Finite Group Databasebeta

← #141981

Subgroups — #141981

Order 512.

584 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 100 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#14161C211yesmaximal, characteristic
2#8560C211yesmaximal, characteristic
3#8153C211yesmaximal, characteristic
4#7189C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×C4×C4⋊C4C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8#1797—21no—
9#1749—21no—
10C2×#324C2212yes—
11C2×#324C2212yes—
12#1446—21no—
13#1446—21no—
14C4⋊#95—21no—
15C4⋊#95—21no—
16C4⋊#95C412yes—
17C4⋊#95C2212yes—
18#1280—21no—
19#1280—21no—
20C23×Q8—21no—
21C2×C4⋊Q8—24no—
22C2×C4⋊Q8—42no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—41no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C412yes—
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×#97—41no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C4×C4⋊C4—42no—
41C2×#95—21no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95D411yescharacteristic
47C2×#95D411yescharacteristic
48#336—41no—
49#336—41no—
50#324—22no—
51#324—24no—
52#324Q814yescomplemented
53C22×Q8—24no—
54C22×Q8—42no—
55C22×Q8—42no—
56C22×Q8—82no—
57C22×Q8—82no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C22×C8—42no—
68C22×C8—42no—
69C4⋊Q8—48no—
70C4⋊Q8—48no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4M1614yes—
87C2×C4⋊C4C2×Q812yes—
88C2×C4⋊C4#4112yes—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—24no—
98C2×C42—42no—
99C2×C42—42no—
100C2×C42—42no—
101C2×C42—42no—
102C4⋊C8—82no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—82no—
108#95—41no—
109#95—41no—
110#95—42no—
111#95—82no—
112#95—82no—
113#95—82no—
114#95—82no—
115C24C2×C22⋊C411yescharacteristic
116C2×Q8—48no—
117C2×Q8—88no—
118C2×Q8—84no—
119C2×Q8—84no—
120C2×Q8—88no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4#9911yescharacteristic
165C22×C4#9911yescharacteristic
166C22×C4#10012yes—
167C22×C4#10112yes—
168C22×C4C4×Q812yes—
169C22×C4C22⋊Q812yes—
170C22×C4#12411yescharacteristic
171C22×C4C4⋊Q811yescharacteristic
172C22×C4C2×M1612yes—
173C22×C4#13112yes—
174C2×C8—44no—
175C2×C8—82no—
176C2×C8—82no—
177C2×C8—82no—
178C2×C8—82no—
179C2×C8—82no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—48no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C42—24no—
189C42—24no—
190C42—42no—
191C42—42no—
192C42—81no—
193C42—81no—
194C42—84no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23#39111yescharacteristic
200C23#40711yescharacteristic
201C23#40811yescharacteristic
202C23C2×#9911yescharacteristic
203C23#41011yescharacteristic
204C23C2×#10011yescharacteristic
205C23C2×#10111yescharacteristic
206Q8—816no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4#35412yes—
261C2×C4#35612yes—
262C2×C4#44614yes—
263C8—84no—
264C8—162no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22#146211yescharacteristic
278C22#146711yescharacteristic
279C22#146911yescharacteristic
280C22#147211yescharacteristic
281C22#198511yescharacteristic
282C22#199611yescharacteristic
283C22#199711yescharacteristic
284C22C2×#35411yescharacteristic
285C22#212811yescharacteristic
286C22C2×#35611yescharacteristic
287C22#213411yescharacteristic
288C4—24no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—42no—
296C4—84no—
297C4—81no—
298C4—84no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—21no—
303C2#988211yescharacteristic
304C2#1150611yescharacteristic
305C2#1150911yescharacteristic
306C2#1156911yescharacteristic
307C2#1157411yescharacteristic
308C2#1329911yescharacteristic
309C2#1330211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
52. #324Q8semidirect

Maximal subgroups

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

subgroupindex
#141612
#85602
#81532
#71892