Finite Group Databasebeta

← #101109

Subgroups — #101109

Order 512.

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

normal subgroups only

This group has 720 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#15014C211yesmaximal, characteristic
2#14349C211yesmaximal, characteristic
3#8712C212yesmaximal
4#8698C211yesmaximal, characteristic
5#7035C212yesmaximal
6C2×#385—21no—
7C2×#380C2211yescharacteristic
8C22×#95C2211yescharacteristic
9#1779C2212yes—
10#1445C412yescomplemented
11#1444C412yescomplemented
12#1443—24no—
13#1442—22no—
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⋊C4—22no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33#385—44no—
34#380—22no—
35#380—22no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—21no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49C2×#95C2×C412yes—
50C2×#95D412yes—
51C2×#95D414yes—
52C25C2×D411yescharacteristic
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—24no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—81no—
68C23×C4—81no—
69C23×C4—81no—
70C23×C4—81no—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C412yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D412yes—
75C23×C4C2×D411yescharacteristic
76C23×C4#4112yes—
77C22×C8—42no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—82no—
83C2×C4⋊C4—82no—
84C2×C4⋊C4—81no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—44no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C22⋊C8—44no—
97C22⋊C8—44no—
98C22⋊C8—82no—
99#95—24no—
100#95—24no—
101#95—42no—
102#95—44no—
103#95—44no—
104#95—44no—
105#95—42no—
106#95—81no—
107#95—82no—
108#95—82no—
109#95—82no—
110#95—82no—
111#95—82no—
112#95—82no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—41no—
118C24—41no—
119C24C2×C22⋊C411yescharacteristic
120C24#12011yescharacteristic
121C24#12311yescharacteristic
122C24#13512yes—
123C24AGL(1,ℤ/8)12yes—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—42no—
139C22×C4—41no—
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—42no—
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—44no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—84no—
183C22×C4—88no—
184C22×C4—88no—
185C22×C4—84no—
186C22×C4C4×D412yes—
187C22×C4#12112yes—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—88no—
194C22⋊C4—48no—
195C22⋊C4—84no—
196C22⋊C4—84no—
197C22⋊C4—82no—
198C22⋊C4—82no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—24no—
208C23—24no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—24no—
216C23—24no—
217C23—42no—
218C23—42no—
219C23—44no—
220C23—44no—
221C23—44no—
222C23—44no—
223C23#38611yescharacteristic
224C23#41012yes—
225C23#41611yescharacteristic
226C23#41811yescharacteristic
227C23#42112yes—
228C23#44912yes—
229C23#45612yes—
230C23#45812yes—
231C23#48212yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4—82no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C8—162no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—48no—
299C22—48no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22#151412yes—
305C22#152112yes—
306C22#160914yes—
307C22#187511yescharacteristic
308C22#188611yescharacteristic
309C22#189011yescharacteristic
310C22#189711yescharacteristic
311C22#189911yescharacteristic
312C22#191111yescharacteristic
313C22#191311yescharacteristic
314C4—42no—
315C4—42no—
316C4—84no—
317C4—82no—
318C4—82no—
319C4—164no—
320C4—161no—
321C4—161no—
322C2—22no—
323C2—24no—
324C2—22no—
325C2—42no—
326C2#902611yescharacteristic
327C2#915711yescharacteristic
328C2#920511yescharacteristic
329C2#941211yescharacteristic
330C2#946111yescharacteristic
331C2#948111yescharacteristic
332C2#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
10. #1445C4semidirect
11. #1444C4semidirect

Maximal subgroups

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

subgroupindex
#150142
#143492
#87122
#86982
#70352