Finite Group Databasebeta

← #280991

Subgroups — #280991

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28227C211yesmaximal, characteristic
2#25010C212yesmaximal, complemented
3C4⋊C4×D8C211yesmaximal, characteristic, complemented
4#10615C212yesmaximal, complemented
5#10430C212yesmaximal, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8882C212yesmaximal, complemented
9#8818C211yesmaximal, characteristic, complemented
10#8644C211yesmaximal, characteristic, complemented
11C8×#102C211yesmaximal, characteristic, complemented
12C2×#480—22no—
13C2×#466—21no—
14C2×C4×D8—21no—
15C2×C4×D8C2211yescharacteristic
16C2×C8×D4C2211yescharacteristic
17#2537C2212yes—
18D4×C4⋊C4C2212yes—
19#2075—22no—
20#2033—22no—
21#2033—22no—
22#1954C2212yes—
23#1947C2212yes—
24#1942—21no—
25#1862—22no—
26#1852—22no—
27#1845C2211yescharacteristic
28#1845C2212yes—
29#1845C2211yescharacteristic
30#1845C2212yes—
31#1808—21no—
32#1802C2212yes—
33#1802C2212yes—
34#1802C2212yes—
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×D4—22no—
50C2×C4×D4C2312yes—
51C2×C4×D4C2311yescharacteristic
52C22×C4⋊C4—22no—
53#480—48no—
54#466—44no—
55C4×D8—22no—
56C4×D8—22no—
57C4×D8—44no—
58C8×D4—22no—
59C8×D4—22no—
60C2×#107—22no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—42no—
68C2×#107D411yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102D412yes—
76C2×#102D411yescharacteristic
77C2×#102D411yescharacteristic
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#394—42no—
86#389—22no—
87#385—42no—
88#384C2312yes—
89#384C2312yes—
90#380—22no—
91#380—22no—
92#380—22no—
93C4×C4⋊C4C2311yescharacteristic
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340D411yescharacteristic
99#340D411yescharacteristic
100#336—21no—
101#335—22no—
102D4⋊C8—22no—
103D4⋊C8—24no—
104C82—22no—
105C22×D4—42no—
106C22×D4C2×D412yes—
107C22×D4C2×D411yescharacteristic
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—42no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8D814yes—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125#121—44no—
126#121—44no—
127#121—82no—
128#121—82no—
129C4×D4—21no—
130C4×D4—22no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—21no—
137C4×D4—44no—
138C4×D4—82no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C22⋊C4—22no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C42—21no—
158C2×C42C2411yescharacteristic
159#107—22no—
160#107—42no—
161#107—42no—
162#107—44no—
163#107—44no—
164#107—82no—
165#107—81no—
166#107—81no—
167#107—84no—
168C4⋊C8—22no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174#102—24no—
175#102—22no—
176#102—44no—
177#102—44no—
178#102—44no—
179#102D814yescomplemented
180C22⋊C8—42no—
181C22⋊C8—44no—
182C22⋊C8—44no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—42no—
187C4×C8#4112yes—
188C4×C8#4112yes—
189C4×C8#4112yes—
190#95—21no—
191#95—42no—
192#95—42no—
193C24—22no—
194C24—21no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—84no—
206C2×D4—84no—
207C2×D4—162no—
208C2×D4C2×D812yes—
209C2×D4C2×D814yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—162no—
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4H(5,𝔽2)11yescharacteristic
234D8—28no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—22no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—84no—
254C2×C8#12312yes—
255C2×C8#12312yes—
256C2×C8C2×D812yes—
257C2×C8C2×D812yes—
258C2×C8#13512yes—
259C2×C8#13712yes—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—84no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4C2×D812yes—
279C22⋊C4—42no—
280C22⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—84no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C42—41no—
287C42—41no—
288C42C2×#4111yescharacteristic
289C42C2×#4111yescharacteristic
290C42C2×#4111yescharacteristic
291C422−1+411yescharacteristic
292C23—24no—
293C23—22no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—42no—
301C23—41no—
302C23—42no—
303C23—41no—
304C23—42no—
305C23—41no—
306C23D4211yescharacteristic
307D4—24no—
308D4—22no—
309D4—24no—
310D4—22no—
311D4—41no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—168no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—168no—
345C2×C4C2×#12311yescharacteristic
346C2×C4#53211yescharacteristic
347C2×C4#54211yescharacteristic
348C2×C4#54711yescharacteristic
349C2×C4#54711yescharacteristic
350C2×C4#55311yescharacteristic
351C2×C4C22×D811yescharacteristic
352C2×C4C22×D811yescharacteristic
353C2×C4C2×#13511yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4#57611yescharacteristic
356C2×C4#57611yescharacteristic
357C2×C4#57811yescharacteristic
358C2×C4#57811yescharacteristic
359C8—24no—
360C8—22no—
361C8—22no—
362C8—22no—
363C8—81no—
364C8—81no—
365C8—82no—
366C8#48014yescomplemented
367C22—44no—
368C22—42no—
369C22—44no—
370C22—42no—
371C22—44no—
372C22—42no—
373C22—81no—
374C22—82no—
375C22—81no—
376C22—82no—
377C22—82no—
378C22—81no—
379C22—81no—
380C22—82no—
381C22#242211yescharacteristic
382C22#327611yescharacteristic
383C22D4×D811yescharacteristic
384C22D4×D811yescharacteristic
385C22#328311yescharacteristic
386C22#329211yescharacteristic
387C22#329311yescharacteristic
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—82no—
394C4—162no—
395C4—162no—
396C4C2×#48011yescharacteristic
397C4#309811yescharacteristic
398C4#316911yescharacteristic
399C4#317511yescharacteristic
400C4#333411yescharacteristic
401C4#333411yescharacteristic
402C4#333911yescharacteristic
403C4#335711yescharacteristic
404C2—84no—
405C2—82no—
406C2#1988711yescharacteristic
407C2#1995711yescharacteristic
408C2D8211yescharacteristic
409C2#1997111yescharacteristic
410C2#2501011yescharacteristic
411C2#2503411yescharacteristic
412C2#2505311yescharacteristic
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. #25010C2semidirect
3. C4⋊C4×D8C2semidirect
4. #10615C2semidirect
5. #10430C2semidirect
6. #9657C2semidirect
7. #9614C2semidirect
8. #8882C2semidirect
9. #8818C2semidirect
10. #8644C2semidirect
11. C8×#102C2semidirect
179. #102D8semidirect
366. C8#480semidirect

Maximal subgroups

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

subgroupindex
#282272
#250102
C4⋊C4×D82
#106152
#104302
#96572
#96142
#88822
#88182
#86442
C8×#1022