Finite Group Databasebeta

← #10981927

Subgroups — #10981927

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×#520C211yesmaximal, characteristic
2C2×#5567C211yesmaximal, characteristic, complemented
3#84093C212yesmaximal
4#82468C212yesmaximal, complemented
5#82463C211yesmaximal, characteristic, complemented
6#82391C212yesmaximal, complemented
7#82388C211yesmaximal, characteristic, complemented
8#82382C211yesmaximal, characteristic, complemented
9#82366C214yesmaximal, complemented
10#82348C212yesmaximal, complemented
11#82344C212yesmaximal, complemented
12#82262C211yesmaximal, characteristic, complemented
13#82261C212yesmaximal, complemented
14#82258C212yesmaximal, complemented
15#82245C211yesmaximal, characteristic, complemented
16#82174C214yesmaximal, complemented
17#82162C212yesmaximal, complemented
18#35712—31nomaximal, Sylow
19C22×C3×#41C2211yescharacteristic
20C3×#520—28no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C2×C3×C4×D4C2212yes—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×#117C2211yescharacteristic
28C2×C3×#117C2212yes—
29C22×C3×C4⋊C4C2211yescharacteristic
30C22×C3×C4⋊C4C2212yes—
31C2×#944C2212yes—
32C2×#936C2212yes—
33C2×#924C2212yes—
34#5567—28no—
35C2×#889C2214yes—
36C2×#887C2212yes—
37C2×#870C2212yes—
38C2×#869C2211yescharacteristic
39C2×C4×Dic6C2211yescharacteristic
40C2×C4×Dic6C2212yes—
41#5229C2212yes—
42#5222C2214yes—
43#5222C2214yes—
44#5221C2212yes—
45#5212C2214yes—
46#5210C2214yes—
47#5210C2214yes—
48#5210C2214yes—
49#5210C2214yes—
50#5210C2214yes—
51C4×#217C2212yes—
52C4×#217C2214yes—
53C4×#217C2214yes—
54#4986C2211yescharacteristic
55#4985C2214yes—
56#4983C2214yes—
57#4982C2212yes—
58#4981C2212yes—
59#4981C2212yes—
60#4980C2214yes—
61#4979C2211yescharacteristic
62#4978C2214yes—
63#4976C2214yes—
64Dic3×C4⋊C4C2212yes—
65#4973C2212yes—
66#4972C2212yes—
67#4953C2212yes—
68#4953C2212yes—
69#4953C2212yes—
70#4953C2212yes—
71#4949C2211yescharacteristic
72#4949C2212yes—
73#4948C2212yes—
74#4947C2211yescharacteristic
75#4947C2212yes—
76#4945C2214yes—
77#4944C2214yes—
78#4937C2212yes—
79#4936C2211yescharacteristic
80C4×C4⋊Dic3C2211yescharacteristic
81#4933C2214yes—
82#4933C2214yes—
83#4932C2212yes—
84C4×Dic3⋊C4C2212yes—
85#4929C2212yes—
86C2×#557—31no—
87C2×#520D311yescharacteristic
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—
103C2×C3×#41—24no—
104C2×C3×#41—24no—
105C2×C3×#41—24no—
106C22×C3×Q8C2311yescharacteristic
107C22×C3×D4C2312yes—
108C22×C3×D4C2311yescharacteristic
109C23×C3×C4C2312yes—
110C23×C3×C4C2311yescharacteristic
111C23×C3×C4C2312yes—
112C23×C3×C4C2312yes—
113C23×C3×C4C2312yes—
114C22×Dic6C2312yes—
115C22×Dic6C2311yescharacteristic
116C3×C4×Q8—24no—
117C3×C4×Q8D418yes—
118C3×C4×D4—24no—
119C3×C4×D4—28no—
120C3×C4×D4—24no—
121C3×C4×D4—28no—
122C3×#117—24no—
123C3×#117—28no—
124C2×C3×C4⋊C4—24no—
125C2×C3×C4⋊C4—28no—
126C2×C3×C4⋊C4C2311yescharacteristic
127C2×C3×C4⋊C4C2312yes—
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C4⋊C4C2311yescharacteristic
130C2×C3×C4⋊C4C2312yes—
131C2×C3×C4⋊C4C2311yescharacteristic
132C2×C3×C4⋊C4C2312yes—
133C2×C3×C22⋊C4C2312yes—
134C2×C3×C22⋊C4C2311yescharacteristic
135C2×C3×C22⋊C4C2312yes—
136C2×C3×C22⋊C4C2311yescharacteristic
137C2×C3×C42C2311yescharacteristic
138C2×C3×C42C2312yes—
139C2×C3×C42C2312yes—
140C2×C3×C42C2311yescharacteristic
141C2×#217C2314yes—
142C2×#217C2314yes—
143C2×#217C2314yes—
144C2×#217C2312yes—
145C2×#217C2314yes—
146C2×#217C2314yes—
147C2×#217C2312yes—
148#944—28no—
149#936—28no—
150C2×C4⋊Dic3C2312yes—
151C2×C4⋊Dic3C2311yescharacteristic
152C2×C4⋊Dic3C2312yes—
153C2×C4⋊Dic3C2312yes—
154C2×C4⋊Dic3C2312yes—
155#924—28no—
156C2×Dic3⋊C4C2312yes—
157C2×Dic3⋊C4C2314yes—
158C2×Dic3⋊C4C2314yes—
159C2×Dic3⋊C4C2314yes—
160C2×Dic3⋊C4C2312yes—
161C2×Dic3⋊C4C2314yes—
162C2×Dic3⋊C4C2312yes—
163C2×Dic3⋊C4C2314yes—
164C2×C4×Dic3C2312yes—
165C2×C4×Dic3C2312yes—
166C2×C4×Dic3C2314yes—
167C2×C4×Dic3C2314yes—
168#889—216no—
169#887—28no—
170#870—28no—
171#869—24no—
172C4×Dic6—28no—
173C4×Dic6—24no—
174C3⋊#95C2314yes—
175C3⋊#95C2314yes—
176C3⋊#95C2312yes—
177C3⋊#95C2312yes—
178C3⋊#95C2311yescharacteristic
179C3⋊#95C2314yes—
180C3⋊#95C2312yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2311yescharacteristic
186C3⋊#95C2312yes—
187C3⋊#95C2314yes—
188C3⋊#95C2312yes—
189C3⋊#95C2312yes—
190C22×#41C2×D311yescharacteristic
191#557—68no—
192C2×C4⋊Q8—32no—
193C2×C4⋊Q8—31no—
194C2×#125—34no—
195C2×#125—32no—
196C2×#124—32no—
197C2×C22⋊Q8—32no—
198#520—28no—
199C2×C4×Q8—32no—
200C2×C4×Q8—31no—
201C2×C4×Q8—32no—
202C2×C4×Q8C2×D311yescharacteristic
203C2×C4×Q8C2×D311yescharacteristic
204C2×C4×D4C2×D311yescharacteristic
205C2×C4×D4C2×D311yescharacteristic
206C2×C4×D4C2×D312yes—
207C2×C4×D4C2×D312yes—
208C2×#117C2×D312yes—
209C2×#117C2×D311yescharacteristic
210C22×C4⋊C4C2×D311yescharacteristic
211C22×C4⋊C4C2×D312yes—
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—32no—
235#389—34no—
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—
257C24×C3C2412yes—
258C24×C3C2411yescharacteristic
259C3×#41—232no—
260C2×C3×Q8—22no—
261C2×C3×Q8—22no—
262C2×C3×Q8C2×D414yes—
263C2×C3×D4—24no—
264C2×C3×D4—24no—
265C2×C3×D4—24no—
266C2×C3×D4—22no—
267C2×C3×D4—22no—
268C2×C3×D4#4114yes—
269C22×C3×C4—24no—
270C22×C3×C4—24no—
271C22×C3×C4—22no—
272C22×C3×C4—24no—
273C22×C3×C4—22no—
274C22×C3×C4—24no—
275C22×C3×C4—24no—
276C22×C3×C4—24no—
277C22×C3×C4—44no—
278C22×C3×C4—44no—
279C22×C3×C4—44no—
280C22×C3×C4#4114yes—
281C22×C3×C4C2411yescharacteristic
282C22×C3×C4C2411yescharacteristic
283C22×C3×C4C2412yes—
284C22×C3×C4C2412yes—
285C22×C3×C4C2411yescharacteristic
286C22×C3×C4C2412yes—
287C22×C3×C4C2411yescharacteristic
288C22×C3×C4C2411yescharacteristic
289C22×C3×C4C2411yescharacteristic
290C22×Dic3C2412yes—
291C22×Dic3C2412yes—
292C22×Dic3C2414yes—
293C22×Dic3C2414yes—
294C22×Dic3C2412yes—
295C22×Dic3C2412yes—
296C2×Dic6—22no—
297C2×Dic6—24no—
298C2×Dic6—24no—
299C2×Dic6—24no—
300C2×Dic6—24no—
301C3×C4⋊C4—24no—
302C3×C4⋊C4—22no—
303C3×C4⋊C4—24no—
304C3×C4⋊C4—22no—
305C3×C4⋊C4—22no—
306C3×C4⋊C4C2×D414yes—
307C3×C4⋊C4C2×D418yes—
308C3×C22⋊C4—24no—
309C3×C22⋊C4—22no—
310C3×C22⋊C4—22no—
311C3×C22⋊C4—24no—
312C3×C42—24no—
313C3×C42—22no—
314C3×C42C2×D414yes—
315C3×C42C2×D418yes—
316#217—28no—
317#217—28no—
318#217—28no—
319#217—28no—
320#217—24no—
321#217—44no—
322#217—42no—
323C4⋊Dic3—24no—
324C4⋊Dic3—22no—
325C4⋊Dic3—24no—
326C4⋊Dic3—24no—
327C4⋊Dic3—42no—
328Dic3⋊C4—24no—
329Dic3⋊C4—28no—
330Dic3⋊C4—28no—
331Dic3⋊C4—28no—
332Dic3⋊C4—28no—
333Dic3⋊C4—24no—
334Dic3⋊C4—44no—
335Dic3⋊C4—42no—
336C4×Dic3—28no—
337C4×Dic3—24no—
338C4×Dic3—24no—
339C4×Dic3—44no—
340C2×#41—24no—
341C2×#41—24no—
342C2×#41—24no—
343C22×Q8—31no—
344C22×Q8—32no—
345C22×Q8C22×D311yescharacteristic
346C22×D4C22×D312yes—
347C22×D4C22×D311yescharacteristic
348C23×C4C22×D312yes—
349C23×C4C22×D311yescharacteristic
350C23×C4C22×D312yes—
351C23×C4C22×D312yes—
352C23×C4C22×D312yes—
353C4⋊Q8—68no—
354C4⋊Q8—64no—
355#125—616no—
356#125—68no—
357#124—68no—
358C22⋊Q8—68no—
359C4×Q8—24no—
360C4×Q8—68no—
361C4×Q8—68no—
362C4×Q8—64no—
363C4×Q8#6718yes—
364C4×D4—24no—
365C4×D4—28no—
366C4×D4—28no—
367C4×D4—24no—
368#117—28no—
369#117—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—31no—
375C2×C4⋊C4—34no—
376C2×C4⋊C4—32no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—34no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—34no—
381C2×C4⋊C4—32no—
382C2×C4⋊C4—32no—
383C2×C4⋊C4—32no—
384C2×C4⋊C4—34no—
385C2×C4⋊C4C22×D312yes—
386C2×C4⋊C4C22×D311yescharacteristic
387C2×C4⋊C4C22×D311yescharacteristic
388C2×C4⋊C4C22×D312yes—
389C2×C4⋊C4C22×D311yescharacteristic
390C2×C4⋊C4C22×D311yescharacteristic
391C2×C4⋊C4C22×D312yes—
392C2×C22⋊C4—34no—
393C2×C22⋊C4—34no—
394C2×C22⋊C4—34no—
395C2×C22⋊C4—32no—
396C2×C22⋊C4—34no—
397C2×C22⋊C4—32no—
398C2×C22⋊C4—34no—
399C2×C22⋊C4C22×D312yes—
400C2×C22⋊C4C22×D312yes—
401C2×C22⋊C4C22×D311yescharacteristic
402C2×C22⋊C4C22×D311yescharacteristic
403C2×C42—34no—
404C2×C42—32no—
405C2×C42—32no—
406C2×C42—34no—
407C2×C42C22×D311yescharacteristic
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×C3—22no—
428C23×C3—21no—
429C23×C3—22no—
430C23×C3—21no—
431C23×C3—22no—
432C23×C3—24no—
433C23×C3—22no—
434C23×C3—24no—
435C23×C3C2×#4114yes—
436C23×C3C2×#4112yes—
437C23×C3C2511yescharacteristic
438C3×Q8—28no—
439C3×D4—216no—
440C3×D4—28no—
441C2×C3×C4—216no—
442C2×C3×C4—21no—
443C2×C3×C4—21no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—24no—
448C2×C3×C4—22no—
449C2×C3×C4—28no—
450C2×C3×C4—21no—
451C2×C3×C4—22no—
452C2×C3×C4—21no—
453C2×C3×C4—22no—
454C2×C3×C4—24no—
455C2×C3×C4—22no—
456C2×C3×C4—84no—
457C2×C3×C4—84no—
458C2×C3×C4—84no—
459C2×C3×C4C22×D414yes—
460C2×C3×C4C22×D414yes—
461C2×C3×C4C22×D412yes—
462C2×C3×C4C22×D412yes—
463C2×C3×C4C22×D412yes—
464C2×C3×C4C2×#4112yes—
465C2×C3×C4C2×#4112yes—
466C2×C3×C4C2×#4112yes—
467C2×Dic3—22no—
468C2×Dic3—22no—
469C2×Dic3—22no—
470C2×Dic3—24no—
471C2×Dic3—22no—
472C2×Dic3—22no—
473C2×Dic3—24no—
474C2×Dic3—24no—
475C2×Dic3—24no—
476C2×Dic3—22no—
477C2×Dic3—24no—
478C2×Dic3—22no—
479C2×Dic3—22no—
480C2×Dic3—24no—
481C2×Dic3—22no—
482C2×Dic3—22no—
483C2×Dic3—22no—
484C2×Dic32−1+414yes—
485Dic6—28no—
486Dic6—48no—
487C24C23×D312yes—
488C24C23×D311yescharacteristic
489#41—232no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—64no—
493C2×Q8—64no—
494C2×Q8—64no—
495C2×Q8—62no—
496C2×Q8—64no—
497C2×Q8C2×#6714yes—
498C2×D4—24no—
499C2×D4—24no—
500C2×D4—22no—
501C2×D4—24no—
502C2×D4—22no—
503C2×D4#23514yes—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—22no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—32no—
513C22×C4—32no—
514C22×C4—32no—
515C22×C4—34no—
516C22×C4—34no—
517C22×C4—32no—
518C22×C4—44no—
519C22×C4—44no—
520C22×C4—44no—
521C22×C4#23514yes—
522C22×C4C23×D311yescharacteristic
523C22×C4C23×D311yescharacteristic
524C22×C4C23×D312yes—
525C22×C4C23×D311yescharacteristic
526C22×C4C23×D311yescharacteristic
527C22×C4C23×D312yes—
528C22×C4C23×D312yes—
529C22×C4C23×D311yescharacteristic
530C22×C4C23×D311yescharacteristic
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—64no—
537C4⋊C4—68no—
538C4⋊C4—64no—
539C4⋊C4—64no—
540C4⋊C4—68no—
541C4⋊C4—62no—
542C4⋊C4—68no—
543C4⋊C4—64no—
544C4⋊C4—68no—
545C4⋊C4—64no—
546C4⋊C4—124no—
547C4⋊C4—122no—
548C4⋊C4—122no—
549C4⋊C4C2×#6714yes—
550C4⋊C4C2×#6718yes—
551C22⋊C4—24no—
552C22⋊C4—22no—
553C22⋊C4—24no—
554C22⋊C4—22no—
555C22⋊C4—64no—
556C22⋊C4—68no—
557C22⋊C4—68no—
558C22⋊C4—68no—
559C22⋊C4—68no—
560C22⋊C4—124no—
561C22⋊C4—122no—
562C42—24no—
563C42—22no—
564C42—64no—
565C42—68no—
566C42—64no—
567C42—124no—
568C42C2×#6718yes—
569C42C2×#6714yes—
570C22×C3—24no—
571C22×C3—22no—
572C22×C3—24no—
573C22×C3—28no—
574C22×C3—41no—
575C22×C3—42no—
576C22×C3—41no—
577C22×C3—42no—
578C22×C3—44no—
579C22×C3—42no—
580C22×C3C23×D411yescharacteristic
581C22×C3C22×#4111yescharacteristic
582C22×C3C22×#4111yescharacteristic
583C22×C3C2×2−1+412yes—
584C22×C3#58512yes—
585C3×C4—22no—
586C3×C4—24no—
587C3×C4—22no—
588C3×C4—24no—
589C3×C4—22no—
590C3×C4—41no—
591C3×C4—42no—
592C3×C4—41no—
593C3×C4#52514yes—
594Dic3—24no—
595Dic3—44no—
596Dic3—42no—
597Dic3—44no—
598Dic3—42no—
599Dic3—42no—
600C23—22no—
601C23—21no—
602C23—22no—
603C23—24no—
604C23—22no—
605C23—24no—
606C23—22no—
607C23—21no—
608C23C2×#23514yes—
609C23C2×#23512yes—
610C23C24×D311yescharacteristic
611Q8—28no—
612Q8—68no—
613Q8—128no—
614D4—216no—
615D4—28no—
616C2×C4—216no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—34no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—64no—
639C2×C4—62no—
640C2×C4—64no—
641C2×C4—62no—
642C2×C4—62no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—62no—
648C2×C4—64no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4C2×#23512yes—
653C2×C4C2×#23512yes—
654C2×C4C2×#23512yes—
655C2×C4C22×#6714yes—
656C2×C4C22×#6714yes—
657C2×C4C22×#6712yes—
658C2×C4C22×#6712yes—
659C2×C4C22×#6712yes—
660C2×C3—42no—
661C2×C3—44no—
662C2×C3C2×#52511yescharacteristic
663C2×C3#348611yescharacteristic
664C2×C3#350011yescharacteristic
665C2×C3#350712yes—
666C2×C3#351512yes—
667C22—24no—
668C22—28no—
669C22—24no—
670C22—22no—
671C22—44no—
672C22—42no—
673C22—41no—
674C22—42no—
675C22—41no—
676C22—42no—
677C22C22×#23511yescharacteristic
678C22C22×#23511yescharacteristic
679C22#596612yes—
680C22D3×2−1+412yes—
681C22C23×#6711yescharacteristic
682C4—22no—
683C4—24no—
684C4—24no—
685C4—22no—
686C4—22no—
687C4—42no—
688C4—41no—
689C4—41no—
690C4—64no—
691C4—124no—
692C4—122no—
693C4—122no—
694C4—124no—
695C4—122no—
696C4#579114yes—
697C3#3571211yescharacteristic, complemented, Sylow
698C2—42no—
699C2—44no—
700C2#8589212yes—
701C2#8589512yes—
702C2C2×#579111yescharacteristic
703C2#8728311yescharacteristic
704C2#8729411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#520C2
prime radicalC22×C3×#41C22
hypercenterC2×#520D3
hypocenterC3#35712
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. C2×#5567C2semidirect
4. #82468C2semidirect
5. #82463C2semidirect
6. #82391C2semidirect
7. #82388C2semidirect
8. #82382C2semidirect
9. #82366C2semidirect
10. #82348C2semidirect
11. #82344C2semidirect
12. #82262C2semidirect
13. #82261C2semidirect
14. #82258C2semidirect
15. #82245C2semidirect
16. #82174C2semidirect
17. #82162C2semidirect
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×#5202
C2×#55672
#840932
#824682
#824632
#823912
#823882
#823822
#823662
#823482
#823442
#822622
#822612
#822582
#822452
#821742
#821622
#357123