Finite Group Databasebeta

← #502351

Subgroups — #502351

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2240C211yesmaximal, characteristic, complemented
2C2×#2206C211yesmaximal, characteristic
3#28527C211yesmaximal, characteristic, complemented
4#23185C211yesmaximal, characteristic, complemented
5#13480C212yesmaximal, complemented
6#13448C212yesmaximal, complemented
7#13275C211yesmaximal, characteristic, complemented
8#13159C211yesmaximal, characteristic, complemented
9#13012C212yesmaximal, complemented
10#12811C212yesmaximal, complemented
11C4×#357C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C2×#500C2211yescharacteristic
14C2×#493C2211yescharacteristic
15C2×#474—21no—
16C2×C4×D8C2211yescharacteristic
17C2×C4×D8C2212yes—
18#2542—21no—
19#2366—21no—
20#2240—24no—
21#2206—24no—
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2212yes—
29C2×C4×C16C2211yescharacteristic
30#2093—22no—
31#1982—21no—
32#1970C2212yes—
33#1970C2212yes—
34#1879—22no—
35#1845C2212yes—
36C4×#107C2211yescharacteristic
37C8×C4⋊C4C2211yescharacteristic
38#1380C2211yescharacteristic
39#1380C2212yes—
40#1380C2211yescharacteristic
41#1380C2212yes—
42#1374C2212yes—
43C22×D8—21no—
44C22×D8C2312yes—
45C22×D8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#127—21no—
48C2×C22⋊Q8—41no—
49C2×C22⋊Q8—41no—
50C2×C4×D4—21no—
51C2×C4×D4—22no—
52C2×C4×D4—41no—
53C2×D16—24no—
54C2×D16—24no—
55C2×D16—42no—
56C22×C16C2311yescharacteristic
57C22×C16C2311yescharacteristic
58C22×C16C2312yes—
59#500—22no—
60#500—22no—
61#493—22no—
62#493D414yes—
63#474—44no—
64C4×D8—24no—
65C4×D8—22no—
66C4×D8—22no—
67C4×D8—42no—
68C2×#107—21no—
69C2×#107—22no—
70C2×#107C2311yescharacteristic
71C2×#107C2312yes—
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×#102—21no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×C4×C8C2312yes—
80C2×C4×C8C2311yescharacteristic
81#399—42no—
82#390—22no—
83#390—22no—
84#390—41no—
85#386—42no—
86#384—22no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D411yescharacteristic
89C4⋊C16—22no—
90C4⋊C16—42no—
91#357—24no—
92#357—24no—
93#357—24no—
94#357—22no—
95#357—22no—
96C4×C16—22no—
97#340—21no—
98#336D412yes—
99C22×Q8—41no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—41no—
103C22×D4—41no—
104C23×C4—41no—
105C23×C4—42no—
106C2×D8—24no—
107C2×D8—21no—
108C2×D8—21no—
109C2×D8—21no—
110C2×D8—22no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8—21no—
115C2×D8—22no—
116C2×D8—21no—
117C2×D8—81no—
118C2×D8—81no—
119C2×D8C2×D414yes—
120C2×D8#4112yes—
121C22×C8C2×D412yes—
122C22×C8C2×D411yescharacteristic
123C22×C8C2411yescharacteristic
124C4⋊Q8—42no—
125C4⋊Q8—42no—
126#127—24no—
127#127—42no—
128C22⋊Q8—84no—
129C22⋊Q8—84no—
130C4×D4—44no—
131C4×D4—42no—
132C4×D4—42no—
133C4×D4—82no—
134C4×D4—84no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4D811yescharacteristic
143C2×C4⋊C4D811yescharacteristic
144C2×C22⋊C4—42no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C42—41no—
151C2×C42C2×D412yes—
152C2×C42C2×D411yescharacteristic
153D16—48no—
154C2×C16—21no—
155C2×C16—21no—
156C2×C16—22no—
157C2×C16—21no—
158C2×C16—21no—
159C2×C16—22no—
160C2×C16—21no—
161C2×C16—22no—
162C2×C16—21no—
163#107—21no—
164#107—21no—
165#107—22no—
166#107—21no—
167#107—21no—
168#107—22no—
169#107—21no—
170#107—44no—
171#107—42no—
172#107#4112yes—
173C4⋊C8—22no—
174#102—44no—
175#102—44no—
176#102—42no—
177#102—82no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8C2×D412yes—
181C4×C8#4112yes—
182#95—41no—
183#95D812yes—
184C24—41no—
185C24—42no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×Q8—82no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—161no—
206C2×D4—161no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—84no—
212C22×C4—82no—
213C22×C4—84no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—162no—
217C22×C4C2×D812yes—
218C22×C4C2×D811yescharacteristic
219C22×C4C2×D812yes—
220C22×C4C2×D811yescharacteristic
221C22×C4C22×D411yescharacteristic
222D8—22no—
223D8—22no—
224D8—24no—
225D8—24no—
226D8—42no—
227D8—41no—
228D8—42no—
229D8—41no—
230D8—84no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8#12112yes—
237C2×C8C22×D411yescharacteristic
238C2×C8C2×#4111yescharacteristic
239C2×C8C2×#4111yescharacteristic
240C2×C8C2×#4111yescharacteristic
241C2×C8H(5,𝔽2)11yescharacteristic
242C2×C82−1+411yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—84no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C4⋊C4D1614yescomplemented
256C22⋊C4—84no—
257C22⋊C4—84no—
258C22⋊C4—82no—
259C22⋊C4—84no—
260C22⋊C4—84no—
261C22⋊C4—162no—
262C42—22no—
263C42—22no—
264C42—82no—
265C42#12112yes—
266C42AGL(1,ℤ/8)12yes—
267C16—41no—
268C16—42no—
269C16—41no—
270C23—42no—
271C23—44no—
272C23—82no—
273C23—81no—
274C23—82no—
275C23—81no—
276C23—82no—
277C23—82no—
278C23—82no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—82no—
283C23—81no—
284C23C22×D811yescharacteristic
285Q8—164no—
286D4—42no—
287D4—42no—
288D4—44no—
289D4—44no—
290D4—82no—
291D4—81no—
292D4—82no—
293D4—88no—
294D4—81no—
295D4—164no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—168no—
316C2×C4—164no—
317C2×C4#45912yes—
318C2×C4#45912yes—
319C2×C4C2×D1614yes—
320C2×C4C2×D1612yes—
321C2×C4#50912yes—
322C2×C4C2×#12111yescharacteristic
323C2×C4#54611yescharacteristic
324C2×C4#54811yescharacteristic
325C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
326C2×C4#57711yescharacteristic
327C2×C4#57711yescharacteristic
328C8—22no—
329C8—42no—
330C8#52911yescharacteristic
331C8#54211yescharacteristic
332C8#54611yescharacteristic
333C8#54711yescharacteristic
334C22—84no—
335C22—84no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—84no—
340C22—161no—
341C22—161no—
342C22—162no—
343C22—161no—
344C22—161no—
345C22—162no—
346C22—162no—
347C22—162no—
348C22C2×#45911yescharacteristic
349C22#329411yescharacteristic
350C22#335811yescharacteristic
351C22C22×D1611yescharacteristic
352C22C2×#50911yescharacteristic
353C22#341611yescharacteristic
354C22#341611yescharacteristic
355C4—22no—
356C4—24no—
357C4—42no—
358C4—82no—
359C4—161no—
360C4—162no—
361C4—161no—
362C4#220614yes—
363C4#242711yescharacteristic
364C4#304211yescharacteristic
365C4#333211yescharacteristic
366C4#337311yescharacteristic
367C2—164no—
368C2—162no—
369C2#2531811yescharacteristic
370C2C2×#220611yescharacteristic
371C2#3358611yescharacteristic
372C2#3386111yescharacteristic
373C2#3389911yescharacteristic
374C2#3392511yescharacteristic
375C2#3394011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#2206C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2240C2semidirect
3. #28527C2semidirect
4. #23185C2semidirect
5. #13480C2semidirect
6. #13448C2semidirect
7. #13275C2semidirect
8. #13159C2semidirect
9. #13012C2semidirect
10. #12811C2semidirect
11. C4×#357C2semidirect
255. C4⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#22402
C2×#22062
#285272
#231852
#134802
#134482
#132752
#131592
#130122
#128112
C4×#3572