Finite Group Databasebeta

← #499571

Subgroups — #499571

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#466C211yesmaximal, characteristic
2#20817C211yesmaximal, characteristic, complemented
3#13537C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13492C212yesmaximal, complemented
7#13480C211yesmaximal, characteristic, complemented
8#13380C212yesmaximal
9C4×#366C212yesmaximal, complemented
10C4×#357C212yesmaximal, complemented
11C4×C22⋊C16C211yesmaximal, characteristic, complemented
12C2×#466C2211yescharacteristic
13C2×C4×D8C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C4×#121—21no—
16#2285—21no—
17C2×#366C2212yes—
18C2×#366C2212yes—
19C2×#357C2212yes—
20C2×#357C2212yes—
21C2×C22⋊C16C2212yes—
22C2×C4×C16C2212yes—
23#1970C2211yescharacteristic
24#1942C2211yescharacteristic
25#1934—21no—
26#1893—21no—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1834—21no—
30#1808—21no—
31#1808C2211yescharacteristic
32#1808C2212yes—
33C4×#107—21no—
34C4×#107C2211yescharacteristic
35C4×#107C2212yes—
36C4×#102—21no—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×D8C2311yescharacteristic
44C23×C8D411yescharacteristic
45C23×C8C2311yescharacteristic
46C2×#121—21no—
47C2×C4×D4—21no—
48C2×C4×D4—41no—
49C2×#117—41no—
50C22×C42D411yescharacteristic
51C22×C16C2312yes—
52C22×C16C2312yes—
53#466—22no—
54#466—22no—
55C4×D8—24no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107C2312yes—
59C2×#107C2312yes—
60C2×#107C2311yescharacteristic
61C2×#107C2311yescharacteristic
62C2×#107C2311yescharacteristic
63C2×#102—21no—
64C2×#102—21no—
65C2×#102—21no—
66C2×C4×C8—24no—
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D411yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#390—21no—
72#389—21no—
73#388—41no—
74#386—41no—
75#384—21no—
76#384—21no—
77#382—41no—
78#380—22no—
79#380—21no—
80#380—41no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—41no—
84C4×C22⋊C4—41no—
85#366—24no—
86#366—24no—
87#357—24no—
88#357—24no—
89C22⋊C16—24no—
90C4×C16—24no—
91#340—22no—
92#340—22no—
93#340—21no—
94#340—21no—
95#340—21no—
96C22×D4—21no—
97C22×D4—41no—
98C23×C4—41no—
99C23×C4D811yescharacteristic
100C23×C4D811yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—21no—
105C2×D8#4112yes—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—41no—
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2411yescharacteristic
117#121—24no—
118#121—42no—
119C4×D4—42no—
120C4×D4—42no—
121C4×D4—84no—
122#117—84no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—42no—
142C2×C42—41no—
143C2×C42D811yescharacteristic
144C2×C42D811yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—22no—
152C2×C16#4114yes—
153#107—21no—
154#107—22no—
155#107—22no—
156#107—22no—
157#107—22no—
158#107—22no—
159#107—22no—
160#107—42no—
161#107—42no—
162#107#4112yes—
163#102—24no—
164#102—42no—
165#102—42no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8#4114yes—
170#95—41no—
171#95—41no—
172#95—42no—
173#95—41no—
174#95—42no—
175C24—41no—
176C24C2×D811yescharacteristic
177C2×D4—22no—
178C2×D4—44no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—82no—
185C2×D4—82no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4D1614yes—
206C22×C4C2×D811yescharacteristic
207C22×C4C2×D811yescharacteristic
208C22×C4C2×D811yescharacteristic
209C22×C4C2×D811yescharacteristic
210C22×C4C2×D811yescharacteristic
211C22×C4C22×D411yescharacteristic
212D8—24no—
213D8—42no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8#12312yes—
225C2×C8#13512yes—
226C2×C8C2×#4112yes—
227C2×C8C2×#4111yescharacteristic
228C2×C8C2×#4111yescharacteristic
229C2×C8C2×#4112yes—
230C4⋊C4—22no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C22⋊C4—44no—
246C22⋊C4—84no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—21no—
252C42—21no—
253C42—22no—
254C42—22no—
255C42—84no—
256C42—82no—
257C42#12312yes—
258C42#13712yes—
259C16—24no—
260C16—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—42no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23C2×D1612yes—
275C23C22×D811yescharacteristic
276D4—42no—
277D4—42no—
278D4—88no—
279D4—81no—
280D4—81no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—28no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—164no—
312C2×C4#48012yes—
313C2×C4#48012yes—
314C2×C4C2×D1612yes—
315C2×C4C2×D1612yes—
316C2×C4#50812yes—
317C2×C4#51012yes—
318C2×C4C2×#12311yescharacteristic
319C2×C4#52511yescharacteristic
320C2×C4#53211yescharacteristic
321C2×C4C2×#13511yescharacteristic
322C2×C4C2×#13711yescharacteristic
323C2×C4#57511yescharacteristic
324C8—22no—
325C8—22no—
326C8—41no—
327C8#52912yes—
328C8#52912yes—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—161no—
340C22—161no—
341C22—161no—
342C22—161no—
343C22C2×#48011yescharacteristic
344C22#310011yescharacteristic
345C22#316911yescharacteristic
346C22C22×D1611yescharacteristic
347C22C2×#50811yescharacteristic
348C22C2×#51011yescharacteristic
349C22#341411yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—82no—
355C4—161no—
356C4—162no—
357C4—161no—
358C4—162no—
359C4#223214yes—
360C4#239511yescharacteristic
361C4#309111yescharacteristic
362C4#310711yescharacteristic
363C4#317211yescharacteristic
364C2—42no—
365C2—162no—
366C2#2375411yescharacteristic
367C2C2×#223211yescharacteristic
368C2#3364511yescharacteristic
369C2#3366011yescharacteristic
370C2#3366711yescharacteristic
371C2#3373311yescharacteristic
372C2#3373811yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutator, Frattini, cosocleC22×C8C24
socleC23C22×D8
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20817C2semidirect
3. #13537C2semidirect
4. #13526C2semidirect
5. #13505C2semidirect
6. #13492C2semidirect
7. #13480C2semidirect
9. C4×#366C2semidirect
10. C4×#357C2semidirect
11. C4×C22⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4662
#208172
#135372
#135262
#135052
#134922
#134802
#133802
C4×#3662
C4×#3572
C4×C22⋊C162