Finite Group Databasebeta

← #281009

Subgroups — #281009

Order 512.

682 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28227C211yesmaximal, characteristic
2#25013C212yesmaximal, complemented
3#22807C211yesmaximal, characteristic, complemented
4#10619C212yesmaximal, complemented
5#10431C212yesmaximal, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8897C212yesmaximal, complemented
9#8814C211yesmaximal, characteristic, complemented
10#8644C211yesmaximal, characteristic, complemented
11C8×#102C211yesmaximal, characteristic, complemented
12C2×#482—22no—
13C2×#466—21no—
14C2×C4×D8C2211yescharacteristic
15C2×C8×D4C2211yescharacteristic
16#2537C2212yes—
17#2354C2212yes—
18#2077—22no—
19#2034—22no—
20#2033—22no—
21#1970—21no—
22#1954C2212yes—
23#1947C2212yes—
24#1925—22no—
25#1854—22no—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1844C2212yes—
29#1834C2212yes—
30#1808—21no—
31#1802C2212yes—
32#1802C2212yes—
33#1802C2212yes—
34C4×#107—21no—
35C8×C4⋊C4C2211yescharacteristic
36C4×#102C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×D4⋊C8C2211yescharacteristic
39C2×D4⋊C8C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1277C2211yescharacteristic
45C22×D8D411yescharacteristic
46C23×C8—21no—
47C2×#121—22no—
48C2×#121—22no—
49C2×C4×D4C2312yes—
50C2×C4×D4C2311yescharacteristic
51C2×#117—22no—
52#482—48no—
53#466—44no—
54C4×D8—22no—
55C4×D8—22no—
56C8×D4—22no—
57C8×D4—22no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—22no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107D411yescharacteristic
64C2×#106—22no—
65C2×#106—22no—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102D412yes—
73C2×#102D411yescharacteristic
74C2×#102D411yescharacteristic
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82#394—42no—
83#390—22no—
84#388—42no—
85#384C2312yes—
86#384C2312yes—
87#380—22no—
88#380—22no—
89#380—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2311yescharacteristic
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—42no—
97#340D411yescharacteristic
98#340D411yescharacteristic
99#336—21no—
100#335—22no—
101D4⋊C8—22no—
102D4⋊C8—24no—
103C82—22no—
104C22×D4—42no—
105C22×D4C2×D412yes—
106C22×D4C2×D411yescharacteristic
107C23×C4—22no—
108C23×C4—21no—
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—22no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123#121—44no—
124#121—44no—
125#121—82no—
126#121—82no—
127C4×D4—21no—
128C4×D4—22no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—21no—
134C4×D4—22no—
135#117—44no—
136#117—82no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D411yescharacteristic
146C2×C4⋊C4C2×D411yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42C2411yescharacteristic
156#107—22no—
157#107—42no—
158#107—42no—
159#107—44no—
160#107—81no—
161#107—81no—
162#106—44no—
163#106—82no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170#102—24no—
171#102—22no—
172#102—44no—
173#102—44no—
174#102—82no—
175#102D814yescomplemented
176C22⋊C8—42no—
177C22⋊C8—44no—
178C22⋊C8—44no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—42no—
183C4×C8#4112yes—
184C4×C8#4112yes—
185C4×C8#4112yes—
186#95—21no—
187#95—42no—
188#95—42no—
189C24—21no—
190C24—22no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—21no—
202C2×D4—84no—
203C2×D4—84no—
204C2×D4—162no—
205C2×D4C2×D812yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4H(5,𝔽2)11yescharacteristic
227D8—44no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—22no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—84no—
247C2×C8#12312yes—
248C2×C8#12312yes—
249C2×C8C2×D812yes—
250C2×C8#13512yes—
251C2×C8#13512yes—
252C2×C8AGL(1,ℤ/8)12yes—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4C2×D812yes—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—84no—
276C22⋊C4—84no—
277C22⋊C4—84no—
278C22⋊C4—162no—
279C42—41no—
280C42—41no—
281C42—84no—
282C42C2×#4111yescharacteristic
283C42C2×#4111yescharacteristic
284C42C2×#4111yescharacteristic
285C422−1+411yescharacteristic
286C23—22no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23D4211yescharacteristic
301D4—22no—
302D4—22no—
303D4—41no—
304D4—42no—
305D4—42no—
306D4—42no—
307D4—42no—
308D4—41no—
309D4—168no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4C2×#12311yescharacteristic
339C2×C4#53211yescharacteristic
340C2×C4#54211yescharacteristic
341C2×C4#54711yescharacteristic
342C2×C4#54711yescharacteristic
343C2×C4#55311yescharacteristic
344C2×C4C22×D811yescharacteristic
345C2×C4C2×#13511yescharacteristic
346C2×C4C2×#13511yescharacteristic
347C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C4#57611yescharacteristic
349C2×C4#57711yescharacteristic
350C2×C4#57711yescharacteristic
351C2×C4#57811yescharacteristic
352C8—22no—
353C8—22no—
354C8—22no—
355C8—42no—
356C8—81no—
357C8—81no—
358C8—82no—
359C8#48214yescomplemented
360C22—42no—
361C22—42no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22#242211yescharacteristic
375C22#327211yescharacteristic
376C22D4×D811yescharacteristic
377C22#328311yescharacteristic
378C22#328411yescharacteristic
379C22#329011yescharacteristic
380C22#329011yescharacteristic
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4—81no—
387C4—162no—
388C4—162no—
389C4C2×#48211yescharacteristic
390C4#309911yescharacteristic
391C4#316911yescharacteristic
392C4#317511yescharacteristic
393C4#333411yescharacteristic
394C4#333911yescharacteristic
395C4#334011yescharacteristic
396C4#335411yescharacteristic
397C2—82no—
398C2—162no—
399C2#1988311yescharacteristic
400C2#1995311yescharacteristic
401C2#1997211yescharacteristic
402C2#2001611yescharacteristic
403C2#2501311yescharacteristic
404C2#2503411yescharacteristic
405C2#2505211yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#28227C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #25013C2semidirect
3. #22807C2semidirect
4. #10619C2semidirect
5. #10431C2semidirect
6. #9657C2semidirect
7. #9614C2semidirect
8. #8897C2semidirect
9. #8814C2semidirect
10. #8644C2semidirect
11. C8×#102C2semidirect
175. #102D8semidirect
359. C8#482semidirect

Maximal subgroups

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

subgroupindex
#282272
#250132
#228072
#106192
#104312
#96572
#96142
#88972
#88142
#86442
C8×#1022