Finite Group Databasebeta

← #98254

Subgroups — #98254

Order 512.

751 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 132 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#36594C211yesmaximal, characteristic
2#9896C212yesmaximal, complemented
3#8585C211yesmaximal, characteristic
4#8564C212yesmaximal, complemented
5#7569C211yesmaximal, characteristic
6C2×#556C412yes—
7#2868C2211yescharacteristic
8#2858—21no—
9#2547C412yes—
10#2545—22no—
11#2538—22no—
12#2512—24no—
13#2354—22no—
14C4×C4⋊Q8—21no—
15C4×#127C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#1842C2212yes—
18C4×C4⋊C8C2212yes—
19C2×#331—21no—
20C2×Q8⋊C8—22no—
21#1296—21no—
22#1296—24no—
23#1279—22no—
24#1270—22no—
25#556—44no—
26#556—44no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8C2×C412yes—
30C2×#127D411yescharacteristic
31C2×#126—24no—
32C2×#124—22no—
33C2×#124—22no—
34C2×C22⋊Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D412yes—
38C2×C4×D4—22no—
39C2×C4×D4C2×C412yes—
40C2×#117—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4×C8—22no—
46#397—24no—
47#396—28no—
48#391—22no—
49#390—24no—
50#390—24no—
51#390—22no—
52#390D412yes—
53#389—24no—
54#388—28no—
55#384—22no—
56#384D411yescharacteristic
57#384D414yes—
58#380—24no—
59C4×C4⋊C4—24no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—21no—
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4D412yes—
64C43—21no—
65C43C2311yescharacteristic
66#336—22no—
67#331—42no—
68Q8⋊C8—44no—
69C22×Q8—22no—
70C22×Q8C22⋊C411yescharacteristic
71C22×D4—22no—
72C22×D4C22⋊C412yes—
73C23×C4—22no—
74C22×C8—42no—
75C22×C8—42no—
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—82no—
82#127—22no—
83#127—21no—
84#127—21no—
85#126—416no—
86#124—48no—
87#124—84no—
88C22⋊Q8—44no—
89C22⋊Q8—44no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—48no—
97C4×D4—42no—
98C4×D4—42no—
99C4×D4—42no—
100C4×D4—42no—
101C4×D4—42no—
102C4×D4—42no—
103#117—44no—
104#117—44no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4C22⋊C412yes—
113C2×C4⋊C4C22⋊C411yescharacteristic
114C2×C4⋊C4C22⋊C412yes—
115C2×C4⋊C4C22⋊C414yes—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—28no—
121C2×C42—21no—
122C2×C42—24no—
123C2×C42—21no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C4⋊C8—24no—
132C4⋊C8—42no—
133C4⋊C8—44no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136C4⋊C8—42no—
137C4⋊C8—42no—
138C4×C8—42no—
139C4×C8—42no—
140#95—28no—
141#95—21no—
142#95—22no—
143C24—22no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8#10012yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4#12012yes—
175C22×C4#12012yes—
176C2×C8—44no—
177C2×C8—82no—
178C2×C8—82no—
179C2×C8—82no—
180C2×C8—82no—
181C2×C8—82no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—48no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—84no—
198C4⋊C4#10012yes—
199C22⋊C4—48no—
200C22⋊C4—44no—
201C22⋊C4—48no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—88no—
205C42—24no—
206C42—21no—
207C42—21no—
208C42—22no—
209C42—22no—
210C42—41no—
211C42—48no—
212C42—42no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42—42no—
217C42#13512yes—
218C42#13512yes—
219C42#13512yes—
220C42AGL(1,ℤ/8)12yes—
221C42AGL(1,ℤ/8)11yescharacteristic
222C42#13712yes—
223C42#13711yescharacteristic
224C23—44no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—42no—
229C23#37911yescharacteristic
230Q8—24no—
231Q8—41no—
232Q8—41no—
233Q8—84no—
234Q8—84no—
235D4—82no—
236D4—82no—
237D4—82no—
238D4—82no—
239D4—82no—
240D4—82no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—82no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4C2×#10011yescharacteristic
263C2×C4C2×#10011yescharacteristic
264C2×C4#41611yescharacteristic
265C2×C4#41612yes—
266C2×C4#41711yescharacteristic
267C2×C4#41712yes—
268C2×C4#44912yes—
269C2×C4#44912yes—
270C2×C4#44912yes—
271C2×C4#45212yes—
272C2×C4#45212yes—
273C2×C4#45212yes—
274C2×C4#45412yes—
275C2×C4#45612yes—
276C8—84no—
277C8—162no—
278C22—84no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—84no—
283C22#178311yescharacteristic
284C22#178811yescharacteristic
285C22#178812yes—
286C22#179312yes—
287C22#179511yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—84no—
297C4—82no—
298C4#154311yescharacteristic
299C4#155112yes—
300C4#156011yescharacteristic
301C4#162812yes—
302C4#163812yes—
303C4#165012yes—
304C4#165212yes—
305C2—162no—
306C2#834411yescharacteristic
307C2#838511yescharacteristic
308C2#842212yes—
309C2#844912yes—
310C2#847111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9896C2semidirect
4. #8564C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#98962
#85852
#85642
#75692