Finite Group Databasebeta

← #10967708

Subgroups — #10967708

Order 768.

2474 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2C2×C4×#235C211yesmaximal, characteristic
3C3×C4×#117C211yesmaximal, characteristic, complemented
4#82277C214yesmaximal, complemented
5#73802C211yesmaximal, characteristic, complemented
6#73800C211yesmaximal, characteristic, complemented
7C4×#897C214yesmaximal, complemented
8#73722C211yesmaximal, characteristic, complemented
9#73711C211yesmaximal, characteristic, complemented
10#73583C214yesmaximal, complemented
11#73514C212yesmaximal, complemented
12#73489C212yesmaximal
13#73427C214yesmaximal, complemented
14C4×#869C212yesmaximal, complemented
15C4×#875C212yesmaximal, complemented
16#34979—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C2×C3×#117C2212yes—
19C22×C3×C42C2211yescharacteristic
20C2×C4×#67C2212yes—
21C2×C4×#67C2212yes—
22C2×#926C2211yescharacteristic
23C2×#926C2212yes—
24C2×#924C2212yes—
25C2×#924C2212yes—
26C2×#897C2214yes—
27C2×#875C2212yes—
28C4×#235—28no—
29C2×C4×D12C2211yescharacteristic
30C2×C4×D12C2211yescharacteristic
31C2×#872C2212yes—
32C2×C42×D3C2212yes—
33#5470—28no—
34C2×#870C2212yes—
35C2×#869C2212yes—
36C2×#869C2211yescharacteristic
37C2×#869C2211yescharacteristic
38C2×C4×Dic6C2212yes—
39C2×C4×Dic6C2212yes—
40C2×C4×Dic6C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42C3×C4×C4⋊C4C2212yes—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C22⋊C4C2212yes—
46C3×C4×C22⋊C4C2212yes—
47C3×#376C2212yes—
48C3×C43C2212yes—
49#4987C2214yes—
50#4984C2214yes—
51#4973C2214yes—
52#4961C2218yes—
53#4953C2218yes—
54#4940C2218yes—
55#4939C2212yes—
56C4×(C2×D3)⋊C4C2218yes—
57#4936C2212yes—
58#4935C2212yes—
59C4×C4⋊Dic3C2218yes—
60#4929C2212yes—
61#4929C2212yes—
62#4929C2212yes—
63#4928C2218yes—
64#4677C2214yes—
65#4674C2214yes—
66#4668C2218yes—
67#4661C2214yes—
68#4659C2214yes—
69#4653C2218yes—
70C2×#533—31no—
71C2×C4×#41—31no—
72#2611—34no—
73#2568—31no—
74#2547—31no—
75#2546—31no—
76#2542—31no—
77#2499—32no—
78#2432—32no—
79#2415—34no—
80#2395—34no—
81C4×C4⋊Q8—32no—
82C4×#124—32no—
83C4×C22⋊Q8—34no—
84C4×#117D311yescharacteristic, complemented
85C23×C3×C4C2311yescharacteristic
86C23×C3×C4C2311yescharacteristic
87C23×C3×C4C2311yescharacteristic
88C22×#67C2312yes—
89C2×#235—24no—
90C2×#235—24no—
91C2×#235—24no—
92C22×D12C2311yescharacteristic
93C22×C4×D3C2312yes—
94C22×C4×D3C2312yes—
95C22×C4×D3C2312yes—
96C22×Dic6C2312yes—
97C22×Dic6C2312yes—
98C22×Dic6C2311yescharacteristic
99C3×#117—28no—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4C2314yes—
103C2×C3×C42D418yes—
104C2×C3×C42C2314yes—
105C2×C3×C42C2312yes—
106C2×C3×C42C2312yes—
107C2×C3×C42C2314yes—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×#217C2312yes—
113C2×#217C2314yes—
114C4×#67—28no—
115C4×#67—28no—
116C2×(C2×D3)⋊C4C2318yes—
117C2×(C2×D3)⋊C4C2318yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2312yes—
120#926—28no—
121#926—24no—
122C2×C4⋊Dic3C2318yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2318yes—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2312yes—
129C2×C4⋊Dic3C2312yes—
130#924—28no—
131#924—28no—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2312yes—
135C2×Dic3⋊C4C2314yes—
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2314yes—
139C2×C4×Dic3C2312yes—
140#897—216no—
141#875—28no—
142C4×D12—24no—
143C4×D12—24no—
144#872—28no—
145C42×D3—28no—
146#870—28no—
147#869—28no—
148#869—24no—
149#869—24no—
150C4×Dic6—28no—
151C4×Dic6—28no—
152C4×Dic6—24no—
153C4×Dic6—24no—
154C3×#95C2314yes—
155C3×#95C2312yes—
156C3×#95C2312yes—
157C3⋊#95C2318yes—
158C3⋊#95C2318yes—
159C3⋊#95C2318yes—
160C3⋊#95C2318yes—
161C22×#41—31no—
162#533—68no—
163C2×C4⋊Q8—32no—
164C2×C4⋊Q8—31no—
165C2×C4⋊Q8—31no—
166C2×#125—32no—
167C2×#124—32no—
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8—32no—
170C2×C22⋊Q8—34no—
171C4×#41—68no—
172C2×C4×Q8—31no—
173C2×C4×Q8—32no—
174C2×C4×Q8—31no—
175C2×C4×Q8—32no—
176C2×C4×D4—32no—
177C2×C4×D4—32no—
178C2×C4×D4—31no—
179C2×C4×D4—31no—
180C2×#117—32no—
181C2×#117—31no—
182C2×#117—32no—
183C2×#117C2×D312yes—
184C22×C42—32no—
185C22×C42C2×D311yescharacteristic
186#400—34no—
187#399—34no—
188#398—34no—
189#397—38no—
190#397—34no—
191#394—38no—
192#391—34no—
193#391—32no—
194#391—32no—
195#391—32no—
196#390—32no—
197#389—34no—
198#389—38no—
199#388—38no—
200#387—38no—
201#385—38no—
202#384—32no—
203#382—32no—
204#380—34no—
205C4×C4⋊C4—38no—
206C4×C4⋊C4C2×D311yescharacteristic
207C4×C4⋊C4C2×D311yescharacteristic
208C4×C4⋊C4C2×D312yes—
209C4×C22⋊C4—38no—
210C4×C22⋊C4C2×D312yes—
211C4×C22⋊C4C2×D312yes—
212#376C2×D312yes—
213C43C2×D312yes—
214C24×C3C2411yescharacteristic
215C23×D3C2412yes—
216C22×C3×C4—24no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—24no—
220C22×C3×C4C2×D414yes—
221C22×C3×C4C2×D414yes—
222C22×C3×C4C2×D414yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2414yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C2×#67—24no—
232C2×#67—24no—
233C2×#67—24no—
234C22×Dic3C2412yes—
235C22×Dic3C2412yes—
236C22×Dic3C2414yes—
237C22×Dic3C2414yes—
238C22×Dic3C2412yes—
239#235—232no—
240C2×D12—22no—
241C2×D12—22no—
242C2×D12#4114yes—
243C2×C4×D3—24no—
244C2×C4×D3—24no—
245C2×C4×D3—24no—
246C2×C4×D3—24no—
247C2×C4×D3—24no—
248C2×C4×D3—28no—
249C2×C4×D3—28no—
250C2×Dic6—22no—
251C2×Dic6—22no—
252C2×Dic6—24no—
253C2×Dic6—24no—
254C2×Dic6—24no—
255C2×Dic6—24no—
256C2×Dic6—28no—
257C2×Dic6#4114yes—
258C3×C4⋊C4—24no—
259C3×C4⋊C4#4118yescomplemented
260C3×C22⋊C4—28no—
261C3×C42—24no—
262C3×C42—24no—
263C3×C42—28no—
264C3×C42—28no—
265C3×C42C2×D414yes—
266C3×C42C2×D414yes—
267C3×C42C2×D414yes—
268C3×C42C2×D414yes—
269#217—28no—
270#217—24no—
271(C2×D3)⋊C4—216no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—216no—
275C4⋊Dic3—24no—
276C4⋊Dic3—24no—
277C4⋊Dic3—22no—
278C4⋊Dic3—22no—
279C4⋊Dic3—24no—
280C4⋊Dic3—24no—
281C4⋊Dic3—216no—
282C4⋊Dic3—216no—
283Dic3⋊C4—28no—
284Dic3⋊C4—28no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287C4×Dic3—28no—
288C4×Dic3—24no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C2×#41—64no—
292C2×#41—64no—
293C2×#41—64no—
294C22×Q8—31no—
295C22×Q8—32no—
296C22×Q8—32no—
297C22×D4—32no—
298C22×D4—31no—
299C23×C4—32no—
300C23×C4—32no—
301C23×C4—32no—
302C23×C4C22×D311yescharacteristic
303C23×C4C22×D311yescharacteristic
304C23×C4C22×D311yescharacteristic
305C4⋊Q8—68no—
306C4⋊Q8—64no—
307C4⋊Q8—64no—
308#125—68no—
309#124—68no—
310C22⋊Q8—68no—
311C22⋊Q8—68no—
312C22⋊Q8—616no—
313C4×Q8—64no—
314C4×Q8—64no—
315C4×Q8—68no—
316C4×Q8—68no—
317C4×D4—68no—
318C4×D4—68no—
319C4×D4—64no—
320C4×D4—64no—
321#117—28no—
322#117—64no—
323#117—68no—
324#117—68no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—38no—
336C2×C4⋊C4—38no—
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D312yes—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—38no—
344C2×C22⋊C4—38no—
345C2×C22⋊C4C22×D314yes—
346C2×C42—32no—
347C2×C42—32no—
348C2×C42—34no—
349C2×C42—32no—
350C2×C42—68no—
351C2×C42D1218yes—
352C2×C42C22×D314yes—
353C2×C42C22×D312yes—
354C2×C42C22×D312yes—
355C2×C42C22×D314yes—
356C2×C42C22×D311yescharacteristic
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D311yescharacteristic
359C2×C42C22×D311yescharacteristic
360#95—38no—
361#95—38no—
362#95—38no—
363#95—38no—
364#95C22×D314yes—
365#95C22×D312yes—
366#95C22×D312yes—
367C23×C3—22no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—22no—
371C23×C3C22×D412yes—
372C23×C3C2511yescharacteristic
373C22×D3—22no—
374C22×D3—24no—
375C22×D3—22no—
376C22×D3—24no—
377C22×D3C2×#4114yes—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—28no—
385C2×C3×C4—28no—
386C2×C3×C4—28no—
387C2×C3×C4—24no—
388C2×C3×C4—24no—
389C2×C3×C4—24no—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C22×D412yes—
394C2×C3×C4C22×D412yes—
395C2×C3×C4C22×D412yes—
396C2×C3×C4C2×#4118yes—
397C2×C3×C4C2×#4114yes—
398C2×C3×C4C2×#4112yes—
399C2×C3×C4C2×#4112yes—
400C2×C3×C42−1+414yes—
401#67—216no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—22no—
406C2×Dic3—24no—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3—24no—
410C2×Dic3—28no—
411C2×Dic3—24no—
412C2×Dic3—24no—
413C2×Dic3C2×#4114yes—
414D12—28no—
415C4×D3—216no—
416C4×D3—48no—
417C4×D3—48no—
418Dic6—28no—
419Dic6—48no—
420Dic6—48no—
421C24—32no—
422C24C23×D311yescharacteristic
423#41—632no—
424C2×Q8—34no—
425C2×Q8—62no—
426C2×Q8—64no—
427C2×Q8—64no—
428C2×Q8—68no—
429C2×Q8—62no—
430C2×Q8—64no—
431C2×Q8—64no—
432C2×D4—34no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4—62no—
437C2×D4—62no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—34no—
446C22×C4—34no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4C2×D1214yes—
455C22×C4C2×D1214yes—
456C22×C4C2×D1214yes—
457C22×C4C23×D314yes—
458C22×C4C23×D311yescharacteristic
459C22×C4C23×D311yescharacteristic
460C22×C4C23×D311yescharacteristic
461C22×C4C23×D314yes—
462C22×C4C23×D311yescharacteristic
463C22×C4C23×D311yescharacteristic
464C22×C4C23×D311yescharacteristic
465C4⋊C4—24no—
466C4⋊C4—616no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—68no—
474C4⋊C4—68no—
475C4⋊C4—64no—
476C4⋊C4—64no—
477C4⋊C4—616no—
478C4⋊C4#23518yescomplemented
479C22⋊C4—28no—
480C22⋊C4—616no—
481C22⋊C4—64no—
482C22⋊C4—68no—
483C22⋊C4—64no—
484C22⋊C4—64no—
485C22⋊C4—616no—
486C42—28no—
487C42—24no—
488C42—24no—
489C42—28no—
490C42—64no—
491C42—64no—
492C42—64no—
493C42—68no—
494C42C2×D1214yes—
495C42C2×D1214yes—
496C42C2×D1214yes—
497C42C2×D1214yes—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3C23×D411yescharacteristic
504C22×C3C22×#4112yes—
505C22×C3C22×#4111yescharacteristic
506C22×C3C22×#4111yescharacteristic
507C22×C3C2×2−1+412yes—
508C2×D3—24no—
509C2×D3—28no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—44no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—28no—
517C3×C4—44no—
518C3×C4#52514yes—
519C3×C4#53214yes—
520C3×C4#53614yes—
521Dic3—24no—
522Dic3—44no—
523Dic3—44no—
524Dic3—42no—
525Dic3—42no—
526C23—22no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—34no—
531C23—62no—
532C23—64no—
533C23—62no—
534C23—64no—
535C23C22×D1212yes—
536C23C24×D311yescharacteristic
537Q8—68no—
538Q8—128no—
539Q8—128no—
540D4—616no—
541D4—68no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—34no—
555C2×C4—616no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—68no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—128no—
568C2×C4—128no—
569C2×C4C22×D1212yes—
570C2×C4C22×D1212yes—
571C2×C4C22×D1212yes—
572C2×C4C22×D1212yes—
573C2×C4C22×D1212yes—
574C2×C4C22×D1212yes—
575C2×C4C2×#23518yes—
576C2×C4C2×#23514yes—
577C2×C4D3×#4112yes—
578C2×C4D3×#4112yes—
579C2×C4#101014yes—
580C2×C3—42no—
581C2×C3C2×#52511yescharacteristic
582C2×C3C2×#53211yescharacteristic
583C2×C3C2×#53611yescharacteristic
584C2×C3#348212yes—
585C2×C3#348212yes—
586D3—44no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—42no—
591C22—42no—
592C22—68no—
593C22—64no—
594C22—124no—
595C22—122no—
596C22—122no—
597C22C23×D1211yescharacteristic
598C22C22×#23512yes—
599C22C2×D3×#4111yescharacteristic
600C22C2×D3×#4111yescharacteristic
601C22C2×#101012yes—
602C4—28no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—44no—
607C4—64no—
608C4—122no—
609C4—124no—
610C4—122no—
611C4—124no—
612C4#547714yes—
613C4#552414yes—
614C4#552614yes—
615C3#3497911yescharacteristic, complemented, Sylow
616C2—42no—
617C2—124no—
618C2C2×#547711yescharacteristic
619C2C2×#552611yescharacteristic
620C2C2×#552411yescharacteristic
621C2#8582612yes—
622C2#8584412yes—
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC3×C4×#117C2
prime radicalC22×#235C22
hypercenterC4×#117D3
hypocenterC3#34979
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
3. C3×C4×#117C2semidirect
4. #82277C2semidirect
5. #73802C2semidirect
6. #73800C2semidirect
7. C4×#897C2semidirect
8. #73722C2semidirect
9. #73711C2semidirect
10. #73583C2semidirect
11. #73514C2semidirect
13. #73427C2semidirect
14. C4×#869C2semidirect
15. C4×#875C2semidirect
84. C4×#117D3semidirect
259. C3×C4⋊C4#41semidirect
478. C4⋊C4#235semidirect
615. C3#34979semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C2×C4×#2352
C3×C4×#1172
#822772
#738022
#738002
C4×#8972
#737222
#737112
#735832
#735142
#734892
#734272
C4×#8692
C4×#8752
#349793