Finite Group Databasebeta

← #374595

Subgroups — #374595

Order 512.

1581 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic, complemented
4#21306C211yesmaximal, characteristic, complemented
5C4×#418C211yesmaximal, characteristic
6#11213C214yesmaximal, complemented
7#10855C211yesmaximal, characteristic, complemented
8#10836C211yesmaximal, characteristic, complemented
9#10785C214yesmaximal
10C2×#533C2211yescharacteristic
11C2×#518C2212yes—
12C22×C4×D4C2211yescharacteristic
13#3077—22no—
14#3077—22no—
15#3077—22no—
16#3077—22no—
17C2×#478C414yescomplemented
18C2×#476C412yescomplemented
19C2×#474C412yescomplemented
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#418C2212yes—
23C2×#418C2212yes—
24C2×C4×M16C2211yescharacteristic
25#2349—22no—
26#2349—22no—
27D4×C4⋊C4—22no—
28D4×C4⋊C4—22no—
29#2316C2212yes—
30#2315—22no—
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33C2×#380—22no—
34C4×#117C2211yescharacteristic
35#1996—22no—
36#1996—22no—
37#1986—22no—
38#1986—22no—
39#1762—22no—
40C4×#102—22no—
41C2×#344C2214yes—
42#1740C2214yes—
43#1736C2212yes—
44#1733C2214yes—
45C2×H(3,ℤ/4)C2214yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C22×M16C2312yes—
49C22×M16C2312yes—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58C2×C4⋊Q8C2×C411yescharacteristic, complemented
59C2×C4⋊Q8C2×C411yescharacteristic, complemented
60C2×#125C2×C412yescomplemented
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63#518—28no—
64#518—44no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—24no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—24no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×C4×D4C2×C412yes—
77C2×C4×D4D412yes—
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×C4×D4D412yes—
81C2×#117C2312yes—
82C2×#117C2312yes—
83C2×#117C2312yes—
84C2×#117C2312yes—
85C22×C4⋊C4—24no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—24no—
89C22×C42C2311yescharacteristic
90#478—28no—
91#478—28no—
92#476—24no—
93#476—24no—
94#474—24no—
95#474—24no—
96#428—24no—
97#428—24no—
98C2×#107C2×C414yescomplemented
99C2×#106C2×C414yescomplemented
100#423—21no—
101#423—21no—
102#423—21no—
103#423—21no—
104C2×C4⋊C8C2×C412yescomplemented
105C2×C4⋊C8C2×C412yescomplemented
106#418—24no—
107#418—24no—
108#418—24no—
109#418Q818yescomplemented
110C2×#102—24no—
111C2×#102C2×C414yes—
112C2×#102C2×C414yes—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16—21no—
116C4×M16—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119#389—24no—
120#389—24no—
121#385—28no—
122#385—28no—
123#384—21no—
124#384—21no—
125#382—21no—
126#382—21no—
127#380—24no—
128#380—24no—
129#380—21no—
130#380—21no—
131#380—24no—
132#380D412yes—
133#380D412yes—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4D412yes—
139C4×C4⋊C4D412yes—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143C2×#95—24no—
144C43—21no—
145#344—44no—
146#342—28no—
147#340—24no—
148#339—28no—
149H(3,ℤ/4)—44no—
150C25—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4C22×C412yes—
159C22×D4C2×D412yes—
160C22×D4C2×D412yes—
161C22×D4C2×Q814yes—
162C22×D4#4114yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—28no—
171C23×C4—84no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—24no—
179C2×M16—24no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16C2×Q814yes—
185C22×C8—22no—
186C22×C8C22×C414yes—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195#125—22no—
196#125—22no—
197#125—22no—
198#125—22no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—416no—
228C4×D4—48no—
229C4×D4—48no—
230C4×D4C22⋊C414yes—
231C4×D4C22⋊C414yes—
232C4×D4C22⋊C414yes—
233C4×D4C22⋊C414yes—
234#117—24no—
235#117—24no—
236#117—22no—
237#117—24no—
238#117—24no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117C2×Q814yes—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—88no—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—416no—
278C2×C22⋊C4C2×D414yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D414yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4111yescharacteristic
294C2×C42#4111yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42#4111yescharacteristic
297#107—28no—
298#106—28no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303#102—28no—
304#102—28no—
305#102—216no—
306#97—41no—
307#97—41no—
308C4×C8—41no—
309C4×C8—41no—
310#95—28no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—41no—
316#95—41no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—82no—
323C24—82no—
324C24C22×D411yescharacteristic
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4C2×C22⋊C414yes—
345C2×D4C4×Q818yes—
346C2×D4C22⋊Q818yes—
347C2×D4#12414yes—
348C2×D4C4⋊Q814yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—416no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—81no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—82no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4#12011yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12011yescharacteristic
380C22×C4#12011yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
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—24no—
396M16—24no—
397M16—42no—
398M16—42no—
399M16—44no—
400M16—42no—
401M16—42no—
402C2×C8—24no—
403C2×C8—28no—
404C2×C8—24no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—88no—
432C4⋊C4C2×C22⋊C414yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—28no—
438C22⋊C4—24no—
439C22⋊C4—416no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C42—28no—
445C42—24no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—82no—
452C42—82no—
453C42C2×C22⋊C411yescharacteristic
454C42C2×C22⋊C411yescharacteristic
455C42C2×C22⋊C411yescharacteristic
456C42C2×C22⋊C411yescharacteristic
457C42C4×D411yescharacteristic
458C42C4×D411yescharacteristic
459C42C4×D411yescharacteristic
460C42C4×D411yescharacteristic
461C42#12111yescharacteristic
462C42#12111yescharacteristic
463C42#12111yescharacteristic
464C42#12111yescharacteristic
465C42#12311yescharacteristic
466C42#12311yescharacteristic
467C42#12311yescharacteristic
468C42#12311yescharacteristic
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—41no—
475C23—41no—
476C23—44no—
477C23—44no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—88no—
483C23—88no—
484C23C2×C4×D411yescharacteristic
485C23C2×#12011yescharacteristic
486C23C2×#12111yescharacteristic
487C23#54612yes—
488C23D4×Q812yes—
489Q8—82no—
490Q8—82no—
491Q8—82no—
492Q8—82no—
493D4—24no—
494D4—28no—
495D4—28no—
496D4—24no—
497D4—24no—
498D4#39118yes—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—1616no—
533C2×C4#38611yescharacteristic
534C2×C4#38611yescharacteristic
535C2×C4#38611yescharacteristic
536C2×C4#38611yescharacteristic
537C2×C4#38611yescharacteristic
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4#38611yescharacteristic
541C2×C4C22×C22⋊C411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×Q812yes—
544C2×C4#52012yes—
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×C22⋊Q812yes—
547C2×C4C2×#12311yescharacteristic
548C2×C4C2×#12411yescharacteristic
549C2×C4C2×C4⋊Q811yescharacteristic
550C2×C4#53511yescharacteristic
551C2×C4#53611yescharacteristic
552C2×C4#53712yes—
553C2×C4#54612yes—
554C2×C4D4×Q812yes—
555C2×C4#55212yes—
556C2×C4#55212yes—
557C2×C4#55412yes—
558C2×C4#55412yes—
559C8—44no—
560C8—44no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—28no—
568C22—42no—
569C22—42no—
570C22—48no—
571C22—48no—
572C22—44no—
573C22—44no—
574C22—88no—
575C22—88no—
576C22C2×#38611yescharacteristic
577C22#236012yes—
578C22#241712yes—
579C22#243411yescharacteristic
580C22#243511yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—44no—
589C4—84no—
590C4—84no—
591C4—84no—
592C4—82no—
593C4—82no—
594C4#190412yes—
595C4C2×#39111yescharacteristic
596C4#231111yescharacteristic
597C4#236012yes—
598C4#241712yes—
599C4#250211yescharacteristic
600C4#250311yescharacteristic
601C2—21no—
602C2—22no—
603C2—21no—
604C2—22no—
605C2—48no—
606C2#1413911yescharacteristic
607C2C2×#190411yescharacteristic
608C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21332C2semidirect
4. #21306C2semidirect
6. #11213C2semidirect
7. #10855C2semidirect
8. #10836C2semidirect
17. C2×#478C4semidirect
18. C2×#476C4semidirect
19. C2×#474C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
60. C2×#125C2×C4semidirect
98. C2×#107C2×C4semidirect
99. C2×#106C2×C4semidirect
104. C2×C4⋊C8C2×C4semidirect
105. C2×C4⋊C8C2×C4semidirect
109. #418Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#341812
#213322
#213062
C4×#4182
#112132
#108552
#108362
#107852