Finite Group Databasebeta

← #98251

Subgroups — #98251

Order 512.

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

normal subgroups only

This group has 721 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#36188C211yesmaximal, characteristic
2#9898C212yesmaximal, complemented
3#8587C211yesmaximal, characteristic, complemented
4#8563C212yesmaximal
5#7571C211yesmaximal, characteristic, complemented
6C2×#556C412yes—
7#2868C2211yescharacteristic
8#2857—21no—
9#2568C412yes—
10#2547—22no—
11#2499—22no—
12#2496—24no—
13#2365—22no—
14C4×C4⋊Q8C2211yescharacteristic
15#2285—21no—
16C42×D4C2211yescharacteristic
17#1842C2212yes—
18C4×C4⋊C8C2212yes—
19C2×#333—21no—
20C2×D4⋊C8—22no—
21#1296—21no—
22#1296—24no—
23#1279—22no—
24#1272—22no—
25#556—44no—
26#556—44no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8C2×C412yes—
29C2×C4⋊Q8D411yescharacteristic
30C2×#126—24no—
31C2×#124—22no—
32C2×C22⋊Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8C2×C412yes—
35C2×C4×D4C2×C411yescharacteristic
36C2×C4×D4D412yes—
37C2×#117—22no—
38C22×C42—21no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4×C8—22no—
44#400—24no—
45#397—28no—
46#391—22no—
47#391—24no—
48#391D412yes—
49#389—24no—
50#389—22no—
51#387—28no—
52#384—21no—
53#384—21no—
54#384D411yescharacteristic
55#384D414yes—
56#382—22no—
57#380—24no—
58#380—24no—
59C4×C4⋊C4—24no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4C2×C411yescharacteristic
62C4×C4⋊C4D412yes—
63C4×C22⋊C4—22no—
64C4×C22⋊C4—21no—
65C43C2311yescharacteristic
66#336—22no—
67#333—42no—
68D4⋊C8—44no—
69C22×Q8—22no—
70C22×Q8C22⋊C412yes—
71C22×D4C22⋊C411yescharacteristic
72C23×C4—21no—
73C23×C4—22no—
74C22×C8—42no—
75C22×C8—42no—
76C4⋊Q8—22no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—42no—
80C4⋊Q8—42no—
81C4⋊Q8—42no—
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C4⋊Q8—42no—
85#126—416no—
86#124—44no—
87#124—44no—
88C22⋊Q8—48no—
89C4×Q8—42no—
90C4×Q8—42no—
91C4×Q8—44no—
92C4×Q8—44no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99#117—48no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C22⋊C411yescharacteristic
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C414yes—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C42—24no—
116C2×C42—22no—
117C2×C42—21no—
118C2×C42—22no—
119C2×C42—42no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42C22×C411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C4⋊C8—24no—
128C4⋊C8—42no—
129C4⋊C8—44no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C4×C8—42no—
135C4×C8—42no—
136#95—28no—
137#95—28no—
138#95—21no—
139#95—22no—
140#95—24no—
141C24—21no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×D4—21no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4#10012yes—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4#12012yes—
173C22×C4#12012yes—
174C2×C8—44no—
175C2×C8—82no—
176C2×C8—82no—
177C2×C8—82no—
178C2×C8—82no—
179C2×C8—82no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4#10012yes—
197C22⋊C4—24no—
198C22⋊C4—48no—
199C22⋊C4—48no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—21no—
203C42—21no—
204C42—22no—
205C42—22no—
206C42—41no—
207C42—48no—
208C42—42no—
209C42—42no—
210C42—41no—
211C42—41no—
212C42—42no—
213C42—42no—
214C42#13512yes—
215C42#13512yes—
216C42#13512yes—
217C42AGL(1,ℤ/8)12yes—
218C42AGL(1,ℤ/8)11yescharacteristic
219C42#13711yescharacteristic
220C42#13712yes—
221C23—22no—
222C23—42no—
223C23—41no—
224C23—42no—
225C23—41no—
226C23#37911yescharacteristic
227Q8—82no—
228Q8—82no—
229Q8—82no—
230Q8—84no—
231Q8—82no—
232D4—24no—
233D4—41no—
234D4—41no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—48no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4C2×#10011yescharacteristic
261C2×C4C2×#10011yescharacteristic
262C2×C4#41611yescharacteristic
263C2×C4#41612yes—
264C2×C4#41711yescharacteristic
265C2×C4#41712yes—
266C2×C4#44912yes—
267C2×C4#44912yes—
268C2×C4#44912yes—
269C2×C4#45212yes—
270C2×C4#45212yes—
271C2×C4#45212yes—
272C2×C4#45412yes—
273C2×C4#45612yes—
274C8—84no—
275C8—162no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22—82no—
281C22#178311yescharacteristic
282C22#178811yescharacteristic
283C22#178812yes—
284C22#179312yes—
285C22#179611yescharacteristic
286C4—21no—
287C4—21no—
288C4—22no—
289C4—22no—
290C4—41no—
291C4—42no—
292C4—82no—
293C4—82no—
294C4—84no—
295C4—82no—
296C4#154311yescharacteristic
297C4#155112yes—
298C4#156011yescharacteristic
299C4#162812yes—
300C4#163812yes—
301C4#165012yes—
302C4#165212yes—
303C2—82no—
304C2#834411yescharacteristic
305C2#839311yescharacteristic
306C2#842712yes—
307C2#844912yes—
308C2#847111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
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. #9898C2semidirect
3. #8587C2semidirect
5. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#98982
#85872
#85632
#75712