Finite Group Databasebeta

← #294910

Subgroups — #294910

Order 512.

873 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 170 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#39385C211yesmaximal, characteristic
2#33225C211yesmaximal, characteristic, complemented
3#29868C211yesmaximal, characteristic
4#29226C211yesmaximal, characteristic, complemented
5#10060C214yesmaximal
6#9871C211yesmaximal, characteristic, complemented
7#9705C211yesmaximal, characteristic, complemented
8#9065C214yesmaximal, complemented
9#8177C211yesmaximal, characteristic
10C2×#541—21no—
11C2×#487C2214yes—
12C2×#476C2212yes—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2721—24no—
16#2621—21no—
17#2552—22no—
18#2542C2212yes—
19#2512—22no—
20#2429—22no—
21#2365C2212yes—
22C4×#125C2211yescharacteristic
23#2293C2211yescharacteristic
24#2099—22no—
25#2084—22no—
26#2071—22no—
27#1992—21no—
28#1987C2214yes—
29#1987C2212yes—
30#1956—24no—
31#1836—21no—
32#1831C2211yescharacteristic
33#1767C2211yescharacteristic
34C42×C8C2211yescharacteristic
35C2×#332C2214yes—
36C2×D4⋊C8C2212yes—
37#1279C2214yes—
38#1279C2212yes—
39#541—44no—
40#541—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2312yes—
45C2×C4⋊Q8C2312yes—
46C2×#126—24no—
47C2×#125—24no—
48C2×#125D411yescharacteristic
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×D4C2311yescharacteristic
53C2×#117—21no—
54#487—28no—
55#487—44no—
56#476—44no—
57C2×#106—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—28no—
61C2×#102—22no—
62C2×#97—21no—
63C2×C4×C8—22no—
64C2×C4×C8C2312yes—
65#399—24no—
66#398—24no—
67#398—24no—
68#397—24no—
69#396—24no—
70#391—22no—
71#390—22no—
72#390D412yes—
73#389—24no—
74#389—22no—
75#389D412yes—
76#389D412yes—
77#388—24no—
78#388—22no—
79#387—24no—
80#387—22no—
81#384—22no—
82#384—22no—
83#384—22no—
84#384—21no—
85#384—21no—
86#384D414yes—
87#382D411yescharacteristic
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4C2312yes—
91C4×C4⋊C4C2311yescharacteristic
92C4×C22⋊C4—21no—
93C43C2311yescharacteristic
94#340—24no—
95#340—24no—
96#340—24no—
97#336—22no—
98#336—21no—
99#332—44no—
100D4⋊C8—24no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×D4C2×D411yescharacteristic
104C23×C4—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—24no—
112C4⋊Q8—48no—
113C4⋊Q8—42no—
114C4⋊Q8—42no—
115C4⋊Q8—82no—
116#126—416no—
117#125—22no—
118#125—416no—
119#125SD1614yes—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—84no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×D4—22no—
126C4×D4—22no—
127#117—42no—
128#117—42no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D412yes—
144C2×C4⋊C4C2×D414yes—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C42—24no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2411yescharacteristic
160#106—44no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—24no—
164#103—416no—
165#102—44no—
166#97—42no—
167C4×C8—22no—
168C4×C8—44no—
169C4×C8#4114yes—
170#95—24no—
171#95—24no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—21no—
176#95—22no—
177C24—21no—
178C2×Q8—48no—
179C2×Q8—44no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4#12012yes—
198C22×C4#12012yes—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C22×D411yescharacteristic
202C2×C8—24no—
203C2×C8—44no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—44no—
209C4⋊C4—22no—
210C4⋊C4—28no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—28no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—48no—
217C4⋊C4—48no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C4⋊C4—48no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C4⋊C4C2×SD1614yes—
231C22⋊C4—48no—
232C22⋊C4—42no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C22⋊C4—84no—
236C42—22no—
237C42—24no—
238C42—24no—
239C42—48no—
240C42—41no—
241C42—41no—
242C42—44no—
243C42C2×SD1612yes—
244C42AGL(1,ℤ/8)12yes—
245C42C2×#4112yes—
246C42H(5,𝔽2)12yes—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23C2×#12011yescharacteristic
253Q8—88no—
254Q8—84no—
255Q8—84no—
256D4—42no—
257D4—42no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—84no—
282C2×C4#45014yes—
283C2×C4#45414yes—
284C2×C4#53211yescharacteristic
285C2×C4#53511yescharacteristic
286C2×C4#54612yes—
287C2×C4#55012yes—
288C2×C4C22×SD1611yescharacteristic
289C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
290C2×C4#57712yes—
291C2×C4#57814yes—
292C8—44no—
293C8—82no—
294C8—82no—
295C22—82no—
296C22—81no—
297C22—81no—
298C22—82no—
299C22—81no—
300C22#243111yescharacteristic
301C22C2×#45011yescharacteristic
302C22#300911yescharacteristic
303C22#301112yes—
304C22C2×#45412yes—
305C4—24no—
306C4—24no—
307C4—22no—
308C4—44no—
309C4—82no—
310C4—81no—
311C4—84no—
312C4—82no—
313C4—162no—
314C4#317712yes—
315C4#322312yes—
316C4#339212yes—
317C4#339712yes—
318C2—161no—
319C2#1864112yes—
320C2#1869512yes—
321C2#2559111yescharacteristic
322C2#2560911yescharacteristic
323C2#2562711yescharacteristic
subgroupG/H
center, socleC23C2×#120
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
2. #33225C2semidirect
4. #29226C2semidirect
6. #9871C2semidirect
7. #9705C2semidirect
8. #9065C2semidirect

Maximal subgroups

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

subgroupindex
#393852
#332252
#298682
#292262
#100602
#98712
#97052
#90652
#81772