Finite Group Databasebeta

← #379126

Subgroups — #379126

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23627C211yesmaximal, characteristic
2#21144C211yesmaximal, characteristic, complemented
3C2×#1938C211yesmaximal, characteristic
4C2×#1937C211yesmaximal, characteristic
5#20384C211yesmaximal, characteristic
6#11359C211yesmaximal, characteristic, complemented
7#11269C211yesmaximal, characteristic, complemented
8#11053C211yesmaximal, characteristic
9#10883C211yesmaximal, characteristic, complemented
10#10882C211yesmaximal, characteristic, complemented
11#10809C211yesmaximal, characteristic, complemented
12#10809C211yesmaximal, characteristic, complemented
13#10791C211yesmaximal, characteristic
14#10791C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30#2285C2211yescharacteristic
31#2074C412yescomplemented
32#2073C412yescomplemented
33#2072C412yescomplemented
34#2071C412yescomplemented
35#1940—22no—
36#1940—22no—
37#1938—24no—
38#1937—24no—
39#1753—22no—
40#1752—22no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C4×#95C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C2×#131—22no—
63C2×#131—22no—
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—41no—
67C2×#131—41no—
68C22×M16—21no—
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8—21no—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#424—22no—
85#424—22no—
86#424—22no—
87#424—22no—
88#423—41no—
89#423—41no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—22no—
96C2×#104—22no—
97C2×#104—41no—
98C2×#104—41no—
99#416—22no—
100#416—22no—
101#416—22no—
102#416—22no—
103C2×#103C2×C412yes—
104C2×#103C2×C412yes—
105C2×#102C2×C412yes—
106C2×#102C2×C412yes—
107#408—22no—
108#408D414yes—
109C2×C22⋊C8—22no—
110#405—22no—
111#405—22no—
112C2×#97—22no—
113C2×#97—21no—
114C2×#97—22no—
115C2×#97—41no—
116C2×#97—41no—
117C2×#97—41no—
118C2×#97D412yes—
119C2×#97D412yes—
120C2×C4×C8—22no—
121C2×C4×C8—21no—
122C2×C4×C8—22no—
123C2×C4×C8—41no—
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#386—22no—
129#384C2×C412yes—
130#382C2×C412yes—
131#380—22no—
132C4×C4⋊C4—22no—
133C4×C22⋊C4—22no—
134C2×#95—22no—
135#341—22no—
136#341—22no—
137#341—41no—
138#341—41no—
139#340C2×C412yes—
140#340C2×C412yes—
141#340C2×C412yes—
142#340C2×C412yes—
143#339—22no—
144#339—22no—
145#339—22no—
146#339—22no—
147#339—22no—
148#339—22no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—22no—
154#336—22no—
155#336—22no—
156#336—22no—
157#336C2×C412yes—
158#336C2×C412yes—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41C2×D412yes—
170C2×#41#4112yes—
171C22×Q8C22×C411yescharacteristic
172C22×D4—21no—
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179#131—44no—
180#131—44no—
181#131—44no—
182#131—44no—
183C2×M16—24no—
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—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—42no—
207C2×M16—42no—
208C2×M16C2×D412yes—
209C2×M16#4112yes—
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—24no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C22×C8C2×D412yes—
240C22×C8C2×D412yes—
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8#4112yes—
244C22×C8#4112yes—
245C22×C8#4112yes—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42#4112yes—
290C2×C42#4112yes—
291C4⋊C8—44no—
292C4⋊C8—44no—
293C4⋊C8—44no—
294#104—44no—
295#104—44no—
296#103—24no—
297#103—24no—
298#102—24no—
299#102—24no—
300C22⋊C8—28no—
301#97—42no—
302#97—42no—
303#97—44no—
304#97—42no—
305C4×C8—44no—
306C4×C8—42no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—44no—
311C24—21no—
312C24C2×#4111yescharacteristic
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8C4×D412yes—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4C4×D412yes—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4C4×D412yes—
373C22×C4C4×D412yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C4H(5,𝔽2)11yescharacteristic
386C22×C42−1+411yescharacteristic
387C22×C42−1+411yescharacteristic
388M16—22no—
389M16—22no—
390M16—22no—
391M16—22no—
392M16—48no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—48no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8#12114yes—
447C2×C8#12412yes—
448C2×C8#12412yes—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23C2×C4×D411yescharacteristic
495C23#52911yescharacteristic
496C23#52911yescharacteristic
497C23#52911yescharacteristic
498C23#53211yescharacteristic
499C23#53911yescharacteristic
500C23#53911yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505D4—22no—
506D4—22no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—41no—
512D4—41no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C4×#4111yescharacteristic
550C2×C4C4×#4111yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4#52011yescharacteristic
553C2×C4#52011yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4#52511yescharacteristic
556C2×C4C2×#12411yescharacteristic
557C2×C4#53211yescharacteristic
558C2×C4#53411yescharacteristic
559C2×C4#53611yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54611yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4#54711yescharacteristic
564C2×C4#55111yescharacteristic
565C2×C4#55111yescharacteristic
566C2×C4#55111yescharacteristic
567C2×C4#55111yescharacteristic
568C2×C4#55211yescharacteristic
569C2×C4#55211yescharacteristic
570C2×C4#55211yescharacteristic
571C2×C4#55211yescharacteristic
572C2×C4#55311yescharacteristic
573C2×C4#55311yescharacteristic
574C2×C4#55611yescharacteristic
575C2×C4#55611yescharacteristic
576C8—22no—
577C8—22no—
578C8—24no—
579C8—42no—
580C8—42no—
581C8—42no—
582C8—42no—
583C8—81no—
584C8—81no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—44no—
594C22—42no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22C4×#12111yescharacteristic
599C22C4×#12411yescharacteristic
600C22#232211yescharacteristic
601C22#255311yescharacteristic
602C22#255311yescharacteristic
603C22#257311yescharacteristic
604C22#257311yescharacteristic
605C22#295612yes—
606C22#295712yes—
607C4—44no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—81no—
615C4—81no—
616C4#235911yescharacteristic
617C4#235911yescharacteristic
618C4#237311yescharacteristic
619C4#237311yescharacteristic
620C4#254311yescharacteristic
621C4#254711yescharacteristic
622C4#256111yescharacteristic
623C4#261111yescharacteristic
624C2—21no—
625C2—21no—
626C2—21no—
627C2—21no—
628C2—82no—
629C2#1422911yescharacteristic
630C2#2151912yes—
631C2#2165911yescharacteristic
632C2#2166211yescharacteristic
633C2#2209712yes—
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 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21144C2semidirect
6. #11359C2semidirect
7. #11269C2semidirect
9. #10883C2semidirect
10. #10882C2semidirect
11. #10809C2semidirect
12. #10809C2semidirect
15. #10696C2semidirect
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
#211442
C2×#19382
C2×#19372
#203842
#113592
#112692
#110532
#108832
#108822
#108092
#107912
#106962