Finite Group Databasebeta

← #362122

Subgroups — #362122

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2906C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1815C212yesmaximal
5#20648C211yesmaximal, characteristic
6C4×#417C211yesmaximal, characteristic
7#11242C212yesmaximal, complemented
8#11238C212yesmaximal, complemented
9#10857C211yesmaximal, characteristic, complemented
10#10847C211yesmaximal, characteristic, complemented
11#10774C212yesmaximal
12C22×#131C2212yes—
13C2×C4×#41C2211yescharacteristic
14C22×#108C2212yes—
15C22×#106C2212yes—
16#2906—24no—
17#2906C418yescomplemented
18C2×#424—22no—
19C2×#424C2212yes—
20C2×#423—21no—
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8—21no—
24C22×C4⋊C8C2211yescharacteristic
25C2×#417C2212yes—
26C2×#417C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1950—24no—
32#1948—21no—
33#1948—21no—
34#1948—22no—
35#1815—28no—
36#1803—21no—
37#1803—21no—
38#1802—21no—
39#1802—21no—
40#1763—21no—
41#1762—21no—
42C4×#103—21no—
43C4×#102—21no—
44C2×#343C2212yes—
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2212yes—
48C2×#341C2212yes—
49#1736C2212yes—
50#1736C2211yescharacteristic
51#1736C2211yescharacteristic
52C2×#340C2212yes—
53C22×#41C2311yescharacteristic
54C2×#131—24no—
55C2×#131—24no—
56C2×#131—24no—
57C2×#131—42no—
58C2×#131—42no—
59C2×#131C2×C418yes—
60C22×M16—22no—
61C22×M16C2312yes—
62C22×M16C2312yes—
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C23×C8—22no—
66C23×C8C2312yes—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41C2×C414yes—
70C4×#41Q814yes—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×#117—21no—
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C2×#108—24no—
88C2×#108—24no—
89C2×#108—42no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—24no—
93C2×#106—42no—
94#424—24no—
95#424—216no—
96#424C2×C418yes—
97#423—22no—
98#423—28no—
99#423—22no—
100#423—22no—
101#423C2×C414yes—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—28no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2×C414yes—
110C2×C4⋊C8D411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D412yes—
113C2×C4⋊C8D414yes—
114#417—22no—
115#417—22no—
116#417—24no—
117#417—24no—
118#417D414yes—
119#417Q814yes—
120C2×#103—21no—
121C2×#103—21no—
122C2×#103—22no—
123C2×#102—21no—
124C2×#102—21no—
125C2×#102—22no—
126C4×M16—22no—
127C4×M16—22no—
128C2×#97—21no—
129C2×C4×C8—21no—
130#384—21no—
131#384—21no—
132#384—21no—
133#384—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C2×#95—21no—
139#343—24no—
140#343—24no—
141#343—42no—
142#343—42no—
143#341—24no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41C22×C412yes—
159C2×#41C2×D412yes—
160C2×#41C2×Q812yes—
161C2×#41C2×Q812yes—
162C2×#41#4112yes—
163C2×#41#4112yes—
164C22×Q8C2×D411yescharacteristic
165C22×D4—21no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4#4111yescharacteristic
174C23×C4#4111yescharacteristic
175C23×C4C2411yescharacteristic
176#131—216no—
177#131—48no—
178#131—48no—
179#131—48no—
180C2×M16—22no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—28no—
184C2×M16—22no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—22no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16C22×C414yes—
205C2×M16C2×D412yes—
206C2×M16C2×Q812yes—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—28no—
215C22×C8—22no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8C22×C414yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D414yes—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8C4⋊C414yes—
225C4×Q8C4⋊C414yes—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C4⋊C414yes—
233C4×D4C4⋊C414yes—
234#117—21no—
235#117—21no—
236#117—24no—
237#117—24no—
238#117C22×C412yes—
239#117C2×Q812yes—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×Q812yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—82no—
279C2×C42C22×C412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×Q812yes—
286C2×C42#4112yes—
287#108—48no—
288#106—44no—
289#106—44no—
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8C22⋊C414yes—
297C4⋊C8C22⋊C414yes—
298C4⋊C8C22⋊C418yes—
299#103—24no—
300#103—24no—
301#103—44no—
302#102—24no—
303#102—24no—
304#102—44no—
305#97—42no—
306C4×C8—42no—
307#95—41no—
308#95—41no—
309C24—21no—
310C24C2×#4111yescharacteristic
311#41—22no—
312#41—22no—
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—22no—
322#41—22no—
323#41C4×D414yes—
324#41C4×Q814yes—
325#41#12514yes—
326#41C4⋊Q814yes—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8C2×C4⋊C412yes—
331C2×Q8#12112yes—
332C2×Q8C22⋊Q812yes—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4C2×C4⋊C412yes—
343C2×D4#12112yes—
344C2×D4C22⋊Q812yes—
345C22×C4—21no—
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—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4#11712yes—
380C22×C4#11712yes—
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4C22⋊Q812yes—
386C22×C4C22⋊Q812yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×Q811yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C42−1+411yescharacteristic
400C22×C42−1+411yescharacteristic
401M16—24no—
402M16—24no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—22no—
407M16—416no—
408M16—42no—
409M16—41no—
410M16—42no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—42no—
415M16—42no—
416M16—42no—
417M16—42no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—416no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8C2×C22⋊C414yes—
440C2×C8C2×C22⋊C414yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4C2×C4⋊C412yes—
462C4⋊C4C2×C4⋊C412yes—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—82no—
472C42—82no—
473C42C2×C22⋊C412yes—
474C42C2×C22⋊C412yes—
475C42C2×C4⋊C412yes—
476C42C2×C4⋊C412yes—
477C42#12112yes—
478C42#12112yes—
479C42C22⋊Q812yes—
480C42C22⋊Q812yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×#11711yescharacteristic
493C23C2×#12111yescharacteristic
494C23C2×C22⋊Q811yescharacteristic
495C23#54711yescharacteristic
496C23#55211yescharacteristic
497C23#55311yescharacteristic
498C23#55411yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—22no—
502Q8#38914yes—
503D4—24no—
504D4—24no—
505D4—24no—
506D4—24no—
507D4—22no—
508D4—22no—
509D4—22no—
510D4#38914yes—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—162no—
550C2×C4—162no—
551C2×C4#38012yes—
552C2×C4#38012yes—
553C2×C4#38012yes—
554C2×C4#38012yes—
555C2×C4#47112yes—
556C2×C4#47112yes—
557C2×C4C22×C22⋊C411yescharacteristic
558C2×C4C22×C4⋊C411yescharacteristic
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×C4×Q811yescharacteristic
561C2×C4#51811yescharacteristic
562C2×C4#51911yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×C22⋊Q811yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4C2×#12511yescharacteristic
568C2×C4C2×C4⋊Q811yescharacteristic
569C2×C4#53711yescharacteristic
570C2×C4#53711yescharacteristic
571C2×C4#54411yescharacteristic
572C2×C4#54411yescharacteristic
573C2×C4D4211yescharacteristic
574C2×C4D4211yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54611yescharacteristic
577C2×C4#54711yescharacteristic
578C2×C4#54811yescharacteristic
579C2×C4#54811yescharacteristic
580C2×C4D4×Q811yescharacteristic
581C2×C4D4×Q811yescharacteristic
582C2×C4#55211yescharacteristic
583C2×C4#55311yescharacteristic
584C2×C4#55411yescharacteristic
585C8—24no—
586C8—24no—
587C8—44no—
588C8—42no—
589C8—42no—
590C8—82no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22C2×#38011yescharacteristic
611C22#234411yescharacteristic
612C22#234511yescharacteristic
613C22#241611yescharacteristic
614C22#241711yescharacteristic
615C22#244011yescharacteristic
616C22#244211yescharacteristic
617C22C2×#47111yescharacteristic
618C22C2×#47111yescharacteristic
619C22#306811yescharacteristic
620C22#306911yescharacteristic
621C4—24no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4—162no—
630C4#181514yescomplemented
631C4C2×#38911yescharacteristic
632C4#231611yescharacteristic
633C4D4×C22⋊C411yescharacteristic
634C4Q8×C22⋊C411yescharacteristic
635C4#241611yescharacteristic
636C4#241711yescharacteristic
637C4#244011yescharacteristic
638C4#244211yescharacteristic
639C2—21no—
640C2—21no—
641C2—21no—
642C2—21no—
643C2—44no—
644C2#1390711yescharacteristic
645C2C2×#181511yescharacteristic
646C2#2067911yescharacteristic
647C2#2477111yescharacteristic
648C2#2477311yescharacteristic
649C2#2485611yescharacteristic
650C2#2485811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
3. C2×#1948C2semidirect
7. #11242C2semidirect
8. #11238C2semidirect
9. #10857C2semidirect
10. #10847C2semidirect
17. #2906C4semidirect
630. C4#1815semidirect

Maximal subgroups

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

subgroupindex
C2×#29062
C2×#19502
C2×#19482
C2×#18152
#206482
C4×#4172
#112422
#112382
#108572
#108472
#107742