Finite Group Databasebeta

← #507428

Subgroups — #507428

Order 512.

528 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33208C211yesmaximal, characteristic, complemented
2#21544C211yesmaximal, characteristic
3#13542C211yesmaximal, characteristic, complemented
4#13534C211yesmaximal, characteristic, complemented
5#13458C211yesmaximal, characteristic, complemented
6#13372C211yesmaximal, characteristic, complemented
7#13116C212yesmaximal, complemented
8#13065C212yesmaximal
9#12950C212yesmaximal, complemented
10#12831C212yesmaximal, complemented
11C2×#1376C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#499C2211yescharacteristic
14#3050C2211yescharacteristic
15C2×#443C2211yescharacteristic
16#2929—21no—
17C22×C4×C8C2211yescharacteristic
18#2851—21no—
19C2×#365C2212yes—
20C2×#365C2212yes—
21C2×#360C2212yes—
22C2×#360C2212yes—
23#1991—21no—
24#1974C2211yescharacteristic
25#1945C2211yescharacteristic
26#1942—21no—
27#1941C2212yes—
28#1937—21no—
29#1896—21no—
30#1846C2211yescharacteristic
31#1810C2211yescharacteristic
32#1808—21no—
33#1807C2212yes—
34#1772C2211yescharacteristic
35#1764—21no—
36#1386C2212yes—
37#1386C2212yes—
38#1383C2212yes—
39#1383C2212yes—
40#1376C2212yes—
41#1376C2211yescharacteristic
42#1376C2211yescharacteristic
43C2×#135C2311yescharacteristic
44C2×#131—21no—
45C23×C8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#125—21no—
48#525—21no—
49C22×C42D411yescharacteristic
50C2×M32—22no—
51C2×M32—24no—
52C2×M32—42no—
53C2×M32C2312yes—
54C2×M32C2312yes—
55C22×C16—22no—
56#499—42no—
57#467Q811yescharacteristic
58#467Q811yescharacteristic
59#466Q811yescharacteristic
60#466Q811yescharacteristic
61#465—21no—
62#465—21no—
63#443—42no—
64#435—41no—
65#435—41no—
66C8×D4—41no—
67C8×D4—41no—
68C2×#108C2311yescharacteristic
69C2×#108C2311yescharacteristic
70#427C2311yescharacteristic
71C2×#107—21no—
72C2×#107—21no—
73C2×#106C2312yes—
74C2×#106C2312yes—
75#424—21no—
76C2×#104—21no—
77C2×#104—21no—
78#416—21no—
79#408—21no—
80#407—21no—
81#405—21no—
82C2×C4×C8—42no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#389—22no—
88#387—42no—
89#385—42no—
90#382—21no—
91#380—22no—
92#365—24no—
93#365—42no—
94#360—24no—
95#360—42no—
96#341—21no—
97#340—22no—
98#340—22no—
99#339—21no—
100#339—21no—
101C2×#41—21no—
102C23×C4—21no—
103C23×C4C2×D411yescharacteristic
104#135—22no—
105#135—22no—
106C2×Q16C2×Q811yescharacteristic
107C2×SD16—21no—
108C2×D8C2×Q811yescharacteristic
109#131—42no—
110#131—42no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—41no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8SD1611yescharacteristic
120C22×C8SD1611yescharacteristic
121C22×C8SD1611yescharacteristic
122C22×C8SD1611yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8C2×Q811yescharacteristic
127C22×C8C2×Q811yescharacteristic
128C22×C8#4111yescharacteristic
129C22×C8#4111yescharacteristic
130C22×C8C2411yescharacteristic
131#125—42no—
132#125—42no—
133#123—41no—
134C22⋊Q8—21no—
135C22⋊Q8—21no—
136#121—21no—
137#121—21no—
138#120—41no—
139C4×D4—41no—
140C4×D4—41no—
141#117—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C42—22no—
148C2×C42—21no—
149C2×C42—41no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152M32—22no—
153M32—22no—
154M32—22no—
155M32—44no—
156M32#4114yes—
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—44no—
160#108—22no—
161#108—22no—
162#107—42no—
163#107—42no—
164#107C2×Q811yescharacteristic
165#107C2×Q811yescharacteristic
166#106—21no—
167#106—22no—
168#106—22no—
169#106—22no—
170#106—22no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173#104—42no—
174#104—42no—
175#103—21no—
176#103—21no—
177#102—21no—
178#102—21no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183#97—41no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189#95—42no—
190#95—42no—
191#95—42no—
192C24AGL(1,ℤ/8)11yescharacteristic
193#41—41no—
194#41—41no—
195#41—41no—
196#41—41no—
197C2×Q8—21no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C22⋊Q811yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C2×SD1611yescharacteristic
217C22×C4C2×SD1611yescharacteristic
218C22×C4AGL(1,ℤ/8)11yescharacteristic
219C22×C4C22×D411yescharacteristic
220Q16—22no—
221SD16—42no—
222D8—22no—
223M16—41no—
224M16—41no—
225M16—41no—
226M16—41no—
227M16—82no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—82no—
245C2×C8C22⋊Q811yescharacteristic
246C2×C8C22⋊Q811yescharacteristic
247C2×C8C2×SD1611yescharacteristic
248C2×C8C2×SD1611yescharacteristic
249C2×C8C2×SD1611yescharacteristic
250C2×C8C2×SD1611yescharacteristic
251C2×C8C22×Q811yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8C2×#4112yes—
254C2×C8H(5,𝔽2)12yes—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—84no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—41no—
276C42#13512yes—
277C16—24no—
278C16—42no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23#47611yescharacteristic
285C23#47611yescharacteristic
286C23C2×AGL(1,ℤ/8)11yescharacteristic
287Q8—41no—
288Q8—41no—
289D4—41no—
290D4—41no—
291D4—82no—
292D4—81no—
293D4—81no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—48no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#47611yescharacteristic
324C2×C4#47611yescharacteristic
325C2×C4C2×C22⋊Q811yescharacteristic
326C2×C4#53211yescharacteristic
327C2×C4#53411yescharacteristic
328C2×C4C22×SD1611yescharacteristic
329C2×C4C2×#13511yescharacteristic
330C2×C4#57511yescharacteristic
331C8—21no—
332C8—21no—
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C8—81no—
338C8—81no—
339C8#55112yes—
340C8#55412yes—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22#223112yes—
354C22C2×#47611yescharacteristic
355C22#318011yescharacteristic
356C22#318811yescharacteristic
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—162no—
365C4—162no—
366C4#256411yescharacteristic
367C4#307611yescharacteristic
368C4#316711yescharacteristic
369C4#318711yescharacteristic
370C2—21no—
371C2—21no—
372C2—22no—
373C2—41no—
374C2—161no—
375C2#2915611yescharacteristic
376C2C2×#223111yescharacteristic
377C2#3363711yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33208C2semidirect
3. #13542C2semidirect
4. #13534C2semidirect
5. #13458C2semidirect
6. #13372C2semidirect
7. #13116C2semidirect
9. #12950C2semidirect
10. #12831C2semidirect
11. C2×#1376C2semidirect

Maximal subgroups

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

subgroupindex
#332082
#215442
#135422
#135342
#134582
#133722
#131162
#130652
#129502
#128312
C2×#13762