Finite Group Databasebeta

← #145782

Subgroups — #145782

Order 512.

421 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 88 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#30292C211yesmaximal, characteristic
2#12920C211yesmaximal, characteristic
3#12919C211yesmaximal, characteristic
4#11328C211yesmaximal, characteristic
5#7258C211yesmaximal, characteristic
6#7258C211yesmaximal, characteristic
7#7183C211yesmaximal, characteristic
8C22×#107C2211yescharacteristic
9#2851—21no—
10C2×#389—21no—
11C2×#367—21no—
12C2×#366—21no—
13C2×C22⋊C16C2211yescharacteristic
14C2×C22⋊C16C2211yescharacteristic
15#2085—21no—
16#2085—21no—
17#2059—21no—
18#2059—21no—
19#1808C2211yescharacteristic
20#1808C2211yescharacteristic
21#1807—21no—
22C2×#340—21no—
23C2×#340C2211yescharacteristic
24C2×#336C2211yescharacteristic
25#1380—21no—
26#1380—21no—
27C23×C8D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×#125—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—41no—
33C22×C42D411yescharacteristic
34C22×C16—21no—
35C22×C16—21no—
36C2×#107—21no—
37C2×#107—21no—
38C2×#107—42no—
39C2×#107D411yescharacteristic
40C2×#107D411yescharacteristic
41C2×#107D411yescharacteristic
42C2×#107D411yescharacteristic
43C2×#106—21no—
44C2×#106—21no—
45#389—21no—
46#389—21no—
47#389—42no—
48#389—42no—
49#387—41no—
50#387—41no—
51#385—41no—
52#385—41no—
53#382—21no—
54#380—21no—
55#380—21no—
56C2×#95—41no—
57#367—42no—
58#366—42no—
59C22⋊C16—21no—
60C22⋊C16—21no—
61C22⋊C16Q812yes—
62C22⋊C16Q812yes—
63#340—21no—
64#340—21no—
65#340—21no—
66#340—21no—
67#340—21no—
68#340—42no—
69#340Q812yes—
70#336—21no—
71#336—21no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4D811yescharacteristic
75C23×C4D811yescharacteristic
76C23×C4C2×D411yescharacteristic
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8Q1612yes—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2×Q811yescharacteristic
87C22×C8C2×Q811yescharacteristic
88C22×C8C2×Q811yescharacteristic
89C22×C8#4111yescharacteristic
90#125—81no—
91#125—81no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4—82no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C16—22no—
122C2×C16—22no—
123C2×C16—41no—
124C2×C16—41no—
125C2×C16—41no—
126C2×C16—41no—
127#107—22no—
128#107—22no—
129#107—41no—
130#107—41no—
131#107—41no—
132#107—41no—
133#107—41no—
134#107—41no—
135#106—81no—
136#106—81no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95—44no—
142#95—41no—
143#95—41no—
144C24C2×D811yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4#12011yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C4⋊Q811yescharacteristic
180C22×C4C2×Q1611yescharacteristic
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182C22×C4#13711yescharacteristic
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—22no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—81no—
196C2×C8C22⋊Q811yescharacteristic
197C2×C8C22⋊Q811yescharacteristic
198C2×C8C4⋊Q811yescharacteristic
199C2×C8C4⋊Q811yescharacteristic
200C4⋊C4—24no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—81no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—88no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C4⋊C4—81no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C42—41no—
226C42—41no—
227C42—81no—
228C16—42no—
229C16—42no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23#44711yescharacteristic
235C23#47411yescharacteristic
236C23#50011yescharacteristic
237C23#50811yescharacteristic
238C23#50811yescharacteristic
239C23#50911yescharacteristic
240C23#51011yescharacteristic
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—161no—
284C2×C4—161no—
285C2×C4—161no—
286C2×C4—161no—
287C2×C4#39511yescharacteristic
288C2×C4#45111yescharacteristic
289C2×C4#45311yescharacteristic
290C2×C4#45611yescharacteristic
291C2×C4B(4,𝔽2)11yescharacteristic
292C2×C4#45811yescharacteristic
293C2×C4#47711yescharacteristic
294C2×C4#50111yescharacteristic
295C8—21no—
296C8—21no—
297C8—21no—
298C8—21no—
299C8—42no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22#159511yescharacteristic
313C22#161011yescharacteristic
314C22#205711yescharacteristic
315C22#218611yescharacteristic
316C22#218911yescharacteristic
317C22#219011yescharacteristic
318C22#219111yescharacteristic
319C22#222811yescharacteristic
320C22#222911yescharacteristic
321C22#225311yescharacteristic
322C22#225511yescharacteristic
323C4—21no—
324C4—21no—
325C4—84no—
326C4—81no—
327C4—81no—
328C4—161no—
329C4—161no—
330C4—161no—
331C4—162no—
332C4—161no—
333C4#160511yescharacteristic
334C4#161411yescharacteristic
335C4#206011yescharacteristic
336C4#206211yescharacteristic
337C2—21no—
338C2—21no—
339C2—21no—
340C2—21no—
341C2#1056711yescharacteristic
342C2#1210411yescharacteristic
343C2#1210711yescharacteristic
344C2#1229011yescharacteristic
345C2#1229311yescharacteristic
346C2#1345311yescharacteristic
347C2#1345411yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#302922
#129202
#129192
#113282
#72582
#71832