Finite Group Databasebeta

← #502382

Subgroups — #502382

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×D4×C16C211yesmaximal, characteristic, complemented
3#28704C211yesmaximal, characteristic, complemented
4#28227C211yesmaximal, characteristic
5#13492C212yesmaximal, complemented
6#13380C212yesmaximal, complemented
7#13282C211yesmaximal, characteristic, complemented
8#13176C211yesmaximal, characteristic, complemented
9#13059C212yesmaximal, complemented
10#12834C212yesmaximal, complemented
11C4×#357C211yesmaximal, characteristic, complemented
12C23×C16C2212yes—
13C2×#500C2211yescharacteristic
14C2×#466C2212yes—
15C2×C4×D8C2211yescharacteristic
16C2×C8×D4C2211yescharacteristic
17#2547—21no—
18#2537—21no—
19#2224—24no—
20D4×C16—24no—
21C2×#366C2212yes—
22C2×#366C2212yes—
23C2×C4⋊C16C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2212yes—
26C2×#357C2211yescharacteristic
27C2×C22⋊C16C2212yes—
28C2×C4×C16C2211yescharacteristic
29#2075—22no—
30#2033—22no—
31#1988—21no—
32#1845C2211yescharacteristic
33#1808C2212yes—
34#1808C2212yes—
35#1808C2212yes—
36#1802—21no—
37C4×#107C2211yescharacteristic
38#1380C2212yes—
39#1380C2212yes—
40#1380C2211yescharacteristic
41#1380C2212yes—
42#1380C2211yescharacteristic
43C22×D8C2311yescharacteristic
44C23×C8C2312yes—
45C2×C4⋊Q8—21no—
46C2×#124—41no—
47C2×#121—22no—
48C2×#121—41no—
49C2×C4×D4—21no—
50C2×C4×D4D411yescharacteristic
51C22×C16—24no—
52C22×C16—24no—
53C22×C16—42no—
54C22×C16C2311yescharacteristic
55C22×C16C2312yes—
56C22×C16C2311yescharacteristic
57#500—22no—
58#500—22no—
59#466—24no—
60#466—42no—
61C4×D8—22no—
62C4×D8—22no—
63C8×D4—22no—
64C8×D4—22no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2312yes—
72C2×C4⋊C8D411yescharacteristic
73C2×#102—22no—
74C2×#102—21no—
75C2×C22⋊C8D412yes—
76C2×C4×C8C2311yescharacteristic
77#397—42no—
78#394—42no—
79#391—41no—
80#384—21no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—41no—
85C4×C4⋊C4—21no—
86#366—24no—
87#366—42no—
88C4⋊C16D414yescomplemented
89#357—22no—
90#357—24no—
91#357—22no—
92C22⋊C16—24no—
93C4×C16—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—21no—
98C22×Q8—41no—
99C22×D4—21no—
100C22×D4—42no—
101C22×D4D811yescharacteristic
102C23×C4—41no—
103C23×C4C2×D412yes—
104C2×D8—21no—
105C2×D8—21no—
106C2×D8—21no—
107C2×D8—21no—
108C2×D8—21no—
109C2×D8#4112yes—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8C2×D412yes—
117C22×C8C2×D411yescharacteristic
118C22×C8C2411yescharacteristic
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121#124—84no—
122#121—44no—
123#121—82no—
124#121—84no—
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—42no—
128C4×D4—42no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4D811yescharacteristic
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4D812yes—
144C2×C42—41no—
145C2×C42C2×D411yescharacteristic
146C2×C16—21no—
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—21no—
150C2×C16—21no—
151C2×C16—21no—
152C2×C16—21no—
153C2×C16—48no—
154C2×C16C2×D414yes—
155C2×C16#4112yes—
156#107—22no—
157#107—21no—
158#107—21no—
159#107—21no—
160#107—21no—
161#107—22no—
162#107—22no—
163#107—21no—
164#107—22no—
165#107—21no—
166#107—42no—
167#107—81no—
168C4⋊C8—22no—
169#102—44no—
170#102—42no—
171C22⋊C8—24no—
172C4×C8C2×D412yes—
173C4×C8#4112yes—
174#95—41no—
175#95—42no—
176#95—42no—
177#95—42no—
178C24—41no—
179C24C2×D812yes—
180C2×Q8—82no—
181C2×Q8—82no—
182C2×Q8—82no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—84no—
192C2×D4—84no—
193C2×D4—162no—
194C2×D4D1614yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4C2×D812yes—
210C22×C4C2×D811yescharacteristic
211C22×C4C2×D811yescharacteristic
212C22×C4C22×D411yescharacteristic
213D8—22no—
214D8—22no—
215D8—41no—
216D8—41no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8#12112yes—
227C2×C8C22×D411yescharacteristic
228C2×C8C2×#4111yescharacteristic
229C2×C8C2×#4111yescharacteristic
230C2×C8C2×#4111yescharacteristic
231C2×C8H(5,𝔽2)11yescharacteristic
232C2×C82−1+411yescharacteristic
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—161no—
247C22⋊C4—24no—
248C22⋊C4—84no—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C22⋊C4—82no—
253C42—82no—
254C42#12112yes—
255C42#13512yes—
256C16—22no—
257C16—24no—
258C16—41no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23C2×D1614yes—
273C23C22×D811yescharacteristic
274Q8—164no—
275D4—28no—
276D4—42no—
277D4—42no—
278D4—81no—
279D4—81no—
280D4—168no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—164no—
306C2×C4#46612yes—
307C2×C4#46612yes—
308C2×C4C2×D1612yes—
309C2×C4#50812yes—
310C2×C4C2×#12111yescharacteristic
311C2×C4#54611yescharacteristic
312C2×C4#54711yescharacteristic
313C2×C4C2×#13511yescharacteristic
314C2×C4#57711yescharacteristic
315C2×C4#57811yescharacteristic
316C8—22no—
317C8—42no—
318C8#52911yescharacteristic
319C8#54211yescharacteristic
320C8#54611yescharacteristic
321C8#54811yescharacteristic
322C22—24no—
323C22—24no—
324C22—24no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—82no—
330C22—82no—
331C22—82no—
332C22—161no—
333C22—161no—
334C22—161no—
335C22—161no—
336C22C2×#46611yescharacteristic
337C22#329411yescharacteristic
338C22#333411yescharacteristic
339C22C22×D1611yescharacteristic
340C22C2×#50811yescharacteristic
341C22#341611yescharacteristic
342C22#341711yescharacteristic
343C4—22no—
344C4—42no—
345C4—161no—
346C4—162no—
347C4—161no—
348C4—162no—
349C4—161no—
350C4#221514yes—
351C4#242711yescharacteristic
352C4#306311yescharacteristic
353C4#330211yescharacteristic
354C4#334011yescharacteristic
355C2—44no—
356C2—162no—
357C2#2533811yescharacteristic
358C2C2×#221511yescharacteristic
359C2#3361311yescharacteristic
360C2#3386111yescharacteristic
361C2#3386911yescharacteristic
362C2#3390111yescharacteristic
363C2#3390711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28227C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2224C2semidirect
2. C2×D4×C16C2semidirect
3. #28704C2semidirect
5. #13492C2semidirect
6. #13380C2semidirect
7. #13282C2semidirect
8. #13176C2semidirect
9. #13059C2semidirect
10. #12834C2semidirect
11. C4×#357C2semidirect
88. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×D4×C162
#287042
#282272
#134922
#133802
#132822
#131762
#130592
#128342
C4×#3572