yes—
274C2×C4#46112yes—
275C2×C4#46511yescharacteristic
276C2×C4#46711yescharacteristic
277C2×C4#46811yescharacteristic
278C2×C4#47011yescharacteristic
279C2×C4#47212yes—
280C2×C4#49211yescharacteristic
281C2×C4#49711yescharacteristic
282C8—44no—
283C8—44no—
284C8—82no—
285C8—82no—
286C22#200011yescharacteristic
287C22#200111yescharacteristic
288C22#200511yescharacteristic
289C22#200711yescharacteristic
290C22#200811yescharacteristic
291C22#201311yescharacteristic
292C22#201811yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—41no—
300C4—82no—
301C4—82no—
302C4—161no—
303C4—162no—
304C4—161no—
305C4#164011yescharacteristic
306C4#164511yescharacteristic
307C4#164811yescharacteristic
308C4#165112yes—
309C4#165312yes—
310C4#165311yescharacteristic
311C4#167512yes—
312C4#169512yes—
313C2#1007511yescharacteristic
314C2#1011811yescharacteristic
315C2#1017811yescharacteristic
316C2#1019711yescharacteristic
317C2#1023411yescharacteristic
318C2#1025311yescharacteristic
319C2#1027811yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#292872
#292062
#286892
#98962
#96732
#75712
#70312