Finite Group Databasebeta

← #330309

Subgroups — #330309

Order 512.

560 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 138 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#30279C211yesmaximal, characteristic, complemented
2#29868C211yesmaximal, characteristic
3#29865C211yesmaximal, characteristic
4#22013C211yesmaximal, characteristic
5#10543C212yesmaximal, complemented
6#10541C212yesmaximal, complemented
7#10278C212yesmaximal
8#10275C212yesmaximal
9#9709C211yesmaximal, characteristic, complemented
10#9702C211yesmaximal, characteristic, complemented
11#8788C211yesmaximal, characteristic, complemented
12C2×#500C2212yes—
13C2×#487C2212yes—
14C2×#476C2212yes—
15C2×#476C2211yescharacteristic
16C2×#474C2211yescharacteristic
17#2868C2211yescharacteristic
18#2862C2211yescharacteristic
19#2293C2211yescharacteristic
20#2095—21no—
21#2095—21no—
22#2095—22no—
23#2085—22no—
24#2084—22no—
25#2073—22no—
26#2071—22no—
27#1988C2211yescharacteristic
28#1987C2212yes—
29#1987C2211yescharacteristic
30#1954C2212yes—
31#1939C2212yes—
32#1924—22no—
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1818—21no—
36#1798—21no—
37#1767C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#333C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#332C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×C4×D4C2311yescharacteristic
50C2×#117—21no—
51#500—42no—
52#500—42no—
53#487—42no—
54#487—42no—
55#476—42no—
56#476—42no—
57#476—42no—
58#474—42no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—21no—
62C2×#106—22no—
63C2×#106—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×#103—22no—
70C2×#103—22no—
71C2×#102—22no—
72C2×#102—21no—
73C2×#102—21no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×#97—21no—
77C2×C4×C8C2312yes—
78#391—21no—
79#391—21no—
80#391—22no—
81#390D412yes—
82#389D412yes—
83#388—22no—
84#384—22no—
85#384—21no—
86#384—21no—
87#382—22no—
88C4×C4⋊C4C2311yescharacteristic
89C4×C22⋊C4—21no—
90C43C2311yescharacteristic
91#340—24no—
92#340—22no—
93#340—22no—
94#336—22no—
95#336—21no—
96#336—22no—
97#336—22no—
98#336D411yescharacteristic
99#336D411yescharacteristic
100#336D412yes—
101#333—42no—
102#333—42no—
103#333—42no—
104#332—42no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—21no—
108C22×D4C2×D411yescharacteristic
109C23×C4—21no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8C2×D412yes—
115C22×C8C2×D412yes—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—84no—
125C4×D4—22no—
126C4×D4—22no—
127#117—82no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D412yes—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C42—21no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42#4112yes—
147C2×C42C2411yescharacteristic
148#107—42no—
149#107—44no—
150#107—44no—
151#106—44no—
152#106—42no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—82no—
160#103—44no—
161#103—44no—
162#102—42no—
163#102—44no—
164#102—42no—
165C22⋊C8—82no—
166C22⋊C8—82no—
167#97—81no—
168C4×C8—22no—
169C4×C8—22no—
170#95—22no—
171#95—21no—
172#95—42no—
173#95—42no—
174C24—21no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—84no—
183C2×Q8—82no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4#12112yes—
198C22×C4#12112yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C22⋊C4—42no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C42—21no—
239C42—22no—
240C42—21no—
241C42—22no—
242C42—22no—
243C42—42no—
244C42—42no—
245C42H(5,𝔽2)12yes—
246C42H(5,𝔽2)12yes—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23C2×#12111yescharacteristic
253Q8—82no—
254Q8—82no—
255Q8—162no—
256Q8—162no—
257Q8—162no—
258D4—41no—
259D4—41no—
260D4—42no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4—82no—
283C2×C4#53512yes—
284C2×C4#53511yescharacteristic
285C2×C4#53511yescharacteristic
286C2×C4#54012yes—
287C2×C4#57711yescharacteristic
288C2×C4#57712yes—
289C2×C4#57812yes—
290C2×C4#57812yes—
291C2×C4#57811yescharacteristic
292C8—42no—
293C8—42no—
294C8—82no—
295C8—82no—
296C8—81no—
297C8—81no—
298C22—82no—
299C22—82no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#261911yescharacteristic
304C22#304211yescharacteristic
305C22#304611yescharacteristic
306C22#305912yes—
307C22#306812yes—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—82no—
312C4—81no—
313C4—161no—
314C4—162no—
315C4—161no—
316C4#322112yes—
317C4#322312yes—
318C4#322311yescharacteristic
319C4#322311yescharacteristic
320C4#322512yes—
321C2—161no—
322C2#1951512yes—
323C2#1954712yes—
324C2#2980411yescharacteristic
325C2#2981211yescharacteristic
326C2#2982211yescharacteristic
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. #30279C2semidirect
5. #10543C2semidirect
6. #10541C2semidirect
9. #9709C2semidirect
10. #9702C2semidirect
11. #8788C2semidirect

Maximal subgroups

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

subgroupindex
#302792
#298682
#298652
#220132
#105432
#105412
#102782
#102752
#97092
#97022
#87882