Finite Group Databasebeta

← #325500

Subgroups — #325500

Order 512.

549 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#33263C211yesmaximal, characteristic, complemented
2#29140C211yesmaximal, characteristic
3#29122C211yesmaximal, characteristic
4#21633C211yesmaximal, characteristic
5#10626C211yesmaximal, characteristic, complemented
6#10625C211yesmaximal, characteristic
7#10623C211yesmaximal, characteristic, complemented
8#10622C211yesmaximal, characteristic
9#9901C211yesmaximal, characteristic, complemented
10#9899C211yesmaximal, characteristic, complemented
11#8653C211yesmaximal, characteristic, complemented
12#8651C211yesmaximal, characteristic, complemented
13#8644C211yesmaximal, characteristic
14#8642C211yesmaximal, characteristic
15#8275C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2859C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×#124C2211yescharacteristic
21#2101—21no—
22#2042—21no—
23#2026—21no—
24#2025—21no—
25#1984C2211yescharacteristic
26#1958—21no—
27#1954C2211yescharacteristic
28#1944—21no—
29#1944—21no—
30#1943—21no—
31#1943—21no—
32#1934—21no—
33#1933—21no—
34#1924—21no—
35#1924—21no—
36#1845C2211yescharacteristic
37#1844C2211yescharacteristic
38#1835—21no—
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1831C2211yescharacteristic
42#1803C2211yescharacteristic
43#1803C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52#1752—21no—
53#1752—21no—
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×#125—21no—
69C2×#125—21no—
70C2×#124D411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#446—22no—
74#446—22no—
75C8×D4—22no—
76C8×D4—22no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×#97—21no—
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×#97D411yescharacteristic
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391D411yescharacteristic
109#390D411yescharacteristic
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#388—21no—
119#388—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125#382—21no—
126#382D411yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C43C2311yescharacteristic
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—
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#334—22no—
150C22×Q8C2×D411yescharacteristic
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—42no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166#125—81no—
167#125—81no—
168#125—82no—
169#124—22no—
170#124—41no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—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—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#107—42no—
212#107—42no—
213#106—42no—
214#106—42no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—42no—
228#103—42no—
229#103—81no—
230#103—81no—
231#103—81no—
232#102—42no—
233#102—81no—
234#102—81no—
235#102—81no—
236C22⋊C8—42no—
237C22⋊C8—42no—
238C22⋊C8—42no—
239#97—22no—
240#97—22no—
241#97—41no—
242#97—41no—
243#97—42no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247C4×C8—81no—
248C4×C8#4112yes—
249C4×C8#4112yes—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258C24—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4#13712yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
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#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
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—84no—
318C2×C8#12412yes—
319C2×C8#12412yes—
320C2×C8#13512yes—
321C2×C8#13512yes—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
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—81no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—81no—
357C22⋊C4—81no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C422−1+411yescharacteristic
373C422−1+411yescharacteristic
374C23—22no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23C2×#12111yescharacteristic
382Q8—41no—
383Q8—41no—
384Q8—41no—
385Q8—41no—
386D4—22no—
387D4—22no—
388D4—41no—
389D4—41no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
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×C4C2×#12411yescharacteristic
430C2×C4#53611yescharacteristic
431C2×C4#54711yescharacteristic
432C2×C4#54711yescharacteristic
433C2×C4#55611yescharacteristic
434C2×C4#55611yescharacteristic
435C2×C4C2×#13511yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4C2×#13711yescharacteristic
438C2×C4#57511yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57711yescharacteristic
443C8—22no—
444C8—41no—
445C8—41no—
446C8—81no—
447C8—81no—
448C8—81no—
449C8—81no—
450C8—81no—
451C8—81no—
452C8#48814yes—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#254711yescharacteristic
461C22#303511yescharacteristic
462C22#303511yescharacteristic
463C22#305611yescharacteristic
464C22#306211yescharacteristic
465C22#306811yescharacteristic
466C22#306911yescharacteristic
467C4—21no—
468C4—21no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4C2×#48811yescharacteristic
481C4#314211yescharacteristic
482C4#323311yescharacteristic
483C4#323311yescharacteristic
484C4#334411yescharacteristic
485C4#334611yescharacteristic
486C4#334711yescharacteristic
487C4#334811yescharacteristic
488C2—82no—
489C2#1917911yescharacteristic
490C2#1918111yescharacteristic
491C2#1924611yescharacteristic
492C2#1924811yescharacteristic
493C2#2871511yescharacteristic
494C2#2872111yescharacteristic
495C2#2873511yescharacteristic
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. #33263C2semidirect
5. #10626C2semidirect
7. #10623C2semidirect
9. #9901C2semidirect
10. #9899C2semidirect
11. #8653C2semidirect
12. #8651C2semidirect
15. #8275C2semidirect

Maximal subgroups

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

subgroupindex
#332632
#291402
#291222
#216332
#106262
#106252
#106232
#106222
#99012
#98992
#86532
#86512
#86442
#86422
#82752