Finite Group Databasebeta

← #11166400

Subgroups — #11166400

Order 768.

2054 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic, complemented
2C2×#5532C211yesmaximal, characteristic
3#83263C212yesmaximal, complemented
4#83261C212yesmaximal, complemented
5#83256C212yesmaximal, complemented
6#83236C212yesmaximal, complemented
7#83071C212yesmaximal, complemented
8#83064C212yesmaximal, complemented
9#83061C212yesmaximal, complemented
10#83036C212yesmaximal, complemented
11#82800C214yesmaximal, complemented
12#82792C214yesmaximal, complemented
13#82288C211yesmaximal, characteristic, complemented
14#82274C211yesmaximal, characteristic, complemented
15#82239C211yesmaximal, characteristic, complemented
16#82196C211yesmaximal, characteristic, complemented
17#82180C211yesmaximal, characteristic, complemented
18#42965—31nomaximal, Sylow
19C3×#544—28no—
20C2×C3×C4⋊Q8C2212yes—
21C2×C3×C4⋊Q8C2212yes—
22C2×C3×#125C2212yes—
23C2×C3×#125C2212yes—
24C2×C3×C22⋊Q8C2214yes—
25C2×C3×#117C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C2×#946C2214yes—
29C2×#930C2211yescharacteristic
30C2×#929C2212yes—
31C2×C4×#67C2211yescharacteristic
32#5532—28no—
33C2×#895C2211yescharacteristic
34C2×#893C2211yescharacteristic
35C2×#892C2211yescharacteristic
36C2×#888C2211yescharacteristic
37C2×#885C2212yes—
38C2×#884C2212yes—
39C2×#875C2212yes—
40C2×#872C2212yes—
41#5231C2214yes—
42#5231C2214yes—
43#5229C2214yes—
44#5213C2214yes—
45#5212C2211yescharacteristic
46#5211C2212yes—
47C4×#217C2211yescharacteristic
48#4994C2214yes—
49#4994C2214yes—
50#4993C2211yescharacteristic
51#4993C2212yes—
52#4992C2212yes—
53#4992C2212yes—
54#4992C2212yes—
55#4992C2212yes—
56#4992C2214yes—
57#4992C2211yescharacteristic
58#4991C2214yes—
59#4990C2214yes—
60#4989C2214yes—
61#4988C2212yes—
62#4988C2214yes—
63#4987C2212yes—
64#4985C2211yescharacteristic
65#4984C2212yes—
66#4983C2212yes—
67#4983C2211yescharacteristic
68#4983C2214yes—
69#4979C2214yes—
70#4978C2214yes—
71#4977C2212yes—
72#4976C2214yes—
73Dic3×C4⋊C4C2212yes—
74#4973C2214yes—
75#4972C2212yes—
76#4959C2212yes—
77#4959C2214yes—
78#4952C2212yes—
79#4951C2214yes—
80#4939C2212yes—
81C4×(C2×D3)⋊C4C2212yes—
82#4936C2212yes—
83#4932C2212yes—
84C4×Dic3⋊C4C2212yes—
85#4928C2212yes—
86C2×#555—31no—
87C2×#544D311yescharacteristic, complemented
88#2820—32no—
89#2700—34no—
90#2681—34no—
91#2640—32no—
92#2624—31no—
93#2621—32no—
94#2611—32no—
95#2550—32no—
96#2504—31no—
97#2490—32no—
98#2486—31no—
99#2452—32no—
100#2439—32no—
101#2395—31no—
102#2346—31no—
103C22×C3×Q8C2314yes—
104C23×C3×C4C2311yescharacteristic
105C23×C3×C4C2312yes—
106C22×#67C2311yescharacteristic
107C22×#67C2312yes—
108C22×D12C2311yescharacteristic
109C22×D12C2312yes—
110C22×C4×D3C2311yescharacteristic
111C22×C4×D3C2312yes—
112C22×Dic6C2312yes—
113C3×C4⋊Q8—28no—
114C3×C4⋊Q8—28no—
115C3×#125—28no—
116C3×#125—28no—
117C3×C22⋊Q8—216no—
118C3×#117—24no—
119C3×#117D418yes—
120C2×C3×C4⋊C4—24no—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2314yes—
123C2×C3×C4⋊C4C2314yes—
124C2×C3×C4⋊C4C2314yes—
125C2×C3×C4⋊C4C2312yes—
126C2×C3×C4⋊C4C2311yescharacteristic
127C2×C3×C4⋊C4C2311yescharacteristic
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C4⋊C4C2311yescharacteristic
130C2×C3×C22⋊C4C2312yes—
131C2×C3×C22⋊C4C2312yes—
132C2×C3×C42C2312yes—
133C2×C3×C42C2312yes—
134C2×#217C2311yescharacteristic
135C2×#217C2312yes—
136C2×#217C2312yes—
137#946—48no—
138#930—24no—
139#929—44no—
140C4×#67—24no—
141C2×(C2×D3)⋊C4C2312yes—
142C2×(C2×D3)⋊C4C2312yes—
143C2×(C2×D3)⋊C4C2311yescharacteristic
144C2×(C2×D3)⋊C4C2312yes—
145C2×(C2×D3)⋊C4C2312yes—
146C2×(C2×D3)⋊C4C2312yes—
147C2×(C2×D3)⋊C4C2312yes—
148C2×(C2×D3)⋊C4C2314yes—
149C2×(C2×D3)⋊C4C2314yes—
150C2×(C2×D3)⋊C4C2314yes—
151C2×(C2×D3)⋊C4C2314yes—
152C2×(C2×D3)⋊C4C2314yes—
153C2×C4⋊Dic3C2311yescharacteristic
154C2×C4⋊Dic3C2312yes—
155C2×C4⋊Dic3C2314yes—
156C2×Dic3⋊C4C2312yes—
157C2×Dic3⋊C4C2311yescharacteristic
158C2×Dic3⋊C4C2312yes—
159C2×Dic3⋊C4C2314yes—
160C2×Dic3⋊C4C2314yes—
161C2×Dic3⋊C4C2312yes—
162C2×C4×Dic3C2311yescharacteristic
163C2×C4×Dic3C2312yes—
164C2×C4×Dic3C2312yes—
165C2×C4×Dic3C2314yes—
166#895—24no—
167#893—24no—
168#892—24no—
169#888—24no—
170#885—28no—
171#884—28no—
172#875—28no—
173#872—28no—
174C3⋊#95C2314yes—
175C3⋊#95C2312yes—
176C3⋊#95C2312yes—
177C3⋊#95C2311yescharacteristic
178C3⋊#95C2312yes—
179C3⋊#95C2314yes—
180C3⋊#95C2311yescharacteristic
181C3⋊#95C2314yes—
182C3⋊#95C2312yes—
183C3⋊#95C2312yes—
184C3⋊#95C2314yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188#555—68no—
189#544—28no—
190C2×C4⋊Q8—31no—
191C2×C4⋊Q8C2×D312yes—
192C2×C4⋊Q8C2×D312yes—
193C2×#125C2×D312yes—
194C2×#125C2×D312yes—
195C2×#124—32no—
196C2×#124—32no—
197C2×#123—32no—
198C2×C22⋊Q8—34no—
199C2×C22⋊Q8C2×D314yes—
200C2×#121—31no—
201C2×#121—31no—
202C2×#121—32no—
203C2×C4×D4—31no—
204C2×C4×D4—31no—
205C2×#117—31no—
206C2×#117—32no—
207C2×#117C2×D311yescharacteristic
208C2×#117C2×D311yescharacteristic
209C22×C4⋊C4C2×D311yescharacteristic
210#400—31no—
211#399—31no—
212#399—32no—
213#398—32no—
214#398—31no—
215#398—34no—
216#397—32no—
217#397—32no—
218#397—32no—
219#397—32no—
220#397—34no—
221#397—31no—
222#397—32no—
223#396—34no—
224#396—34no—
225#395—34no—
226#394—34no—
227#394—34no—
228#394—32no—
229#393—34no—
230#393—34no—
231#391—34no—
232#391—34no—
233#390—34no—
234#390—32no—
235#390—32no—
236#389—32no—
237#389—34no—
238#389—32no—
239#389—32no—
240#388—34no—
241#388—34no—
242#387—34no—
243#386—34no—
244#385—34no—
245#382—32no—
246#382—32no—
247#381—32no—
248#380—31no—
249#380—32no—
250C4×C4⋊C4—32no—
251C4×C4⋊C4—32no—
252C4×C22⋊C4—31no—
253C4×C22⋊C4—32no—
254#376—32no—
255C24×C3C2411yescharacteristic
256C23×D3C2412yes—
257C23×D3C2411yescharacteristic
258C2×C3×Q8—28no—
259C2×C3×Q8—44no—
260C2×C3×Q8—44no—
261C22×C3×C4—24no—
262C22×C3×C4—24no—
263C22×C3×C4—42no—
264C22×C3×C4—42no—
265C22×C3×C4C2×D414yes—
266C22×C3×C4C2412yes—
267C22×C3×C4C2414yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2411yescharacteristic
270C22×C3×C4C2412yes—
271C22×C3×C4C2412yes—
272C22×C3×C4C2411yescharacteristic
273C2×#67—22no—
274C2×#67—24no—
275C2×#67—22no—
276C2×#67—41no—
277C2×#67—44no—
278C22×Dic3C2412yes—
279C22×Dic3C2411yescharacteristic
280C22×Dic3C2412yes—
281C22×Dic3C2412yes—
282C22×Dic3C2414yes—
283C22×Dic3C2411yescharacteristic
284C22×Dic3C2411yescharacteristic
285C2×D12—22no—
286C2×D12—24no—
287C2×D12—24no—
288C2×D12—44no—
289C2×C4×D3—24no—
290C2×C4×D3—22no—
291C2×C4×D3—24no—
292C2×C4×D3—24no—
293C2×C4×D3—42no—
294C2×Dic6—24no—
295C2×Dic6—44no—
296C3×C4⋊C4—24no—
297C3×C4⋊C4—24no—
298C3×C4⋊C4—22no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4—28no—
301C3×C4⋊C4—28no—
302C3×C4⋊C4—28no—
303C3×C4⋊C4C2×D414yes—
304C3×C4⋊C4C2×D414yes—
305C3×C22⋊C4—24no—
306C3×C22⋊C4C2×D418yes—
307C3×C42—24no—
308C3×C42C2×D418yes—
309#217—22no—
310#217—42no—
311#217—42no—
312(C2×D3)⋊C4—24no—
313(C2×D3)⋊C4—24no—
314(C2×D3)⋊C4—22no—
315(C2×D3)⋊C4—24no—
316(C2×D3)⋊C4—24no—
317(C2×D3)⋊C4—44no—
318(C2×D3)⋊C4—42no—
319(C2×D3)⋊C4—44no—
320(C2×D3)⋊C4—42no—
321(C2×D3)⋊C4—44no—
322(C2×D3)⋊C4—44no—
323(C2×D3)⋊C4—44no—
324C4⋊Dic3—22no—
325C4⋊Dic3—24no—
326C4⋊Dic3—44no—
327Dic3⋊C4—22no—
328Dic3⋊C4—24no—
329Dic3⋊C4—28no—
330Dic3⋊C4—24no—
331Dic3⋊C4—44no—
332Dic3⋊C4—42no—
333C4×Dic3—24no—
334C4×Dic3—22no—
335C4×Dic3—24no—
336C4×Dic3—44no—
337C22×Q8—32no—
338C22×Q8C22×D314yes—
339C22×D4—31no—
340C22×D4—32no—
341C22×D4—31no—
342C22×D4—32no—
343C23×C4—31no—
344C23×C4—32no—
345C23×C4C22×D311yescharacteristic
346C23×C4C22×D312yes—
347C4⋊Q8—28no—
348C4⋊Q8—28no—
349C4⋊Q8—64no—
350#125—28no—
351#125—28no—
352#124—68no—
353#124—68no—
354#123—124no—
355C22⋊Q8—216no—
356C22⋊Q8—128no—
357#121—64no—
358#121—64no—
359#121—68no—
360C4×D4—64no—
361C4×D4—64no—
362#117—24no—
363#117—64no—
364#117—68no—
365#117#6718yes—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—31no—
369C2×C4⋊C4—32no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—34no—
372C2×C4⋊C4—32no—
373C2×C4⋊C4—34no—
374C2×C4⋊C4—32no—
375C2×C4⋊C4—34no—
376C2×C4⋊C4C22×D312yes—
377C2×C4⋊C4C22×D312yes—
378C2×C4⋊C4C22×D311yescharacteristic
379C2×C4⋊C4C22×D311yescharacteristic
380C2×C4⋊C4C22×D314yes—
381C2×C4⋊C4C22×D314yes—
382C2×C4⋊C4C22×D314yes—
383C2×C4⋊C4C22×D311yescharacteristic
384C2×C4⋊C4C22×D311yescharacteristic
385C2×C22⋊C4—31no—
386C2×C22⋊C4—32no—
387C2×C22⋊C4—32no—
388C2×C22⋊C4—32no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—31no—
391C2×C22⋊C4—32no—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—34no—
394C2×C22⋊C4—34no—
395C2×C22⋊C4—34no—
396C2×C22⋊C4—32no—
397C2×C22⋊C4—32no—
398C2×C22⋊C4—34no—
399C2×C22⋊C4—34no—
400C2×C22⋊C4C22×D312yes—
401C2×C22⋊C4C22×D312yes—
402C2×C42—32no—
403C2×C42—31no—
404C2×C42—32no—
405C2×C42—34no—
406C2×C42C22×D312yes—
407C2×C42C22×D312yes—
408#95—31no—
409#95—32no—
410#95—34no—
411#95—31no—
412#95—32no—
413#95—32no—
414#95—34no—
415#95—32no—
416#95—32no—
417#95—34no—
418#95—34no—
419#95—34no—
420#95—34no—
421#95—34no—
422C23×C3—21no—
423C23×C3—22no—
424C23×C3—22no—
425C23×C3—21no—
426C23×C3C22×D412yes—
427C23×C3C2511yescharacteristic
428C22×D3—21no—
429C22×D3—22no—
430C22×D3—22no—
431C22×D3—21no—
432C22×D3—22no—
433C22×D3—24no—
434C22×D3—22no—
435C22×D3—21no—
436C22×D3—22no—
437C22×D3—24no—
438C3×Q8—88no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—24no—
442C2×C3×C4—22no—
443C2×C3×C4—28no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—24no—
447C2×C3×C4—24no—
448C2×C3×C4—22no—
449C2×C3×C4—22no—
450C2×C3×C4—22no—
451C2×C3×C4—24no—
452C2×C3×C4—22no—
453C2×C3×C4—84no—
454C2×C3×C4C22×D414yes—
455C2×C3×C4C22×D412yes—
456C2×C3×C4C22×D414yes—
457C2×C3×C4C22×D412yes—
458C2×C3×C42−1+414yes—
459#67—44no—
460#67—84no—
461C2×Dic3—21no—
462C2×Dic3—21no—
463C2×Dic3—21no—
464C2×Dic3—22no—
465C2×Dic3—21no—
466C2×Dic3—22no—
467C2×Dic3—24no—
468C2×Dic3—22no—
469C2×Dic3—22no—
470C2×Dic3—24no—
471C2×Dic3—22no—
472C2×Dic3—22no—
473C2×Dic3—24no—
474C2×Dic3—22no—
475C2×Dic3—22no—
476C2×Dic3—24no—
477C2×Dic3—22no—
478C2×Dic3H(5,𝔽2)12yes—
479D12—44no—
480D12—84no—
481C4×D3—48no—
482C4×D3—44no—
483Dic6—84no—
484C24—31no—
485C24—32no—
486C24C23×D311yescharacteristic
487C2×Q8—28no—
488C2×Q8—44no—
489C2×Q8—44no—
490C2×Q8—64no—
491C2×Q8—124no—
492C2×D4—62no—
493C2×D4—62no—
494C2×D4—64no—
495C2×D4—64no—
496C2×D4—62no—
497C2×D4—64no—
498C2×D4—124no—
499C2×D4—121no—
500C2×D4—124no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—31no—
504C22×C4—32no—
505C22×C4—31no—
506C22×C4—32no—
507C22×C4—34no—
508C22×C4—31no—
509C22×C4—32no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—64no—
513C22×C4—64no—
514C22×C4—64no—
515C22×C4—62no—
516C22×C4—122no—
517C22×C4C2×#6714yes—
518C22×C4C23×D312yes—
519C22×C4C23×D314yes—
520C22×C4C23×D312yes—
521C22×C4C23×D311yescharacteristic
522C22×C4C23×D312yes—
523C22×C4C23×D312yes—
524C22×C4C23×D311yescharacteristic
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—28no—
532C4⋊C4—64no—
533C4⋊C4—68no—
534C4⋊C4—64no—
535C4⋊C4—62no—
536C4⋊C4—64no—
537C4⋊C4—62no—
538C4⋊C4—124no—
539C4⋊C4—122no—
540C4⋊C4—124no—
541C4⋊C4C2×#6714yes—
542C4⋊C4C2×#6714yes—
543C22⋊C4—24no—
544C22⋊C4—62no—
545C22⋊C4—64no—
546C22⋊C4—64no—
547C22⋊C4—64no—
548C22⋊C4—64no—
549C22⋊C4—62no—
550C22⋊C4—124no—
551C22⋊C4—124no—
552C22⋊C4—124no—
553C22⋊C4—124no—
554C22⋊C4—122no—
555C22⋊C4—124no—
556C22⋊C4—122no—
557C22⋊C4—122no—
558C22⋊C4—122no—
559C22⋊C4C2×#6718yes—
560C42—24no—
561C42—64no—
562C42—64no—
563C42—62no—
564C42—124no—
565C42C2×#6718yes—
566C22×C3—22no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—42no—
570C22×C3—42no—
571C22×C3C23×D411yescharacteristic
572C22×C3C2×H(5,𝔽2)11yescharacteristic
573C22×C3C2×2−1+412yes—
574C22×C3#58511yescharacteristic
575C22×C3#58512yes—
576C2×D3—41no—
577C2×D3—42no—
578C2×D3—42no—
579C2×D3—41no—
580C2×D3—41no—
581C2×D3—42no—
582C2×D3—44no—
583C2×D3—42no—
584C2×D3—42no—
585C2×D3—44no—
586C3×C4—22no—
587C3×C4—24no—
588C3×C4—24no—
589C3×C4—44no—
590C3×C4—42no—
591C3×C4—42no—
592C3×C4#53614yes—
593Dic3—22no—
594Dic3—41no—
595Dic3—41no—
596Dic3—42no—
597Dic3—42no—
598Dic3—44no—
599Dic3—42no—
600C23—22no—
601C23—21no—
602C23—21no—
603C23—22no—
604C23—62no—
605C23—64no—
606C23—62no—
607C23—61no—
608C23—62no—
609C23—61no—
610C23—62no—
611C23—64no—
612C23—61no—
613C23—62no—
614C23C22×#6712yes—
615C23C24×D311yescharacteristic
616Q8—88no—
617Q8—244no—
618D4—124no—
619D4—124no—
620D4—244no—
621D4—244no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—32no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—62no—
640C2×C4—62no—
641C2×C4—64no—
642C2×C4—62no—
643C2×C4—64no—
644C2×C4—62no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—64no—
648C2×C4—62no—
649C2×C4—64no—
650C2×C4—62no—
651C2×C4—61no—
652C2×C4—61no—
653C2×C4—62no—
654C2×C4—84no—
655C2×C4—128no—
656C2×C4—124no—
657C2×C4#100714yes—
658C2×C4C22×#6714yes—
659C2×C4C22×#6712yes—
660C2×C4C22×#6714yes—
661C2×C4C22×#6712yes—
662C2×C3—42no—
663C2×C3C2×#53611yescharacteristic
664C2×C3#348811yescharacteristic
665C2×C3#350011yescharacteristic
666C2×C3#354712yes—
667C2×C3#356512yes—
668D3—81no—
669D3—82no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—42no—
674C22—42no—
675C22—122no—
676C22—124no—
677C22—122no—
678C22—124no—
679C22—121no—
680C22—122no—
681C22—122no—
682C22—121no—
683C22—121no—
684C22—122no—
685C22C2×#100712yes—
686C22D3×H(5,𝔽2)11yescharacteristic
687C22#596611yescharacteristic
688C22#596812yes—
689C22C23×#6711yescharacteristic
690C4—22no—
691C4—24no—
692C4—24no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—62no—
697C4—122no—
698C4—124no—
699C4—122no—
700C4—121no—
701C4—122no—
702C4—121no—
703C4#581614yes—
704C3#4296511yescharacteristic, complemented, Sylow
705C2—42no—
706C2—241no—
707C2—242no—
708C2#8658912yes—
709C2#8659412yes—
710C2C2×#581611yescharacteristic
711C2#8728511yescharacteristic
712C2#8729311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#544C2
prime radicalC2×#5532C2
hypercenterC2×#544D3
hypocenterC3#42965
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×C3×#544C2semidirect
3. #83263C2semidirect
4. #83261C2semidirect
5. #83256C2semidirect
6. #83236C2semidirect
7. #83071C2semidirect
8. #83064C2semidirect
9. #83061C2semidirect
10. #83036C2semidirect
11. #82800C2semidirect
12. #82792C2semidirect
13. #82288C2semidirect
14. #82274C2semidirect
15. #82239C2semidirect
16. #82196C2semidirect
17. #82180C2semidirect
87. C2×#544D3semidirect
704. C3#42965semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×#55322
#832632
#832612
#832562
#832362
#830712
#830642
#830612
#830362
#828002
#827922
#822882
#822742
#822392
#821962
#821802
#429653