Finite Group Databasebeta

← #134543

Subgroups — #134543

Order 512.

442 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C4×#467C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C2×#1350C212yesmaximal
4#10864C211yesmaximal, characteristic
5#7259C212yesmaximal
6C2×#467C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic
8C22×#108—21no—
9C22×C4×C8C2211yescharacteristic
10C4×C22⋊Q8—21no—
11C2×C4⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#1969C411yescharacteristic
14#1942C411yescharacteristic
15#1933—21no—
16#1892—21no—
17#1808C2211yescharacteristic
18C4×#107C411yescharacteristic
19C4×#103—21no—
20C2×#341C2212yes—
21#1728—21no—
22#1374—21no—
23#1350—24no—
24C22×Q16C2×C411yescharacteristic
25C22×M16—22no—
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C22⋊Q8—21no—
29C2×C4×Q8—21no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#467—21no—
33#467—21no—
34#467D412yes—
35#467Q812yes—
36C4×Q16—42no—
37C2×#108—24no—
38C2×#108—42no—
39C2×#108—42no—
40C2×#107C2×C411yescharacteristic
41C2×#107C2×C411yescharacteristic
42C2×#107C2×C411yescharacteristic
43C2×#103—21no—
44C2×#103—21no—
45C2×#103—21no—
46C2×C4×C8—42no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#391—21no—
52#389—21no—
53#387—41no—
54#385—41no—
55#380—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—41no—
58C4×C22⋊C4—41no—
59C4⋊C16—42no—
60C22⋊C16—22no—
61C22⋊C16—22no—
62#341—22no—
63#341—22no—
64#340—21no—
65#340—21no—
66#340—21no—
67#336—41no—
68#336—41no—
69C22×Q8—21no—
70C23×C4SD1611yescharacteristic
71C23×C4SD1611yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×Q16—21no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×Q16C4⋊C412yes—
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—42no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22×C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×Q811yescharacteristic
96C22×C8#4111yescharacteristic
97C22×C8#4111yescharacteristic
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100C22⋊Q8—41no—
101C22⋊Q8—41no—
102C4×Q8—42no—
103C4×Q8—42no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42C22⋊C411yescharacteristic
117C2×C42C22⋊C411yescharacteristic
118C2×C42D811yescharacteristic
119C2×C42D811yescharacteristic
120C2×C16—42no—
121C2×C16—42no—
122C2×C16—42no—
123#108—48no—
124#107—21no—
125#107—41no—
126#107—41no—
127#107C4⋊C412yes—
128#103—22no—
129#103—22no—
130#103—42no—
131#103—42no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—41no—
135C4×C8—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141C24C2×SD1611yescharacteristic
142C2×Q8—22no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4#10211yescharacteristic
163C22×C4#10211yescharacteristic
164C22×C4#10211yescharacteristic
165C22×C4#10211yescharacteristic
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4C2×D811yescharacteristic
170C22×C4AGL(1,ℤ/8)11yescharacteristic
171C22×C4#13711yescharacteristic
172Q16—42no—
173Q16—42no—
174M16—44no—
175M16—44no—
176M16—82no—
177M16—82no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8#9912yes—
196C2×C8#10412yes—
197C2×C8C2×C4⋊C411yescharacteristic
198C2×C8#11711yescharacteristic
199C2×C8#12111yescharacteristic
200C2×C8C22⋊Q811yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—41no—
219C42—41no—
220C42—82no—
221C42—82no—
222C16—82no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23C2×#10211yescharacteristic
230C23#46111yescharacteristic
231C23#47611yescharacteristic
232Q8—42no—
233Q8—42no—
234Q8—81no—
235Q8—81no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
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—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4#35912yes—
271C2×C4#36212yes—
272C2×C4#38011yescharacteristic
273C2×C4C2×#9911yescharacteristic
274C2×C4#41011yescharacteristic
275C2×C4#41711yescharacteristic
276C2×C4C2×#10411yescharacteristic
277C2×C4#42111yescharacteristic
278C2×C4#45911yescharacteristic
279C2×C4#47411yescharacteristic
280C8—21no—
281C8—21no—
282C8—21no—
283C8—21no—
284C8—22no—
285C8—22no—
286C8—84no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22#150211yescharacteristic
298C22#151911yescharacteristic
299C22#180211yescharacteristic
300C22C2×#35911yescharacteristic
301C22#214311yescharacteristic
302C22C2×#36211yescharacteristic
303C22#214811yescharacteristic
304C4—22no—
305C4—22no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—82no—
311C4—82no—
312C4—161no—
313C4—161no—
314C4#150411yescharacteristic
315C4#151211yescharacteristic
316C4#179911yescharacteristic
317C4#180611yescharacteristic
318C2—41no—
319C2—41no—
320C2#857711yescharacteristic
321C2#1174811yescharacteristic
322C2#1175011yescharacteristic
323C2#1189311yescharacteristic
324C2#1189611yescharacteristic
325C2#1281511yescharacteristic
326C2#1281911yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ 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
C4×#4672
#127832
C2×#13502
#108642
#72592