Finite Group Databasebeta

← #10944117

Subgroups — #10944117

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C2×#5239C212yesmaximal
3C2×#5031C211yesmaximal, characteristic, complemented
4C2×#5019C211yesmaximal, characteristic, complemented
5#82262C211yesmaximal, characteristic, complemented
6#71632C212yesmaximal, complemented
7#71627C211yesmaximal, characteristic, complemented
8#71612C212yesmaximal, complemented
9#71605C211yesmaximal, characteristic, complemented
10#71583C211yesmaximal, characteristic, complemented
11#71559C212yesmaximal, complemented
12#32916—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C3×#520—24no—
15C3×#520—24no—
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×#216C2212yes—
27C22×#214C2212yes—
28C2×#924C2212yes—
29C2×#869C2211yescharacteristic
30#5239—28no—
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×#830C2212yes—
34#5031—24no—
35#5019—24no—
36#4981—21no—
37#4981—21no—
38C2×#810C2212yes—
39C2×#810C2212yes—
40#4953—21no—
41#4953—21no—
42#4936C2211yescharacteristic
43#4929C2212yes—
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4529C2212yes—
47#4529C2212yes—
48#4524C2212yes—
49C2×#520D311yescharacteristic
50C2×#462—31no—
51C2×#461—31no—
52C2×#452—32no—
53#2621—31no—
54#1878—32no—
55#1877—32no—
56#1872—32no—
57#1805—31no—
58#1804—31no—
59#1798—31no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41D414yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2311yescharacteristic
74C22×Dic6C2312yes—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8D414yes—
79C3×C4×D4—28no—
80C3×C4×D4—28no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4D414yes—
85C3×#117—28no—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—28no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×#217—22no—
107C2×#216—24no—
108C2×#216—24no—
109C2×#216—24no—
110C2×#214—24no—
111C2×#214—24no—
112C2×#214—24no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2312yes—
116#924—24no—
117#924—24no—
118C2×Dic3⋊C4—22no—
119C22×C3⋊C8C2312yes—
120C22×C3⋊C8C2312yes—
121#869—22no—
122#869—22no—
123#835—24no—
124#832—24no—
125C3⋊#95—22no—
126C3⋊#95—22no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#830—24no—
130#810—24no—
131#810—24no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—22no—
134C22×#41C2×D311yescharacteristic
135C22×Q16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139#520—24no—
140#520—24no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4C2×D311yescharacteristic
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—21no—
148C2×#117C2×D311yescharacteristic
149C22×C4⋊C4—21no—
150C22×C4⋊C4C2×D311yescharacteristic
151#462—64no—
152#461—64no—
153#452—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8—31no—
156C2×#103—32no—
157C2×#103—32no—
158C2×#103—32no—
159C2×#102—32no—
160C2×C22⋊C8—32no—
161#398—61no—
162#398—61no—
163#397—61no—
164#397—61no—
165#391—32no—
166#382—31no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—21no—
171C24×C3C2×D411yescharacteristic
172C3×#41—28no—
173C3×#41—28no—
174C3×#41—28no—
175C3×#41—28no—
176C2×C3×Q8—21no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8C2×D412yes—
181C2×C3×Q8#4112yes—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4C2×D412yes—
189C2×C3×D4C2×D412yes—
190C2×C3×D4#4112yes—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—24no—
204C22×C3×C4—82no—
205C22×C3×C4—82no—
206C22×C3×C4—82no—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4#4112yes—
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—22no—
216C22×Dic3—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4C2×D412yes—
231C3×C4⋊C4C2×D412yes—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C42—24no—
237C3×C42—24no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42C2×D412yes—
241C3×C42C2×D412yes—
242#217—44no—
243#216—48no—
244#214—48no—
245C4⋊Dic3—22no—
246C4⋊Dic3—22no—
247C4⋊Dic3—42no—
248C4⋊Dic3—42no—
249Dic3⋊C4—44no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41D1214yes—
262C22×Q8—32no—
263C22×Q8C22×D311yescharacteristic
264C22×D4—21no—
265C22×D4C22×D311yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C22×D311yescharacteristic
271C2×Q16—64no—
272C2×Q16—64no—
273C2×Q16—64no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—64no—
277C22×C8—32no—
278C22×C8—32no—
279C4⋊Q8—62no—
280C4⋊Q8—62no—
281C22⋊Q8—64no—
282C22⋊Q8—64no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8#6714yes—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4#6714yes—
293#117—28no—
294#117—22no—
295#117—22no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4#6711yescharacteristic
307C2×C4⋊C4#6711yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—62no—
313C2×C22⋊C4#6711yescharacteristic
314C2×C22⋊C4#6711yescharacteristic
315C2×C42—21no—
316C2×C42—21no—
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—64no—
322#103—64no—
323#103—64no—
324#102—64no—
325C22⋊C8—64no—
326#95—61no—
327#95—61no—
328#95—62no—
329#95—62no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3#12112yes—
340C23×C3C22×D411yescharacteristic
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8#12114yes—
345C3×D4—216no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—22no—
349C3×D4#12114yes—
350C2×C3×C4—216no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—84no—
371C2×C3×C4—84no—
372C2×C3×C4—84no—
373C2×C3×C4#12012yes—
374C2×C3×C4#12012yes—
375C2×C3×C4#12112yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388Dic6—42no—
389Dic6—42no—
390Dic6—42no—
391Dic6—42no—
392C3⋊C8—42no—
393C3⋊C8—42no—
394C24—21no—
395C24C2×#6711yescharacteristic
396#41—28no—
397#41—28no—
398#41—28no—
399#41—28no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×Q8—62no—
409C2×Q8C2×D1212yes—
410C2×Q8#23512yes—
411C2×Q8C2×#6712yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×D1212yes—
419C2×D4#23512yes—
420C2×D4C2×#6712yes—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C2×D1212yes—
440C22×C4#23512yes—
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C23×D311yescharacteristic
447Q16—128no—
448SD16—128no—
449C2×C8—62no—
450C2×C8—62no—
451C2×C8—62no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—124no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4C2×#6712yes—
468C4⋊C4C2×#6712yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—124no—
474C42—24no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42C2×#6712yes—
479C42C2×#6712yes—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—24no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—41no—
486C22×C3—42no—
487C22×C3—41no—
488C22×C3—42no—
489C22×C3—44no—
490C22×C3C2×#12011yescharacteristic
491C22×C3C2×#12111yescharacteristic
492C22×C3#52511yescharacteristic
493C22×C3#57712yes—
494C22×C3#57812yes—
495C3×C4—22no—
496C3×C4—24no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—41no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4C2×#12111yescharacteristic
504C3×C4C2×#12111yescharacteristic
505C3×C4#52511yescharacteristic
506C3×C4#52511yescharacteristic
507Dic3—82no—
508Dic3—82no—
509C23—22no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23#93012yes—
519C23C22×#6711yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—122no—
524Q8—122no—
525Q8—122no—
526Q8—122no—
527Q8#93014yes—
528D4—216no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4#93014yes—
533C2×C4—216no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4#93012yes—
563C2×C4#95312yes—
564C2×C4#95312yes—
565C2×C4C22×D1211yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×#23511yescharacteristic
569C2×C4C22×#6711yescharacteristic
570C2×C4C22×#6711yescharacteristic
571C8—122no—
572C8—122no—
573C2×C3—42no—
574C2×C3—44no—
575C2×C3#284711yescharacteristic
576C2×C3#300911yescharacteristic
577C2×C3#301111yescharacteristic
578C2×C3#304612yes—
579C2×C3#312012yes—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—41no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—41no—
589C22—42no—
590C22C2×#93011yescharacteristic
591C22#579111yescharacteristic
592C22#583612yes—
593C22#583812yes—
594C22C2×#95311yescharacteristic
595C4—22no—
596C4—24no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—242no—
604C4—242no—
605C4C2×#93011yescharacteristic
606C4C2×#93011yescharacteristic
607C4#579111yescharacteristic
608C4#579111yescharacteristic
609C3#3291611yescharacteristic, complemented, Sylow
610C2—42no—
611C2—44no—
612C2#8257212yes—
613C2#8257512yes—
614C2#8403211yescharacteristic
615C2#8410511yescharacteristic
616C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#520C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#520D3
hypocenterC3#32916
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. C2×#5031C2semidirect
4. C2×#5019C2semidirect
5. #82262C2semidirect
6. #71632C2semidirect
7. #71627C2semidirect
8. #71612C2semidirect
9. #71605C2semidirect
10. #71583C2semidirect
11. #71559C2semidirect
609. C3#32916semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×#52392
C2×#50312
C2×#50192
#822622
#716322
#716272
#716122
#716052
#715832
#715592
#329163