Finite Group Databasebeta

← #134541

Subgroups — #134541

Order 512.

486 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#467C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3C2×#1348C212yesmaximal
4#10862C211yesmaximal, characteristic
5#7258C212yesmaximal
6C2×#467C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic, complemented
8C22×#107—21no—
9C22×C4×C8C2211yescharacteristic
10C4×C22⋊Q8—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×#340C2212yes—
21C4⋊#95—21no—
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×#107—24no—
38C2×#107—42no—
39C2×#107—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—
59C2×#95—41no—
60C2×#95—41no—
61C4⋊C16—42no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64#340—21no—
65#340—21no—
66#340—21no—
67#340—22no—
68#340—22no—
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—22no—
81C22×C8—22no—
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—41no—
113C2×C42—41no—
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—48no—
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—
141#95—82no—
142#95—82no—
143C24C2×D811yescharacteristic
144C2×Q8—22no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—44no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4#10211yescharacteristic
172C22×C4#10211yescharacteristic
173C22×C4#10211yescharacteristic
174C22×C4#10211yescharacteristic
175C22×C4SD3212yes—
176C22×C4Q3212yes—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182C22×C4#13711yescharacteristic
183Q16—42no—
184Q16—42no—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8#9912yes—
200C2×C8#10412yes—
201C2×C8C2×C4⋊C411yescharacteristic
202C2×C8#11711yescharacteristic
203C2×C8#12111yescharacteristic
204C2×C8C22⋊Q811yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—41no—
228C42—41no—
229C42—82no—
230C42—82no—
231C16—82no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23C2×#10211yescharacteristic
237C23#45911yescharacteristic
238C23#47411yescharacteristic
239C23C2×SD3211yescharacteristic
240C23C2×Q3211yescharacteristic
241C23#50911yescharacteristic
242C23#51011yescharacteristic
243Q8—42no—
244Q8—42no—
245Q8—81no—
246Q8—81no—
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—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—162no—
287C2×C4—162no—
288C2×C4—162no—
289C2×C4—162no—
290C2×C4#35812yes—
291C2×C4#35812yes—
292C2×C4#38011yescharacteristic
293C2×C4C2×#9911yescharacteristic
294C2×C4#41011yescharacteristic
295C2×C4#41711yescharacteristic
296C2×C4C2×#10411yescharacteristic
297C2×C4#42111yescharacteristic
298C2×C4#46111yescharacteristic
299C2×C4#47611yescharacteristic
300C8—21no—
301C8—21no—
302C8—21no—
303C8—21no—
304C8—22no—
305C8—22no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22#150211yescharacteristic
319C22#151911yescharacteristic
320C22#180211yescharacteristic
321C22C2×#35811yescharacteristic
322C22#213911yescharacteristic
323C22#214511yescharacteristic
324C22#214511yescharacteristic
325C22#220711yescharacteristic
326C22#220911yescharacteristic
327C22#222511yescharacteristic
328C22#222711yescharacteristic
329C4—22no—
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—82no—
336C4—82no—
337C4—164no—
338C4—161no—
339C4—161no—
340C4#150411yescharacteristic
341C4#151211yescharacteristic
342C4#179911yescharacteristic
343C4#180611yescharacteristic
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2#857711yescharacteristic
349C2#1174311yescharacteristic
350C2#1174611yescharacteristic
351C2#1188711yescharacteristic
352C2#1189011yescharacteristic
353C2#1281211yescharacteristic
354C2#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
#108622
#72582