Finite Group Databasebeta

← #330358

Subgroups — #330358

Order 512.

582 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 142 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#30280C211yesmaximal, characteristic, complemented
2#29868C211yesmaximal, characteristic
3#29829C211yesmaximal, characteristic
4#22009C211yesmaximal, characteristic
5#10544C212yesmaximal, complemented
6#10541C212yesmaximal, complemented
7#10275C212yesmaximal
8#10271C212yesmaximal
9#9700C211yesmaximal, characteristic, complemented
10#9691C211yesmaximal, characteristic, complemented
11#8789C211yesmaximal, characteristic, complemented
12C2×#498C2212yes—
13C2×#487C2212yes—
14C2×#476C2212yes—
15C2×#474C2211yescharacteristic
16C2×#461C2211yescharacteristic
17#2868C2211yescharacteristic
18#2862C2211yescharacteristic
19#2293C2211yescharacteristic
20#2096—21no—
21#2096—21no—
22#2096—22no—
23#2084—22no—
24#2071—22no—
25#2068—22no—
26#2055—22no—
27#1988C2211yescharacteristic
28#1987C2212yes—
29#1981C2211yescharacteristic
30#1954C2212yes—
31#1939C2212yes—
32#1923—22no—
33#1842C2211yescharacteristic
34#1831C2211yescharacteristic
35#1818—21no—
36#1798—21no—
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#333C2212yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C22×SD16—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2312yes—
50C2×C4×D4C2311yescharacteristic
51C2×#117—21no—
52#498—42no—
53#498—42no—
54#487—42no—
55#487—42no—
56#476—42no—
57#476—42no—
58#474—42no—
59#461—42no—
60C2×#107—21no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—21no—
73C2×#103—22no—
74C2×#102—21no—
75C2×#102—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#391—22no—
81#391—21no—
82#391—21no—
83#390D412yes—
84#389D412yes—
85#388—22no—
86#384—22no—
87#384—21no—
88#384—21no—
89#382—22no—
90C4×C4⋊C4C2311yescharacteristic
91C4×C22⋊C4—21no—
92C43C2311yescharacteristic
93#340—22no—
94#340—24no—
95#336—22no—
96#336—21no—
97#336—22no—
98#336D411yescharacteristic
99#336D411yescharacteristic
100#336D412yes—
101#333—42no—
102#333—42no—
103#332—42no—
104#332—42no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×D4C2×D411yescharacteristic
109C23×C4—21no—
110C2×SD16—42no—
111C2×SD16—82no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—84no—
127C4×D4—22no—
128C4×D4—22no—
129#117—82no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×D412yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—22no—
144C2×C42—21no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42#4112yes—
149C2×C42C2411yescharacteristic
150#107—42no—
151#106—44no—
152#106—44no—
153#106—44no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—82no—
161C4⋊C8—82no—
162#103—42no—
163#103—44no—
164#103—44no—
165#103—82no—
166#102—42no—
167#102—44no—
168C22⋊C8—82no—
169C22⋊C8—82no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—81no—
173#95—22no—
174#95—21no—
175#95—42no—
176#95—42no—
177C24—21no—
178C2×Q8—42no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—42no—
188C2×Q8—84no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4#12112yes—
203C22×C4#12112yes—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C2×#4111yescharacteristic
207SD16—84no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—82no—
238C4⋊C4—84no—
239C4⋊C4—84no—
240C22⋊C4—42no—
241C22⋊C4—44no—
242C22⋊C4—82no—
243C42—21no—
244C42—22no—
245C42—22no—
246C42—21no—
247C42—22no—
248C42—24no—
249C42—42no—
250C42H(5,𝔽2)12yes—
251C42H(5,𝔽2)12yes—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23C2×#12111yescharacteristic
258Q8—82no—
259Q8—81no—
260Q8—81no—
261Q8—82no—
262Q8—82no—
263Q8—162no—
264Q8—162no—
265D4—41no—
266D4—41no—
267D4—42no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—84no—
289C2×C4#47114yes—
290C2×C4#53512yes—
291C2×C4#53511yescharacteristic
292C2×C4#53511yescharacteristic
293C2×C4#54012yes—
294C2×C4#57711yescharacteristic
295C2×C4#57712yes—
296C2×C4#57712yes—
297C2×C4#57812yes—
298C2×C4#57811yescharacteristic
299C8—42no—
300C8—42no—
301C8—82no—
302C8—81no—
303C8—81no—
304C8—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—82no—
310C22#261911yescharacteristic
311C22#304211yescharacteristic
312C22#304611yescharacteristic
313C22#306312yes—
314C22C2×#47112yes—
315C4—22no—
316C4—24no—
317C4—42no—
318C4—82no—
319C4—81no—
320C4—161no—
321C4—161no—
322C4—162no—
323C4#322112yes—
324C4#322211yescharacteristic
325C4#322311yescharacteristic
326C4#322512yes—
327C4#322512yes—
328C2—161no—
329C2#1955912yes—
330C2#1958812yes—
331C2#2980411yescharacteristic
332C2#2981711yescharacteristic
333C2#2981911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30280C2semidirect
5. #10544C2semidirect
6. #10541C2semidirect
9. #9700C2semidirect
10. #9691C2semidirect
11. #8789C2semidirect

Maximal subgroups

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

subgroupindex
#302802
#298682
#298292
#220092
#105442
#105412
#102752
#102712
#97002
#96912
#87892