Finite Group Databasebeta

← #530810

Subgroups — #530810

Order 512.

2718 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
1C22×#389C211yesmaximal, characteristic
2#14684C211yesmaximal, characteristic
3#14490C212yesmaximal, complemented
4#14487C212yesmaximal, complemented
5#14460C214yesmaximal, complemented
6#14286C212yesmaximal, complemented
7#14057C214yesmaximal, complemented
8#13904C212yesmaximal
9#13882C212yesmaximal
10#13874C212yesmaximal
11#13870C212yesmaximal
12#13870C212yesmaximal
13#13799C212yesmaximal
14#13792C212yesmaximal, complemented
15C4×#381C211yesmaximal, characteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C42C2211yescharacteristic
21C2×#389C4116yescomplemented
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#381C2212yes—
32C2×#381C2211yescharacteristic
33C2×#381C2212yes—
34C2×#381C2211yescharacteristic
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C22×#95C2212yes—
40#1445C2214yes—
41#1445C2214yes—
42#1445C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2214yes—
59C4⋊#95C2214yes—
60#1434C2214yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#117—28no—
82C2×#117—44no—
83C2×#117—44no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2×C418yes—
90C22×C4⋊C4C2×C418yes—
91C22×C4⋊C4C2×C4116yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42—42no—
109C22×C42—42no—
110C22×C42C2×C418yes—
111C22×C42C2314yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#389C2×C4116yes—
120#389C2×C418yes—
121#389C2×C418yes—
122#389C2×C4116yes—
123#389C2×C418yes—
124#389C2×C418yes—
125#387—216no—
126#381—28no—
127#381—28no—
128#381—42no—
129#381—44no—
130#381—44no—
131#381—42no—
132#380—44no—
133C4×C22⋊C4—42no—
134C4×C22⋊C4—44no—
135C4×C22⋊C4—42no—
136C2×#95C2×C4116yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C25C2411yescharacteristic
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—28no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4C22×C418yes—
191C23×C4C22×C414yes—
192C23×C4C22×C414yes—
193C23×C4C22×C418yes—
194C23×C4C22×C414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209#117—432no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—232no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—416no—
216C2×C4⋊C4—432no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4C22×C418yes—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C418yes—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4#4118yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4#4118yes—
258C2×C42—28no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—416no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42C22×C414yes—
280C2×C42C22×C414yes—
281C2×C42C22×C414yes—
282C2×C42C22×C414yes—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C22×C418yes—
310#95C22×C418yes—
311#95C22×C418yes—
312#95C22×C418yes—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24C23×C412yes—
318C24C2×#4114yes—
319C24H(5,𝔽2)12yes—
320C24H(5,𝔽2)14yes—
321C24H(5,𝔽2)12yes—
322C24C2511yescharacteristic
323C22×C4—216no—
324C22×C4—216no—
325C22×C4—28no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
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—22no—
381C22×C4—24no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
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—48no—
406C22×C4—44no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4C23×C412yes—
418C22×C4C23×C414yes—
419C22×C4C23×C414yes—
420C22×C4C23×C414yes—
421C22×C4C23×C412yes—
422C22×C4C23×C412yes—
423C22×C4C23×C414yes—
424C22×C4C23×C412yes—
425C22×C4C23×C412yes—
426C22×C4C23×C412yes—
427C22×C4C2×#4118yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4118yes—
430C22×C42−1+414yes—
431C22×C42−1+414yes—
432C4⋊C4—216no—
433C4⋊C4—28no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—416no—
442C4⋊C4—88no—
443C4⋊C4#117116yescomplemented
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C42—44no—
449C42—44no—
450C42—44no—
451C42—44no—
452C42—416no—
453C42—88no—
454C42—84no—
455C42—84no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23C4×#4114yes—
469C23#51814yes—
470C23#51814yes—
471C23#53818yes—
472C23#54018yes—
473C23C24×C411yescharacteristic
474C23C22×#4112yes—
475C23C22×#4112yes—
476C23C2×H(5,𝔽2)11yescharacteristic
477C23C2×H(5,𝔽2)12yes—
478C23C2×H(5,𝔽2)11yescharacteristic
479C23C2×2−1+412yes—
480C23C2×2−1+412yes—
481C23#58512yes—
482C2×C4—232no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—416no—
521C2×C4—432no—
522C2×C4—416no—
523C2×C4—432no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
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—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4C2×#11714yes—
566C2×C4C2×#11718yes—
567C2×C4#51914yes—
568C2×C4#51914yes—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—216no—
576C22—216no—
577C22#233318yes—
578C22C22×#11711yescharacteristic
579C22C2×C4×#4111yescharacteristic
580C22C2×#51811yescharacteristic
581C22C2×#51811yescharacteristic
582C22C2×#51911yescharacteristic
583C22C2×#51911yescharacteristic
584C22#342811yescharacteristic
585C22C2×#53812yes—
586C22C2×#54012yes—
587C22#348214yes—
588C22#349212yes—
589C22#349212yes—
590C22#351314yes—
591C22#351514yes—
592C22#353512yes—
593C22#353512yes—
594C22#354414yes—
595C4—24no—
596C4—24no—
597C4—24no—
598C4—28no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—44no—
603C4—42no—
604C4—44no—
605C4—48no—
606C4—84no—
607C4—84no—
608C4—84no—
609C2—28no—
610C2C2×#233311yescharacteristic
611C2#3412811yescharacteristic
612C2#3417811yescharacteristic
613C2#3419011yescharacteristic
614C2#3423911yescharacteristic
615C2#3435911yescharacteristic
616C2#3436511yescharacteristic
617C2#3860514yes—
618C2#3860714yes—
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
3. #14490C2semidirect
4. #14487C2semidirect
5. #14460C2semidirect
6. #14286C2semidirect
7. #14057C2semidirect
14. #13792C2semidirect
21. C2×#389C4semidirect
443. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#146842
#144902
#144872
#144602
#142862
#140572
#139042
#138822
#138742
#138702
#137992
#137922
C4×#3812