Finite Group Databasebeta

← #519247

Subgroups — #519247

Order 512.

3298 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#2293C212yesmaximal
3#14707C211yesmaximal, characteristic, complemented
4#14699C212yesmaximal, complemented
5#14150C211yesmaximal, characteristic, complemented
6#14132C212yesmaximal, complemented
7#14108C214yesmaximal, complemented
8#14063C214yesmaximal, complemented
9#13983C214yesmaximal, complemented
10#13941C211yesmaximal, characteristic
11#13916C212yesmaximal
12#13804C211yesmaximal, characteristic, complemented
13#13799C214yesmaximal, complemented
14C4×#390C211yesmaximal, characteristic
15C42×C4⋊C4C211yesmaximal, characteristic, complemented
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2214yes—
19#2349C4116yescomplemented
20#2293—216no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#385C2214yes—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#381C2214yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2214yes—
43C22×#95C2212yes—
44C2×C43C2212yes—
45C2×C43C2211yescharacteristic
46#1445C2214yes—
47#1444C2212yes—
48#1442C2212yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1437C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68#1434C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C2×C4×D4—28no—
81C2×C4×D4—28no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4C2×C4116yes—
85C2×C4×D4C2×C418yes—
86C2×#117—216no—
87C2×#117—216no—
88C2×#117—48no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#390—28no—
120#390—24no—
121#390D418yes—
122#389—28no—
123#389—28no—
124#389C2×C4116yes—
125#388—216no—
126#388—216no—
127#387—48no—
128#385C2×C4132yes—
129#382—24no—
130#382C2×C418yes—
131#381—216no—
132#380C2×C4116yes—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4C2×C418yes—
139C4×C22⋊C4—216no—
140C4×C22⋊C4—48no—
141C2×#95C2×C4116yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C43—28no—
169C43—24no—
170C25C2412yes—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4C22×C414yes—
176C22×D4C2×D414yes—
177C22×D4#4114yes—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C22×C4116yes—
189C23×C4C22×C418yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C4×D4—264no—
205C4×D4—232no—
206#117—464no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C418yes—
236C2×C4⋊C4C22×C4116yes—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C22×C4116yes—
254C2×C22⋊C4C22×C418yes—
255C2×C22⋊C4C2×D4116yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—44no—
288C2×C42C22×C418yes—
289C2×C42C22×C414yes—
290C2×C42C2×D414yes—
291C2×C42#4118yes—
292C2×C42#4114yes—
293C2×C42#4114yes—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C22×C414yes—
315#95C22×C418yes—
316#95C22×C4116yes—
317#95C22×C414yes—
318#95C22×C418yes—
319#95#4118yes—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24C23×C414yes—
329C24C22×D414yes—
330C24C2×#4114yes—
331C24C2511yescharacteristic
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4C4×D4116yes—
339C22×C4—22no—
340C22×C4—22no—
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—232no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—232no—
391C22×C4—216no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4C23×C418yes—
414C22×C4C23×C414yes—
415C22×C4C23×C414yes—
416C22×C4C23×C412yes—
417C22×C4C23×C412yes—
418C22×C4C23×C414yes—
419C22×C4C23×C412yes—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—28no—
445C4⋊C4—416no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C22⋊C4—232no—
451C22⋊C4—216no—
452C22⋊C4—232no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C42—216no—
457C42—216no—
458C42—216no—
459C42—28no—
460C42—48no—
461C42—48no—
462C42—48no—
463C42—416no—
464C42—48no—
465C42—48no—
466C42—44no—
467C42—84no—
468C42#117116yescomplemented
469C23—24no—
470C23—28no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—42no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23C2×C4×D418yes—
492C23C24×C411yescharacteristic
493C23C23×D411yescharacteristic
494C23C22×#4111yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4112yes—
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23#58512yes—
503D4—232no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—464no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—816no—
583C2×C4C2×#11714yes—
584C2×C4C2×#11714yes—
585C2×C4C2×#11714yes—
586C2×C4C2×C4×D414yes—
587C2×C4C4×#4118yes—
588C2×C4#51914yes—
589C2×C4#54618yes—
590C2×C4#54714yes—
591C2×C4#54714yes—
592C2×C4#55214yes—
593C2×C4#55214yes—
594C22—28no—
595C22—216no—
596C22—48no—
597C22—44no—
598C22—48no—
599C22—44no—
600C22—44no—
601C22—48no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22C22×#11711yescharacteristic
606C22C22×C4×D411yescharacteristic
607C22C2×C4×#4112yes—
608C22C2×#51911yescharacteristic
609C22C4×H(5,𝔽2)12yes—
610C22#344411yescharacteristic
611C22#344412yes—
612C22#344411yescharacteristic
613C22C2×#54612yes—
614C22C2×#54711yescharacteristic
615C22C2×#54711yescharacteristic
616C22D4×#4112yes—
617C22C2×#55211yescharacteristic
618C22C2×#55211yescharacteristic
619C22#347512yes—
620C22#348112yes—
621C22#349412yes—
622C22#350712yes—
623C22#350811yescharacteristic
624C22#351011yescharacteristic
625C22#351112yes—
626C22#351512yes—
627C22#352212yes—
628C4—28no—
629C4—28no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—48no—
638C4—44no—
639C4—82no—
640C4—84no—
641C4#234418yes—
642C2—48no—
643C2#3402611yescharacteristic
644C2C2×#234411yescharacteristic
645C2C4×#54612yes—
646C2#3425511yescharacteristic
647C2#3431712yes—
648C2#3613511yescharacteristic
649C2#3613912yes—
650C2#3614911yescharacteristic
651C2#3637011yescharacteristic
652C2#3637212yes—
653C2#3639411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14707C2semidirect
4. #14699C2semidirect
5. #14150C2semidirect
6. #14132C2semidirect
7. #14108C2semidirect
8. #14063C2semidirect
9. #13983C2semidirect
12. #13804C2semidirect
13. #13799C2semidirect
15. C42×C4⋊C4C2semidirect
19. #2349C4semidirect
468. C42#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#22932
#147072
#146992
#141502
#141322
#141082
#140632
#139832
#139412
#139162
#138042
#137992
C4×#3902
C42×C4⋊C42