Finite Group Databasebeta

← #10700593

Subgroups — #10700593

Order 768.

1298 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#235C211yesmaximal, characteristic
2C2×#5132C211yesmaximal, characteristic
3C2×#5006C211yesmaximal, characteristic
4C2×C3×#339C211yesmaximal, characteristic, complemented
5C2×#4557C211yesmaximal, characteristic, complemented
6C2×#4531C211yesmaximal, characteristic, complemented
7C2×#4523C211yesmaximal, characteristic, complemented
8#10758—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C2×C4×#67—22no—
14C2×#926C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C4×#235—22no—
18C4×#235—22no—
19C4×#235—22no—
20C4×#235—22no—
21C2×C4×D12—21no—
22C2×#872C412yescomplemented
23C2×C42×D3C412yescomplemented
24C2×C4×Dic6—21no—
25C2×#824—21no—
26#5132—22no—
27#5132C414yescomplemented
28C2×#823—21no—
29#5006—22no—
30#5006C414yescomplemented
31C2×#806—22no—
32C3×#339—22no—
33#4557—22no—
34#4531—22no—
35#4523—22no—
36C2×C4×#41—31no—
37C2×#413—31no—
38C2×#410—31no—
39C2×#339—31no—
40C2×#339D311yescharacteristic, complemented
41C2×H(3,ℤ/4)—31no—
42C2×H(3,ℤ/4)—31no—
43C23×C3×C4—21no—
44C23×C3×C4C2311yescharacteristic
45C22×#67D412yes—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235—21no—
53C2×#235—21no—
54C2×#235C2×C412yes—
55C2×#235C2×C412yes—
56C2×#235D412yes—
57C2×#235Q812yes—
58C22×D12D411yescharacteristic
59C22×C4×D3—22no—
60C22×C4×D3C2×C412yes—
61C22×Dic6D411yescharacteristic
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16C2×C412yescomplemented
68C22×C3×C8—21no—
69C3×#117—21no—
70C3×#117—21no—
71C3×#117—21no—
72C3×#117C2×C412yescomplemented
73C2×C3×C4⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×C3×C42—21no—
78C2×C3×C42—21no—
79C2×C3×C42C2×C411yescharacteristic
80C2×C3×C42C2×C411yescharacteristic
81C2×C3×C42D412yes—
82C2×C3×C42D412yes—
83C2×#217—21no—
84C2×#217—21no—
85C4×#67—28no—
86C4×#67—28no—
87C2×(C2×D3)⋊C4—22no—
88#926—21no—
89#926—21no—
90#926—21no—
91#926—21no—
92#926—21no—
93#926C2×C412yescomplemented
94#926D412yes—
95#926D412yes—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2×C412yes—
102C2×#208—21no—
103C2×#208—21no—
104C2×#208—21no—
105C2×#208—21no—
106C2×#208—21no—
107C2×#208C2×C412yescomplemented
108C22×C3⋊C8—21no—
109C4×D12—24no—
110C4×D12—24no—
111#872—24no—
112#872—24no—
113C42×D3—24no—
114C42×D3—24no—
115C4×Dic6—24no—
116C4×Dic6—24no—
117#824—24no—
118#823—24no—
119#806—28no—
120C22×#41—31no—
121C22×M16—31no—
122C22×M16C2×D311yescharacteristic
123C4×#41—62no—
124C4×#41—62no—
125C4×#41—62no—
126C4×#41—62no—
127C2×C4×Q8—61no—
128C2×C4×D4—61no—
129C2×C4×D4—62no—
130C2×#117—31no—
131C2×#117—32no—
132C2×#117—31no—
133C2×#117C2×D311yescharacteristic
134C22×C42—32no—
135C22×C42C2×D311yescharacteristic
136#413—34no—
137#413—62no—
138C2×#101—61no—
139C2×#100—61no—
140#410—34no—
141#410—62no—
142C2×#99—62no—
143#339—22no—
144#339—62no—
145H(3,ℤ/4)—62no—
146H(3,ℤ/4)—62no—
147C24×C3C2×D411yescharacteristic
148C23×D3C22⋊C412yes—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—44no—
154C22×C3×C4—41no—
155C22×C3×C4C22×C411yescharacteristic
156C22×C3×C4C22×C411yescharacteristic
157C22×C3×C4C22×C411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×Q811yescharacteristic
165C22×C3×C4#4111yescharacteristic
166C22×C3×C4#4111yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C22×C3×C4#4111yescharacteristic
169C2×#67—22no—
170C2×#67—22no—
171C2×#67—22no—
172C2×#67—22no—
173C2×#67C22⋊C414yes—
174C2×#67C4⋊C414yes—
175C22×Dic3—22no—
176C22×Dic3—41no—
177C22×Dic3C22⋊C412yes—
178#235—22no—
179#235—22no—
180#235—22no—
181#235—22no—
182#235—22no—
183#235—22no—
184#235—22no—
185#235—22no—
186#235—22no—
187#235—22no—
188#235—22no—
189#235—22no—
190#235—22no—
191#235—22no—
192#235—22no—
193#235—22no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12C22⋊C412yes—
199C2×D12C4⋊C412yes—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—28no—
205C2×C4×D3—48no—
206C2×C4×D3C4214yes—
207C2×C4×D3C4⋊C414yes—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C22⋊C412yes—
213C2×Dic6C4⋊C412yes—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—41no—
217C3×M16—41no—
218C2×C3×C8—22no—
219C2×C3×C8—41no—
220C2×C3×C8—41no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—41no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—41no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—41no—
231C3×C42C22⋊C412yes—
232C3×C42C22⋊C412yes—
233#217—22no—
234#217—22no—
235#217—22no—
236#217—41no—
237(C2×D3)⋊C4—28no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—41no—
242Dic3⋊C4—28no—
243C4×Dic3—28no—
244C4×Dic3—22no—
245C4×Dic3—22no—
246C4×Dic3—41no—
247C4×Dic3C22⋊C414yes—
248#208—22no—
249#208—22no—
250#208—41no—
251#208—41no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—41no—
254C2×C3⋊C8—41no—
255C2×#41—32no—
256C2×#41—32no—
257C2×#41—32no—
258C2×#41—32no—
259C2×#41—61no—
260C2×#41—61no—
261C2×#41—61no—
262C2×#41—61no—
263C2×#41—61no—
264C2×#41—61no—
265C2×#41—61no—
266C2×#41—61no—
267C22×Q8—31no—
268C22×D4—32no—
269C22×D4—31no—
270C23×C4—21no—
271C23×C4—32no—
272C23×C4—62no—
273C23×C4C22×D311yescharacteristic
274C2×M16—21no—
275C2×M16—21no—
276C2×M16—21no—
277C2×M16—21no—
278C2×M16—21no—
279C2×M16—32no—
280C2×M16—61no—
281C2×M16—61no—
282C2×M16—61no—
283C2×M16—61no—
284C2×M16—61no—
285C2×M16C4×D312yescomplemented
286C22×C8—21no—
287C22×C8—61no—
288C4×Q8—64no—
289C4×Q8—64no—
290C4×D4—64no—
291C4×D4—64no—
292C4×D4—68no—
293C4×D4—68no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—32no—
298#117—32no—
299#117—32no—
300#117—64no—
301#117—61no—
302#117—61no—
303#117—61no—
304#117—64no—
305#117—61no—
306#117—61no—
307#117C4×D312yescomplemented
308C2×C4⋊C4—21no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—62no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—32no—
321C2×C42—64no—
322C2×C42—62no—
323C2×C42—64no—
324C2×C42—61no—
325C2×C42C4×D311yescharacteristic
326C2×C42C4×D311yescharacteristic
327C2×C42D1212yes—
328C2×C42#6712yes—
329#101—64no—
330#100—64no—
331#99—68no—
332C23×C3—42no—
333C23×C3C2×C22⋊C411yescharacteristic
334C23×C3C2×C22⋊C411yescharacteristic
335C23×C3C2×C4⋊C411yescharacteristic
336C23×C3#12011yescharacteristic
337C23×C3#12011yescharacteristic
338C23×C3#12311yescharacteristic
339C23×C3#12311yescharacteristic
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—44no—
343C22×D3#9514yes—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—44no—
348C2×C3×C4—44no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—82no—
352C2×C3×C4C2×C4211yescharacteristic
353C2×C3×C4C2×C22⋊C411yescharacteristic
354C2×C3×C4C2×C22⋊C411yescharacteristic
355C2×C3×C4C2×C22⋊C411yescharacteristic
356C2×C3×C4C2×C4⋊C411yescharacteristic
357C2×C3×C4C2×C4⋊C411yescharacteristic
358C2×C3×C4#11711yescharacteristic
359C2×C3×C4#11711yescharacteristic
360C2×C3×C4C4×D411yescharacteristic
361C2×C3×C4C4×D411yescharacteristic
362C2×C3×C4C4×D411yescharacteristic
363C2×C3×C4C4×D411yescharacteristic
364C2×C3×C4C4×D411yescharacteristic
365C2×C3×C4C4×D411yescharacteristic
366C2×C3×C4C4×D411yescharacteristic
367C2×C3×C4C4×D411yescharacteristic
368C2×C3×C4#12011yescharacteristic
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4#12111yescharacteristic
371C2×C3×C4#12111yescharacteristic
372C2×C3×C4#12111yescharacteristic
373C2×C3×C4#12111yescharacteristic
374C2×C3×C4C22⋊Q811yescharacteristic
375C2×C3×C4C22⋊Q811yescharacteristic
376C2×C3×C4C22⋊Q811yescharacteristic
377C2×C3×C4C22⋊Q811yescharacteristic
378C2×C3×C4#12311yescharacteristic
379C2×C3×C4#12311yescharacteristic
380#67—24no—
381#67—24no—
382#67—24no—
383#67—24no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—44no—
388C2×Dic3—42no—
389C2×Dic3—82no—
390C2×Dic3#9514yes—
391D12—22no—
392D12—22no—
393D12—22no—
394D12—22no—
395C4×D3—24no—
396C4×D3—24no—
397C4×D3—24no—
398C4×D3—24no—
399C4×D3—416no—
400Dic6—22no—
401Dic6—22no—
402Dic6—22no—
403Dic6—22no—
404C3×C8—42no—
405C3⋊C8—42no—
406C24—32no—
407C24D3×D411yescharacteristic
408#41—62no—
409#41—62no—
410#41—62no—
411#41—62no—
412#41—62no—
413#41—62no—
414#41—62no—
415#41—62no—
416#41—62no—
417#41—62no—
418#41—62no—
419#41—62no—
420#41—62no—
421#41—62no—
422#41—62no—
423#41—62no—
424C2×Q8—32no—
425C2×Q8—32no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—34no—
431C2×D4—34no—
432C2×D4—32no—
433C2×D4—32no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—32no—
447C22×C4—34no—
448C22×C4—34no—
449C22×C4—44no—
450C22×C4—41no—
451C22×C4—68no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—128no—
458C22×C4—121no—
459C22×C4C2×C4×D311yescharacteristic
460C22×C4C2×C4×D311yescharacteristic
461C22×C4C2×C4×D311yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4#23511yescharacteristic
464C22×C4#23511yescharacteristic
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4#23711yescharacteristic
470C22×C4D3×Q811yescharacteristic
471C22×C4#23911yescharacteristic
472C22×C4C2×#6711yescharacteristic
473M16—22no—
474M16—22no—
475M16—41no—
476M16—41no—
477M16—62no—
478M16—62no—
479M16—121no—
480M16—121no—
481C2×C8—22no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—62no—
485C2×C8—121no—
486C2×C8—121no—
487C4⋊C4—22no—
488C4⋊C4—41no—
489C4⋊C4—68no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—121no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—68no—
497C22⋊C4—62no—
498C22⋊C4—62no—
499C22⋊C4—62no—
500C22⋊C4—121no—
501C42—21no—
502C42—21no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—34no—
507C42—41no—
508C42—68no—
509C42—62no—
510C42—62no—
511C42—121no—
512C42(C2×D3)⋊C412yes—
513C42(C2×D3)⋊C412yes—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3C2×#9511yescharacteristic
522C22×C3#37911yescharacteristic
523C22×C3#38111yescharacteristic
524C22×C3#38511yescharacteristic
525C22×C3#38511yescharacteristic
526C22×C3#38611yescharacteristic
527C22×C3#38611yescharacteristic
528C2×D3—24no—
529C2×D3—44no—
530C2×D3—44no—
531C2×D3—44no—
532C3×C4—24no—
533C3×C4—42no—
534C3×C4—82no—
535C3×C4C4×C22⋊C411yescharacteristic
536C3×C4C4×C22⋊C411yescharacteristic
537C3×C4#38011yescharacteristic
538C3×C4#38011yescharacteristic
539C3×C4#38511yescharacteristic
540C3×C4#38511yescharacteristic
541C3×C4#38611yescharacteristic
542C3×C4#38611yescharacteristic
543Dic3—24no—
544Dic3—44no—
545Dic3—82no—
546C23—34no—
547C23—42no—
548C23—62no—
549C23—62no—
550C23—124no—
551C23D3×C22⋊C411yescharacteristic
552C23D3×C22⋊C411yescharacteristic
553C23#88211yescharacteristic
554C23#88311yescharacteristic
555C23D3×C4⋊C411yescharacteristic
556C23#89411yescharacteristic
557C23#93711yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—62no—
561Q8—62no—
562D4—64no—
563D4—64no—
564D4—64no—
565D4—64no—
566D4—62no—
567D4—62no—
568D4—62no—
569D4—62no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—34no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—82no—
586C2×C4—1216no—
587C2×C4—124no—
588C2×C4—122no—
589C2×C4—242no—
590C2×C4C42×D311yescharacteristic
591C2×C4#87211yescharacteristic
592C2×C4C4×D1211yescharacteristic
593C2×C4C4×D1211yescharacteristic
594C2×C4D3×C22⋊C411yescharacteristic
595C2×C4D3×C22⋊C411yescharacteristic
596C2×C4#67⋊C411yescharacteristic
597C2×C4#67⋊C411yescharacteristic
598C2×C4#88211yescharacteristic
599C2×C4#88311yescharacteristic
600C2×C4#88411yescharacteristic
601C2×C4#88411yescharacteristic
602C2×C4D3×C4⋊C411yescharacteristic
603C2×C4D3×C4⋊C411yescharacteristic
604C2×C4#89211yescharacteristic
605C2×C4#89311yescharacteristic
606C2×C4#89311yescharacteristic
607C2×C4#89411yescharacteristic
608C2×C4#89511yescharacteristic
609C2×C4#89611yescharacteristic
610C2×C4#89611yescharacteristic
611C2×C4#89711yescharacteristic
612C2×C4C2×(C2×D3)⋊C411yescharacteristic
613C2×C4C4×#6711yescharacteristic
614C2×C4C4×#6711yescharacteristic
615C2×C4#93711yescharacteristic
616C2×C4#93811yescharacteristic
617C2×C4#94611yescharacteristic
618C8—42no—
619C8—122no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3C22⋊#9511yescharacteristic
627C2×C3#182712yes—
628D3—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—64no—
637C22—124no—
638C22—124no—
639C22—124no—
640C22D3×#9511yescharacteristic
641C22#466411yescharacteristic
642C22#466511yescharacteristic
643C22#466811yescharacteristic
644C22#495511yescharacteristic
645C22#495611yescharacteristic
646C22#498911yescharacteristic
647C4—24no—
648C4—42no—
649C4—64no—
650C4—82no—
651C4—124no—
652C4—242no—
653C4#466511yescharacteristic
654C4#466611yescharacteristic
655C4#466711yescharacteristic
656C4#466811yescharacteristic
657C4C4×(C2×D3)⋊C411yescharacteristic
658C4#495611yescharacteristic
659C4#498711yescharacteristic
660C4#498911yescharacteristic
661C3#1075811yescharacteristic, complemented, Sylow
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—124no—
669C2#6873712yes—
670C2(C2×D3)⋊#9511yescharacteristic
subgroupG/H
centerC2×C4#892
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#339C2
socleC22×C3#381
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#339D3
hypocenterC3#10758
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. C2×C3×#339C2semidirect
5. C2×#4557C2semidirect
6. C2×#4531C2semidirect
7. C2×#4523C2semidirect
22. C2×#872C4semidirect
23. C2×C42×D3C4semidirect
27. #5132C4semidirect
30. #5006C4semidirect
40. C2×#339D3semidirect
67. C2×C3×M16C2×C4semidirect
72. C3×#117C2×C4semidirect
93. #926C2×C4semidirect
107. C2×#208C2×C4semidirect
285. C2×M16C4×D3semidirect
307. #117C4×D3semidirect
661. C3#10758semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
C2×#51322
C2×#50062
C2×C3×#3392
C2×#45572
C2×#45312
C2×#45232
#107583