Finite Group Databasebeta

← #515374

Subgroups — #515374

Order 512.

3154 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic
4#14083C214yesmaximal, complemented
5#14081C214yesmaximal, complemented
6#14035C214yesmaximal, complemented
7#14033C214yesmaximal, complemented
8#13882C212yesmaximal
9#13878C212yesmaximal
10#13874C212yesmaximal
11#13870C212yesmaximal
12#13776C211yesmaximal, characteristic
13#13776C211yesmaximal, characteristic
14C4×#380C211yesmaximal, characteristic
15#13753C211yesmaximal, characteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2313—28no—
23C2×#389C4116yescomplemented
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#381C2212yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36#2274—28no—
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C4⋊C4C2214yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69#1434C2212yes—
70#1434C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C4×#95C2212yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#117—28no—
86C2×#117—28no—
87C2×#117—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2×C418yes—
95C22×C4⋊C4C2×C418yes—
96C22×C4⋊C4C2×C418yes—
97C22×C4⋊C4C2×C418yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—44no—
114C22×C42C2×C418yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#389C2×C4116yes—
126#389C2×C4116yes—
127#389C2×C4116yes—
128#389C2×C4116yes—
129#387—216no—
130#387—216no—
131#387—48no—
132#381—28no—
133#381—28no—
134#381—28no—
135#380—28no—
136#380—28no—
137C4×C4⋊C4—216no—
138C4×C4⋊C4—216no—
139C4×C4⋊C4—44no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4D418yes—
145#376—44no—
146C2×#95C2×C418yes—
147C2×#95C2×C418yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25C2411yescharacteristic
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C22×C414yes—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C22×C414yes—
205C23×C4C22×C414yes—
206C23×C4C2×D414yes—
207C23×C4#4118yes—
208C23×C4#4114yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223#117—264no—
224C2×C4⋊C4—232no—
225C2×C4⋊C4—232no—
226C2×C4⋊C4—232no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C418yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C418yes—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C418yes—
258C2×C4⋊C4C22×C418yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4C2×D418yes—
270C2×C22⋊C4#4118yes—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—416no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C22×C414yes—
294C2×C42C22×C414yes—
295C2×C42C22×C414yes—
296C2×C42C22×C414yes—
297C2×C42C2×D418yes—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C22×C418yes—
321#95C22×C418yes—
322#95C22×C418yes—
323#95C22×C418yes—
324#95C2×D418yes—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24C23×C412yes—
330C24C22×D412yes—
331C24C2×#4114yes—
332C24C2×#4112yes—
333C24C2×#4114yes—
334C24C2511yescharacteristic
335C22×C4—232no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4#117116yes—
424C22×C4C4×D4116yes—
425C22×C4C23×C412yes—
426C22×C4C23×C414yes—
427C22×C4C23×C414yes—
428C22×C4C23×C414yes—
429C22×C4C23×C414yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C22×D418yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+414yes—
445C22×C42−1+414yes—
446C4⋊C4—232no—
447C4⋊C4—216no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—48no—
455C4⋊C4—416no—
456C4⋊C4—88no—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C42—232no—
462C42—28no—
463C42—28no—
464C42—44no—
465C42—44no—
466C42—48no—
467C42—48no—
468C42—84no—
469C42—84no—
470C42—88no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23C2×#11714yes—
484C23C2×C4×D414yes—
485C23C4×#4114yes—
486C23#52518yes—
487C23#52918yes—
488C23C24×C411yescharacteristic
489C23C23×D411yescharacteristic
490C23C22×#4112yes—
491C23C22×#4111yescharacteristic
492C23C22×#4112yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×2−1+412yes—
495C23C2×2−1+412yes—
496C23#58512yes—
497C2×C4—232no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—232no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—416no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—416no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4C2×#11718yes—
577C2×C4C2×C4×D414yes—
578C2×C4C2×C4×D414yes—
579C2×C4#51814yes—
580C2×C4#51914yes—
581C2×C4#51914yes—
582C22—216no—
583C22—216no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22#229018yes—
591C22C22×#11711yescharacteristic
592C22C22×C4×D411yescharacteristic
593C22C2×C4×#4111yescharacteristic
594C22C2×#51811yescharacteristic
595C22C2×#51911yescharacteristic
596C22C2×#51911yescharacteristic
597C22#342811yescharacteristic
598C22C2×#52512yes—
599C22C2×#52912yes—
600C22#349214yes—
601C22#349612yes—
602C22#349912yes—
603C22#350512yes—
604C22#350714yes—
605C22#350912yes—
606C22#351514yes—
607C22#352014yes—
608C4—24no—
609C4—24no—
610C4—24no—
611C4—24no—
612C4—24no—
613C4—28no—
614C4—48no—
615C4—48no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—44no—
620C4—84no—
621C4—84no—
622C2—28no—
623C2C2×#229011yescharacteristic
624C2#3417811yescharacteristic
625C2#3418711yescharacteristic
626C2#3420511yescharacteristic
627C2#3420911yescharacteristic
628C2#3423911yescharacteristic
629C2#3424511yescharacteristic
630C2#3536314yes—
631C2#3538014yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14083C2semidirect
5. #14081C2semidirect
6. #14035C2semidirect
7. #14033C2semidirect
23. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
C22×#3892
C2×#22742
#140832
#140812
#140352
#140332
#138822
#138782
#138742
#138702
#137762
C4×#3802
#137532