Finite Group Databasebeta

← C2×#71817 #10796728

Subgroups — C2×#71817 #10796728

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5819C211yesmaximal, characteristic, complemented
3C2×#5240C211yesmaximal, characteristic, complemented
4C2×#5236C212yesmaximal, complemented
5#71817C218yesmaximal, direct product factor
6C2×#4523C212yesmaximal, complemented
7C2×#1884—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×C3×#41C2211yescharacteristic
10C3×#532—22no—
11C3×#532—22no—
12C3×#532C414yescomplemented
13C3×#532C2214yes—
14C2×C3×C4⋊Q8C411yescharacteristic, complemented
15C2×C3×#127C411yescharacteristic, complemented
16C2×C3×#124C411yescharacteristic, complemented
17C2×C3×#124C411yescharacteristic, complemented
18C2×C3×#121—22no—
19C2×C3×C4×D4—22no—
20C22×C3×C42C2211yescharacteristic
21#5819—22no—
22#5819—24no—
23#5819C2214yescomplemented
24C2×#948—24no—
25C2×#926C2212yes—
26C22×#208—22no—
27C22×#208C2212yes—
28#5240—22no—
29#5240C2214yescomplemented
30#5236—24no—
31#5236C2218yescomplemented
32C2×#837—24no—
33C2×#834—22no—
34#4523C2218yescomplemented
35C2×#568—31no—
36C2×#532D311yescharacteristic
37C2×#421—32no—
38C2×#410—31no—
39#1884—38no—
40C2×#339—32no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—22no—
46C2×C3×#41—22no—
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41C2×C412yes—
52C2×C3×#41C2×C412yes—
53C2×C3×#41D412yes—
54C2×C3×#41D412yes—
55C2×C3×#41C2312yes—
56C2×C3×#41C2312yes—
57C22×C3×Q8C2×C411yescharacteristic
58C22×C3×Q8C2×C411yescharacteristic
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4C2×C411yescharacteristic
62C22×C3×D4C2×C411yescharacteristic
63C22×C3×D4D412yes—
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4D412yes—
67C23×C3×C4C2311yescharacteristic
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8C2×C414yescomplemented
70C3×#127—22no—
71C3×#127C2×C414yescomplemented
72C3×#124—22no—
73C3×#124—22no—
74C3×#124C2×C414yescomplemented
75C3×#124C2×C414yescomplemented
76C3×#121—28no—
77C3×#121—28no—
78C3×C4×D4—28no—
79C3×C4×D4—28no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42—21no—
86C2×C3×C42C2×C411yescharacteristic
87C2×C3×C42C2×C411yescharacteristic
88C2×C3×C42C2×C412yes—
89C2×C3×C42C2312yes—
90C2×#217—22no—
91#948—216no—
92#948—216no—
93#926—22no—
94#926—22no—
95#926—22no—
96#926C2314yes—
97C2×C4⋊Dic3—22no—
98C2×C4×Dic3—22no—
99C2×#208—24no—
100C2×#208—28no—
101C2×#208—22no—
102C2×#208—28no—
103C2×#208—22no—
104C2×#208—22no—
105C2×#208D414yescomplemented
106C2×#208D414yescomplemented
107C2×#208C2314yes—
108C22×C3⋊C8—24no—
109C22×C3⋊C8—22no—
110#837—216no—
111#834—28no—
112C22×#41C2×D311yescharacteristic
113C22×#41C2×D311yescharacteristic
114#568—34no—
115#568—64no—
116#568—62no—
117C2×#131—64no—
118C22×M16—32no—
119C22×M16—62no—
120#532—22no—
121#532—22no—
122#532Dic314yescomplemented
123#532C2×D314yes—
124C2×C4⋊Q8Dic311yescharacteristic, complemented
125C2×#127Dic311yescharacteristic, complemented
126C2×#124Dic311yescharacteristic, complemented
127C2×#124Dic311yescharacteristic, complemented
128C2×#121—22no—
129C2×C4×D4—22no—
130C2×#117—32no—
131C22×C42C2×D311yescharacteristic
132#421—38no—
133#421—64no—
134C2×#104—64no—
135#410—34no—
136#410—62no—
137C2×#99—62no—
138#339—38no—
139C24×C3—21no—
140C24×C3C22⋊C412yes—
141C24×C3C2×D411yescharacteristic
142C3×#41—24no—
143C3×#41—24no—
144C3×#41—24no—
145C3×#41—24no—
146C3×#41—24no—
147C3×#41—24no—
148C3×#41—24no—
149C3×#41—24no—
150C3×#41—24no—
151C3×#41—24no—
152C3×#41—24no—
153C3×#41—24no—
154C3×#41—42no—
155C3×#41—42no—
156C3×#41—42no—
157C3×#41—42no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8C22⋊C412yes—
164C2×C3×Q8C22×C412yes—
165C2×C3×Q8C22×C412yes—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—28no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—48no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4C22⋊C414yes—
180C2×C3×D4C22⋊C412yes—
181C2×C3×D4C22×C412yes—
182C2×C3×D4C22×C412yes—
183C2×C3×D4C2×D414yes—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—44no—
196C22×C3×C4—41no—
197C22×C3×C4—41no—
198C22×C3×C4—41no—
199C22×C3×C4—41no—
200C22×C3×C4C22⋊C412yes—
201C22×C3×C4C22⋊C414yes—
202C22×C3×C4C22×C411yescharacteristic
203C22×C3×C4C22×C411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4#4112yes—
211C22×C3×C4#4112yes—
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—42no—
214C3×C4⋊C4—28no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—28no—
217C3×C42—22no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—22no—
221C3×C42C22×C412yes—
222C3×C42C22×C412yes—
223#217—24no—
224#217—42no—
225C4⋊Dic3—24no—
226C4⋊Dic3—42no—
227C4×Dic3—24no—
228C4×Dic3—42no—
229#208—216no—
230#208—24no—
231#208—24no—
232#208—24no—
233#208C2×D418yescomplemented
234C2×C3⋊C8—216no—
235C2×C3⋊C8—24no—
236C2×C3⋊C8—24no—
237C2×C3⋊C8—24no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—21no—
243C2×#41—21no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41C2×Dic312yes—
249C2×#41C2×Dic312yes—
250C2×#41#6712yes—
251C2×#41#6712yes—
252C2×#41C22×D312yes—
253C2×#41C22×D312yes—
254C22×Q8C2×Dic311yescharacteristic
255C22×Q8C2×Dic311yescharacteristic
256C22×D4—22no—
257C22×D4—21no—
258C22×D4C2×Dic311yescharacteristic
259C22×D4C2×Dic311yescharacteristic
260C22×D4#6712yes—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4#6712yes—
264C23×C4C22×D311yescharacteristic
265#131—616no—
266#131—616no—
267C2×M16—34no—
268C2×M16—34no—
269C2×M16—34no—
270C2×M16—64no—
271C2×M16—68no—
272C2×M16—62no—
273C2×M16—62no—
274C2×M16—62no—
275C2×M16—68no—
276C22×C8—64no—
277C22×C8—62no—
278C4⋊Q8—22no—
279C4⋊Q8C2×Dic314yescomplemented
280#127—22no—
281#127C2×Dic314yescomplemented
282#124—22no—
283#124—22no—
284#124C2×Dic314yescomplemented
285#124C2×Dic314yescomplemented
286#121—28no—
287#121—28no—
288C4×D4—28no—
289C4×D4—28no—
290#117—34no—
291#117—62no—
292#117—62no—
293#117—62no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—62no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—62no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—62no—
303C2×C42C2×Dic311yescharacteristic
304C2×C42C2×Dic311yescharacteristic
305C2×C42C2×Dic312yes—
306C2×C42C22×D312yes—
307#104—616no—
308#99—68no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—42no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3C2×C22⋊C411yescharacteristic
318C23×C3C2×C22⋊C414yes—
319C23×C3C2×C22⋊C411yescharacteristic
320C23×C3#12112yes—
321C23×C3#12112yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×Q8—42no—
327C3×D4—28no—
328C3×D4—28no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—24no—
332C3×D4—416no—
333C3×D4—44no—
334C3×D4—44no—
335C3×D4—42no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—44no—
347C2×C3×C4—44no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4C2×C22⋊C414yes—
356C2×C3×C4C2×C22⋊C411yescharacteristic
357C2×C3×C4C2×C22⋊C411yescharacteristic
358C2×C3×C4C4×D412yes—
359C2×C3×C4C4×D412yes—
360C2×C3×C4C4×D412yes—
361C2×C3×C4C4×D412yes—
362C2×C3×C4#12012yes—
363C2×C3×C4#12011yescharacteristic
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12112yes—
366C2×C3×C4#12112yes—
367C2×C3×C4#12311yescharacteristic
368C2×C3×C4#12311yescharacteristic
369C2×C3×C4#12312yes—
370C2×C3×C4C23×C411yescharacteristic
371C2×C3×C4C22×D412yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4112yes—
374C2×Dic3—44no—
375C2×Dic3—84no—
376C3⋊C8—28no—
377C24—21no—
378C24#21712yes—
379C24C2×#6711yescharacteristic
380#41—24no—
381#41—24no—
382#41—24no—
383#41—24no—
384#41—24no—
385#41—24no—
386#41—24no—
387#41—24no—
388#41—24no—
389#41—24no—
390#41—24no—
391#41—24no—
392#41—42no—
393#41—42no—
394#41—42no—
395#41—42no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8#21712yes—
402C2×Q8C22×Dic312yes—
403C2×Q8C22×Dic312yes—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—48no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4#21714yes—
418C2×D4#21712yes—
419C2×D4C22×Dic312yes—
420C2×D4C22×Dic312yes—
421C2×D4C2×#6714yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—122no—
439C22×C4#21712yes—
440C22×C4#21714yes—
441C22×C4D3×D412yes—
442C22×C4D3×D412yes—
443C22×C4#23712yes—
444C22×C4#23712yes—
445C22×C4C22×Dic311yescharacteristic
446C22×C4C22×Dic311yescharacteristic
447C22×C4C2×#6714yes—
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452M16—38no—
453M16—616no—
454M16—64no—
455M16—64no—
456M16—64no—
457C2×C8—616no—
458C2×C8—64no—
459C2×C8—64no—
460C2×C8—64no—
461C4⋊C4—28no—
462C4⋊C4—64no—
463C4⋊C4—122no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—64no—
467C22⋊C4—122no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—22no—
472C42—64no—
473C42—122no—
474C42C22×Dic312yes—
475C42C22×Dic312yes—
476C22×C3—24no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—44no—
483C22×C3—44no—
484C22×C3—44no—
485C22×C3—42no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3#38611yescharacteristic
490C22×C3#38612yes—
491C22×C3#38611yescharacteristic
492C22×C3C22×C22⋊C411yescharacteristic
493C22×C3C2×#12112yes—
494C3×C4—24no—
495C3×C4—24no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4#38611yescharacteristic
499C3×C4#38611yescharacteristic
500C3×C4#38612yes—
501C3×C4C2×C4×D412yes—
502C3×C4C2×#12011yescharacteristic
503C3×C4C2×#12311yescharacteristic
504Dic3—84no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—41no—
512C23—41no—
513C23#93912yes—
514C23#93912yes—
515C23C2×#21711yescharacteristic
516C23C2×#21714yes—
517C23C2×#21711yescharacteristic
518C23C22×#6711yescharacteristic
519Q8—24no—
520Q8—24no—
521Q8—24no—
522Q8—42no—
523D4—28no—
524D4—28no—
525D4—24no—
526D4—24no—
527D4—24no—
528D4—416no—
529D4—44no—
530D4—44no—
531D4—42no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—124no—
552C2×C4—244no—
553C2×C4D4×Dic312yes—
554C2×C4D4×Dic312yes—
555C2×C4D4×Dic312yes—
556C2×C4D4×Dic312yes—
557C2×C4#93511yescharacteristic
558C2×C4#93511yescharacteristic
559C2×C4#93512yes—
560C2×C4#93712yes—
561C2×C4#93711yescharacteristic
562C2×C4#93711yescharacteristic
563C2×C4#93912yes—
564C2×C4#93912yes—
565C2×C4C2×#21714yes—
566C2×C4C2×#21711yescharacteristic
567C2×C4C2×#21711yescharacteristic
568C2×C4C2×D3×D412yes—
569C2×C4C2×#23712yes—
570C2×C4C23×Dic311yescharacteristic
571C2×C4C22×#6711yescharacteristic
572C8—68no—
573C2×C3—22no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—22no—
577C2×C3—44no—
578C2×C3—82no—
579C2×C3#188412yescomplemented
580C2×C3C2×#38611yescharacteristic
581C22—22no—
582C22—24no—
583C22—24no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22#522011yescharacteristic
595C22#522012yes—
596C22#522011yescharacteristic
597C22C2×#93912yes—
598C22C22×#21711yescharacteristic
599C4—24no—
600C4—24no—
601C4—42no—
602C4—42no—
603C4—244no—
604C4#522011yescharacteristic
605C4#522011yescharacteristic
606C4#522012yes—
607C4C2×D4×Dic312yes—
608C4C2×#93511yescharacteristic
609C4C2×#93711yescharacteristic
610C3C2×#188411yescharacteristic, complemented, Sylow
611C2—21no—
612C2—22no—
613C2—21no—
614C2—22no—
615C2—44no—
616C2—82no—
617C2#7181712yesdirect product factor
618C2C2×#522011yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC22×C3#386
cosocleC22×C3×C4C24
hypercenterC2×#532D3
hypocenterC3C2×#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5819C2semidirect
3. C2×#5240C2semidirect
4. C2×#5236C2semidirect
5. #71817C2direct (N × H)
6. C2×#4523C2semidirect
12. C3×#532C4semidirect
14. C2×C3×C4⋊Q8C4semidirect
15. C2×C3×#127C4semidirect
16. C2×C3×#124C4semidirect
17. C2×C3×#124C4semidirect
23. #5819C22semidirect
29. #5240C22semidirect
31. #5236C22semidirect
34. #4523C22semidirect
69. C3×C4⋊Q8C2×C4semidirect
71. C3×#127C2×C4semidirect
74. C3×#124C2×C4semidirect
75. C3×#124C2×C4semidirect
105. C2×#208D4semidirect
106. C2×#208D4semidirect
122. #532Dic3semidirect
124. C2×C4⋊Q8Dic3semidirect
125. C2×#127Dic3semidirect
126. C2×#124Dic3semidirect
127. C2×#124Dic3semidirect
233. #208C2×D4semidirect
279. C4⋊Q8C2×Dic3semidirect
281. #127C2×Dic3semidirect
284. #124C2×Dic3semidirect
285. #124C2×Dic3semidirect
579. C2×C3#1884semidirect
610. C3C2×#1884semidirect
617. C2#71817direct (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
C2×#58192
C2×#52402
C2×#52362
#718172
C2×#45232
C2×#18843