Finite Group Databasebeta

← #103779

Subgroups — #103779

Order 512.

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

normal subgroups only

This group has 489 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#29286C211yesmaximal, characteristic
2#29208C211yesmaximal, characteristic
3#28511C211yesmaximal, characteristic
4#9694C211yesmaximal, characteristic
5#9673C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#477—22no—
9#2868C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12#2095—22no—
13#2086—22no—
14#2057—22no—
15#1990—21no—
16#1988—21no—
17#1988—21no—
18#1956—21no—
19#1947—22no—
20#1947—21no—
21#1947—21no—
22#1845—21no—
23#1845—21no—
24#1842C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#332—22no—
29C2×Q8⋊C8—22no—
30#1271—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#477—44no—
39#477—44no—
40C2×#107—24no—
41C2×#107—24no—
42C2×#107—22no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D412yes—
48C2×C4⋊C8D412yes—
49C2×#103—24no—
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53#391—21no—
54#391—21no—
55#389—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#384—21no—
60#384—21no—
61#384—21no—
62#384D411yescharacteristic
63#384D411yescharacteristic
64#384D412yes—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C43C2311yescharacteristic
70#340—24no—
71#340—22no—
72#340—22no—
73#340—22no—
74#336—21no—
75#336—21no—
76#336D412yes—
77#332—44no—
78Q8⋊C8—44no—
79C22×Q8—21no—
80C22×Q8—21no—
81C22×Q8—41no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—41no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—82no—
97C4⋊Q8Q1612yes—
98C4⋊Q8Q1612yes—
99C4×Q8—42no—
100C4×Q8—42no—
101C4×Q8—42no—
102C4×Q8—42no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4111yescharacteristic
130#107—48no—
131#107—84no—
132#107—82no—
133C4⋊C8—24no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—42no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8D814yes—
143#103—48no—
144C4×C8—42no—
145C4×C8—41no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—82no—
162C2×Q8—82no—
163C2×Q8—81no—
164C2×Q8—81no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4#12011yescharacteristic
171C22×C4#12011yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12711yescharacteristic
177C2×C8—24no—
178C2×C8—24no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—81no—
188C2×C8—81no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—81no—
208C4⋊C4—82no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C42—21no—
217C42—21no—
218C42—22no—
219C42—22no—
220C42—41no—
221C42—82no—
222C42—82no—
223C42C2×D812yes—
224C42C2×Q1611yescharacteristic
225C42C2×Q1611yescharacteristic
226C42#13511yescharacteristic
227C42#13511yescharacteristic
228C42AGL(1,ℤ/8)12yes—
229C42#13711yescharacteristic
230C42#13711yescharacteristic
231C42#13711yescharacteristic
232C42#13711yescharacteristic
233C23#39211yescharacteristic
234Q8—44no—
235Q8—44no—
236Q8—82no—
237Q8—82no—
238Q8—162no—
239Q8—162no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4#44712yes—
265C2×C4#45111yescharacteristic
266C2×C4#45111yescharacteristic
267C2×C4#45111yescharacteristic
268C2×C4#45111yescharacteristic
269C2×C4#45211yescharacteristic
270C2×C4#45211yescharacteristic
271C2×C4#45211yescharacteristic
272C2×C4#45211yescharacteristic
273C2×C4#45311yescharacteristic
274C2×C4#45411yescharacteristic
275C2×C4#45612yes—
276C2×C4#45912yes—
277C2×C4#46711yescharacteristic
278C2×C4#46711yescharacteristic
279C2×C4#47011yescharacteristic
280C2×C4#47011yescharacteristic
281C2×C4#47312yes—
282C2×C4#49311yescharacteristic
283C2×C4#49611yescharacteristic
284C8—44no—
285C8—44no—
286C8—82no—
287C8—82no—
288C22#200111yescharacteristic
289C22#200111yescharacteristic
290C22#200511yescharacteristic
291C22#200611yescharacteristic
292C22#200811yescharacteristic
293C22#201111yescharacteristic
294C22#202011yescharacteristic
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—82no—
303C4—82no—
304C4—161no—
305C4—161no—
306C4—162no—
307C4#163011yescharacteristic
308C4#163311yescharacteristic
309C4#163611yescharacteristic
310C4#163611yescharacteristic
311C4#164712yes—
312C4#164912yes—
313C4#166912yes—
314C4#168912yes—
315C2#1007611yescharacteristic
316C2#1011711yescharacteristic
317C2#1018311yescharacteristic
318C2#1019211yescharacteristic
319C2#1023711yescharacteristic
320C2#1025311yescharacteristic
321C2#1027411yescharacteristic
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
#292862
#292082
#285112
#96942
#96732
#75702
#70312