Finite Group Databasebeta

← #10964091

Subgroups — #10964091

Order 768.

2662 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C2×#5522C211yesmaximal, characteristic, complemented
3C3×#2297C211yesmaximal, characteristic, complemented
4#84086C211yesmaximal, characteristic, complemented
5#82226C211yesmaximal, characteristic, complemented
6#82070C211yesmaximal, characteristic, complemented
7C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
8#76506C212yesmaximal, complemented
9#76504C212yesmaximal, complemented
10#76321C212yesmaximal, complemented
11#76318C212yesmaximal, complemented
12#74707C214yesmaximal, complemented
13#74704C214yesmaximal, complemented
14#73582C214yesmaximal, complemented
15C4×#883C214yesmaximal, complemented
16#34744—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C22×#235C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21#5832—28no—
22C2×#947C2211yescharacteristic
23C2×#944C2211yescharacteristic
24C2×#940C2211yescharacteristic
25C2×#939C2214yes—
26C2×#936C2211yescharacteristic
27C2×C4×#67C2214yes—
28C2×#926C2211yescharacteristic
29C22×C4⋊Dic3C2212yes—
30C22×C4×Dic3C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32#5522—28no—
33C2×#893C2212yes—
34C2×#887C2212yes—
35C2×#883C2214yes—
36C2×#67⋊C4C2214yes—
37C2×#872C2212yes—
38C2×C42×D3C2212yes—
39C3×#391C2212yes—
40C3×#390C2212yes—
41C3×#386C2212yes—
42C3×#386C2212yes—
43C3×C4×C22⋊C4C2214yes—
44#5229C2212yes—
45#5221C2212yes—
46#5220C2214yes—
47#5213C2212yes—
48#5213C2212yes—
49C4×#217C2212yes—
50C4×#217C2212yes—
51C2×C3⋊#95C2212yes—
52#4972C2212yes—
53#4972C2212yes—
54#4961C2214yes—
55#4959C2214yes—
56#4956C2214yes—
57#4952C2214yes—
58#4951C2214yes—
59#4949C2214yes—
60#4946C2214yes—
61#4945C2214yes—
62#4939C2212yes—
63C4×(C2×D3)⋊C4C2214yes—
64C4×C4⋊Dic3C2212yes—
65C4×C4⋊Dic3C2212yes—
66C4×C4⋊Dic3C2212yes—
67C4×C4⋊Dic3C2212yes—
68C4×Dic3⋊C4C2214yes—
69#4929C2212yes—
70#4675C2214yes—
71#4674C2214yes—
72#4668C2214yes—
73#4664C2214yes—
74#4658C2214yes—
75#4657C2214yes—
76#4654C2214yes—
77#4652C2214yes—
78C2×#532—31no—
79C2×C4×#41—31no—
80#2448—32no—
81#2447—32no—
82#2446—32no—
83#2445—32no—
84#2444—34no—
85#2443—34no—
86#2395—34no—
87C4×#123—34no—
88#2297—31no—
89#2297—31no—
90#2297D311yescharacteristic, complemented
91#2296—31no—
92C2×C4×C4⋊C4—31no—
93C2×C3×#41—24no—
94C2×C3×#41—24no—
95C2×C3×#41—24no—
96C22×C3×Q8C2311yescharacteristic
97C22×C3×D4C2312yes—
98C22×C3×D4C2311yescharacteristic
99C23×C3×C4C2312yes—
100C23×C3×C4C2311yescharacteristic
101C23×C3×C4C2311yescharacteristic
102C23×C3×C4C2311yescharacteristic
103C22×#67C2314yes—
104C22×#67C2312yes—
105C23×Dic3C2312yes—
106C23×Dic3C2312yes—
107C2×#235—24no—
108C2×#235—24no—
109C2×#235D418yes—
110C22×D12C2311yescharacteristic
111C22×C4×D3C2314yes—
112C22×C4×D3C2312yes—
113C22×Dic6C2311yescharacteristic
114C3×#117—24no—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C22⋊C4C2312yes—
117C2×C3×C22⋊C4C2314yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C42—24no—
120C2×C3×C42C2312yes—
121C2×C3×C42C2312yes—
122C2×C3×C42C2312yes—
123C2×#217C2314yes—
124C2×#217C2314yes—
125C2×#217C2312yes—
126C2×#217C2314yes—
127#947—24no—
128#944—24no—
129#940—24no—
130#939—216no—
131#936—24no—
132C4×#67—216no—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137#926—24no—
138C2×C4⋊Dic3—28no—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2312yes—
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2312yes—
143C2×C4⋊Dic3C2312yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2314yes—
148C2×C4×Dic3—24no—
149C2×C4×Dic3—24no—
150C2×C4×Dic3C2314yes—
151C2×C4×Dic3C2312yes—
152C2×C4×Dic3C2312yes—
153C2×C4×Dic3C2311yescharacteristic
154C2×C4×Dic3C2311yescharacteristic
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2311yescharacteristic
157C2×C4×Dic3C2311yescharacteristic
158#893—28no—
159#887—28no—
160#883—216no—
161#67⋊C4—216no—
162#872—28no—
163C42×D3—28no—
164C3×#95C2314yes—
165C3×#95C2314yes—
166C3⋊#95C2312yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C3⋊#95C2312yes—
174C3⋊#95C2314yes—
175C3⋊#95C2312yes—
176C22×#41—31no—
177C22×#41C2×D311yescharacteristic
178#532—68no—
179C2×C4⋊Q8—31no—
180C2×#127—31no—
181C2×#124—31no—
182C2×#124—31no—
183C2×#123—34no—
184C2×#121—34no—
185C4×#41—68no—
186C2×C4×Q8—32no—
187C2×C4×D4—34no—
188C2×C4×D4—32no—
189C2×C4×D4—34no—
190C2×#117—31no—
191C2×#117—32no—
192C2×#117C2×D311yescharacteristic
193C22×C4⋊C4—32no—
194C22×C42—31no—
195C22×C42—31no—
196C22×C42—32no—
197C22×C42C2×D311yescharacteristic
198#400—34no—
199#399—34no—
200#398—34no—
201#397—34no—
202#397—34no—
203#397—34no—
204#394—34no—
205#394—34no—
206#391—32no—
207#391—32no—
208#391C2×D312yes—
209#390—32no—
210#390—32no—
211#390C2×D312yes—
212#389—32no—
213#389—32no—
214#388—34no—
215#388—34no—
216#387—34no—
217#387—34no—
218#386—34no—
219#386—32no—
220#386—32no—
221#386—34no—
222#386C2×D312yes—
223#386C2×D312yes—
224#385—34no—
225#385—34no—
226#381—34no—
227C4×C4⋊C4—34no—
228C4×C4⋊C4—32no—
229C4×C4⋊C4—32no—
230C4×C4⋊C4—32no—
231C4×C4⋊C4—32no—
232C4×C22⋊C4—32no—
233C4×C22⋊C4—32no—
234C4×C22⋊C4—34no—
235C4×C22⋊C4C2×D314yes—
236C2×#95—32no—
237C24×C3C2412yes—
238C24×C3C2411yescharacteristic
239C23×D3C2412yes—
240C3×#41—416no—
241C2×C3×Q8—22no—
242C2×C3×Q8—22no—
243C2×C3×Q8—22no—
244C2×C3×D4—24no—
245C2×C3×D4—24no—
246C2×C3×D4—24no—
247C2×C3×D4—22no—
248C2×C3×D4—22no—
249C2×C3×D4—22no—
250C22×C3×C4—22no—
251C22×C3×C4—24no—
252C22×C3×C4—22no—
253C22×C3×C4—24no—
254C22×C3×C4—24no—
255C22×C3×C4—24no—
256C22×C3×C4—42no—
257C22×C3×C4C2×D414yes—
258C22×C3×C4#4114yes—
259C22×C3×C4#4114yes—
260C22×C3×C4C2412yes—
261C22×C3×C4C2412yes—
262C22×C3×C4C2414yes—
263C22×C3×C4C2412yes—
264C22×C3×C4C2411yescharacteristic
265C22×C3×C4C2411yescharacteristic
266C2×#67—24no—
267C2×#67—24no—
268C2×#67—28no—
269C2×#67—28no—
270C2×#67—44no—
271C2×#67C2×D418yes—
272C22×Dic3—24no—
273C22×Dic3—24no—
274C22×Dic3—24no—
275C22×Dic3—24no—
276C22×Dic3—42no—
277C22×Dic3—42no—
278C22×Dic3C2414yes—
279C22×Dic3C2412yes—
280C22×Dic3C2412yes—
281C22×Dic3C2412yes—
282C22×Dic3C2412yes—
283C22×Dic3C2412yes—
284#235—232no—
285C2×D12—22no—
286C2×D12—22no—
287C2×D12C2×D414yes—
288C2×C4×D3—24no—
289C2×C4×D3—24no—
290C2×C4×D3—28no—
291C2×C4×D3—28no—
292C2×C4×D3—44no—
293C2×C4×D3C2×D418yes—
294C2×Dic6—22no—
295C2×Dic6—22no—
296C2×Dic6C2×D414yes—
297C3×C4⋊C4—24no—
298C3×C22⋊C4—24no—
299C3×C22⋊C4—28no—
300C3×C22⋊C4—44no—
301C3×C42—24no—
302C3×C42—24no—
303C3×C42—24no—
304#217—24no—
305#217—28no—
306#217—28no—
307#217—44no—
308(C2×D3)⋊C4—28no—
309(C2×D3)⋊C4—28no—
310(C2×D3)⋊C4—28no—
311(C2×D3)⋊C4—44no—
312C4⋊Dic3—24no—
313C4⋊Dic3—24no—
314C4⋊Dic3—24no—
315C4⋊Dic3—24no—
316C4⋊Dic3#4118yescomplemented
317Dic3⋊C4—28no—
318Dic3⋊C4—28no—
319Dic3⋊C4—28no—
320Dic3⋊C4—44no—
321C4×Dic3—22no—
322C4×Dic3—22no—
323C4×Dic3—24no—
324C4×Dic3—24no—
325C4×Dic3—22no—
326C4×Dic3—22no—
327C4×Dic3—24no—
328C4×Dic3—28no—
329C2×#41—24no—
330C2×#41—24no—
331C2×#41—24no—
332C2×#41—38no—
333C2×#41—64no—
334C2×#41—64no—
335C22×Q8—31no—
336C22×Q8C22×D311yescharacteristic
337C22×D4—32no—
338C22×D4—34no—
339C22×D4—31no—
340C22×D4C22×D312yes—
341C22×D4C22×D311yescharacteristic
342C23×C4—32no—
343C23×C4—32no—
344C23×C4—32no—
345C23×C4—34no—
346C23×C4C22×D311yescharacteristic
347C23×C4C22×D312yes—
348C23×C4C22×D311yescharacteristic
349C23×C4C22×D311yescharacteristic
350C4⋊Q8—64no—
351#127—64no—
352#124—64no—
353#124—64no—
354#123—616no—
355#121—616no—
356C4×Q8—68no—
357C4×D4—616no—
358C4×D4—68no—
359C4×D4—616no—
360#117—24no—
361#117—68no—
362#117—64no—
363C2×C4⋊C4—32no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—34no—
368C2×C4⋊C4—32no—
369C2×C4⋊C4—34no—
370C2×C4⋊C4—34no—
371C2×C4⋊C4—34no—
372C2×C4⋊C4—68no—
373C2×C4⋊C4C22×D312yes—
374C2×C22⋊C4—32no—
375C2×C22⋊C4—34no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4—34no—
378C2×C22⋊C4—34no—
379C2×C22⋊C4—34no—
380C2×C22⋊C4—34no—
381C2×C22⋊C4—34no—
382C2×C22⋊C4C22×D312yes—
383C2×C22⋊C4C22×D314yes—
384C2×C22⋊C4C22×D314yes—
385C2×C42—24no—
386C2×C42—32no—
387C2×C42—34no—
388C2×C42—31no—
389C2×C42—31no—
390C2×C42—31no—
391C2×C42—32no—
392C2×C42—32no—
393C2×C42—31no—
394C2×C42—68no—
395C2×C42—64no—
396C2×C42—64no—
397C2×C42C22×D312yes—
398C2×C42C22×D312yes—
399C2×C42C22×D312yes—
400#95—32no—
401#95—32no—
402#95—32no—
403#95—32no—
404#95—34no—
405#95—34no—
406#95—34no—
407#95—34no—
408#95—34no—
409#95—34no—
410#95C22×D314yes—
411#95C22×D314yes—
412C23×C3—22no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3—24no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—24no—
419C23×C3C22×D412yes—
420C23×C3C2×#4112yes—
421C23×C3C2×#4112yes—
422C23×C3C2511yescharacteristic
423C22×D3—24no—
424C22×D3—22no—
425C22×D3—22no—
426C22×D3—24no—
427C22×D3C22×D414yes—
428C3×Q8—44no—
429C3×D4—48no—
430C3×D4—44no—
431C2×C3×C4—28no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—24no—
435C2×C3×C4—22no—
436C2×C3×C4—24no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—24no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—44no—
444C2×C3×C4—48no—
445C2×C3×C4#123116yescomplemented
446C2×C3×C4C22×D412yes—
447C2×C3×C4C22×D412yes—
448C2×C3×C4C2×#4112yes—
449C2×C3×C4C2×#4112yes—
450C2×C3×C4C2×#4112yes—
451C2×C3×C4C2×#4112yes—
452C2×C3×C4C2×#4114yes—
453#67—216no—
454#67—416no—
455C2×Dic3—22no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3—22no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3—22no—
463C2×Dic3—24no—
464C2×Dic3—24no—
465C2×Dic3—22no—
466C2×Dic3—22no—
467C2×Dic3—22no—
468C2×Dic3—24no—
469C2×Dic3—22no—
470C2×Dic3—48no—
471C2×Dic3—48no—
472C2×Dic3C22×D414yes—
473C2×Dic3C2×#4114yes—
474C2×Dic3C2×#4114yes—
475D12—28no—
476C4×D3—216no—
477C4×D3—416no—
478Dic6—28no—
479C24—32no—
480C24C23×D312yes—
481C24C23×D311yescharacteristic
482#41—416no—
483#41—632no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—34no—
488C2×Q8—62no—
489C2×Q8—62no—
490C2×D4—24no—
491C2×D4—24no—
492C2×D4—24no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—38no—
497C2×D4—34no—
498C2×D4—68no—
499C2×D4—64no—
500C2×D4—64no—
501C2×D4—68no—
502C2×D4—62no—
503C2×D4—62no—
504C2×D4—124no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—32no—
512C22×C4—32no—
513C22×C4—32no—
514C22×C4—38no—
515C22×C4—34no—
516C22×C4—32no—
517C22×C4—32no—
518C22×C4—42no—
519C22×C4—68no—
520C22×C4—68no—
521C22×C4—64no—
522C22×C4—64no—
523C22×C4—64no—
524C22×C4—64no—
525C22×C4—64no—
526C22×C4—64no—
527C22×C4—124no—
528C22×C4—122no—
529C22×C4—122no—
530C22×C4#23514yes—
531C22×C4D3×D414yes—
532C22×C4#23714yes—
533C22×C4C23×D312yes—
534C22×C4C23×D312yes—
535C22×C4C23×D314yes—
536C22×C4C23×D312yes—
537C22×C4C23×D311yescharacteristic
538C22×C4C23×D311yescharacteristic
539C4⋊C4—24no—
540C4⋊C4—38no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4—68no—
544C4⋊C4—68no—
545C4⋊C4—68no—
546C4⋊C4—64no—
547C4⋊C4—64no—
548C4⋊C4—124no—
549C22⋊C4—24no—
550C22⋊C4—28no—
551C22⋊C4—44no—
552C22⋊C4—68no—
553C22⋊C4—68no—
554C22⋊C4—68no—
555C22⋊C4—68no—
556C22⋊C4—64no—
557C22⋊C4—68no—
558C22⋊C4—124no—
559C22⋊C4—124no—
560C42—24no—
561C42—24no—
562C42—24no—
563C42—68no—
564C42—64no—
565C42—62no—
566C42—62no—
567C42—64no—
568C42—64no—
569C42—62no—
570C42—62no—
571C22×C3—22no—
572C22×C3—22no—
573C22×C3—24no—
574C22×C3—24no—
575C22×C3—42no—
576C22×C3—42no—
577C22×C3—42no—
578C22×C3—44no—
579C22×C3—44no—
580C22×C3C2×#12314yes—
581C22×C3C23×D411yescharacteristic
582C22×C3C22×#4111yescharacteristic
583C22×C3C22×#4111yescharacteristic
584C22×C3C22×#4112yes—
585C22×C3#58512yes—
586C2×D3—24no—
587C2×D3—28no—
588C2×D3—42no—
589C2×D3—42no—
590C2×D3—44no—
591C3×C4—24no—
592C3×C4—42no—
593C3×C4—42no—
594C3×C4—44no—
595C3×C4C2×#12314yes—
596C3×C4C2×#12314yes—
597Dic3—24no—
598Dic3—24no—
599Dic3—24no—
600Dic3—42no—
601Dic3—42no—
602Dic3—44no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—24no—
609C23—24no—
610C23—34no—
611C23—62no—
612C23—62no—
613C23—64no—
614C23—64no—
615C23C2×#23512yes—
616C23C2×D3×D412yes—
617C23C2×#23712yes—
618C23C24×D311yescharacteristic
619Q8—44no—
620Q8—68no—
621D4—48no—
622D4—44no—
623D4—616no—
624D4—68no—
625D4—1216no—
626C2×C4—28no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—34no—
639C2×C4—34no—
640C2×C4—34no—
641C2×C4—44no—
642C2×C4—48no—
643C2×C4—62no—
644C2×C4—62no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—62no—
648C2×C4—62no—
649C2×C4—616no—
650C2×C4—64no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—64no—
654C2×C4—62no—
655C2×C4—62no—
656C2×C4—64no—
657C2×C4—62no—
658C2×C4—62no—
659C2×C4—1216no—
660C2×C4—128no—
661C2×C4—128no—
662C2×C4#883116yescomplemented
663C2×C4C2×#23512yes—
664C2×C4C2×#23512yes—
665C2×C4C2×D3×D412yes—
666C2×C4C2×D3×D412yes—
667C2×C4C2×#23712yes—
668C2×C4C2×#23712yes—
669C2×C4D3×#4114yes—
670C2×C3—24no—
671C2×C3—82no—
672C2×C3C22×#12311yescharacteristic
673C2×C3#344411yescharacteristic
674C2×C3#345511yescharacteristic
675C2×C3D4×#4112yes—
676C2×C3#348012yes—
677D3—44no—
678C22—22no—
679C22—22no—
680C22—24no—
681C22—24no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—64no—
688C22—68no—
689C22—122no—
690C22—122no—
691C22—124no—
692C22C2×#88314yes—
693C22C22×#23511yescharacteristic
694C22C22×D3×D411yescharacteristic
695C22C22×#23711yescharacteristic
696C22C2×D3×#4112yes—
697C22#596412yes—
698C4—24no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—64no—
703C4—64no—
704C4—64no—
705C4—124no—
706C4—122no—
707C4—122no—
708C4C2×#88314yes—
709C4C2×#88314yes—
710C3#3474411yescharacteristic, complemented, Sylow
711C2—24no—
712C2—82no—
713C2—124no—
714C2C22×#88311yescharacteristic
715C2#8582512yes—
716C2#8601811yescharacteristic
717C2#8601911yescharacteristic
718C2#8623812yes—
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC3×#2297C2
prime radicalC2×#5832C2
hypercenter#2297D3
hypocenterC3#34744
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. C2×#5522C2semidirect
3. C3×#2297C2semidirect
4. #84086C2semidirect
5. #82226C2semidirect
6. #82070C2semidirect
7. C2×C4×C4⋊Dic3C2semidirect
8. #76506C2semidirect
9. #76504C2semidirect
10. #76321C2semidirect
11. #76318C2semidirect
12. #74707C2semidirect
13. #74704C2semidirect
14. #73582C2semidirect
15. C4×#883C2semidirect
90. #2297D3semidirect
316. C4⋊Dic3#41semidirect
445. C2×C3×C4#123semidirect
662. C2×C4#883semidirect
710. C3#34744semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
C2×#55222
C3×#22972
#840862
#822262
#820702
C2×C4×C4⋊Dic32
#765062
#765042
#763212
#763182
#747072
#747042
#735822
C4×#8832
#347443