Finite Group Databasebeta

← #339507

Subgroups — #339507

Order 512.

546 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28836C211yesmaximal, characteristic, complemented
2#27822C211yesmaximal, characteristic
3#26236C211yesmaximal, characteristic
4#26107C211yesmaximal, characteristic
5#10675C211yesmaximal, characteristic, complemented
6#10619C211yesmaximal, characteristic
7#10590C211yesmaximal, characteristic, complemented
8#10579C211yesmaximal, characteristic, complemented
9#10420C211yesmaximal, characteristic
10#10353C211yesmaximal, characteristic, complemented
11#9622C211yesmaximal, characteristic, complemented
12#9570C211yesmaximal, characteristic, complemented
13#9546C211yesmaximal, characteristic
14#9537C211yesmaximal, characteristic, complemented
15#8882C211yesmaximal, characteristic
16C2×#484—21no—
17#2571C2211yescharacteristic
18#2552C2211yescharacteristic
19#2542C2211yescharacteristic
20D4×C4⋊C4C2211yescharacteristic
21#2101—21no—
22#2100—21no—
23#2088—21no—
24#2085—21no—
25#2084—21no—
26#2036—21no—
27#2036—21no—
28#2025—21no—
29#1992—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1988C2211yescharacteristic
33#1981C2211yescharacteristic
34#1956C2211yescharacteristic
35#1954C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1944—21no—
41#1939C2211yescharacteristic
42#1936—21no—
43#1872—21no—
44#1864—21no—
45#1847—21no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1804—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1770C2211yescharacteristic
54#1770C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C4⋊C4—21no—
74#484—44no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—41no—
78C2×#107D411yescharacteristic
79C2×#107D411yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102D411yescharacteristic
96C2×#102D411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—41no—
105#399—41no—
106#398—41no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#385—41no—
114#384—21no—
115#384—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#382—21no—
121#380—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125#340—21no—
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—41no—
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#334—22no—
146#332—22no—
147D4⋊C8—22no—
148C22×Q8—21no—
149C22×D4C2×D411yescharacteristic
150C23×C4—21no—
151C23×C4—41no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165#125—42no—
166#125—81no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—81no—
170C22⋊Q8—81no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—42no—
176C4×D4—81no—
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—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C2411yescharacteristic
212#107—22no—
213#107—41no—
214#107—42no—
215#107—81no—
216#107—82no—
217#106—42no—
218#106—81no—
219#106—81no—
220#106—81no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—81no—
229C4⋊C8#4112yes—
230#103—42no—
231#103—42no—
232#103—81no—
233#103—81no—
234#102—22no—
235#102—41no—
236#102—42no—
237#102—81no—
238C22⋊C8—42no—
239C22⋊C8—81no—
240#97—42no—
241#97—81no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4#13712yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—161no—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8AGL(1,ℤ/8)12yes—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—161no—
358C4⋊C4—161no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42C2×#4111yescharacteristic
371C422−1+411yescharacteristic
372C422−1+411yescharacteristic
373C422−1+411yescharacteristic
374C23—22no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23D4211yescharacteristic
382Q8—81no—
383Q8—81no—
384Q8—81no—
385Q8—81no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—164no—
430C2×C4#53611yescharacteristic
431C2×C4#53611yescharacteristic
432C2×C4#54211yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#54811yescharacteristic
435C2×C4#56311yescharacteristic
436C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
437C2×C4C2×#13711yescharacteristic
438C2×C4#57511yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57811yescharacteristic
444C8—22no—
445C8—42no—
446C8—41no—
447C8—41no—
448C8—42no—
449C8—41no—
450C8—81no—
451C8—81no—
452C8—81no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#267411yescharacteristic
461C22#326711yescharacteristic
462C22#327011yescharacteristic
463C22#327311yescharacteristic
464C22#327411yescharacteristic
465C22#328911yescharacteristic
466C22#329111yescharacteristic
467C4—21no—
468C4—21no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4#323411yescharacteristic
481C4#323611yescharacteristic
482C4#324111yescharacteristic
483C4#324311yescharacteristic
484C4#334611yescharacteristic
485C4#334911yescharacteristic
486C4#336811yescharacteristic
487C4#337011yescharacteristic
488C2—82no—
489C2#2009211yescharacteristic
490C2#2020811yescharacteristic
491C2#2024211yescharacteristic
492C2#2030311yescharacteristic
493C2#3079511yescharacteristic
494C2#3082711yescharacteristic
495C2#3083811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
1. #28836C2semidirect
5. #10675C2semidirect
7. #10590C2semidirect
8. #10579C2semidirect
10. #10353C2semidirect
11. #9622C2semidirect
12. #9570C2semidirect
14. #9537C2semidirect

Maximal subgroups

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

subgroupindex
#288362
#278222
#262362
#261072
#106752
#106192
#105902
#105792
#104202
#103532
#96222
#95702
#95462
#95372
#88822