Finite Group Databasebeta

← #351248

Subgroups — #351248

Order 512.

609 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 154 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#33201C211yesmaximal, characteristic
2#33068C211yesmaximal, characteristic
3#30206C211yesmaximal, characteristic, complemented
4#20567C211yesmaximal, characteristic
5#9904C211yesmaximal, characteristic, complemented
6#9709C211yesmaximal, characteristic, complemented
7#9567C212yesmaximal, complemented
8#9562C212yesmaximal, complemented
9#9227C212yesmaximal
10#9223C212yesmaximal
11#8789C211yesmaximal, characteristic, complemented
12C2×#476C2211yescharacteristic
13C2×#474C2211yescharacteristic
14C2×#436C2212yes—
15#2862C2211yescharacteristic
16#2859C2211yescharacteristic
17C42×D4C2211yescharacteristic
18#2074—21no—
19#2074—21no—
20#2026—22no—
21#1990C2212yes—
22#1988C2211yescharacteristic
23#1987C2211yescharacteristic
24#1956C2212yes—
25#1949—22no—
26#1948C2212yes—
27#1920—22no—
28#1864—22no—
29#1842C2211yescharacteristic
30#1831C2211yescharacteristic
31#1804—22no—
32#1802C2212yes—
33C8×C4⋊C4C2212yes—
34#1767C2211yescharacteristic
35#1753—22no—
36C4×C22⋊C8—21no—
37#1744C2211yescharacteristic
38#1728—21no—
39C2×#333C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C22×M16—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×D4D412yes—
50C2×C4×D4C2311yescharacteristic
51C22×C42—21no—
52#476—24no—
53#474—24no—
54#436—24no—
55#436—24no—
56C2×#107—21no—
57C2×#106—22no—
58C2×#106—21no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#102—22no—
67C2×#102—21no—
68C2×#102—21no—
69C2×C22⋊C8—22no—
70C2×#97—21no—
71C2×C4×C8—22no—
72C2×C4×C8C2312yes—
73#391—21no—
74#391—21no—
75#391—22no—
76#389—22no—
77#389—22no—
78#384C2312yes—
79#382—22no—
80#382—21no—
81C4×C4⋊C4D412yes—
82C4×C4⋊C4C2311yescharacteristic
83C4×C22⋊C4—21no—
84C4×C22⋊C4—22no—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#336—22no—
93#336—22no—
94#336—21no—
95#336D411yescharacteristic
96#336D411yescharacteristic
97#336D412yes—
98#333—24no—
99#332—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×D4C2×D411yescharacteristic
103C23×C4—21no—
104C23×C4—22no—
105C2×M16—28no—
106C2×M16—82no—
107C2×M16—82no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118#125—81no—
119#125—81no—
120C4×D4—24no—
121C4×D4—22no—
122C4×D4—42no—
123C4×D4#4114yes—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—22no—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42—82no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4112yes—
143C2×C42C2411yescharacteristic
144#107—42no—
145#106—42no—
146#106—44no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—44no—
153C4⋊C8#4114yes—
154#103—82no—
155#103—82no—
156#102—44no—
157#102—42no—
158#102—42no—
159C22⋊C8—28no—
160#97—81no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—82no—
164#95—22no—
165#95—21no—
166#95—42no—
167#95—42no—
168#95—42no—
169C24—21no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—82no—
176C2×D4—21no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4#12412yes—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—82no—
192C22×C4#12112yes—
193C22×C4#12112yes—
194C22×C4C22×D411yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4C2×#4111yescharacteristic
197M16—88no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—24no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—84no—
228C4⋊C4#12412yes—
229C4⋊C4#13714yes—
230C22⋊C4—24no—
231C22⋊C4—44no—
232C42—22no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—21no—
237C42—42no—
238C42—42no—
239C42—82no—
240C42C2×#4112yes—
241C42C2×#4112yes—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23C2×#12111yescharacteristic
248Q8—82no—
249Q8—82no—
250Q8—162no—
251D4—22no—
252D4—22no—
253D4—42no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—88no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#52512yes—
279C2×C4C2×#12411yescharacteristic
280C2×C4#52912yes—
281C2×C4#53211yescharacteristic
282C2×C4C2×#13712yes—
283C2×C4#57512yes—
284C2×C4#57611yescharacteristic
285C2×C4#57712yes—
286C2×C4#57711yescharacteristic
287C8—42no—
288C8—44no—
289C8—42no—
290C8—81no—
291C8—81no—
292C8—82no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—82no—
297C22—82no—
298C22#285011yescharacteristic
299C22#304211yescharacteristic
300C22#304511yescharacteristic
301C22#305612yes—
302C22#306712yes—
303C4—22no—
304C4—42no—
305C4—44no—
306C4—42no—
307C4—42no—
308C4—161no—
309C4—162no—
310C4—161no—
311C4#311812yes—
312C4#312512yes—
313C4#313911yescharacteristic
314C4#314211yescharacteristic
315C4#317212yes—
316C2—82no—
317C2#1946612yes—
318C2#1949312yes—
319C2#3303411yescharacteristic
320C2#3305911yescharacteristic
321C2#3306211yescharacteristic
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
3. #30206C2semidirect
5. #9904C2semidirect
6. #9709C2semidirect
7. #9567C2semidirect
8. #9562C2semidirect
11. #8789C2semidirect

Maximal subgroups

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

subgroupindex
#332012
#330682
#302062
#205672
#99042
#97092
#95672
#95622
#92272
#92232
#87892