Finite Group Databasebeta

← #134439

Subgroups — #134439

Order 512.

478 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1348C212yesmaximal
4#10842C211yesmaximal, characteristic
5#7258C212yesmaximal
6C2×#467C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic, complemented
8C22×C4×C8C2211yescharacteristic
9C4×C22⋊Q8—21no—
10C2×#382—21no—
11C2×C4⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#1969C411yescharacteristic, complemented
14#1942C411yescharacteristic, complemented
15#1933—21no—
16#1892—21no—
17#1808C2211yescharacteristic
18C4×#107C411yescharacteristic, complemented
19C4×#103—21no—
20C2×#340—21no—
21C2×#340C2212yes—
22#1374—21no—
23#1348—24no—
24C22×Q16C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C23×C8C2311yescharacteristic
27C2×C22⋊Q8—21no—
28C2×C4×Q8—21no—
29C22×C4⋊C4—22no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#467—21no—
33#467—21no—
34#467D412yes—
35#467Q812yes—
36C4×Q16—42no—
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43C2×C4×C8—42no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#391—21no—
49#389—21no—
50#387—41no—
51#385—41no—
52#382—24no—
53#382—42no—
54#380—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4—41no—
57C4×C22⋊C4—41no—
58C2×#95—41no—
59C2×#95—41no—
60C4⋊C16—42no—
61C22⋊C16—22no—
62C22⋊C16—22no—
63#340—21no—
64#340—21no—
65#340—21no—
66#340—22no—
67#340—22no—
68#340—42no—
69C22×Q8—21no—
70C23×C4—42no—
71C23×C4D811yescharacteristic
72C23×C4D811yescharacteristic
73C23×C4C2×D411yescharacteristic
74C2×Q16—21no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16C4⋊C412yes—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22×C411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×Q811yescharacteristic
90C22×C8#4111yescharacteristic
91C22×C8#4111yescharacteristic
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C4×Q8—42no—
97C4×Q8—42no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42C22⋊C411yescharacteristic
117C2×C42C22⋊C411yescharacteristic
118C2×C42SD1611yescharacteristic
119C2×C42SD1611yescharacteristic
120C2×C16—42no—
121C2×C16—42no—
122C2×C16—42no—
123#107—21no—
124#107—41no—
125#107—41no—
126#107C4⋊C412yes—
127#103—22no—
128#103—22no—
129#103—42no—
130#103—42no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—41no—
134C4×C8—41no—
135#95—44no—
136#95—44no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95—41no—
142C24C2×D811yescharacteristic
143C2×Q8—22no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—44no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4#10211yescharacteristic
171C22×C4#10211yescharacteristic
172C22×C4#10211yescharacteristic
173C22×C4#10211yescharacteristic
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C2×SD1611yescharacteristic
178C22×C4AGL(1,ℤ/8)11yescharacteristic
179C22×C4#13711yescharacteristic
180Q16—42no—
181Q16—42no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8C2×C4⋊C411yescharacteristic
199C2×C8#11711yescharacteristic
200C2×C8#12111yescharacteristic
201C2×C8C22⋊Q811yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—82no—
219C42—22no—
220C42—22no—
221C42—21no—
222C42—21no—
223C42—82no—
224C42—82no—
225C42#9912yes—
226C42#10412yes—
227C16—82no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23C2×#10211yescharacteristic
233C23#45911yescharacteristic
234C23#47411yescharacteristic
235C23#50812yes—
236C23#51012yes—
237Q8—42no—
238Q8—42no—
239Q8—81no—
240Q8—81no—
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—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—162no—
279C2×C4—162no—
280C2×C4—162no—
281C2×C4—162no—
282C2×C4#32712yes—
283C2×C4#32712yes—
284C2×C4#38011yescharacteristic
285C2×C4C2×#9911yescharacteristic
286C2×C4#41011yescharacteristic
287C2×C4#41711yescharacteristic
288C2×C4C2×#10411yescharacteristic
289C2×C4#42111yescharacteristic
290C2×C4#46111yescharacteristic
291C2×C4#47611yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8—21no—
296C8—41no—
297C8—41no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22C2×#32711yescharacteristic
307C22#151711yescharacteristic
308C22#180211yescharacteristic
309C22#213811yescharacteristic
310C22#214111yescharacteristic
311C22#214511yescharacteristic
312C22#214511yescharacteristic
313C22#221112yes—
314C22#222912yes—
315C4—21no—
316C4—21no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—82no—
322C4—82no—
323C4—164no—
324C4—161no—
325C4—161no—
326C4#150911yescharacteristic
327C4#151311yescharacteristic
328C4#179911yescharacteristic
329C4#180611yescharacteristic
330C2—22no—
331C2—22no—
332C2#857111yescharacteristic
333C2#1170311yescharacteristic
334C2#1170611yescharacteristic
335C2#1185511yescharacteristic
336C2#1185911yescharacteristic
337C2#1281411yescharacteristic
338C2#1281711yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q16C4semidirect
13. #1969C4semidirect
14. #1942C4semidirect
18. C4×#107C4semidirect

Maximal subgroups

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

subgroupindex
C4×#4672
#127832
C2×#13482
#108422
#72582