Finite Group Databasebeta

← #10981918

Subgroups — #10981918

Order 768.

2262 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×#5574C211yesmaximal, characteristic
3#83261C211yesmaximal, characteristic, complemented
4#83255C212yesmaximal
5#83240C211yesmaximal, characteristic, complemented
6#83234C212yesmaximal, complemented
7#83066C212yesmaximal, complemented
8#83064C212yesmaximal, complemented
9#83059C214yesmaximal, complemented
10#83032C214yesmaximal, complemented
11#82502C212yesmaximal, complemented
12#82501C211yesmaximal, characteristic, complemented
13#82492C212yesmaximal, complemented
14#82491C211yesmaximal, characteristic, complemented
15#82490C212yesmaximal, complemented
16#82466C212yesmaximal, complemented
17#82463C211yesmaximal, characteristic, complemented
18#35712—31nomaximal, Sylow
19C22×#239C2211yescharacteristic
20C3×#557—28no—
21C2×C3×C4⋊Q8C2212yes—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×#125C2214yes—
24C2×C3×#125C2212yes—
25C2×C3×C4×Q8C2212yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2212yes—
28C2×C3×C4×Q8C2211yescharacteristic
29C2×#947C2212yes—
30C2×#946C2212yes—
31C2×Q8×Dic3C2211yescharacteristic
32#5574—28no—
33C2×#893C2211yescharacteristic
34C2×#893C2212yes—
35C2×D3×C4⋊C4C2211yescharacteristic
36C2×D3×C4⋊C4C2212yes—
37C2×C4×D12C2211yescharacteristic
38C2×C4×D12C2212yes—
39C2×#872C2211yescharacteristic
40C2×#872C2212yes—
41#5231C2211yescharacteristic
42#5231C2212yes—
43#5230C2211yescharacteristic
44#5229C2212yes—
45#5229C2212yes—
46#5229C2212yes—
47#4994C2212yes—
48#4993C2211yescharacteristic
49#4993C2212yes—
50#4992C2212yes—
51#4992C2212yes—
52#4992C2212yes—
53#4992C2212yes—
54#4991C2214yes—
55#4991C2214yes—
56#4990C2214yes—
57#4989C2214yes—
58#4989C2214yes—
59#4989C2214yes—
60#4989C2214yes—
61#4989C2214yes—
62#4987C2214yes—
63#4986C2214yes—
64#4985C2211yescharacteristic
65#4983C2212yes—
66#4983C2212yes—
67#4982C2212yes—
68#4981C2214yes—
69#4980C2214yes—
70#4978C2214yes—
71#4978C2214yes—
72#4976C2214yes—
73Dic3×C4⋊C4C2212yes—
74#4940C2214yes—
75#4939C2212yes—
76C4×(C2×D3)⋊C4C2214yes—
77C4×(C2×D3)⋊C4C2212yes—
78C4×(C2×D3)⋊C4C2214yes—
79#4937C2212yes—
80#4936C2211yescharacteristic
81C4×C4⋊Dic3C2211yescharacteristic
82#4933C2214yes—
83#4932C2212yes—
84C4×Dic3⋊C4C2212yes—
85#4928C2212yes—
86C2×#557D311yescharacteristic, complemented
87C2×#520—31no—
88#2633—32no—
89#2621—31no—
90#2611—32no—
91#2599—32no—
92#2580—31no—
93#2547—31no—
94#2473—34no—
95#2460—34no—
96#2446—32no—
97#2432—32no—
98#2398—32no—
99#2396—31no—
100#2367—32no—
101#2360—32no—
102#2356—31no—
103C22×C3×Q8C2312yes—
104C22×C3×Q8C2311yescharacteristic
105C22×C3×Q8C2311yescharacteristic
106C2×#239—24no—
107C2×#239—24no—
108C2×#239—24no—
109C22×D12C2312yes—
110C22×D12C2311yescharacteristic
111C22×C4×D3C2312yes—
112C22×C4×D3C2311yescharacteristic
113C22×C4×D3C2312yes—
114C22×C4×D3C2312yes—
115C22×C4×D3C2312yes—
116C3×C4⋊Q8—28no—
117C3×C4⋊Q8—24no—
118C3×#125—216no—
119C3×#125—28no—
120C3×C4×Q8—28no—
121C3×C4×Q8—28no—
122C3×C4×Q8—24no—
123C3×C4×Q8D418yes—
124C2×C3×C4⋊C4C2314yes—
125C2×C3×C4⋊C4C2314yes—
126C2×C3×C4⋊C4C2314yes—
127C2×C3×C4⋊C4C2312yes—
128C2×C3×C4⋊C4C2312yes—
129C2×C3×C4⋊C4C2312yes—
130C2×C3×C4⋊C4C2311yescharacteristic
131C2×C3×C4⋊C4C2311yescharacteristic
132C2×C3×C4⋊C4C2312yes—
133C2×C3×C42C2312yes—
134C2×C3×C42C2314yes—
135C2×C3×C42C2312yes—
136C2×C3×C42C2311yescharacteristic
137#947—28no—
138#946—28no—
139Q8×Dic3—24no—
140C2×(C2×D3)⋊C4C2312yes—
141C2×(C2×D3)⋊C4C2314yes—
142C2×(C2×D3)⋊C4C2314yes—
143C2×(C2×D3)⋊C4C2314yes—
144C2×(C2×D3)⋊C4C2312yes—
145C2×(C2×D3)⋊C4C2314yes—
146C2×(C2×D3)⋊C4C2311yescharacteristic
147C2×(C2×D3)⋊C4C2312yes—
148C2×(C2×D3)⋊C4C2311yescharacteristic
149C2×(C2×D3)⋊C4C2312yes—
150C2×(C2×D3)⋊C4C2314yes—
151C2×C4⋊Dic3C2311yescharacteristic
152C2×C4⋊Dic3C2312yes—
153C2×C4⋊Dic3C2312yes—
154C2×C4⋊Dic3C2312yes—
155C2×Dic3⋊C4C2312yes—
156C2×Dic3⋊C4C2312yes—
157C2×Dic3⋊C4C2314yes—
158C2×Dic3⋊C4C2311yescharacteristic
159C2×Dic3⋊C4C2311yescharacteristic
160C2×Dic3⋊C4C2314yes—
161C2×Dic3⋊C4C2312yes—
162C2×C4×Dic3C2311yescharacteristic
163C2×C4×Dic3C2312yes—
164C2×C4×Dic3C2312yes—
165C2×C4×Dic3C2314yes—
166#893—24no—
167#893—28no—
168D3×C4⋊C4—24no—
169D3×C4⋊C4—28no—
170C4×D12—28no—
171C4×D12—24no—
172#872—28no—
173#872—24no—
174C3⋊#95C2314yes—
175C3⋊#95C2312yes—
176C3⋊#95C2312yes—
177C3⋊#95C2311yescharacteristic
178C3⋊#95C2312yes—
179C3⋊#95C2314yes—
180C3⋊#95C2312yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2311yescharacteristic
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2312yes—
189C3⋊#95C2314yes—
190C22×#41—31no—
191#557—28no—
192C2×C4⋊Q8C2×D311yescharacteristic
193C2×C4⋊Q8C2×D312yes—
194C2×#125C2×D312yes—
195C2×#125C2×D314yes—
196C2×#124—32no—
197C2×C22⋊Q8—32no—
198#520—68no—
199C2×C4×Q8—31no—
200C2×C4×Q8C2×D311yescharacteristic
201C2×C4×Q8C2×D312yes—
202C2×C4×Q8C2×D312yes—
203C2×C4×Q8C2×D311yescharacteristic
204C2×C4×D4—31no—
205C2×C4×D4—32no—
206C2×C4×D4—31no—
207C2×C4×D4—32no—
208C2×#117—32no—
209C2×#117—31no—
210C22×C4⋊C4—31no—
211C22×C4⋊C4—32no—
212#400—31no—
213#400—34no—
214#399—32no—
215#399—31no—
216#398—32no—
217#398—32no—
218#398—34no—
219#398—32no—
220#397—32no—
221#397—32no—
222#397—32no—
223#397—32no—
224#396—32no—
225#395—31no—
226#394—34no—
227#394—34no—
228#393—32no—
229#393—31no—
230#391—32no—
231#391—32no—
232#391—32no—
233#390—32no—
234#389—34no—
235#389—32no—
236#388—34no—
237#388—34no—
238#387—34no—
239#387—34no—
240#387—34no—
241#387—34no—
242#385—34no—
243#385—34no—
244#385—34no—
245#385—34no—
246#385—34no—
247#383—32no—
248#382—31no—
249#380—34no—
250C4×C4⋊C4—32no—
251C4×C4⋊C4—31no—
252C4×C4⋊C4—32no—
253C4×C22⋊C4—32no—
254C4×C22⋊C4—34no—
255C4×C22⋊C4—34no—
256#376—32no—
257C23×D3C2412yes—
258C23×D3C2411yescharacteristic
259C2×C3×Q8—22no—
260C2×C3×Q8—24no—
261C2×C3×Q8—22no—
262C2×C3×Q8—22no—
263C2×C3×Q8—24no—
264C2×C3×Q8—24no—
265C2×C3×Q8—24no—
266C2×C3×Q8C2×D414yes—
267C22×C3×C4C2414yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2411yescharacteristic
271C22×C3×C4C2412yes—
272C22×C3×C4C2411yescharacteristic
273C22×C3×C4C2412yes—
274C22×C3×C4C2411yescharacteristic
275C22×Dic3C2412yes—
276C22×Dic3C2412yes—
277C22×Dic3C2414yes—
278C22×Dic3C2411yescharacteristic
279C22×Dic3C2412yes—
280C22×Dic3C2411yescharacteristic
281C22×Dic3C2411yescharacteristic
282#239—232no—
283C2×D12—24no—
284C2×D12—24no—
285C2×D12—24no—
286C2×D12—22no—
287C2×D12—22no—
288C2×D12#4114yes—
289C2×C4×D3—24no—
290C2×C4×D3—24no—
291C2×C4×D3—22no—
292C2×C4×D3—22no—
293C2×C4×D3—24no—
294C2×C4×D3—24no—
295C2×C4×D3—24no—
296C2×C4×D3—24no—
297C2×C4×D3—44no—
298C2×C4×D3—44no—
299C2×C4×D3—44no—
300C2×C4×D3#4114yes—
301C3×C4⋊C4—24no—
302C3×C4⋊C4—22no—
303C3×C4⋊C4—28no—
304C3×C4⋊C4—28no—
305C3×C4⋊C4—24no—
306C3×C4⋊C4—28no—
307C3×C4⋊C4—24no—
308C3×C4⋊C4C2×D414yes—
309C3×C4⋊C4C2×D418yes—
310C3×C42—28no—
311C3×C42—24no—
312C3×C42C2×D418yes—
313C3×C42C2×D414yes—
314(C2×D3)⋊C4—28no—
315(C2×D3)⋊C4—28no—
316(C2×D3)⋊C4—28no—
317(C2×D3)⋊C4—24no—
318(C2×D3)⋊C4—22no—
319(C2×D3)⋊C4—24no—
320(C2×D3)⋊C4—22no—
321(C2×D3)⋊C4—24no—
322(C2×D3)⋊C4—28no—
323(C2×D3)⋊C4—44no—
324(C2×D3)⋊C4—42no—
325C4⋊Dic3—24no—
326C4⋊Dic3—22no—
327C4⋊Dic3—24no—
328C4⋊Dic3—42no—
329Dic3⋊C4—22no—
330Dic3⋊C4—24no—
331Dic3⋊C4—22no—
332Dic3⋊C4—24no—
333Dic3⋊C4—28no—
334Dic3⋊C4—44no—
335Dic3⋊C4—42no—
336C4×Dic3—24no—
337C4×Dic3—22no—
338C4×Dic3—24no—
339C4×Dic3—44no—
340C2×#41—64no—
341C2×#41—64no—
342C2×#41—64no—
343C22×Q8C22×D311yescharacteristic
344C22×Q8C22×D312yes—
345C22×Q8C22×D311yescharacteristic
346C22×D4—31no—
347C22×D4—32no—
348C23×C4—31no—
349C23×C4—32no—
350C23×C4—32no—
351C23×C4—32no—
352C23×C4—32no—
353C4⋊Q8—28no—
354C4⋊Q8—24no—
355#125—216no—
356#125—28no—
357#124—68no—
358C22⋊Q8—68no—
359C4×Q8—28no—
360C4×Q8—28no—
361C4×Q8—24no—
362C4×Q8—64no—
363C4×Q8#6718yes—
364C4×D4—64no—
365C4×D4—68no—
366C4×D4—68no—
367C4×D4—64no—
368#117—68no—
369#117—64no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—31no—
372C2×C4⋊C4—31no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—31no—
375C2×C4⋊C4—32no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—32no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—34no—
381C2×C4⋊C4—64no—
382C2×C4⋊C4—68no—
383C2×C4⋊C4C22×D314yes—
384C2×C4⋊C4C22×D312yes—
385C2×C4⋊C4C22×D311yescharacteristic
386C2×C4⋊C4C22×D314yes—
387C2×C4⋊C4C22×D312yes—
388C2×C4⋊C4C22×D312yes—
389C2×C4⋊C4C22×D314yes—
390C2×C4⋊C4C22×D311yescharacteristic
391C2×C4⋊C4C22×D312yes—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—32no—
394C2×C22⋊C4—31no—
395C2×C22⋊C4—31no—
396C2×C22⋊C4—34no—
397C2×C22⋊C4—34no—
398C2×C22⋊C4—34no—
399C2×C22⋊C4—32no—
400C2×C22⋊C4—34no—
401C2×C22⋊C4—32no—
402C2×C22⋊C4—34no—
403C2×C42—31no—
404C2×C42—32no—
405C2×C42—32no—
406C2×C42—34no—
407C2×C42C22×D314yes—
408C2×C42C22×D312yes—
409C2×C42C22×D312yes—
410C2×C42C22×D311yescharacteristic
411#95—31no—
412#95—32no—
413#95—34no—
414#95—32no—
415#95—34no—
416#95—32no—
417#95—32no—
418#95—34no—
419#95—34no—
420#95—31no—
421#95—34no—
422#95—32no—
423#95—34no—
424#95—32no—
425#95—34no—
426#95—34no—
427C23×C3C2511yescharacteristic
428C22×D3—22no—
429C22×D3—21no—
430C22×D3—22no—
431C22×D3—21no—
432C22×D3—22no—
433C22×D3—24no—
434C22×D3—22no—
435C22×D3—24no—
436C22×D3C2×#4114yes—
437C22×D3C2×#4112yes—
438C3×Q8—28no—
439C3×Q8—28no—
440C3×Q8—48no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—24no—
447C2×C3×C4—22no—
448C2×C3×C4—22no—
449C2×C3×C4—24no—
450C2×C3×C4—24no—
451C2×C3×C4—24no—
452C2×C3×C4—22no—
453C2×C3×C4—22no—
454C2×C3×C4C22×D414yes—
455C2×C3×C4C22×D412yes—
456C2×C3×C4C22×D414yes—
457C2×C3×C4C22×D412yes—
458C2×C3×C4C22×D412yes—
459C2×C3×C4C2×#4112yes—
460C2×C3×C4C2×#4112yes—
461C2×C3×C42−1+414yes—
462C2×Dic3—21no—
463C2×Dic3—21no—
464C2×Dic3—21no—
465C2×Dic3—22no—
466C2×Dic3—21no—
467C2×Dic3—22no—
468C2×Dic3—22no—
469C2×Dic3—24no—
470C2×Dic3—22no—
471C2×Dic3—22no—
472C2×Dic3—24no—
473C2×Dic3—22no—
474C2×Dic3—22no—
475C2×Dic3—24no—
476C2×Dic3—22no—
477C2×Dic3—24no—
478C2×Dic3—22no—
479C2×Dic3C2×#4112yes—
480D12—216no—
481D12—28no—
482C4×D3—216no—
483C4×D3—28no—
484C4×D3—84no—
485C4×D3—84no—
486C4×D3—84no—
487C24—31no—
488C24—32no—
489#41—632no—
490C2×Q8—24no—
491C2×Q8—22no—
492C2×Q8—22no—
493C2×Q8—24no—
494C2×Q8—24no—
495C2×Q8—22no—
496C2×Q8—24no—
497C2×Q8C2×#6714yes—
498C2×D4—34no—
499C2×D4—64no—
500C2×D4—64no—
501C2×D4—62no—
502C2×D4—64no—
503C2×D4—62no—
504C22×C4—31no—
505C22×C4—34no—
506C22×C4—32no—
507C22×C4—34no—
508C22×C4—32no—
509C22×C4—31no—
510C22×C4—31no—
511C22×C4—32no—
512C22×C4—64no—
513C22×C4—64no—
514C22×C4—62no—
515C22×C4—64no—
516C22×C4—62no—
517C22×C4—64no—
518C22×C4—64no—
519C22×C4—64no—
520C22×C4—124no—
521C22×C4—124no—
522C22×C4—124no—
523C22×C4C23×D314yes—
524C22×C4C23×D312yes—
525C22×C4C23×D312yes—
526C22×C4C23×D312yes—
527C22×C4C23×D311yescharacteristic
528C22×C4C23×D312yes—
529C22×C4C23×D311yescharacteristic
530C22×C4C23×D311yescharacteristic
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—24no—
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—68no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—62no—
543C4⋊C4—64no—
544C4⋊C4—62no—
545C4⋊C4—62no—
546C4⋊C4—124no—
547C4⋊C4—122no—
548C4⋊C4—122no—
549C4⋊C4C2×#6714yes—
550C4⋊C4C2×#6718yes—
551C22⋊C4—68no—
552C22⋊C4—68no—
553C22⋊C4—68no—
554C22⋊C4—64no—
555C22⋊C4—68no—
556C22⋊C4—64no—
557C22⋊C4—62no—
558C22⋊C4—64no—
559C22⋊C4—62no—
560C22⋊C4—124no—
561C22⋊C4—122no—
562C42—28no—
563C42—24no—
564C42—64no—
565C42—64no—
566C42—62no—
567C42—124no—
568C42C2×#6718yes—
569C42C2×#6714yes—
570C22×C3C23×D411yescharacteristic
571C22×C3C22×#4111yescharacteristic
572C22×C3C22×#4111yescharacteristic
573C22×C3C2×2−1+412yes—
574C22×C3#58512yes—
575C2×D3—24no—
576C2×D3—22no—
577C2×D3—24no—
578C2×D3—28no—
579C2×D3—41no—
580C2×D3—42no—
581C2×D3—41no—
582C2×D3—42no—
583C2×D3—44no—
584C2×D3—42no—
585C3×C4—24no—
586C3×C4—22no—
587C3×C4—24no—
588C3×C4—24no—
589C3×C4—22no—
590C3×C4—44no—
591C3×C4—42no—
592C3×C4#52514yes—
593Dic3—22no—
594Dic3—41no—
595Dic3—41no—
596Dic3—42no—
597Dic3—44no—
598Dic3—42no—
599Dic3—42no—
600C23—32no—
601C23—34no—
602C23—64no—
603C23—62no—
604C23—61no—
605C23—62no—
606C23—64no—
607C23—62no—
608C23—62no—
609C23—61no—
610C23C24×D311yescharacteristic
611Q8—28no—
612Q8—28no—
613Q8—48no—
614D4—616no—
615D4—68no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—32no—
630C2×C4—616no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—68no—
634C2×C4—62no—
635C2×C4—64no—
636C2×C4—62no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—64no—
641C2×C4—62no—
642C2×C4—64no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—62no—
646C2×C4—61no—
647C2×C4—62no—
648C2×C4—61no—
649C2×C4—244no—
650C2×C4—244no—
651C2×C4—244no—
652C2×C4#100714yes—
653C2×C4D3×#4112yes—
654C2×C4D3×#4112yes—
655C2×C4C22×#6714yes—
656C2×C4C22×#6714yes—
657C2×C4C22×#6712yes—
658C2×C4C22×#6712yes—
659C2×C4C22×#6712yes—
660C2×C3C2×#52511yescharacteristic
661C2×C3#348611yescharacteristic
662C2×C3#350011yescharacteristic
663C2×C3#350712yes—
664C2×C3#351512yes—
665D3—42no—
666D3—44no—
667C22—64no—
668C22—62no—
669C22—68no—
670C22—64no—
671C22—122no—
672C22—121no—
673C22—122no—
674C22—124no—
675C22—122no—
676C22—121no—
677C22C2×#100712yes—
678C22C2×D3×#4111yescharacteristic
679C22C2×D3×#4111yescharacteristic
680C22#596812yes—
681C22C23×#6711yescharacteristic
682C4—24no—
683C4—22no—
684C4—24no—
685C4—24no—
686C4—22no—
687C4—44no—
688C4—42no—
689C4—62no—
690C4—121no—
691C4—122no—
692C4—122no—
693C4—124no—
694C4—122no—
695C4—121no—
696C4#582814yes—
697C3#3571211yescharacteristic, complemented, Sylow
698C2—124no—
699C2—122no—
700C2#8702112yes—
701C2#8703312yes—
702C2#8725111yescharacteristic
703C2C2×#582811yescharacteristic
704C2#8729311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#557C2
prime radicalC22×#239C22
hypercenterC2×#557D3
hypocenterC3#35712
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
3. #83261C2semidirect
5. #83240C2semidirect
6. #83234C2semidirect
7. #83066C2semidirect
8. #83064C2semidirect
9. #83059C2semidirect
10. #83032C2semidirect
11. #82502C2semidirect
12. #82501C2semidirect
13. #82492C2semidirect
14. #82491C2semidirect
15. #82490C2semidirect
16. #82466C2semidirect
17. #82463C2semidirect
86. C2×#557D3semidirect
697. C3#35712semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#55742
#832612
#832552
#832402
#832342
#830662
#830642
#830592
#830322
#825022
#825012
#824922
#824912
#824902
#824662
#824632
#357123