Finite Group Databasebeta

← #315698

Subgroups — #315698

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33042C211yesmaximal, characteristic
2#27962C211yesmaximal, characteristic, complemented
3#27907C211yesmaximal, characteristic
4#27891C211yesmaximal, characteristic
5#27889C211yesmaximal, characteristic, complemented
6#20817C211yesmaximal, characteristic, complemented
7#10684C211yesmaximal, characteristic, complemented
8#10679C211yesmaximal, characteristic, complemented
9#10678C211yesmaximal, characteristic, complemented
10#10677C211yesmaximal, characteristic, complemented
11#9830C211yesmaximal, characteristic, complemented
12#9830C211yesmaximal, characteristic, complemented
13#8821C211yesmaximal, characteristic, complemented
14#8820C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#482—22no—
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2856C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#2101—22no—
24#2092—22no—
25#2035—22no—
26#2033—22no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1934—22no—
38#1927—22no—
39#1925—22no—
40#1893—22no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1825—22no—
55#1808—22no—
56C4×#107—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C23×C8D412yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×#127C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#117—21no—
71C2×#117—21no—
72C22×C42C2311yescharacteristic
73#486—22no—
74#486—22no—
75#482—48no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×C22⋊C8D412yes—
91C2×C22⋊C8D412yes—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8D412yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#382—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#336D412yes—
136#336D412yes—
137#322—22no—
138#322—22no—
139C22×Q8—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—24no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160#127—21no—
161#127—21no—
162#127—21no—
163#127—21no—
164C22⋊Q8—82no—
165#121—44no—
166#121—44no—
167#121—82no—
168#117—44no—
169#117—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207#107—44no—
208#107—44no—
209#107—44no—
210#107—82no—
211#106—44no—
212#106—82no—
213#103—44no—
214#103—82no—
215#102—44no—
216#102—44no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—82no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C4×C8—24no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8#4112yes—
235C4×C8#4112yes—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242C24—41no—
243C24#12711yescharacteristic
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—82no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4#12711yescharacteristic
282C22×C4#12711yescharacteristic
283C22×C4#12711yescharacteristic
284C22×C4AGL(1,ℤ/8)14yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—48no—
305C2×C8#12314yes—
306C2×C8#13514yes—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—82no—
343C42—82no—
344C42C2×#4111yescharacteristic
345C42C2×#4111yescharacteristic
346C422−1+411yescharacteristic
347C422−1+411yescharacteristic
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23#49612yes—
362C23C2×#12711yescharacteristic
363Q8—82no—
364Q8—82no—
365D4—82no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—164no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4#49612yes—
408C2×C4C2×#12311yescharacteristic
409C2×C4#53611yescharacteristic
410C2×C4#54711yescharacteristic
411C2×C4#54711yescharacteristic
412C2×C4#54711yescharacteristic
413C2×C4#54711yescharacteristic
414C2×C4C2×#13512yes—
415C2×C4C2×AGL(1,ℤ/8)12yes—
416C2×C4#57512yes—
417C2×C4#57512yes—
418C2×C4#57612yes—
419C2×C4#57812yes—
420C8—24no—
421C8—24no—
422C8—42no—
423C8—42no—
424C8—42no—
425C8—42no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22#250111yescharacteristic
441C22#314711yescharacteristic
442C22C2×#49611yescharacteristic
443C22#315011yescharacteristic
444C22#315311yescharacteristic
445C22#315511yescharacteristic
446C22#315611yescharacteristic
447C4—22no—
448C4—41no—
449C4—41no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4#309312yes—
458C4#324012yes—
459C4#333912yes—
460C4#334112yes—
461C4#334312yes—
462C4#334312yes—
463C2—42no—
464C2—321no—
465C2#1961911yescharacteristic
466C2#1964311yescharacteristic
467C2#1967311yescharacteristic
468C2#1971211yescharacteristic
469C2#2791511yescharacteristic
470C2#2792811yescharacteristic
471C2#2793911yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27962C2semidirect
5. #27889C2semidirect
6. #20817C2semidirect
7. #10684C2semidirect
8. #10679C2semidirect
9. #10678C2semidirect
10. #10677C2semidirect
11. #9830C2semidirect
12. #9830C2semidirect
13. #8821C2semidirect
14. #8820C2semidirect
15. #8145C2semidirect

Maximal subgroups

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

subgroupindex
#330422
#279622
#279072
#278912
#278892
#208172
#106842
#106792
#106782
#106772
#98302
#88212
#88202
#81452