Finite Group Databasebeta

← #835204

Subgroups — #835204

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C212yesmaximal
2#16987C212yesmaximal
3#16111C212yesmaximal, complemented
4#16010C214yesmaximal, complemented
5#16002C212yesmaximal, complemented
6#15205C214yesmaximal
7#15179C212yesmaximal
8#15178C211yesmaximal, characteristic
9#14248C212yesmaximal
10#14142C212yesmaximal, complemented
11#14132C212yesmaximal, complemented
12#14024C212yesmaximal, complemented
13#14017C212yesmaximal, complemented
14#13924C211yesmaximal, characteristic
15C4×#397C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C2×#397C2211yescharacteristic
21#2360—216no—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#387C2214yes—
30C2×#385C2214yes—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2214yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2211yescharacteristic
38#1445C2214yes—
39#1445C2212yes—
40#1445C2212yes—
41#1444C2212yes—
42#1444C2212yes—
43#1444C2214yes—
44#1443C2212yes—
45#1443C2214yes—
46#1442C2211yescharacteristic
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2212yes—
55#1440C2214yes—
56#1439C2211yescharacteristic
57#1439C2212yes—
58#1439C2212yes—
59#1439C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—42no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—44no—
86C2×C4×Q8—44no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2314yes—
114#397Q818yescomplemented
115#391—28no—
116#391—28no—
117#391—28no—
118#391—44no—
119#389—28no—
120#389—216no—
121#389—28no—
122#387—216no—
123#385—216no—
124#380—42no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—42no—
127C4×C22⋊C4—28no—
128C4×C22⋊C4—216no—
129C4×C22⋊C4—42no—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C25C2411yescharacteristic
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8#4118yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4C2×D414yes—
179C23×C4C2×D418yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2414yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C4⋊Q8—416no—
195C22⋊Q8—264no—
196C4×Q8—432no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—416no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C2×D418yes—
228C2×C4⋊C4C2×D418yes—
229C2×C4⋊C4C2×Q814yes—
230C2×C4⋊C4#4118yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C2×Q818yes—
248C2×C22⋊C4C2×Q814yes—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265#95—24no—
266#95—28no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—24no—
277#95—24no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—44no—
287#95—41no—
288#95—44no—
289#95—41no—
290#95—41no—
291#95—42no—
292#95—42no—
293#95—44no—
294#95—44no—
295#95C2×Q814yes—
296#95C2×Q818yes—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—22no—
304C24—22no—
305C24C22×D412yes—
306C24C22×Q812yes—
307C242−1+412yes—
308C24C2511yescharacteristic
309C2×Q8—216no—
310C2×Q8—216no—
311C2×Q8—216no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—48no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—48no—
391C22×C4—416no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—44no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—88no—
398C22×C4C22×D418yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×Q814yes—
402C22×C4C22×Q812yes—
403C22×C4C22×Q814yes—
404C22×C4C22×Q812yes—
405C22×C4C2×#4118yes—
406C22×C4C2×#4118yes—
407C22×C4C2×#4114yes—
408C22×C4C2×#4114yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+412yes—
412C4⋊C4—232no—
413C4⋊C4—216no—
414C4⋊C4—216no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—416no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—88no—
423C4⋊C4—88no—
424C22⋊C4—232no—
425C22⋊C4—216no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—84no—
430C42—416no—
431C42—48no—
432C42—48no—
433C42—44no—
434C42—44no—
435C42—88no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23D4×Q814yes—
459C23C23×D411yescharacteristic
460C23C23×Q811yescharacteristic
461C23C22×#4112yes—
462C23C22×#4112yes—
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)11yescharacteristic
465C23C2×2−1+411yescharacteristic
466C23C2×2−1+411yescharacteristic
467C23#58512yes—
468C23#58512yes—
469C23#58511yescharacteristic
470Q8—432no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—416no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—816no—
554C2×C4#52518yes—
555C2×C4#53414yes—
556C2×C4#55114yes—
557C2×C4#55318yes—
558C2×C4#55914yes—
559C22—24no—
560C22—28no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22C2×#52512yes—
571C22C2×#53411yescharacteristic
572C22C2×D4×Q811yescharacteristic
573C22C2×#55111yescharacteristic
574C22C2×#55312yes—
575C22Q8×#4112yes—
576C22Q8×#4112yes—
577C22C2×#55911yescharacteristic
578C22#348512yes—
579C22#349711yescharacteristic
580C22#350111yescharacteristic
581C22#350311yescharacteristic
582C22#351512yes—
583C22#351612yes—
584C22#351612yes—
585C22#351612yes—
586C22#352112yes—
587C22#352412yes—
588C22#353811yescharacteristic
589C22#354011yescharacteristic
590C22#355111yescharacteristic
591C22#356111yescharacteristic
592C22#356212yes—
593C4—42no—
594C4—42no—
595C4—48no—
596C4—42no—
597C4—48no—
598C4—44no—
599C4—44no—
600C4—48no—
601C4—42no—
602C4—84no—
603C4—81no—
604C4—82no—
605C4—82no—
606C4#260018yes—
607C2—44no—
608C2C2×#260011yescharacteristic
609C2#3580212yes—
610C2#3654712yes—
611C2#3961511yescharacteristic
612C2#4267311yescharacteristic
613C2#4519412yes—
614C2#4522611yescharacteristic
615C2#4542011yescharacteristic
616C2#5962911yescharacteristic
617C2#5969412yes—
618C2#5993311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16111C2semidirect
4. #16010C2semidirect
5. #16002C2semidirect
10. #14142C2semidirect
11. #14132C2semidirect
12. #14024C2semidirect
13. #14017C2semidirect
114. #397Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
#169872
#161112
#160102
#160022
#152052
#151792
#151782
#142482
#141422
#141322
#140242
#140172
#139242
C4×#3972