Finite Group Databasebeta

← #330778

Subgroups — #330778

Order 512.

511 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33291C211yesmaximal, characteristic, complemented
2#29998C211yesmaximal, characteristic
3#29994C211yesmaximal, characteristic
4#21266C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10656C211yesmaximal, characteristic, complemented
8#10655C211yesmaximal, characteristic, complemented
9#10625C211yesmaximal, characteristic
10#10625C211yesmaximal, characteristic
11#10622C211yesmaximal, characteristic
12#10621C211yesmaximal, characteristic
13#9904C211yesmaximal, characteristic, complemented
14#9902C211yesmaximal, characteristic, complemented
15#8784C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2859C2211yescharacteristic
18#2859C2211yescharacteristic
19#2293C2211yescharacteristic
20#2088—21no—
21#2088—21no—
22#2086—21no—
23#2086—21no—
24#2067—21no—
25#2067—21no—
26#2065—21no—
27#2065—21no—
28#1991—21no—
29#1991—21no—
30#1991—21no—
31#1991—21no—
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1923—21no—
41#1923—21no—
42#1842C2211yescharacteristic
43#1842C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1767C2211yescharacteristic
53#1750—21no—
54C42×C8C2211yescharacteristic
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×#125—21no—
68C2×#125—21no—
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71#433—44no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#106—21no—
75C2×#106—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#390D411yescharacteristic
104#390D411yescharacteristic
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389D411yescharacteristic
114#389D411yescharacteristic
115#388—21no—
116#388—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122#382—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C43C2311yescharacteristic
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145C22×D4C2×D411yescharacteristic
146C23×C4—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157#125—42no—
158#125—42no—
159#125—42no—
160#125—42no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—21no—
165#117—42no—
166#117—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
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—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201#107—81no—
202#107—81no—
203#106—81no—
204#106—81no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—81no—
220C4⋊C8—81no—
221C4⋊C8—81no—
222C4⋊C8—81no—
223#102—81no—
224#102—81no—
225#102—81no—
226#102—81no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229C22⋊C8—42no—
230C22⋊C8—42no—
231#97—42no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—41no—
239C4×C8—41no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247C24—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—81no—
329C4⋊C4—81no—
330C4⋊C4—81no—
331C4⋊C4—81no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42#13512yes—
353C42AGL(1,ℤ/8)12yes—
354C422−1+411yescharacteristic
355C422−1+411yescharacteristic
356C422−1+411yescharacteristic
357C422−1+411yescharacteristic
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×#12111yescharacteristic
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
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#46312yes—
408C2×C4#46312yes—
409C2×C4#53611yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#53611yescharacteristic
412C2×C4#53611yescharacteristic
413C2×C4#54111yescharacteristic
414C2×C4#54111yescharacteristic
415C2×C4C2×#13511yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
417C2×C4#57511yescharacteristic
418C2×C4#57511yescharacteristic
419C2×C4#57611yescharacteristic
420C2×C4#57711yescharacteristic
421C2×C4#57711yescharacteristic
422C2×C4#57811yescharacteristic
423C8—41no—
424C8—41no—
425C8—41no—
426C8—41no—
427C8—81no—
428C8—81no—
429C8—81no—
430C8—81no—
431C8—81no—
432C8—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22#262111yescharacteristic
441C22C2×#46311yescharacteristic
442C22#304011yescharacteristic
443C22#306011yescharacteristic
444C22#306111yescharacteristic
445C22#306711yescharacteristic
446C22#306711yescharacteristic
447C4—22no—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—41no—
452C4—41no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4#323311yescharacteristic
461C4#323311yescharacteristic
462C4#323511yescharacteristic
463C4#323511yescharacteristic
464C4#324111yescharacteristic
465C4#324111yescharacteristic
466C4#324211yescharacteristic
467C4#324211yescharacteristic
468C2—161no—
469C2#1920011yescharacteristic
470C2#1920111yescharacteristic
471C2#1936611yescharacteristic
472C2#1936711yescharacteristic
473C2#2986511yescharacteristic
474C2#2988211yescharacteristic
475C2#2988811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #33291C2semidirect
5. #10658C2semidirect
6. #10657C2semidirect
7. #10656C2semidirect
8. #10655C2semidirect
13. #9904C2semidirect
14. #9902C2semidirect
15. #8784C2semidirect

Maximal subgroups

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

subgroupindex
#332912
#299982
#299942
#212662
#106582
#106572
#106562
#106552
#106252
#106222
#106212
#99042
#99022
#87842