Finite Group Databasebeta

← C2×#83296 #11662001

Subgroups — C2×#83296 #11662001

Order 768.

2874 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C22×#950C212yesmaximal, complemented
3C2×C4×#235C211yesmaximal, characteristic, complemented
4C2×#5233C212yesmaximal, complemented
5#83296C2116yesmaximal, direct product factor
6C2×#5090C211yesmaximal, characteristic, complemented
7C2×#5083C211yesmaximal, characteristic, complemented
8C2×#5075C211yesmaximal, characteristic, complemented
9C2×#5072C211yesmaximal, characteristic, complemented
10C2×#5054C212yesmaximal, complemented
11C2×#5057C212yesmaximal, complemented
12C22×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
13C2×#3035—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C22×#235C2211yescharacteristic
16C3×#532—24no—
17C3×#532C2218yescomplemented
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124C2212yes—
21C2×C3×#121—21no—
22C2×C3×#121—21no—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4—21no—
25C22×C3×C42C2211yescharacteristic
26C2×#950—216no—
27C2×#950C22116yescomplemented
28C22×#216C2212yes—
29C22×#215C2212yes—
30C22×#214C2212yes—
31C22×#213C2212yes—
32C2×C4×#67—21no—
33C2×C4×#67—21no—
34C2×#926C2211yescharacteristic
35C23×C3⋊C8C2212yes—
36C4×#235—24no—
37C4×#235C2218yescomplemented
38C2×C4×D12C2211yescharacteristic
39C2×C4×D12C2211yescharacteristic
40C2×#872—21no—
41C2×C42×D3—21no—
42C2×C4×Dic6C2211yescharacteristic
43C2×C4×Dic6C2211yescharacteristic
44#5233C22116yescomplemented
45C2×#835C2212yes—
46C2×#835C2212yes—
47C2×#832C2212yes—
48C2×#832C2212yes—
49#5090C2218yescomplemented
50#5083C2218yescomplemented
51#5075C2218yescomplemented
52#5072C2218yescomplemented
53#5057C22116yescomplemented
54#5054C22116yescomplemented
55C2×C4⋊(C3⋊C8)C2212yes—
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58C2×C4⋊(C3⋊C8)C2218yescomplemented
59C22×#135—32no—
60C2×#532D311yescharacteristic, complemented
61C2×C4×#41—31no—
62#3035—316no—
63C2×#464—32no—
64C2×#462—31no—
65C2×#460—31no—
66C2×#463—32no—
67C2×#461—31no—
68C2×#459—31no—
69C22×C4⋊C8—31no—
70C2×#416—32no—
71C2×C3×#41—24no—
72C2×C3×#41—28no—
73C2×C3×#41—28no—
74C2×C3×#41C2318yes—
75C22×C3×Q8C2312yes—
76C22×C3×D4—22no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2312yes—
80C23×C3×C4—22no—
81C23×C3×C4D411yescharacteristic
82C23×C3×C4D411yescharacteristic
83C23×C3×C4C2311yescharacteristic
84C22×#67—21no—
85C2×#235—22no—
86C2×#235—24no—
87C2×#235D418yescomplemented
88C2×#235C2314yes—
89C22×D12C2311yescharacteristic
90C22×C4×D3—21no—
91C22×C4×D3—21no—
92C22×C4×D3—21no—
93C22×Dic6C2311yescharacteristic
94C3×C4⋊Q8—22no—
95C3×C4⋊Q8C2314yes—
96C3×#127—22no—
97C3×#127C2314yes—
98C3×#124—24no—
99C3×#124C2318yes—
100C3×#121—28no—
101C3×#121—28no—
102C3×C4×D4—28no—
103C3×C4×D4—28no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D411yescharacteristic
110C2×C3×C42D414yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2314yes—
114C2×#217—21no—
115#950—264no—
116C2×#216—28no—
117C2×#216C2318yes—
118C2×#215—28no—
119C2×#215C2318yes—
120C2×#214—28no—
121C2×#214C2318yes—
122C2×#213—28no—
123C2×#213C2318yes—
124C4×#67—28no—
125C4×#67—28no—
126C2×(C2×D3)⋊C4—21no—
127C2×(C2×D3)⋊C4—21no—
128#926—22no—
129#926C2314yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2311yescharacteristic
132C2×Dic3⋊C4—21no—
133C2×Dic3⋊C4—21no—
134C2×C4×Dic3—21no—
135C2×C4×Dic3—21no—
136C2×C4×Dic3—21no—
137C22×C3⋊C8—28no—
138C22×C3⋊C8C2312yes—
139C22×C3⋊C8C2312yes—
140C22×C3⋊C8C2318yes—
141C4×D12—22no—
142C4×D12—22no—
143C4×D12C2314yes—
144C4×D12C2314yes—
145#872—28no—
146C42×D3—28no—
147C4×Dic6—22no—
148C4×Dic6—22no—
149C4×Dic6C2314yes—
150C4×Dic6C2314yes—
151#835C2318yes—
152#835C2318yes—
153#832C2318yes—
154#832C2318yes—
155C4⋊(C3⋊C8)C2318yes—
156C4⋊(C3⋊C8)C2314yes—
157C4⋊(C3⋊C8)C2314yes—
158C22×#41—31no—
159C22×#41C2×D312yes—
160C2×#135—316no—
161C2×#135—616no—
162C22×Q16—32no—
163C22×SD16—32no—
164C22×SD16—32no—
165C22×D8—32no—
166C23×C8—32no—
167#532—24no—
168#532C2×D318yescomplemented
169C2×C4⋊Q8C2×D311yescharacteristic
170C2×#127C2×D311yescharacteristic
171C2×#124C2×D312yes—
172C2×#121—21no—
173C2×#121—21no—
174C4×#41—38no—
175C4×#41—64no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×D4—21no—
179C2×C4×D4—21no—
180C2×C4×D4—31no—
181C2×C4×D4—31no—
182C2×C4×D4—61no—
183C2×C4×D4—61no—
184C2×#117—31no—
185C2×#117—61no—
186C22×C42—61no—
187C22×C42C2×D311yescharacteristic
188#464—316no—
189#463—316no—
190#462—38no—
191#461—38no—
192#460—38no—
193#459—38no—
194C2×C4⋊C8—31no—
195C2×C4⋊C8—31no—
196C2×C4⋊C8—32no—
197C2×C4⋊C8—38no—
198#416—316no—
199C2×#103—32no—
200C2×#103—32no—
201C2×#102—32no—
202C2×#102—32no—
203C24×C3—22no—
204C24×C3C2×D411yescharacteristic
205C23×D3—21no—
206C3×#41—216no—
207C3×#41—216no—
208C3×#41—416no—
209C2×C3×Q8—22no—
210C2×C3×Q8—24no—
211C2×C3×Q8—24no—
212C2×C3×Q8C2414yes—
213C2×C3×D4—28no—
214C2×C3×D4—24no—
215C2×C3×D4—24no—
216C2×C3×D4—24no—
217C2×C3×D4—22no—
218C2×C3×D4—24no—
219C2×C3×D4—44no—
220C2×C3×D4—48no—
221C2×C3×D4—44no—
222C2×C3×D4C2414yes—
223C22×C3×C4—24no—
224C22×C3×C4—22no—
225C22×C3×C4—28no—
226C22×C3×C4—42no—
227C22×C3×C4—48no—
228C22×C3×C4C2×D414yes—
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D414yes—
233C22×C3×C4C2×D414yes—
234C22×C3×C4C2×D412yes—
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4#4114yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2412yes—
240C2×#67—28no—
241C2×#67—24no—
242C22×Dic3—21no—
243C22×Dic3—21no—
244C22×Dic3—21no—
245#235—216no—
246#235—28no—
247#235C2×D4116yescomplemented
248C2×D12—21no—
249C2×D12—22no—
250C2×D12C2×D414yes—
251C2×D12C2412yes—
252C2×C4×D3—28no—
253C2×C4×D3—24no—
254C2×C4×D3—24no—
255C2×C4×D3—24no—
256C2×C4×D3—44no—
257C2×C4×D3—44no—
258C2×Dic6—21no—
259C2×Dic6—22no—
260C2×Dic6C2×D414yes—
261C2×Dic6C2412yes—
262C3×C4⋊C4—24no—
263C3×C4⋊C4—24no—
264C3×C22⋊C4—28no—
265C3×C22⋊C4—28no—
266C3×C42C2×D412yes—
267C3×C42C2×D412yes—
268C3×C42C2×D414yes—
269C3×C42C2×D414yes—
270C3×C42C2412yes—
271C3×C42C2412yes—
272#217—24no—
273#216—216no—
274#215—216no—
275#214—216no—
276#213—216no—
277(C2×D3)⋊C4—24no—
278(C2×D3)⋊C4—24no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3C2412yes—
282C4⋊Dic3C2412yes—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285C4×Dic3—24no—
286C4×Dic3—24no—
287C4×Dic3—24no—
288C2×C3⋊C8—24no—
289C2×C3⋊C8—24no—
290C2×C3⋊C8—216no—
291C2×C3⋊C8C2414yes—
292C2×C3⋊C8C2414yes—
293C2×#41—24no—
294C2×#41—28no—
295C2×#41—28no—
296C2×#41—38no—
297C2×#41—34no—
298C2×#41—64no—
299C2×#41—62no—
300C2×#41C22×D318yes—
301C22×Q8—31no—
302C22×Q8C22×D312yes—
303C22×D4—22no—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—31no—
307C22×D4—61no—
308C22×D4C22×D312yes—
309C23×C4—22no—
310C23×C4—61no—
311C23×C4—61no—
312C23×C4—61no—
313C23×C4#6711yescharacteristic
314C23×C4#6711yescharacteristic
315C23×C4C22×D311yescharacteristic
316#135—664no—
317C2×Q16—38no—
318C2×Q16—68no—
319C2×SD16—38no—
320C2×SD16—38no—
321C2×SD16—68no—
322C2×SD16—68no—
323C2×D8—38no—
324C2×D8—68no—
325C22×C8—32no—
326C22×C8—32no—
327C22×C8—38no—
328C22×C8—68no—
329C4⋊Q8—22no—
330C4⋊Q8C22×D314yes—
331#127—22no—
332#127C22×D314yes—
333#124—24no—
334#124C22×D318yes—
335#121—28no—
336#121—28no—
337C4×Q8—34no—
338C4×Q8—34no—
339C4×Q8—62no—
340C4×Q8—62no—
341C4×D4—28no—
342C4×D4—28no—
343C4×D4—34no—
344C4×D4—34no—
345C4×D4—68no—
346C4×D4—68no—
347C4×D4—62no—
348C4×D4—62no—
349#117—34no—
350#117—62no—
351#117—68no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—31no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4—61no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—61no—
361C2×C22⋊C4—61no—
362C2×C22⋊C4—61no—
363C2×C42—61no—
364C2×C42—61no—
365C2×C42—61no—
366C2×C42—68no—
367C2×C42#6714yes—
368C2×C42#6711yescharacteristic
369C2×C42#6711yescharacteristic
370C2×C42C22×D311yescharacteristic
371C2×C42C22×D311yescharacteristic
372C2×C42C22×D314yes—
373C4⋊C8—34no—
374C4⋊C8—34no—
375C4⋊C8—38no—
376#103—38no—
377#103—38no—
378#102—38no—
379#102—38no—
380C23×C3—22no—
381C23×C3—22no—
382C23×C3—24no—
383C23×C3—44no—
384C23×C3—44no—
385C23×C3—44no—
386C23×C3#12114yes—
387C23×C3C22×D412yes—
388C23×C3C22×D411yescharacteristic
389C22×D3—24no—
390C22×D3—22no—
391C22×D3—42no—
392C22×D3—42no—
393C3×Q8—24no—
394C3×Q8—24no—
395C3×Q8—44no—
396C3×D4—24no—
397C3×D4—24no—
398C3×D4—48no—
399C3×D4—416no—
400C3×D4—44no—
401C3×D4—48no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—28no—
406C2×C3×C4—24no—
407C2×C3×C4—44no—
408C2×C3×C4—44no—
409C2×C3×C4—416no—
410C2×C3×C4#12114yes—
411C2×C3×C4#12118yes—
412C2×C3×C4#13514yescomplemented
413C2×C3×C4C22×D412yes—
414C2×C3×C4C22×D414yes—
415C2×C3×C4C22×D412yes—
416C2×C3×C4C22×D412yes—
417C2×C3×C4C22×D412yes—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D412yes—
421C2×C3×C4C2×#4112yes—
422C2×C3×C4C2×#4114yes—
423C2×C3×C4C2511yescharacteristic
424#67—216no—
425C2×Dic3—24no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—42no—
430C2×Dic3—42no—
431D12—24no—
432D12—22no—
433D12C22×D414yes—
434C4×D3—216no—
435C4×D3—48no—
436C4×D3—48no—
437Dic6—24no—
438Dic6—22no—
439Dic6C22×D414yes—
440C3⋊C8—24no—
441C3⋊C8—24no—
442C24—22no—
443C24—61no—
444C24C2×#6711yescharacteristic
445#41—216no—
446#41—216no—
447#41—316no—
448#41—416no—
449#41—616no—
450#41—68no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—34no—
455C2×Q8—32no—
456C2×Q8—62no—
457C2×Q8—61no—
458C2×Q8C23×D314yes—
459C2×D4—28no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—34no—
466C2×D4—32no—
467C2×D4—44no—
468C2×D4—48no—
469C2×D4—44no—
470C2×D4—68no—
471C2×D4—64no—
472C2×D4—62no—
473C2×D4—61no—
474C2×D4C23×D314yes—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—42no—
479C22×C4—48no—
480C22×C4—61no—
481C22×C4—68no—
482C22×C4—64no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—64no—
486C22×C4—64no—
487C22×C4—124no—
488C22×C4—124no—
489C22×C4D3×D414yes—
490C22×C4#23714yes—
491C22×C4C2×#6711yescharacteristic
492C22×C4C2×#6711yescharacteristic
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6714yes—
495C22×C4C2×#6714yes—
496C22×C4C2×#6712yes—
497C22×C4C2×#6711yescharacteristic
498C22×C4C2×#6711yescharacteristic
499C22×C4C23×D311yescharacteristic
500C22×C4C23×D312yes—
501Q16—616no—
502SD16—616no—
503SD16—616no—
504D8—616no—
505C2×C8—34no—
506C2×C8—34no—
507C2×C8—64no—
508C2×C8—64no—
509C2×C8—616no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—32no—
513C4⋊C4—32no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—64no—
521C22⋊C4—64no—
522C22⋊C4—64no—
523C42—64no—
524C42—64no—
525C42—64no—
526C42C2×#6714yes—
527C42C2×#6714yes—
528C42C2×#6712yes—
529C42C2×#6712yes—
530C42C23×D312yes—
531C42C23×D312yes—
532C22×C3—28no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—44no—
536C22×C3—44no—
537C22×C3—44no—
538C22×C3—88no—
539C22×C3C2×#12114yes—
540C22×C3C2×#12112yes—
541C22×C3C2×#13512yes—
542C22×C3#57512yes—
543C22×C3C23×D411yescharacteristic
544C2×D3—24no—
545C2×D3—48no—
546C2×D3—42no—
547C2×D3—42no—
548C3×C4—22no—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—44no—
552C3×C4C2×#12112yes—
553C3×C4C2×#12114yes—
554C3×C4C2×#13514yescomplemented
555C3×C4C23×D411yescharacteristic
556C3×C4C22×#4111yescharacteristic
557Dic3—24no—
558Dic3—42no—
559Dic3—42no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—62no—
567C23—64no—
568C23—122no—
569C23—122no—
570C23#93814yes—
571C23C22×#6712yes—
572C23C22×#6711yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—34no—
576Q8—44no—
577Q8—62no—
578Q8—64no—
579D4—24no—
580D4—24no—
581D4—34no—
582D4—48no—
583D4—416no—
584D4—44no—
585D4—48no—
586D4—616no—
587D4—62no—
588D4—64no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—416no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—616no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—128no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—128no—
606C2×C4#93814yes—
607C2×C4#93818yes—
608C2×C4#95014yescomplemented
609C2×C4C2×D3×D412yes—
610C2×C4C2×D3×D414yes—
611C2×C4C2×#23712yes—
612C2×C4C2×#23714yes—
613C2×C4C22×#6712yes—
614C2×C4C22×#6712yes—
615C2×C4C22×#6712yes—
616C2×C4C22×#6712yes—
617C2×C4C22×#6711yescharacteristic
618C2×C4C22×#6712yes—
619C2×C4C24×D311yescharacteristic
620C8—64no—
621C8—64no—
622C2×C3—24no—
623C2×C3—84no—
624C2×C3#303514yescomplemented
625C2×C3C22×#12111yescharacteristic
626C2×C3C22×#13511yescharacteristic
627C2×C3C2×#57511yescharacteristic
628D3—44no—
629C22—28no—
630C22—22no—
631C22—22no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—64no—
636C22—88no—
637C22—128no—
638C22—122no—
639C22—122no—
640C22C2×#93814yes—
641C22C2×#93812yes—
642C22C2×#95012yes—
643C22#582212yes—
644C22C23×#6711yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—44no—
649C4—64no—
650C4—122no—
651C4—122no—
652C4C2×#93814yes—
653C4C2×#93812yes—
654C4C2×#95014yescomplemented
655C4C22×D3×D411yescharacteristic
656C4C22×#23711yescharacteristic
657C3C2×#303511yescharacteristic, complemented, Sylow
658C2—24no—
659C2—84no—
660C2—124no—
661C2#8329614yesdirect product factor
662C2C22×#93811yescharacteristic
663C2C22×#95011yescharacteristic
664C2C2×#582211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3C2×#3035
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C22×#950C2semidirect
3. C2×C4×#235C2semidirect
4. C2×#5233C2semidirect
5. #83296C2direct (N × H)
6. C2×#5090C2semidirect
7. C2×#5083C2semidirect
8. C2×#5075C2semidirect
9. C2×#5072C2semidirect
10. C2×#5054C2semidirect
11. C2×#5057C2semidirect
12. C22×C4⋊(C3⋊C8)C2semidirect
17. C3×#532C22semidirect
27. C2×#950C22semidirect
37. C4×#235C22semidirect
44. #5233C22semidirect
49. #5090C22semidirect
50. #5083C22semidirect
51. #5075C22semidirect
52. #5072C22semidirect
53. #5057C22semidirect
54. #5054C22semidirect
58. C2×C4⋊(C3⋊C8)C22semidirect
60. C2×#532D3semidirect
87. C2×#235D4semidirect
168. #532C2×D3semidirect
247. #235C2×D4semidirect
412. C2×C3×C4#135semidirect
554. C3×C4C2×#135semidirect
608. C2×C4#950semidirect
624. C2×C3#3035semidirect
654. C4C2×#950semidirect
657. C3C2×#3035semidirect
661. C2#83296direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C22×#9502
C2×C4×#2352
C2×#52332
#832962
C2×#50902
C2×#50832
C2×#50752
C2×#50722
C2×#50542
C2×#50572
C22×C4⋊(C3⋊C8)2
C2×#30353