Finite Group Databasebeta

← #10701782

Subgroups — #10701782

Order 768.

1274 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5526C211yesmaximal, characteristic
2C2×#5120C211yesmaximal, characteristic, complemented
3#83762C211yesmaximal, characteristic
4C2×#5006C211yesmaximal, characteristic
5C2×#5004C211yesmaximal, characteristic, complemented
6C2×C3×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
7C2×#4531C211yesmaximal, characteristic, complemented
8#10988—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C23×Dic6C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5526—22no—
17#5526—22no—
18#5526—22no—
19#5526—22no—
20C2×#897—22no—
21C2×#886—22no—
22C2×#875C412yescomplemented
23C2×#869C412yescomplemented
24#5129—22no—
25#5126—22no—
26#5120—22no—
27#5120—22no—
28#5110—22no—
29C2×#816—22no—
30#5006—22no—
31#5006—22no—
32#5004—22no—
33#5004—22no—
34C2×#810—22no—
35C2×#806—22no—
36C3×H(3,ℤ/4)—22no—
37#4531—22no—
38C2×#536—31no—
39C2×#419—31no—
40C2×#419—31no—
41C2×#410—31no—
42#2879—31no—
43C2×H(3,ℤ/4)—31no—
44C2×H(3,ℤ/4)D311yescharacteristic, complemented
45C23×C3×C4C2311yescharacteristic
46C22×#67D412yes—
47C23×Dic3—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235—21no—
54C2×#235—21no—
55C2×#235—21no—
56C2×#235—21no—
57C2×#235—21no—
58C2×#235—21no—
59C2×#235—21no—
60C22×D12C2×C411yescharacteristic
61C22×C4×D3D412yes—
62C22×Dic6—21no—
63C22×Dic6—21no—
64C22×Dic6—21no—
65C22×Dic6—21no—
66C22×Dic6—21no—
67C22×Dic6—21no—
68C22×Dic6—21no—
69C22×Dic6—21no—
70C22×Dic6—21no—
71C22×Dic6—21no—
72C22×Dic6—21no—
73C22×Dic6C2×C411yescharacteristic
74C22×Dic6C2×C412yes—
75C22×Dic6D412yes—
76C22×Dic6D412yes—
77C2×C3×M16—21no—
78C2×C3×M16—21no—
79C2×C3×M16—21no—
80C2×C3×M16—21no—
81C2×C3×M16—21no—
82C2×C3×M16—21no—
83C22×C3×C8D412yes—
84C3×#117—21no—
85C3×#117—21no—
86C3×#117—21no—
87C3×#117—21no—
88C3×#117—21no—
89C3×#117D412yes—
90C3×#117D412yes—
91C3×#117D412yes—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2×C412yes—
98C2×#217—21no—
99C2×(C2×D3)⋊C4—24no—
100#926—21no—
101#926—21no—
102#926—21no—
103#926D412yes—
104C2×C4⋊Dic3—24no—
105C2×C4⋊Dic3—21no—
106C2×C4×Dic3—21no—
107C2×#208—21no—
108C2×#208—21no—
109C2×#208—21no—
110C2×#208—21no—
111C2×#208—21no—
112C2×#208—21no—
113C22×C3⋊C8D412yes—
114#897—28no—
115#897—28no—
116#886—28no—
117#886—28no—
118#875—24no—
119#875—24no—
120#869—24no—
121#869—24no—
122(C2×D3)⋊C8—28no—
123#816—28no—
124#810—28no—
125#806—28no—
126C22×#41—31no—
127C23×Q8—31no—
128C22×M16—31no—
129C22×M16C2×D311yescharacteristic
130#536—62no—
131#536—62no—
132#536—62no—
133#536—62no—
134C2×C4⋊Q8—32no—
135C2×#124—32no—
136C2×#123—62no—
137C2×C22⋊Q8—62no—
138C2×#117—31no—
139C2×#117C2×D311yescharacteristic
140C2×#117C2×D311yescharacteristic
141#419—62no—
142#419—62no—
143#419—62no—
144#419—62no—
145C2×#103—62no—
146C2×#103—62no—
147#410—62no—
148#410—62no—
149C2×#99—62no—
150#408—62no—
151#408—62no—
152#407—62no—
153H(3,ℤ/4)—22no—
154H(3,ℤ/4)—62no—
155C24×C3C2×D411yescharacteristic
156C23×D3C22⋊C412yes—
157C22×C3×C4—22no—
158C22×C3×C4—41no—
159C22×C3×C4C22×C411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4#4111yescharacteristic
170C22×C3×C4#4111yescharacteristic
171C22×C3×C4#4111yescharacteristic
172C22×C3×C4#4111yescharacteristic
173C2×#67—22no—
174C2×#67—22no—
175C2×#67—22no—
176C2×#67—22no—
177C2×#67—22no—
178C2×#67—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—41no—
183C22×Dic3—41no—
184C22×Dic3—41no—
185C22×Dic3—41no—
186C22×Dic3—41no—
187C22×Dic3C22⋊C412yes—
188#235—41no—
189#235—41no—
190#235—41no—
191#235—41no—
192#235—41no—
193#235—41no—
194#235—41no—
195#235—41no—
196#235—41no—
197#235—41no—
198#235—41no—
199#235—41no—
200#235—41no—
201#235—41no—
202#235—41no—
203#235—41no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—22no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—22no—
236C2×Dic6—22no—
237C2×Dic6—41no—
238C2×Dic6—41no—
239C2×Dic6—41no—
240C2×Dic6—41no—
241C2×Dic6—41no—
242C2×Dic6—41no—
243C2×Dic6—41no—
244C2×Dic6—41no—
245C2×Dic6—41no—
246C2×Dic6—41no—
247C2×Dic6—41no—
248C2×Dic6—41no—
249C2×Dic6C22⋊C414yes—
250C3×M16—41no—
251C3×M16—41no—
252C3×M16—41no—
253C3×M16—41no—
254C2×C3×C8—22no—
255C2×C3×C8—22no—
256C2×C3×C8—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—41no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—41no—
265C3×C42—22no—
266C3×C42—22no—
267C3×C42—41no—
268C3×C42C22⋊C414yes—
269#217—22no—
270#217—41no—
271(C2×D3)⋊C4—216no—
272C4⋊Dic3—216no—
273C4⋊Dic3—22no—
274C4⋊Dic3—41no—
275C4×Dic3—22no—
276C4×Dic3—41no—
277#208—41no—
278#208—41no—
279#208—41no—
280#208—41no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8—22no—
283C2×C3⋊C8—22no—
284C2×#41—61no—
285C2×#41—61no—
286C2×#41—61no—
287C2×#41—61no—
288C2×#41—61no—
289C2×#41—61no—
290C2×#41—61no—
291C2×#41—61no—
292C2×#41—61no—
293C2×#41—61no—
294C2×#41—61no—
295C2×#41—61no—
296C22×Q8—31no—
297C22×Q8—32no—
298C22×Q8—32no—
299C22×Q8—32no—
300C22×Q8—61no—
301C22×Q8—61no—
302C22×Q8—61no—
303C22×Q8—61no—
304C22×Q8—61no—
305C22×Q8—61no—
306C22×Q8—61no—
307C22×Q8—61no—
308C22×Q8—61no—
309C22×Q8—61no—
310C22×Q8—61no—
311C22×D4—32no—
312C22×D4—31no—
313C23×C4—32no—
314C23×C4—61no—
315C23×C4C22×D311yescharacteristic
316C2×M16—21no—
317C2×M16—21no—
318C2×M16—21no—
319C2×M16—21no—
320C2×M16—21no—
321C2×M16—21no—
322C2×M16—61no—
323C2×M16—61no—
324C2×M16—61no—
325C2×M16—61no—
326C2×M16—61no—
327C2×M16—61no—
328C22×C8—32no—
329C22×C8D1212yes—
330C4⋊Q8—64no—
331C4⋊Q8—64no—
332#124—64no—
333#124—64no—
334#123—68no—
335#123—68no—
336C22⋊Q8—68no—
337C22⋊Q8—68no—
338#117—21no—
339#117—21no—
340#117—21no—
341#117—21no—
342#117—21no—
343#117—32no—
344#117—61no—
345#117—61no—
346#117—61no—
347#117D1212yes—
348#117D1212yes—
349#117#6712yes—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—64no—
353C2×C4⋊C4—61no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—64no—
357C2×C22⋊C4—61no—
358C2×C42—21no—
359C2×C42—61no—
360C2×C42C4×D312yes—
361#103—68no—
362#103—68no—
363#99—68no—
364C22⋊C8—68no—
365C23×C3—42no—
366C23×C3C2×C22⋊C411yescharacteristic
367C23×C3#12011yescharacteristic
368C23×C3#12011yescharacteristic
369C23×C3#12011yescharacteristic
370C23×C3#12111yescharacteristic
371C23×C3#12111yescharacteristic
372C23×C3#12311yescharacteristic
373C22×D3—22no—
374C22×D3—22no—
375C22×D3—22no—
376C22×D3—44no—
377C2×C3×C4—24no—
378C2×C3×C4—44no—
379C2×C3×C4—42no—
380C2×C3×C4—82no—
381C2×C3×C4C2×C22⋊C411yescharacteristic
382C2×C3×C4C2×C22⋊C411yescharacteristic
383C2×C3×C4C4×D411yescharacteristic
384C2×C3×C4C4×D411yescharacteristic
385C2×C3×C4C4×D411yescharacteristic
386C2×C3×C4C4×D411yescharacteristic
387C2×C3×C4#12011yescharacteristic
388C2×C3×C4#12011yescharacteristic
389C2×C3×C4#12011yescharacteristic
390C2×C3×C4#12011yescharacteristic
391C2×C3×C4#12011yescharacteristic
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4#12111yescharacteristic
394C2×C3×C4#12111yescharacteristic
395C2×C3×C4#12111yescharacteristic
396C2×C3×C4#12111yescharacteristic
397C2×C3×C4#12111yescharacteristic
398C2×C3×C4#12111yescharacteristic
399C2×C3×C4#12111yescharacteristic
400C2×C3×C4#12111yescharacteristic
401C2×C3×C4#12111yescharacteristic
402C2×C3×C4#12311yescharacteristic
403C2×C3×C4#12311yescharacteristic
404C2×C3×C4#12311yescharacteristic
405C2×C3×C4#12411yescharacteristic
406C2×C3×C4#12411yescharacteristic
407C2×C3×C4#12711yescharacteristic
408C2×C3×C4#12711yescharacteristic
409#67—42no—
410#67—42no—
411#67—42no—
412#67—42no—
413C2×Dic3—22no—
414C2×Dic3—24no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—42no—
418C2×Dic3—42no—
419C2×Dic3—42no—
420C2×Dic3—42no—
421C2×Dic3—42no—
422C2×Dic3—42no—
423C2×Dic3—42no—
424C2×Dic3—82no—
425D12—41no—
426D12—41no—
427D12—41no—
428D12—41no—
429C4×D3—42no—
430C4×D3—42no—
431C4×D3—42no—
432C4×D3—42no—
433Dic6—24no—
434Dic6—24no—
435Dic6—41no—
436Dic6—41no—
437Dic6—42no—
438Dic6—42no—
439Dic6—41no—
440Dic6—41no—
441Dic6—42no—
442Dic6—42no—
443Dic6—42no—
444Dic6—42no—
445C3×C8—42no—
446C3⋊C8—42no—
447C24—32no—
448C24D3×D411yescharacteristic
449#41—121no—
450#41—121no—
451#41—121no—
452#41—121no—
453#41—121no—
454#41—121no—
455#41—121no—
456#41—121no—
457#41—121no—
458#41—121no—
459#41—121no—
460#41—121no—
461#41—121no—
462#41—121no—
463#41—121no—
464#41—121no—
465C2×Q8—34no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8—61no—
469C2×Q8—62no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—62no—
473C2×Q8—61no—
474C2×Q8—62no—
475C2×Q8—62no—
476C2×Q8—62no—
477C2×Q8—62no—
478C2×Q8—62no—
479C2×Q8—62no—
480C2×Q8—62no—
481C2×Q8—62no—
482C2×Q8—62no—
483C2×Q8—62no—
484C2×Q8—62no—
485C2×Q8—62no—
486C2×Q8—62no—
487C2×Q8—121no—
488C2×Q8—121no—
489C2×Q8—121no—
490C2×Q8—121no—
491C2×Q8—121no—
492C2×Q8—121no—
493C2×Q8—121no—
494C2×Q8—121no—
495C2×Q8—121no—
496C2×Q8—121no—
497C2×Q8—121no—
498C2×Q8—121no—
499C2×D4—62no—
500C2×D4—62no—
501C2×D4—62no—
502C2×D4—62no—
503C2×D4—62no—
504C2×D4—62no—
505C2×D4—61no—
506C2×D4—61no—
507C2×D4—61no—
508C2×D4—61no—
509C2×D4—61no—
510C2×D4—61no—
511C22×C4—22no—
512C22×C4—32no—
513C22×C4—41no—
514C22×C4—62no—
515C22×C4—62no—
516C22×C4—62no—
517C22×C4—62no—
518C22×C4—62no—
519C22×C4—62no—
520C22×C4—62no—
521C22×C4—62no—
522C22×C4—62no—
523C22×C4—121no—
524C22×C4—121no—
525C22×C4—121no—
526C22×C4—121no—
527C22×C4—121no—
528C22×C4C2×C4×D311yescharacteristic
529C22×C4C2×D1211yescharacteristic
530C22×C4C2×D1211yescharacteristic
531C22×C4C2×D1211yescharacteristic
532C22×C4#23511yescharacteristic
533C22×C4#23511yescharacteristic
534C22×C4D3×D411yescharacteristic
535C22×C4D3×D411yescharacteristic
536C22×C4D3×D411yescharacteristic
537C22×C4D3×D411yescharacteristic
538C22×C4D3×D411yescharacteristic
539C22×C4#23911yescharacteristic
540C22×C4#23911yescharacteristic
541C22×C4C2×#6711yescharacteristic
542M16—41no—
543M16—41no—
544M16—41no—
545M16—41no—
546M16—121no—
547M16—121no—
548M16—121no—
549M16—121no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—62no—
554C2×C8—62no—
555C2×C8—62no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—41no—
560C4⋊C4—616no—
561C4⋊C4—62no—
562C4⋊C4—121no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—41no—
567C22⋊C4—616no—
568C22⋊C4—62no—
569C22⋊C4—121no—
570C42—22no—
571C42—22no—
572C42—41no—
573C42—62no—
574C42—121no—
575C42(C2×D3)⋊C414yes—
576C22×C3—22no—
577C22×C3—22no—
578C22×C3—22no—
579C22×C3—41no—
580C22×C3—41no—
581C22×C3—41no—
582C22×C3—41no—
583C22×C3#37911yescharacteristic
584C22×C3#38611yescharacteristic
585C22×C3#38611yescharacteristic
586C22×C3#39211yescharacteristic
587C22×C3#39211yescharacteristic
588C22×C3#39411yescharacteristic
589C22×C3#39411yescharacteristic
590C2×D3—44no—
591C2×D3—44no—
592C2×D3—42no—
593C2×D3—82no—
594C3×C4—44no—
595C3×C4—82no—
596C3×C4#38611yescharacteristic
597C3×C4#38611yescharacteristic
598C3×C4#39011yescharacteristic
599C3×C4#39011yescharacteristic
600C3×C4#39211yescharacteristic
601C3×C4#39211yescharacteristic
602C3×C4#39411yescharacteristic
603C3×C4#39411yescharacteristic
604Dic3—44no—
605Dic3—42no—
606Dic3—82no—
607C23—42no—
608C23—62no—
609C23—62no—
610C23—62no—
611C23—124no—
612C23D3×C22⋊C411yescharacteristic
613C23#88211yescharacteristic
614C23#88211yescharacteristic
615C23#88411yescharacteristic
616C23#89411yescharacteristic
617C23#89511yescharacteristic
618C23#93711yescharacteristic
619Q8—64no—
620Q8—64no—
621Q8—121no—
622Q8—121no—
623Q8—122no—
624Q8—122no—
625Q8—122no—
626Q8—121no—
627Q8—122no—
628Q8—122no—
629Q8—121no—
630Q8—122no—
631D4—122no—
632D4—122no—
633D4—122no—
634D4—122no—
635D4—121no—
636D4—121no—
637D4—121no—
638D4—121no—
639C2×C4—24no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—62no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—62no—
646C2×C4—82no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4—122no—
658C2×C4—242no—
659C2×C4C4×D1211yescharacteristic
660C2×C4C4×D1211yescharacteristic
661C2×C4#87411yescharacteristic
662C2×C4#87511yescharacteristic
663C2×C4D3×C22⋊C411yescharacteristic
664C2×C4#88211yescharacteristic
665C2×C4#88211yescharacteristic
666C2×C4#88211yescharacteristic
667C2×C4#88211yescharacteristic
668C2×C4#88411yescharacteristic
669C2×C4#88411yescharacteristic
670C2×C4#88411yescharacteristic
671C2×C4#89311yescharacteristic
672C2×C4#89311yescharacteristic
673C2×C4#89411yescharacteristic
674C2×C4#89411yescharacteristic
675C2×C4#89411yescharacteristic
676C2×C4#89511yescharacteristic
677C2×C4#89511yescharacteristic
678C2×C4#89511yescharacteristic
679C2×C4#89511yescharacteristic
680C2×C4#89511yescharacteristic
681C2×C4C2×(C2×D3)⋊C411yescharacteristic
682C2×C4#93011yescharacteristic
683C2×C4#93011yescharacteristic
684C2×C4#93711yescharacteristic
685C2×C4#94011yescharacteristic
686C2×C4#94711yescharacteristic
687C8—42no—
688C8—122no—
689C2×C3—21no—
690C2×C3—21no—
691C2×C3—21no—
692C2×C3—21no—
693C2×C3—21no—
694C2×C3—21no—
695C2×C3#143811yescharacteristic
696C2×C3#190912yes—
697D3—82no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—124no—
706C22—122no—
707C22—124no—
708C22—242no—
709C22#466511yescharacteristic
710C22#466511yescharacteristic
711C22#467111yescharacteristic
712C22#467211yescharacteristic
713C22#495511yescharacteristic
714C22#496011yescharacteristic
715C22#499111yescharacteristic
716C4—44no—
717C4—82no—
718C4—122no—
719C4—124no—
720C4—242no—
721C4#466511yescharacteristic
722C4#466511yescharacteristic
723C4#467111yescharacteristic
724C4#467211yescharacteristic
725C4#493911yescharacteristic
726C4#496011yescharacteristic
727C4#498811yescharacteristic
728C4#499111yescharacteristic
729C3#1098811yescharacteristic, complemented, Sylow
730C2—21no—
731C2—21no—
732C2—21no—
733C2—21no—
734C2—21no—
735C2—21no—
736C2—242no—
737C2#6871212yes—
738C2#6942611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×H(3,ℤ/4)C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×H(3,ℤ/4)D3
hypocenterC3#10988
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. C2×#5120C2semidirect
5. C2×#5004C2semidirect
6. C2×C3×H(3,ℤ/4)C2semidirect
7. C2×#4531C2semidirect
22. C2×#875C4semidirect
23. C2×#869C4semidirect
44. C2×H(3,ℤ/4)D3semidirect
729. C3#10988semidirect

Maximal subgroups

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

subgroupindex
C2×#55262
C2×#51202
#837622
C2×#50062
C2×#50042
C2×C3×H(3,ℤ/4)2
C2×#45312
#109883