Finite Group Databasebeta

← #387233

Subgroups — #387233

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1861C211yesmaximal, characteristic
2#11367C211yesmaximal, characteristic, complemented
3#11321C211yesmaximal, characteristic, complemented
4#11263C211yesmaximal, characteristic, complemented
5#11262C211yesmaximal, characteristic, complemented
6#11085C211yesmaximal, characteristic
7#11081C211yesmaximal, characteristic
8#10959C211yesmaximal, characteristic
9#10923C211yesmaximal, characteristic
10#10901C211yesmaximal, characteristic, complemented
11#10894C211yesmaximal, characteristic, complemented
12#10878C211yesmaximal, characteristic, complemented
13#10839C211yesmaximal, characteristic, complemented
14#10780C211yesmaximal, characteristic
15#10751C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#410C2211yescharacteristic
24#2873C2211yescharacteristic
25#2297C2211yescharacteristic
26#2296C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1861C418yescomplemented
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34#1740C2211yescharacteristic
35#1738C2211yescharacteristic
36#1733C2211yescharacteristic
37#1733C2211yescharacteristic
38C2×#339C2211yescharacteristic
39C2×#339C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×H(3,ℤ/4)C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50#1728C2211yescharacteristic
51#1728C2211yescharacteristic
52C22×#41C2311yescharacteristic
53#568—41no—
54#568—41no—
55#568—41no—
56#568C2×C414yes—
57C2×#131—41no—
58C2×#131—41no—
59C22×M16—41no—
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C42—41no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#430—41no—
76#430C2×C414yes—
77C2×#108—41no—
78#423—41no—
79#423C2×C414yes—
80C2×C4⋊C8—41no—
81#421—22no—
82#421—22no—
83#421—41no—
84#421C2×C414yes—
85C2×#104—22no—
86C2×#104—21no—
87C2×#104—21no—
88C2×#104—22no—
89C2×#104—22no—
90C2×#104—41no—
91C2×#104—41no—
92C2×#104—41no—
93C2×#104C2×C414yes—
94#410—22no—
95#410—41no—
96C2×#99—41no—
97#405—41no—
98#405—41no—
99C4×M16—41no—
100#391Q812yes—
101#390Q812yes—
102#389—22no—
103#386—41no—
104#385—41no—
105#384—21no—
106#382—21no—
107#380—41no—
108C4×C4⋊C4—41no—
109C4×C4⋊C4—41no—
110C4×C22⋊C4—41no—
111C43—22no—
112C43—41no—
113#344—22no—
114#344—41no—
115#344—41no—
116#344C2×C414yes—
117#342—41no—
118#341—41no—
119#339—22no—
120#339—22no—
121#339—22no—
122#339—21no—
123#339—21no—
124#339—41no—
125#339—41no—
126#339—41no—
127#339—41no—
128#339—41no—
129#339—41no—
130#339—41no—
131#339C2×C414yes—
132H(3,ℤ/4)—41no—
133H(3,ℤ/4)—41no—
134#336—41no—
135#336—41no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41C22×C412yes—
147C2×#41#4112yes—
148C22×Q8C2×Q811yescharacteristic
149C22×D4—21no—
150C22×D4C2×Q811yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2411yescharacteristic
156#131—44no—
157#131—44no—
158#131—44no—
159#131—44no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
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—44no—
192C2×M16—42no—
193C2×M16—44no—
194C2×M16C22×C412yes—
195C2×M16C22×C412yes—
196C2×M16C22×C412yes—
197C2×M16C22×C412yes—
198C2×M16#4112yes—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—42no—
206#117—21no—
207#117—21no—
208#117—21no—
209#117—21no—
210#117—21no—
211#117—21no—
212#117—21no—
213#117—21no—
214#117—21no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117C22×C412yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×Q812yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—42no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—44no—
258C2×C42—48no—
259C2×C42C22×C412yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265#108—44no—
266C4⋊C8—44no—
267#104—24no—
268#104—24no—
269#104—44no—
270#104—44no—
271#104—44no—
272#99—44no—
273#97—42no—
274C4×C8—42no—
275#95—42no—
276#95—42no—
277#95—42no—
278C24—21no—
279C24H(5,𝔽2)11yescharacteristic
280#41—22no—
281#41—22no—
282#41—22no—
283#41—22no—
284#41—22no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8C4×Q812yes—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4C4×Q812yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C42−1+411yescharacteristic
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—22no—
358M16—22no—
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—48no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—48no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C22⋊C4—22no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—84no—
420C42—22no—
421C42—21no—
422C42—22no—
423C42—21no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—416no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42#11712yes—
444C42#11712yes—
445C23—22no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—82no—
450C23#52011yescharacteristic
451C23#53811yescharacteristic
452C23#53811yescharacteristic
453C23#53811yescharacteristic
454C23#53811yescharacteristic
455C23#54011yescharacteristic
456C23#54311yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—41no—
468D4—41no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4C2×#11711yescharacteristic
503C2×C4C2×C4×Q811yescharacteristic
504C2×C4C4×#4111yescharacteristic
505C2×C4#51811yescharacteristic
506C2×C4#51811yescharacteristic
507C2×C4#51811yescharacteristic
508C2×C4#53911yescharacteristic
509C2×C4#53911yescharacteristic
510C2×C4#53911yescharacteristic
511C2×C4#53911yescharacteristic
512C2×C4#54211yescharacteristic
513C2×C4#54411yescharacteristic
514C2×C4#55111yescharacteristic
515C2×C4#55111yescharacteristic
516C2×C4#55311yescharacteristic
517C2×C4#55311yescharacteristic
518C2×C4#55311yescharacteristic
519C2×C4#55311yescharacteristic
520C2×C4#55411yescharacteristic
521C2×C4#55411yescharacteristic
522C2×C4#55411yescharacteristic
523C2×C4#55411yescharacteristic
524C2×C4#55911yescharacteristic
525C2×C4#55911yescharacteristic
526C2×C4#56011yescharacteristic
527C2×C4#56011yescharacteristic
528C2×C4#56011yescharacteristic
529C2×C4H(3,𝔽4)11yescharacteristic
530C8—42no—
531C8—42no—
532C8—42no—
533C8—42no—
534C8—42no—
535C8—81no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22—161no—
547C22#231811yescharacteristic
548C22#232911yescharacteristic
549C22#233511yescharacteristic
550C22#277811yescharacteristic
551C22#278011yescharacteristic
552C22#279711yescharacteristic
553C22#280411yescharacteristic
554C4—24no—
555C4—42no—
556C4—44no—
557C4—44no—
558C4—42no—
559C4—82no—
560C4—82no—
561C4—161no—
562C4—161no—
563C4—161no—
564C4#234511yescharacteristic
565C4#235711yescharacteristic
566C4#236811yescharacteristic
567C4#237511yescharacteristic
568C4#278211yescharacteristic
569C4#279011yescharacteristic
570C4#279611yescharacteristic
571C4#280711yescharacteristic
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—82no—
579C2#1445711yescharacteristic
580C2#3270412yes—
subgroupG/H
centerC2×C4H(3,𝔽4)
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2804
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11367C2semidirect
3. #11321C2semidirect
4. #11263C2semidirect
5. #11262C2semidirect
10. #10901C2semidirect
11. #10894C2semidirect
12. #10878C2semidirect
13. #10839C2semidirect
29. #1861C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18612
#113672
#113212
#112632
#112622
#110852
#110812
#109592
#109232
#109012
#108942
#108782
#108392
#107802
#107512