Finite Group Databasebeta

← #374575

Subgroups — #374575

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic, complemented
4C2×C4×#104C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic, complemented
6#11217C214yesmaximal, complemented
7#10857C211yesmaximal, characteristic, complemented
8#10839C211yesmaximal, characteristic, complemented
9#10781C214yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C2×#520C2212yes—
12C2×C4×#41C2211yescharacteristic
13#3079—24no—
14#3079C418yescomplemented
15C2×#477—21no—
16C2×#475—21no—
17C2×#476—21no—
18C2×#474—21no—
19C2×#428C2212yes—
20C2×#423C2211yescharacteristic
21C22×#104C2212yes—
22C2×#417C2212yes—
23C2×C4×M16C2211yescharacteristic
24Q8×C4⋊C4—21no—
25Q8×C4⋊C4—21no—
26D4×C4⋊C4—21no—
27D4×C4⋊C4—21no—
28D4×C4⋊C4—22no—
29D4×C4⋊C4—22no—
30#2317—22no—
31#2317C2212yes—
32C2×#391—22no—
33#2285—21no—
34#2285C2211yescharacteristic
35C2×#384—21no—
36C2×#384C2211yescharacteristic
37C2×C43C2211yescharacteristic
38#1993—22no—
39#1990—21no—
40#1990—21no—
41C4×#104—24no—
42C2×#343C2214yes—
43#1736C2214yes—
44C2×#339C2212yes—
45C2×#339C2214yes—
46C4⋊#95—22no—
47C4⋊#95C2214yes—
48C22×#41C2311yescharacteristic
49C23×Q8—21no—
50C22×M16C2312yes—
51C22×M16C2312yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yescomplemented
60#520—28no—
61#520—44no—
62C4×#41—22no—
63C4×#41—22no—
64C4×#41C2×C414yes—
65C4×#41D414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4—24no—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×#117—22no—
76C2×#117—21no—
77C2×#117C2312yes—
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#477—28no—
93#476—28no—
94#475—28no—
95#474—28no—
96#428—24no—
97#428C2×C418yescomplemented
98C2×#107—22no—
99C2×#106—22no—
100#423—22no—
101#423C2×C414yescomplemented
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×#104—24no—
105C2×#104—24no—
106C2×#104Q818yescomplemented
107#417—24no—
108#417C2×C418yes—
109C2×#103—22no—
110C2×#102—22no—
111C4×M16—22no—
112C4×M16—22no—
113C2×#97—21no—
114C2×C4×C8—21no—
115#391—48no—
116#389—24no—
117#389—24no—
118#389—22no—
119#389—22no—
120#385—24no—
121#385—24no—
122#385—24no—
123#384—22no—
124#384—22no—
125#384—22no—
126#384—21no—
127#384—21no—
128#384—21no—
129#384—21no—
130#384D411yescharacteristic
131#384D411yescharacteristic
132#382—22no—
133#382D412yes—
134#380—22no—
135#380—22no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C2×#95—28no—
143C43—22no—
144C43—22no—
145#343—44no—
146#340—24no—
147#340—24no—
148#339—24no—
149#339—44no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41C22×C412yes—
155C2×#41C2×D412yes—
156C2×#41C2×Q814yes—
157C2×#41#4114yes—
158C22×Q8—24no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—42no—
164C22×Q8C2×D411yescharacteristic
165C22×D4—21no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D414yes—
176C23×C4C2411yescharacteristic
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—24no—
183C2×M16—22no—
184C2×M16—24no—
185C2×M16C22×C414yes—
186C2×M16C2×Q814yes—
187C22×C8—22no—
188C22×C8—22no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—24no—
193C4⋊Q8—42no—
194C4⋊Q8—42no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×Q8C22⋊C414yes—
202C4×Q8C22⋊C414yes—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—416no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4C22⋊C414yes—
213C4×D4C22⋊C414yes—
214#117—21no—
215#117—21no—
216#117—24no—
217#117—24no—
218#117—24no—
219#117—24no—
220#117—48no—
221#117C22×C412yes—
222#117C2×D412yes—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—84no—
253C2×C4⋊C4—84no—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42C22×C412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×Q814yes—
289C2×C42#4112yes—
290C2×C42#4112yes—
291#107—28no—
292#106—28no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295#104—216no—
296#103—28no—
297#102—28no—
298#97—42no—
299C4×C8—42no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—22no—
304#95—416no—
305C24—21no—
306C24C22×D411yescharacteristic
307#41—24no—
308#41—24no—
309#41—24no—
310#41—24no—
311#41—24no—
312#41—24no—
313#41C4×Q818yes—
314#41C22⋊Q818yes—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—416no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8—88no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×Q8C2×C22⋊C412yes—
327C2×Q8#12412yes—
328C2×Q8C4⋊Q812yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4C2×C22⋊C412yes—
337C2×D4#12412yes—
338C2×D4C4⋊Q812yes—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4C4×D412yes—
378C22×C4C4×D412yes—
379C22×C4#12011yescharacteristic
380C22×C4#12011yescharacteristic
381C22×C4#12011yescharacteristic
382C22×C4#12011yescharacteristic
383C22×C4#12112yes—
384C22×C4#12112yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q812yes—
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4C2×#4111yescharacteristic
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395M16—28no—
396M16—24no—
397M16—24no—
398M16—44no—
399M16—42no—
400M16—42no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4C2×C22⋊C412yes—
434C4⋊C4C2×C22⋊C412yes—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C42—22no—
442C42—22no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—48no—
448C42—48no—
449C42—48no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42C2×C22⋊C412yes—
455C42C2×C22⋊C412yes—
456C42C4×D412yes—
457C42C4×D412yes—
458C42#12112yes—
459C42#12112yes—
460C42#12312yes—
461C42#12312yes—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23C2×C4×D411yescharacteristic
474C23C2×#12011yescharacteristic
475C23C2×#12111yescharacteristic
476C23#54612yes—
477C23D4×Q812yes—
478Q8—22no—
479Q8—22no—
480Q8—22no—
481Q8—88no—
482Q8—88no—
483Q8#39114yescomplemented
484D4—28no—
485D4—28no—
486D4—22no—
487D4—22no—
488D4—22no—
489D4#39114yescomplemented
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—84no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—88no—
533C2×C4—168no—
534C2×C4#38612yes—
535C2×C4#38612yes—
536C2×C4#38612yes—
537C2×C4#38612yes—
538C2×C4#45312yes—
539C2×C4#45412yes—
540C2×C4#45512yes—
541C2×C4#45612yes—
542C2×C4C22×C22⋊C411yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×C4×Q812yes—
545C2×C4#52012yes—
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×C22⋊Q812yes—
548C2×C4C2×#12311yescharacteristic
549C2×C4C2×#12411yescharacteristic
550C2×C4C2×C4⋊Q811yescharacteristic
551C2×C4#53511yescharacteristic
552C2×C4#53611yescharacteristic
553C2×C4#53712yes—
554C2×C4#54612yes—
555C2×C4D4×Q812yes—
556C2×C4#55212yes—
557C2×C4#55212yes—
558C2×C4#55412yes—
559C2×C4#55412yes—
560C8—44no—
561C8—44no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22C2×#38611yescharacteristic
578C22#236012yes—
579C22#241712yes—
580C22#243411yescharacteristic
581C22#243511yescharacteristic
582C22C2×#45311yescharacteristic
583C22C2×#45511yescharacteristic
584C22C2×#45411yescharacteristic
585C22C2×#45611yescharacteristic
586C4—24no—
587C4—22no—
588C4—22no—
589C4—48no—
590C4—42no—
591C4—42no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—84no—
596C4#188214yes—
597C4#188314yes—
598C4C2×#39111yescharacteristic
599C4#231111yescharacteristic
600C4#236012yes—
601C4#241712yes—
602C4#250211yescharacteristic
603C4#250311yescharacteristic
604C2—22no—
605C2—22no—
606C2—44no—
607C2#1413911yescharacteristic
608C2C2×#188211yescharacteristic
609C2C2×#188311yescharacteristic
610C2#2582911yescharacteristic
611C2#2583011yescharacteristic
612C2#2587511yescharacteristic
613C2#2587611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21324C2semidirect
5. #14714C2semidirect
6. #11217C2semidirect
7. #10857C2semidirect
8. #10839C2semidirect
14. #3079C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
97. #428C2×C4semidirect
101. #423C2×C4semidirect
106. C2×#104Q8semidirect
483. Q8#391semidirect
489. D4#391semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#341832
#213242
C2×C4×#1042
#147142
#112172
#108572
#108392
#107812