Finite Group Databasebeta

← #287196

Subgroups — #287196

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29139C211yesmaximal, characteristic, complemented
2#28509C211yesmaximal, characteristic, complemented
3#28251C211yesmaximal, characteristic
4#23173C211yesmaximal, characteristic, complemented
5#10619C212yesmaximal, complemented
6#10473C212yesmaximal, complemented
7#9898C211yesmaximal, characteristic, complemented
8#9602C211yesmaximal, characteristic, complemented
9#9221C212yesmaximal, complemented
10#8642C212yesmaximal, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#476—21no—
13C2×#476—21no—
14C2×#459—21no—
15C2×#459C2212yes—
16C2×C8×D4—21no—
17C2×C8×D4C2212yes—
18#2868C2211yescharacteristic
19C4×#127C2211yescharacteristic
20C4×#127C2211yescharacteristic
21#2094—22no—
22#2054—22no—
23#2023—22no—
24#1957—21no—
25#1957C2212yes—
26#1954—21no—
27#1954C2212yes—
28#1869—22no—
29#1844C2212yes—
30#1842C2211yescharacteristic
31#1834—21no—
32#1802—21no—
33#1802C2212yes—
34#1802C2212yes—
35#1802C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#331C2211yescharacteristic
41C2×#331C2211yescharacteristic
42#1296C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1277C2212yes—
47C22×D8—22no—
48C23×C8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×#127D411yescharacteristic
53C2×#127C2311yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2312yes—
60#476—44no—
61#476—44no—
62#459—24no—
63#459—42no—
64#459—44no—
65C8×D4—24no—
66C8×D4—42no—
67C8×D4—44no—
68C2×#106—22no—
69C2×#106—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390D412yes—
97#384—21no—
98#384—21no—
99#384—21no—
100#384D411yescharacteristic
101#384C2311yescharacteristic
102#384C2312yes—
103C43C2311yescharacteristic
104#340—22no—
105#340—22no—
106#340—22no—
107#336—21no—
108#336—21no—
109#336D412yes—
110#334—42no—
111#331—22no—
112#331—22no—
113C22×Q8—41no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4C2×D412yes—
120C23×C4—22no—
121C23×C4—41no—
122C2×D8—44no—
123C2×D8—44no—
124C2×D8—82no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—44no—
131C22×C8—44no—
132C22×C8—82no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—81no—
138C4⋊Q8—81no—
139C4⋊Q8—82no—
140#127—22no—
141#127—21no—
142#127—21no—
143#127—21no—
144#127SD1614yes—
145#127#4112yes—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—81no—
157C4×D4—81no—
158C4×D4—82no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42C2411yescharacteristic
184#106—44no—
185#106—44no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—42no—
197C4⋊C8—81no—
198C4⋊C8D814yescomplemented
199C4⋊C8#4112yes—
200#102—44no—
201#102—44no—
202#102—44no—
203#102—44no—
204#102—82no—
205#102—82no—
206C22⋊C8—44no—
207C22⋊C8—44no—
208C22⋊C8—82no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—42no—
212C4×C8—42no—
213C24—22no—
214C24—41no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—81no—
242C2×D4—81no—
243C2×D4—81no—
244C2×D4—81no—
245C2×D4C2×SD1614yes—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265D8—88no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—88no—
287C2×C8C2×D814yes—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—22no—
317C42—41no—
318C42C2×D812yes—
319C42C2×SD1612yes—
320C42#13512yes—
321C42AGL(1,ℤ/8)12yes—
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C42H(5,𝔽2)11yescharacteristic
325C422−1+411yescharacteristic
326C23—24no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23C2×#12111yescharacteristic
341Q8—164no—
342D4—24no—
343D4—24no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—48no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—84no—
352D4—82no—
353D4—82no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—164no—
381C2×C4#45912yes—
382C2×C4#45912yes—
383C2×C4#46512yes—
384C2×C4#46512yes—
385C2×C4#47112yes—
386C2×C4#52911yescharacteristic
387C2×C4#54211yescharacteristic
388C2×C4#54611yescharacteristic
389C2×C4#54611yescharacteristic
390C2×C4#54711yescharacteristic
391C2×C4#54811yescharacteristic
392C2×C4C22×D811yescharacteristic
393C2×C4C22×SD1611yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4#57611yescharacteristic
397C2×C4#57711yescharacteristic
398C2×C4#57711yescharacteristic
399C2×C4#57711yescharacteristic
400C8—24no—
401C8—42no—
402C8—42no—
403C8—81no—
404C8—81no—
405C8—82no—
406C8—81no—
407C22—44no—
408C22—44no—
409C22—44no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22#242711yescharacteristic
422C22C2×#45911yescharacteristic
423C22#304211yescharacteristic
424C22C2×#46511yescharacteristic
425C22#306211yescharacteristic
426C22C2×#47111yescharacteristic
427C22#306811yescharacteristic
428C4—21no—
429C4—21no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—82no—
434C4—162no—
435C4—161no—
436C4#166514yes—
437C4#329411yescharacteristic
438C4#329511yescharacteristic
439C4#330311yescharacteristic
440C4#333211yescharacteristic
441C4#333511yescharacteristic
442C4#334111yescharacteristic
443C4#335811yescharacteristic
444C4#337311yescharacteristic
445C2—84no—
446C2—162no—
447C2C2×#166511yescharacteristic
448C2#1923011yescharacteristic
449C2#1939111yescharacteristic
450C2#1955811yescharacteristic
451C2#2531811yescharacteristic
452C2#2533911yescharacteristic
453C2#2535911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#28251C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29139C2semidirect
2. #28509C2semidirect
4. #23173C2semidirect
5. #10619C2semidirect
6. #10473C2semidirect
7. #9898C2semidirect
8. #9602C2semidirect
9. #9221C2semidirect
10. #8642C2semidirect
11. C4×#331C2semidirect
198. C4⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#291392
#285092
#282512
#231732
#106192
#104732
#98982
#96022
#92212
#86422
C4×#3312