Finite Group Databasebeta

← #379321

Subgroups — #379321

Order 512.

853 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23627C211yesmaximal, characteristic
2#21051C211yesmaximal, characteristic
3#21026C211yesmaximal, characteristic, complemented
4#11317C211yesmaximal, characteristic, complemented
5#11269C211yesmaximal, characteristic, complemented
6#11068C211yesmaximal, characteristic
7#10934C211yesmaximal, characteristic
8#10930C211yesmaximal, characteristic
9#10929C211yesmaximal, characteristic
10#10924C211yesmaximal, characteristic
11#10883C211yesmaximal, characteristic, complemented
12#10840C211yesmaximal, characteristic, complemented
13#10809C211yesmaximal, characteristic, complemented
14#10706C211yesmaximal, characteristic, complemented
15#10694C211yesmaximal, characteristic, complemented
16C2×#424C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C22×#104C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C2×#416C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#408C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2297C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#2074C412yescomplemented
32#2073C412yescomplemented
33#2072C412yescomplemented
34#2071C412yescomplemented
35#1924—22no—
36#1923—22no—
37#1917—22no—
38#1916—22no—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1728C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C22×#41C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#424—22no—
75#424—41no—
76#423—22no—
77#423—41no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#104—22no—
81C2×#104—22no—
82C2×#104—22no—
83C2×#104—22no—
84C2×#104—22no—
85C2×#104—41no—
86#416—41no—
87#416—41no—
88#416—41no—
89#416—41no—
90C2×#103C2×C412yes—
91C2×#103C2×C412yes—
92C2×#102C2×C412yes—
93C2×#102C2×C412yes—
94#408—22no—
95#408—22no—
96#408—22no—
97#408—41no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101#405—22no—
102#405—22no—
103#405—22no—
104#405—41no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—22no—
108C2×#97—41no—
109C2×#97—41no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—41no—
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#386—22no—
120#384C2×C412yes—
121#382C2×C412yes—
122#380—22no—
123C4×C4⋊C4—22no—
124C4×C22⋊C4—22no—
125#376—22no—
126#376—41no—
127C2×#95—22no—
128#341—22no—
129#341—41no—
130#341—41no—
131#341—41no—
132#340C2×C412yes—
133#340C2×C412yes—
134#340C2×C412yes—
135#340C2×C412yes—
136#339—22no—
137#339—22no—
138#339—22no—
139#339—22no—
140#339—41no—
141#339—41no—
142#336—21no—
143#336—22no—
144#336—21no—
145#336—22no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336C2×C412yes—
150#336C2×C412yes—
151#336D412yes—
152#336D412yes—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C22×Q8C22×C411yescharacteristic
166C22×D4—21no—
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16#4112yes—
196C2×M16#4112yes—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8C22×C412yes—
224C22×C8C22×C412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8#4112yes—
229C22×C8#4112yes—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274C2×C42#4112yes—
275C4⋊C8—44no—
276C4⋊C8—44no—
277#104—44no—
278#104—44no—
279#103—42no—
280#103—42no—
281#102—42no—
282#102—42no—
283C22⋊C8—44no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286#97—44no—
287#97—42no—
288#97—42no—
289C4×C8—44no—
290C4×C8—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—44no—
296C24—21no—
297C24C2×#4111yescharacteristic
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4C4×D412yes—
358C22×C4C4×D412yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373M16—22no—
374M16—22no—
375M16—22no—
376M16—22no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8#12412yes—
431C2×C8#12412yes—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—44no—
441C4⋊C4—82no—
442C22⋊C4—44no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—82no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42#12614yes—
470C23—22no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23C2×C4×D411yescharacteristic
478C23#52911yescharacteristic
479C23#52911yescharacteristic
480C23#55111yescharacteristic
481C23#55211yescharacteristic
482C23#55311yescharacteristic
483C23#55511yescharacteristic
484Q8—41no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—81no—
493D4—81no—
494D4—81no—
495D4—81no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4C4×#4111yescharacteristic
532C2×C4C4×#4111yescharacteristic
533C2×C4C4×#4111yescharacteristic
534C2×C4C4×#4111yescharacteristic
535C2×C4#51811yescharacteristic
536C2×C4#52011yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12611yescharacteristic
539C2×C4#52911yescharacteristic
540C2×C4#52911yescharacteristic
541C2×C4#52911yescharacteristic
542C2×C4#53211yescharacteristic
543C2×C4#53811yescharacteristic
544C2×C4#53911yescharacteristic
545C2×C4#54011yescharacteristic
546C2×C4#54211yescharacteristic
547C2×C4#54611yescharacteristic
548C2×C4#54611yescharacteristic
549C2×C4#54711yescharacteristic
550C2×C4#55011yescharacteristic
551C2×C4#55111yescharacteristic
552C2×C4#55111yescharacteristic
553C2×C4#55211yescharacteristic
554C2×C4#55311yescharacteristic
555C2×C4#55311yescharacteristic
556C2×C4#55311yescharacteristic
557C2×C4#55611yescharacteristic
558C2×C4#55911yescharacteristic
559C8—22no—
560C8—22no—
561C8—42no—
562C8—42no—
563C8—42no—
564C8—42no—
565C8—42no—
566C8—81no—
567C8—81no—
568C22—22no—
569C22—22no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—82no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—82no—
581C22C4×#12411yescharacteristic
582C22#235911yescharacteristic
583C22#236311yescharacteristic
584C22#251011yescharacteristic
585C22#252911yescharacteristic
586C22#253111yescharacteristic
587C22#259611yescharacteristic
588C22#295512yes—
589C22#295712yes—
590C4—44no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4C4×#12611yescharacteristic
600C4#233511yescharacteristic
601C4#237311yescharacteristic
602C4#237411yescharacteristic
603C4#248811yescharacteristic
604C4#249011yescharacteristic
605C4#251211yescharacteristic
606C4#255711yescharacteristic
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—161no—
612C2#1423311yescharacteristic
613C2#2165911yescharacteristic
614C2#2166111yescharacteristic
615C2#2295112yes—
616C2#2311512yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21026C2semidirect
4. #11317C2semidirect
5. #11269C2semidirect
11. #10883C2semidirect
12. #10840C2semidirect
13. #10809C2semidirect
14. #10706C2semidirect
15. #10694C2semidirect
31. #2074C4semidirect
32. #2073C4semidirect
33. #2072C4semidirect
34. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#236272
#210512
#210262
#113172
#112692
#110682
#109342
#109302
#109292
#109242
#108832
#108402
#108092
#107062
#106942