Finite Group Databasebeta

← #101101

Subgroups — #101101

Order 512.

684 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 94 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#15015C211yesmaximal, characteristic
2#14349C211yesmaximal, characteristic
3#8713C212yesmaximal
4#8698C211yesmaximal, characteristic
5#7035C212yesmaximal
6C2×#380C2211yescharacteristic
7C22×#95C2211yescharacteristic
8#1779C2212yes—
9#1445C412yescomplemented
10#1443—24no—
11#1443—22no—
12#1442C412yescomplemented
13C22⋊#95—21no—
14C22⋊#95—21no—
15C22⋊#95—21no—
16C22⋊#95—21no—
17C22⋊#95C2211yescharacteristic
18C22⋊#95C2212yes—
19#1294—24no—
20#1280—22no—
21C24×C4D411yescharacteristic
22C24×C4D411yescharacteristic
23C24×C4C2311yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C22⋊C4—21no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—21no—
29C2×C22⋊C8—22no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32#380—22no—
33#380—22no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—21no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95C2×C412yes—
48C2×#95C2×C412yes—
49C2×#95D412yes—
50C2×#95D414yes—
51C25C2×D411yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—22no—
61C23×C4—24no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—81no—
67C23×C4—81no—
68C23×C4—81no—
69C23×C4—81no—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C412yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D412yes—
74C23×C4C2×D411yescharacteristic
75C23×C4#4112yes—
76C22×C8—42no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—81no—
81C2×C22⋊C4—44no—
82C2×C22⋊C4—44no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C22⋊C8—44no—
92C22⋊C8—44no—
93C22⋊C8—82no—
94#95—24no—
95#95—24no—
96#95—42no—
97#95—44no—
98#95—44no—
99#95—42no—
100#95—82no—
101#95—81no—
102#95—82no—
103#95—82no—
104#95—82no—
105#95—82no—
106#95—82no—
107#95—82no—
108#95—82no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—41no—
114C24—41no—
115C24C2×C22⋊C411yescharacteristic
116C24#12011yescharacteristic
117C24#12311yescharacteristic
118C24#13512yes—
119C24#13712yes—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—84no—
177C22×C4—88no—
178C22×C4—88no—
179C22×C4—84no—
180C22×C4C4×D412yes—
181C22×C4#12112yes—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C22⋊C4—84no—
188C22⋊C4—84no—
189C22⋊C4—84no—
190C22⋊C4—82no—
191C22⋊C4—82no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—24no—
201C23—24no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—24no—
210C23—42no—
211C23—42no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23#38611yescharacteristic
217C23#41012yes—
218C23#41611yescharacteristic
219C23#41911yescharacteristic
220C23#42112yes—
221C23#45212yes—
222C23#45412yes—
223C23#45812yes—
224C23#48512yes—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4—84no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4—84no—
246C2×C4—84no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—84no—
251C2×C4—82no—
252C2×C4—84no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4—84no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4—82no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C8—162no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—48no—
290C22—48no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22#151512yes—
296C22#152112yes—
297C22#161514yes—
298C22#187311yescharacteristic
299C22#188811yescharacteristic
300C22#189011yescharacteristic
301C22#189711yescharacteristic
302C22#189911yescharacteristic
303C22#191111yescharacteristic
304C22#191311yescharacteristic
305C4—42no—
306C4—42no—
307C4—82no—
308C4—82no—
309C4—164no—
310C4—162no—
311C4—161no—
312C4—161no—
313C2—24no—
314C2—22no—
315C2—22no—
316C2—42no—
317C2#902211yescharacteristic
318C2#915311yescharacteristic
319C2#920111yescharacteristic
320C2#941411yescharacteristic
321C2#946511yescharacteristic
322C2#948111yescharacteristic
323C2#950211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1445C4semidirect
12. #1442C4semidirect

Maximal subgroups

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

subgroupindex
#150152
#143492
#87132
#86982
#70352