Finite Group Databasebeta

← #361940

Subgroups — #361940

Order 512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#1838C211yesmaximal, characteristic, complemented
3#20647C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C2×#1801C211yesmaximal, characteristic, complemented
6C2×#1800C211yesmaximal, characteristic
7#20438C211yesmaximal, characteristic, complemented
8#11365C212yesmaximal, complemented
9#11028C212yesmaximal
10#10943C212yesmaximal
11#10942C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#440—24no—
16C2×#440—22no—
17C2×#440—22no—
18C2×#440C418yescomplemented
19C22×#107C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#419C2212yes—
24C2×#418C2212yes—
25C22×#103C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#101C2212yes—
28C22×#100C2212yes—
29C2×#405C2212yes—
30C22×#97C2211yescharacteristic
31#2285C2211yescharacteristic
32#1838—24no—
33#1805—22no—
34#1804—22no—
35#1803—22no—
36#1802—22no—
37#1801—24no—
38#1800—24no—
39#1771—22no—
40#1771—22no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#341C2212yes—
44C2×#341C2212yes—
45#1736C2212yes—
46#1736C2212yes—
47#1729C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16—22no—
57C22×SD16C2×C414yes—
58C22×M16C2312yes—
59C22×M16C2312yes—
60C22×M16C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8C2×C414yes—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2×C414yes—
79C2×#117C2312yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#440—216no—
87#440—216no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107C2×C414yes—
91#423—22no—
92#423—22no—
93#423—22no—
94#423—22no—
95#423—22no—
96#423D414yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103#419—24no—
104#419—42no—
105#418—24no—
106#418—42no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103C2×C414yes—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—24no—
114C2×#102C2×C414yes—
115C2×#101—24no—
116C2×#101—24no—
117C2×#101—42no—
118C2×#100—24no—
119C2×#100—24no—
120C2×#100—42no—
121#405—42no—
122#405—42no—
123C2×#97—22no—
124C2×#97C2×C414yes—
125C2×#97D414yes—
126C2×#97D412yes—
127C2×#97D412yes—
128C2×C4×C8—22no—
129#384D412yes—
130#382D412yes—
131#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135#343—24no—
136#343—24no—
137#341—24no—
138#341—24no—
139#340—24no—
140#340—24no—
141#336—24no—
142C25—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8C22×C412yes—
155C22×Q8#4112yes—
156C22×Q8#4112yes—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C22×C412yes—
169C22×D4#4112yes—
170C22×D4#4112yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—24no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—42no—
178C23×C4—84no—
179C23×C4#4111yescharacteristic
180C23×C4#4111yescharacteristic
181C23×C4C2411yescharacteristic
182C2×SD16—28no—
183C2×SD16—28no—
184C2×SD16—28no—
185C2×SD16—28no—
186C2×SD16—28no—
187C2×SD16—28no—
188C2×SD16—28no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16C2×D414yes—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8C22×C412yes—
219C22×C8C22×C412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
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—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C2×D414yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—42no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42C22×C412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42#4112yes—
281#107—28no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—42no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288#103—28no—
289#103—48no—
290#102—28no—
291#102—48no—
292#101—48no—
293#100—48no—
294#97—44no—
295#97C22⋊C418yes—
296#97C22⋊C418yes—
297C4×C8—44no—
298#95—22no—
299#95—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—82no—
308C24—82no—
309C24C2×#4111yescharacteristic
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8#11714yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4#11714yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12412yes—
397C22×C4#12412yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C42−1+411yescharacteristic
411C22×C42−1+411yescharacteristic
412SD16—232no—
413M16—24no—
414M16—24no—
415M16—42no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—42no—
421M16—42no—
422M16—42no—
423M16—42no—
424M16—42no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8C2×C22⋊C414yes—
447C2×C8C2×C22⋊C414yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C42—42no—
467C42—42no—
468C42—44no—
469C42—44no—
470C42C2×C22⋊C412yes—
471C42C2×C22⋊C412yes—
472C42C4×D412yes—
473C42C4×D412yes—
474C42#12112yes—
475C42#12112yes—
476C42#12712yes—
477C42#12712yes—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23—82no—
492C23—82no—
493C23—84no—
494C23—84no—
495C23—82no—
496C23—82no—
497C23C2×C4×D411yescharacteristic
498C23C2×#12111yescharacteristic
499C23C2×#12411yescharacteristic
500C23#55211yescharacteristic
501C23#55211yescharacteristic
502C23#55311yescharacteristic
503C23#55311yescharacteristic
504Q8—24no—
505Q8—24no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512D4—24no—
513D4—24no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
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—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—88no—
541C2×C4—88no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#39012yes—
549C2×C4#39012yes—
550C2×C4#39012yes—
551C2×C4#39012yes—
552C2×C4C22×C22⋊C411yescharacteristic
553C2×C4C2×#11711yescharacteristic
554C2×C4C2×#11711yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4#52011yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4C2×#12111yescharacteristic
559C2×C4#52511yescharacteristic
560C2×C4#52511yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4C2×#12711yescharacteristic
564C2×C4#53511yescharacteristic
565C2×C4#53611yescharacteristic
566C2×C4#53711yescharacteristic
567C2×C4#53711yescharacteristic
568C2×C4#54611yescharacteristic
569C2×C4#54611yescharacteristic
570C2×C4#54611yescharacteristic
571C2×C4#54611yescharacteristic
572C2×C4#54811yescharacteristic
573C2×C4#54811yescharacteristic
574C2×C4#54811yescharacteristic
575C2×C4#54811yescharacteristic
576C2×C4#55211yescharacteristic
577C2×C4#55211yescharacteristic
578C2×C4#55311yescharacteristic
579C2×C4#55311yescharacteristic
580C8—24no—
581C8—24no—
582C8—44no—
583C8—82no—
584C8—82no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—44no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—82no—
599C22—82no—
600C22—84no—
601C22—84no—
602C22C2×#39011yescharacteristic
603C22#235211yescharacteristic
604C22#235211yescharacteristic
605C22#253511yescharacteristic
606C22#253611yescharacteristic
607C22#256211yescharacteristic
608C22#256211yescharacteristic
609C22#306812yes—
610C22#306912yes—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—44no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4#228011yescharacteristic
620C4#231111yescharacteristic
621C4#235211yescharacteristic
622C4#235211yescharacteristic
623C4#253511yescharacteristic
624C4#253611yescharacteristic
625C4#254011yescharacteristic
626C4#254011yescharacteristic
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—84no—
632C2#1390511yescharacteristic
633C2#2124911yescharacteristic
634C2#2125011yescharacteristic
635C2#2817812yes—
636C2#2822112yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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×#1838C2semidirect
3. #20647C2semidirect
5. C2×#1801C2semidirect
7. #20438C2semidirect
8. #11365C2semidirect
11. #10942C2semidirect
18. C2×#440C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#18382
#206472
#206462
C2×#18012
C2×#18002
#204382
#113652
#110282
#109432
#109422