Finite Group Databasebeta

← #10701982

Subgroups — #10701982

Order 768.

1278 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#909C211yesmaximal, characteristic
2C22×#824C211yesmaximal, characteristic, complemented
3C2×#5133C211yesmaximal, characteristic
4C22×#816C211yesmaximal, characteristic, complemented
5C2×#5110C211yesmaximal, characteristic
6C2×C3×#341C211yesmaximal, characteristic, complemented
7C2×#4554C211yesmaximal, characteristic, complemented
8#11021—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C23×Dic6C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C2×#909—22no—
17C2×#909—22no—
18C2×#909—22no—
19C2×#909—22no—
20C2×#909—22no—
21C2×#909C414yescomplemented
22C22×Dic12—21no—
23C22×#204—21no—
24C2×#824—22no—
25C2×#824—22no—
26C2×#824—22no—
27#5133—22no—
28#5133—22no—
29C2×#821—21no—
30#5110—22no—
31#5110—22no—
32C2×(C2×D3)⋊C8—22no—
33C2×#816—22no—
34C2×#816—21no—
35C2×#816—22no—
36C2×#816—22no—
37C3×#341—22no—
38#4554—22no—
39C22×#137—31no—
40C2×#417—31no—
41C22×#103—31no—
42C22×#101—31no—
43C2×#408—31no—
44C2×#343—31no—
45C2×#341D311yescharacteristic, complemented
46C23×C3×C4C2311yescharacteristic
47C22×#67—21no—
48C23×Dic3—21no—
49C23×Dic3—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—
60C2×#235C2×C412yes—
61C2×#235D412yes—
62C22×D12D411yescharacteristic
63C22×C4×D3—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×Dic6—21no—
74C22×Dic6—21no—
75C22×Dic6C2×C412yes—
76C22×Dic6D412yes—
77C22×Dic6D412yes—
78C22×Dic6D411yescharacteristic
79C2×C3×M16—21no—
80C2×C3×M16—21no—
81C2×C3×M16—21no—
82C2×C3×M16—21no—
83C2×C3×M16—21no—
84C2×C3×M16—21no—
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C2×C3×M16C2×C412yes—
88C2×C3×M16D412yes—
89C2×C3×M16D412yes—
90C2×C3×M16D412yes—
91C22×C3×C8—21no—
92C22×C3×C8—21no—
93C22×C3×C8—21no—
94C22×C3×C8—21no—
95C22×C3×C8—21no—
96C22×C3×C8—21no—
97C22×C3×C8—21no—
98C22×C3×C8—21no—
99C22×C3×C8D412yes—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D412yes—
106C2×#208—21no—
107C2×#208—21no—
108C2×#208—21no—
109C2×#208—21no—
110C2×#208—21no—
111C2×#208D412yes—
112C22×C3⋊C8—21no—
113#909—28no—
114#909—28no—
115#909—28no—
116#909—28no—
117C2×Dic12—24no—
118C2×Dic12—24no—
119C2×Dic12—24no—
120C2×#204—24no—
121C2×#204—24no—
122C2×#204—24no—
123#824—28no—
124#821—24no—
125(C2×D3)⋊C8—28no—
126#816—24no—
127#816—28no—
128C22×#41—31no—
129C23×Q8—31no—
130C2×#137—34no—
131C2×#137—62no—
132C2×#137—62no—
133C2×#137—62no—
134C2×#137—62no—
135C2×#137—62no—
136C22×Q16—61no—
137C22×SD16—61no—
138C22×M16—31no—
139C22×M16C2×D311yescharacteristic
140C22×M16C2×D311yescharacteristic
141C23×C8C2×D311yescharacteristic
142C22×C4⋊C4—31no—
143#417—62no—
144#417—62no—
145C2×#103—62no—
146C2×#103—62no—
147C2×#103—61no—
148C2×#103—62no—
149C2×#102—61no—
150C2×#101—62no—
151C2×#101—62no—
152C2×#101—62no—
153#408—62no—
154#408—62no—
155C2×C22⋊C8—62no—
156#343—62no—
157#341—22no—
158C24×C3#4111yescharacteristic
159C23×D3—21no—
160C22×C3×C4C22×C411yescharacteristic
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×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4#4111yescharacteristic
172C22×C3×C4#4111yescharacteristic
173C22×C3×C4#4111yescharacteristic
174C2×#67—22no—
175C2×#67—22no—
176C2×#67—41no—
177C2×#67—41no—
178C2×#67—41no—
179C2×#67—41no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—41no—
186C22×Dic3—41no—
187C22×Dic3—41no—
188C22×Dic3—41no—
189C22×Dic3—41no—
190C22×Dic3—41no—
191C22×Dic3—82no—
192#235—22no—
193#235—22no—
194#235—22no—
195#235—22no—
196#235—22no—
197#235—22no—
198#235—22no—
199#235—22no—
200#235—41no—
201#235—41no—
202#235—41no—
203#235—41no—
204#235—41no—
205#235—41no—
206#235—41no—
207#235—41no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12C22⋊C412yes—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—41no—
217C2×C4×D3—41no—
218C2×C4×D3—41no—
219C2×C4×D3—41no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—21no—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—22no—
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6—21no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—22no—
239C2×Dic6—21no—
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×Dic6—41no—
250C2×Dic6—41no—
251C2×Dic6—41no—
252C2×Dic6C22⋊C412yes—
253C2×Dic6C22⋊C414yes—
254C3×M16—22no—
255C3×M16—22no—
256C3×M16—22no—
257C3×M16—22no—
258C3×M16—22no—
259C3×M16—41no—
260C3×M16—41no—
261C3×M16C22⋊C414yes—
262C2×C3×C8—22no—
263C2×C3×C8—22no—
264C2×C3×C8—22no—
265C2×C3×C8—22no—
266C2×C3×C8—22no—
267C2×C3×C8—22no—
268C2×C3×C8—22no—
269C2×C3×C8—41no—
270C2×C3×C8—41no—
271C2×C3×C8—41no—
272C2×C3×C8—41no—
273C2×C3×C8—41no—
274C2×C3×C8—41no—
275C4⋊Dic3—22no—
276C4⋊Dic3—22no—
277C4⋊Dic3—41no—
278C4⋊Dic3—41no—
279#208—22no—
280#208—22no—
281#208—41no—
282#208—41no—
283C2×C3⋊C8—22no—
284C2×C3⋊C8—41no—
285C2×C3⋊C8—41no—
286Dic12—216no—
287#204—216no—
288C2×#41—32no—
289C2×#41—32no—
290C2×#41—61no—
291C2×#41—61no—
292C2×#41—61no—
293C2×#41—61no—
294C2×#41—61no—
295C2×#41—61no—
296C2×#41—61no—
297C2×#41—61no—
298C2×#41—61no—
299C2×#41—61no—
300C22×Q8—32no—
301C22×Q8—31no—
302C22×Q8—32no—
303C22×Q8—32no—
304C22×Q8—61no—
305C22×Q8—61no—
306C22×Q8—61no—
307C22×Q8—61no—
308C22×Q8—61no—
309C22×Q8—61no—
310C22×Q8—61no—
311C22×Q8—61no—
312C22×Q8—61no—
313C22×Q8—61no—
314C22×Q8—61no—
315C22×D4—31no—
316C22×D4—61no—
317C23×C4—61no—
318C23×C4—61no—
319C23×C4—61no—
320C23×C4C22×D311yescharacteristic
321#137—68no—
322#137—68no—
323#137—68no—
324#137—68no—
325C2×Q16—64no—
326C2×Q16—64no—
327C2×Q16—64no—
328C2×SD16—64no—
329C2×SD16—64no—
330C2×SD16—64no—
331C2×M16—21no—
332C2×M16—21no—
333C2×M16—21no—
334C2×M16—21no—
335C2×M16—21no—
336C2×M16—21no—
337C2×M16—21no—
338C2×M16—21no—
339C2×M16—32no—
340C2×M16—61no—
341C2×M16—61no—
342C2×M16—61no—
343C2×M16—61no—
344C2×M16—61no—
345C2×M16C4×D312yes—
346C2×M16D1212yes—
347C2×M16D1212yes—
348C2×M16#6712yes—
349C22×C8—21no—
350C22×C8—21no—
351C22×C8—21no—
352C22×C8—21no—
353C22×C8—21no—
354C22×C8—21no—
355C22×C8—21no—
356C22×C8—21no—
357C22×C8—61no—
358C22×C8D1212yes—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—61no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4—61no—
364C2×C4⋊C4—61no—
365#103—68no—
366#103—64no—
367#102—64no—
368#101—68no—
369C22⋊C8—68no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3C4×D411yescharacteristic
373C23×C3#12111yescharacteristic
374C23×C3#12111yescharacteristic
375C23×C3#12111yescharacteristic
376C23×C3#12311yescharacteristic
377C23×C3#12311yescharacteristic
378C23×C3#12411yescharacteristic
379C22×D3—22no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—42no—
383C22×D3—42no—
384C2×C3×C4C2×C22⋊C411yescharacteristic
385C2×C3×C4C2×C22⋊C411yescharacteristic
386C2×C3×C4C2×C22⋊C411yescharacteristic
387C2×C3×C4C4×D411yescharacteristic
388C2×C3×C4C4×D411yescharacteristic
389C2×C3×C4C4×D411yescharacteristic
390C2×C3×C4#12011yescharacteristic
391C2×C3×C4#12011yescharacteristic
392C2×C3×C4#12011yescharacteristic
393C2×C3×C4#12011yescharacteristic
394C2×C3×C4#12011yescharacteristic
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4#12011yescharacteristic
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4#12111yescharacteristic
399C2×C3×C4#12111yescharacteristic
400C2×C3×C4#12111yescharacteristic
401C2×C3×C4#12111yescharacteristic
402C2×C3×C4#12111yescharacteristic
403C2×C3×C4#12111yescharacteristic
404C2×C3×C4#12111yescharacteristic
405C2×C3×C4#12111yescharacteristic
406C2×C3×C4#12111yescharacteristic
407C2×C3×C4#12311yescharacteristic
408C2×C3×C4#12311yescharacteristic
409C2×C3×C4#12411yescharacteristic
410C2×C3×C4#12711yescharacteristic
411C2×C3×C4#12711yescharacteristic
412#67—42no—
413#67—42no—
414#67—42no—
415#67—42no—
416C2×Dic3—24no—
417C2×Dic3—22no—
418C2×Dic3—42no—
419C2×Dic3—42no—
420C2×Dic3—42no—
421C2×Dic3—41no—
422C2×Dic3—42no—
423C2×Dic3—42no—
424C2×Dic3—42no—
425C2×Dic3—41no—
426C2×Dic3—42no—
427C2×Dic3—82no—
428C2×Dic3—82no—
429C2×Dic3—82no—
430D12—22no—
431D12—22no—
432D12—41no—
433D12—41no—
434C4×D3—42no—
435C4×D3—42no—
436C4×D3—42no—
437C4×D3—42no—
438Dic6—24no—
439Dic6—22no—
440Dic6—24no—
441Dic6—22no—
442Dic6—42no—
443Dic6—42no—
444Dic6—42no—
445Dic6—42no—
446Dic6—42no—
447Dic6—41no—
448Dic6—42no—
449Dic6—41no—
450C3×C8—24no—
451C3×C8—42no—
452C3×C8—42no—
453C3⋊C8—42no—
454C24—61no—
455C24#23911yescharacteristic
456#41—62no—
457#41—62no—
458#41—62no—
459#41—62no—
460#41—62no—
461#41—62no—
462#41—62no—
463#41—62no—
464#41—121no—
465#41—121no—
466#41—121no—
467#41—121no—
468#41—121no—
469#41—121no—
470#41—121no—
471#41—121no—
472C2×Q8—32no—
473C2×Q8—34no—
474C2×Q8—62no—
475C2×Q8—62no—
476C2×Q8—62no—
477C2×Q8—61no—
478C2×Q8—62no—
479C2×Q8—62no—
480C2×Q8—62no—
481C2×Q8—61no—
482C2×Q8—62no—
483C2×Q8—61no—
484C2×Q8—62no—
485C2×Q8—62no—
486C2×Q8—62no—
487C2×Q8—62no—
488C2×Q8—61no—
489C2×Q8—62no—
490C2×Q8—62no—
491C2×Q8—62no—
492C2×Q8—61no—
493C2×Q8—62no—
494C2×Q8—121no—
495C2×Q8—121no—
496C2×Q8—121no—
497C2×Q8—121no—
498C2×Q8—121no—
499C2×Q8—121no—
500C2×Q8—121no—
501C2×Q8—121no—
502C2×Q8—121no—
503C2×Q8—121no—
504C2×Q8—121no—
505C2×Q8—121no—
506C2×D4—32no—
507C2×D4—62no—
508C2×D4—62no—
509C2×D4—61no—
510C2×D4—61no—
511C2×D4—61no—
512C2×D4—61no—
513C2×D4—61no—
514C2×D4—121no—
515C2×D4—121no—
516C2×D4—121no—
517C2×D4—121no—
518C22×C4—62no—
519C22×C4—62no—
520C22×C4—62no—
521C22×C4—61no—
522C22×C4—62no—
523C22×C4—62no—
524C22×C4—62no—
525C22×C4—121no—
526C22×C4—121no—
527C22×C4—121no—
528C22×C4—121no—
529C22×C4—121no—
530C22×C4—121no—
531C22×C4—121no—
532C22×C4—121no—
533C22×C4—121no—
534C22×C4—121no—
535C22×C4—242no—
536C22×C4C2×C4×D311yescharacteristic
537C22×C4C2×D1211yescharacteristic
538C22×C4C2×D1211yescharacteristic
539C22×C4C2×D1211yescharacteristic
540C22×C4#23511yescharacteristic
541C22×C4#23511yescharacteristic
542C22×C4D3×D411yescharacteristic
543C22×C4D3×D411yescharacteristic
544C22×C4D3×D411yescharacteristic
545C22×C4D3×D411yescharacteristic
546C22×C4D3×D411yescharacteristic
547C22×C4D3×D411yescharacteristic
548C22×C4#23911yescharacteristic
549C22×C4C2×#6711yescharacteristic
550Q16—616no—
551SD16—616no—
552M16—22no—
553M16—22no—
554M16—22no—
555M16—22no—
556M16—22no—
557M16—41no—
558M16—41no—
559M16—62no—
560M16—62no—
561M16—121no—
562M16—121no—
563M16(C2×D3)⋊C414yes—
564C2×C8—22no—
565C2×C8—22no—
566C2×C8—22no—
567C2×C8—22no—
568C2×C8—22no—
569C2×C8—22no—
570C2×C8—22no—
571C2×C8—41no—
572C2×C8—41no—
573C2×C8—41no—
574C2×C8—41no—
575C2×C8—41no—
576C2×C8—41no—
577C2×C8—62no—
578C2×C8—121no—
579C2×C8—121no—
580C4⋊C4—62no—
581C4⋊C4—62no—
582C4⋊C4—121no—
583C4⋊C4—121no—
584C22×C3—22no—
585C22×C3—22no—
586C22×C3—22no—
587C22×C3—22no—
588C22×C3—21no—
589C22×C3—21no—
590C22×C3—21no—
591C22×C3—21no—
592C22×C3#38611yescharacteristic
593C22×C3#38611yescharacteristic
594C22×C3#39011yescharacteristic
595C22×C3#39211yescharacteristic
596C22×C3#39411yescharacteristic
597C22×C3#39411yescharacteristic
598C22×C3#39411yescharacteristic
599C22×C3#47112yes—
600C22×C3#47312yes—
601C2×D3—44no—
602C2×D3—42no—
603C2×D3—82no—
604C2×D3—81no—
605C2×D3—81no—
606C3×C4#37911yescharacteristic
607C3×C4#38611yescharacteristic
608C3×C4#38611yescharacteristic
609C3×C4#39011yescharacteristic
610C3×C4#39211yescharacteristic
611C3×C4#39211yescharacteristic
612C3×C4#39211yescharacteristic
613C3×C4#39411yescharacteristic
614Dic3—44no—
615Dic3—42no—
616Dic3—82no—
617C23—22no—
618C23—22no—
619C23—62no—
620C23—122no—
621C23—122no—
622C23—121no—
623C23—121no—
624C23#89311yescharacteristic
625C23#89411yescharacteristic
626C23#89411yescharacteristic
627C23#89511yescharacteristic
628C23#89511yescharacteristic
629C23#89511yescharacteristic
630C23#94711yescharacteristic
631Q8—64no—
632Q8—62no—
633Q8—64no—
634Q8—62no—
635Q8—122no—
636Q8—122no—
637Q8—122no—
638Q8—121no—
639Q8—122no—
640Q8—122no—
641Q8—122no—
642Q8—121no—
643D4—62no—
644D4—62no—
645D4—122no—
646D4—122no—
647D4—122no—
648D4—122no—
649D4—121no—
650D4—121no—
651C2×C4—62no—
652C2×C4—64no—
653C2×C4—122no—
654C2×C4—121no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4—122no—
658C2×C4—121no—
659C2×C4—122no—
660C2×C4—122no—
661C2×C4—122no—
662C2×C4—122no—
663C2×C4—122no—
664C2×C4—122no—
665C2×C4—122no—
666C2×C4—242no—
667C2×C4—242no—
668C2×C4—242no—
669C2×C4C4×D1211yescharacteristic
670C2×C4C4×D1211yescharacteristic
671C2×C4#87411yescharacteristic
672C2×C4#87511yescharacteristic
673C2×C4D3×C22⋊C411yescharacteristic
674C2×C4D3×C22⋊C411yescharacteristic
675C2×C4#88211yescharacteristic
676C2×C4#88211yescharacteristic
677C2×C4#88211yescharacteristic
678C2×C4#88211yescharacteristic
679C2×C4#88211yescharacteristic
680C2×C4#88211yescharacteristic
681C2×C4#88411yescharacteristic
682C2×C4#88411yescharacteristic
683C2×C4#88411yescharacteristic
684C2×C4#88411yescharacteristic
685C2×C4#89311yescharacteristic
686C2×C4#89411yescharacteristic
687C2×C4#89411yescharacteristic
688C2×C4#89511yescharacteristic
689C2×C4#89511yescharacteristic
690C2×C4#89511yescharacteristic
691C2×C4C2×(C2×D3)⋊C411yescharacteristic
692C2×C4#93011yescharacteristic
693C2×C4#93011yescharacteristic
694C2×C4#93711yescharacteristic
695C2×C4#93711yescharacteristic
696C2×C4#94011yescharacteristic
697C8—24no—
698C8—42no—
699C8—42no—
700C8—122no—
701C2×C3—21no—
702C2×C3—21no—
703C2×C3—21no—
704C2×C3—21no—
705C2×C3#143811yescharacteristic
706C2×C3#190812yes—
707C2×C3#201911yescharacteristic
708C2×C3#202011yescharacteristic
709C2×C3#205212yes—
710D3—82no—
711C22—22no—
712C22—22no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—21no—
717C22—22no—
718C22—22no—
719C22—122no—
720C22—124no—
721C22—242no—
722C22—241no—
723C22—241no—
724C22#466511yescharacteristic
725C22#466511yescharacteristic
726C22#467111yescharacteristic
727C22#467211yescharacteristic
728C22#489612yes—
729C22#489812yes—
730C22#498811yescharacteristic
731C22#499111yescharacteristic
732C22#499111yescharacteristic
733C4—122no—
734C4—124no—
735C4—242no—
736C4#466511yescharacteristic
737C4#466511yescharacteristic
738C4#467111yescharacteristic
739C4#467211yescharacteristic
740C4#493911yescharacteristic
741C4#495511yescharacteristic
742C4#496011yescharacteristic
743C4#496011yescharacteristic
744C3#1102111yescharacteristic, complemented, Sylow
745C2—21no—
746C2—21no—
747C2—21no—
748C2—21no—
749C2—242no—
750C2#6894311yescharacteristic
751C2#6894411yescharacteristic
752C2#6894912yes—
753C2#6942611yescharacteristic
754C2#7071612yes—
subgroupG/H
centerC23#895
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#341C2
socleC23×C3#121
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#341D3
hypocenterC3#11021
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#341

Complements

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

NH ≅ G/Ntype
2. C22×#824C2semidirect
4. C22×#816C2semidirect
6. C2×C3×#341C2semidirect
7. C2×#4554C2semidirect
21. C2×#909C4semidirect
45. C2×#341D3semidirect
744. C3#11021semidirect

Maximal subgroups

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

subgroupindex
C22×#9092
C22×#8242
C2×#51332
C22×#8162
C2×#51102
C2×C3×#3412
C2×#45542
#110213