Finite Group Databasebeta

← #350899

Subgroups — #350899

Order 512.

695 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33277C211yesmaximal, characteristic
2#33200C211yesmaximal, characteristic, complemented
3#33165C211yesmaximal, characteristic
4C4×C8×Q8C211yesmaximal, characteristic, complemented
5#9900C211yesmaximal, characteristic, complemented
6#9698C211yesmaximal, characteristic
7#9566C211yesmaximal, characteristic, complemented
8#9565C211yesmaximal, characteristic, complemented
9#9564C211yesmaximal, characteristic
10#9563C211yesmaximal, characteristic
11#9222C211yesmaximal, characteristic
12#9222C211yesmaximal, characteristic
13#9218C211yesmaximal, characteristic, complemented
14#9218C211yesmaximal, characteristic, complemented
15#8784C211yesmaximal, characteristic, complemented
16C2×#479—21no—
17C2×#464—21no—
18C2×#464—21no—
19C2×#462C2211yescharacteristic
20C2×#460C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23#2859C2211yescharacteristic
24#2858C2211yescharacteristic
25C42×Q8C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1989—21no—
29#1983—21no—
30#1983—21no—
31#1982C2211yescharacteristic
32#1981C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1949—21no—
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1934—21no—
39#1934—21no—
40#1934—21no—
41#1934—21no—
42#1842C2211yescharacteristic
43#1842C2211yescharacteristic
44#1805—21no—
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47C8×C4⋊C4—21no—
48C8×C4⋊C4—21no—
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C4×C4⋊C8—21no—
52C4×C4⋊C8C2211yescharacteristic
53C42×C8—21no—
54C42×C8C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16—21no—
68C22×SD16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#125—21no—
72C2×#124—21no—
73C2×#124—21no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77#479—44no—
78#464—44no—
79#464—44no—
80#462—22no—
81#462—22no—
82#460—22no—
83#460—22no—
84C8×Q8—22no—
85C8×Q8—22no—
86C8×Q8—22no—
87C8×Q8—22no—
88C2×#107—21no—
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8D411yescharacteristic
120C2×C4×C8D411yescharacteristic
121C2×C4×C8D411yescharacteristic
122C2×C4×C8D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#390—21no—
126#390—21no—
127#390—21no—
128#390—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#382—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C43—21no—
143C43C2311yescharacteristic
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#332—22no—
156#332—22no—
157#331—22no—
158#331—22no—
159C22×Q8—21no—
160C22×Q8C2×D411yescharacteristic
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C2×Q16—42no—
165C2×Q16—42no—
166C2×Q16—42no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—42no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184#127—21no—
185#127—21no—
186#127—21no—
187#127—21no—
188#125—42no—
189#125—42no—
190#124—42no—
191#124—42no—
192#124—42no—
193#124—42no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8D814yes—
199C4×Q8SD1614yes—
200C4×Q8#4112yes—
201C4×Q8#4112yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—42no—
236#106—42no—
237C4⋊C8—24no—
238C4⋊C8—24no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8#4112yes—
254C4⋊C8#4112yes—
255#103—42no—
256#103—42no—
257#103—42no—
258#103—42no—
259#103—42no—
260#102—42no—
261#102—42no—
262#102—42no—
263#102—42no—
264#102—42no—
265#102—42no—
266#102—42no—
267C4×C8—24no—
268C4×C8—24no—
269C4×C8—22no—
270C4×C8—22no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—41no—
276C4×C8—41no—
277C4×C8—42no—
278C4×C8—42no—
279C4×C8—42no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—41no—
284#95—41no—
285C24—41no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8#12412yes—
296C2×Q8C2×D812yes—
297C2×Q8C2×SD1612yes—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—81no—
313C2×D4—81no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328Q16—84no—
329SD16—84no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8#13512yes—
359C2×C8#13712yes—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4#12412yes—
384C4⋊C4C2×D812yes—
385C4⋊C4C2×SD1612yes—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—22no—
391C42—22no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—24no—
397C42—24no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C42—41no—
403C42—41no—
404C42C2×D812yes—
405C42C2×SD1612yes—
406C42#13512yes—
407C42#13512yes—
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23C2×#12111yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—22no—
423Q8—81no—
424Q8—81no—
425Q8—81no—
426Q8—81no—
427Q8#48614yes—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—81no—
436D4—161no—
437D4—161no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#46512yes—
470C2×C4#46512yes—
471C2×C4#46612yes—
472C2×C4#46612yes—
473C2×C4#52511yescharacteristic
474C2×C4#52511yescharacteristic
475C2×C4C2×#12411yescharacteristic
476C2×C4#52911yescharacteristic
477C2×C4#52911yescharacteristic
478C2×C4#53211yescharacteristic
479C2×C4C22×D811yescharacteristic
480C2×C4C22×SD1611yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×#13711yescharacteristic
485C2×C4#57511yescharacteristic
486C2×C4#57511yescharacteristic
487C8—22no—
488C8—22no—
489C8—41no—
490C8—42no—
491C8—41no—
492C8—42no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22#285011yescharacteristic
505C22#303511yescharacteristic
506C22#303711yescharacteristic
507C22C2×#46511yescharacteristic
508C22C2×#46611yescharacteristic
509C22#306711yescharacteristic
510C22#306711yescharacteristic
511C4—24no—
512C4—21no—
513C4—21no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4#310211yescharacteristic
525C4#310811yescharacteristic
526C4#311111yescharacteristic
527C4#311711yescharacteristic
528C4C2×#48611yescharacteristic
529C4#313111yescharacteristic
530C4#316711yescharacteristic
531C4#316911yescharacteristic
532C2—321no—
533C2#1912411yescharacteristic
534C2#1912511yescharacteristic
535C2#1926011yescharacteristic
536C2#1926111yescharacteristic
537C2#3302911yescharacteristic
538C2#3304211yescharacteristic
539C2#3306211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #33200C2semidirect
4. C4×C8×Q8C2semidirect
5. #9900C2semidirect
7. #9566C2semidirect
8. #9565C2semidirect
13. #9218C2semidirect
14. #9218C2semidirect
15. #8784C2semidirect

Maximal subgroups

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

subgroupindex
#332772
#332002
#331652
C4×C8×Q82
#99002
#96982
#95662
#95652
#95642
#95632
#92222
#92182
#87842