Finite Group Databasebeta

← C2×#76946 #10967030

Subgroups — C2×#76946 #10967030

Order 768.

3958 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#940C211yesmaximal, characteristic
2C22×#887C211yesmaximal, characteristic, complemented
3C22×#885C212yesmaximal, complemented
4C2×C3×#394C211yesmaximal, characteristic, complemented
5C2×#5222C211yesmaximal, characteristic, complemented
6C2×#4991C211yesmaximal, characteristic, complemented
7C2×#4957C212yesmaximal, complemented
8C2×#4957C212yesmaximal, complemented
9C2×#4957C212yesmaximal, complemented
10#76946C2116yesmaximal, direct product factor
11C2×#4672C211yesmaximal, characteristic, complemented
12C2×#4649C211yesmaximal, characteristic, complemented
13C2×#2548—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×#67C2212yes—
16C23×#67C2212yes—
17C23×D12C2211yescharacteristic
18C23×Dic6C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2212yes—
22C22×#217C2212yes—
23C22×#217C2212yes—
24C2×#940—28no—
25C2×#940C2218yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2211yescharacteristic
33C22×C4×Dic3C2212yes—
34C22×C4×Dic3C2211yescharacteristic
35C22×C4×Dic3C2211yescharacteristic
36C22×C4×Dic3C2211yescharacteristic
37C2×#887—28no—
38C2×#887C2218yescomplemented
39C2×#885—216no—
40C2×#885C22116yescomplemented
41C3×#394C2218yescomplemented
42C2×C3×#95C2211yescharacteristic
43#5222C2218yescomplemented
44C2×C3⋊#95C2211yescharacteristic
45#4991C2218yescomplemented
46#4957C22116yescomplemented
47#4957C22116yescomplemented
48#4957C22116yescomplemented
49#4672C2218yescomplemented
50#4649C2218yescomplemented
51C22×#127—31no—
52C22×#124—32no—
53C22×C4×Q8—31no—
54#2548—316no—
55C2×#394—31no—
56C2×#394—31no—
57C2×#394—31no—
58C2×#394D311yescharacteristic, complemented
59C2×#390—32no—
60C2×#390—32no—
61C2×#390—32no—
62C2×C4×C4⋊C4—31no—
63C25×C3C2312yes—
64C24×D3C2312yes—
65C22×C3×D4—24no—
66C22×C3×D4—44no—
67C22×C3×D4C2314yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C22×#67—28no—
73C22×#67—28no—
74C22×#67—28no—
75C22×#67—28no—
76C22×#67—28no—
77C22×#67—28no—
78C22×#67C2318yes—
79C22×#67C2318yes—
80C23×Dic3C2312yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2311yescharacteristic
84C23×Dic3C2311yescharacteristic
85C22×D12—24no—
86C22×D12—44no—
87C22×D12C2314yes—
88C22×Dic6—24no—
89C22×Dic6—24no—
90C22×Dic6D418yescomplemented
91C22×Dic6C2314yes—
92C2×C3×C4⋊C4—24no—
93C2×C3×C4⋊C4C2314yes—
94C2×C3×C22⋊C4—28no—
95C2×C3×C22⋊C4—44no—
96C2×C3×C22⋊C4C2318yes—
97C2×C3×C22⋊C4C2318yes—
98C2×#217—28no—
99C2×#217—44no—
100C2×#217C2318yes—
101C2×#217C2318yes—
102#940—232no—
103C2×(C2×D3)⋊C4—28no—
104C2×(C2×D3)⋊C4—28no—
105C2×(C2×D3)⋊C4—44no—
106C2×(C2×D3)⋊C4—44no—
107C2×(C2×D3)⋊C4C2318yes—
108C2×(C2×D3)⋊C4C2318yes—
109C2×(C2×D3)⋊C4C2318yes—
110C2×(C2×D3)⋊C4C2318yes—
111C2×Dic3⋊C4—24no—
112C2×Dic3⋊C4—24no—
113C2×Dic3⋊C4—28no—
114C2×Dic3⋊C4C2318yes—
115C2×Dic3⋊C4C2314yes—
116C2×Dic3⋊C4C2314yes—
117C2×C4×Dic3—28no—
118C2×C4×Dic3—24no—
119C2×C4×Dic3—24no—
120C2×C4×Dic3D418yes—
121C2×C4×Dic3C2318yes—
122C2×C4×Dic3C2314yes—
123C2×C4×Dic3C2314yes—
124C2×C4×Dic3C2314yes—
125#887—232no—
126#885—264no—
127C3×#95C2314yes—
128C3⋊#95C2314yes—
129C23×Q8—31no—
130C23×D4—32no—
131C23×D4—32no—
132C23×D4—31no—
133C23×D4C2×D311yescharacteristic
134C2×#127—38no—
135C2×#127—68no—
136C2×#124—316no—
137C2×#124—616no—
138C2×C4×Q8—38no—
139C2×C4×Q8—68no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—32no—
143C22×C4⋊C4C2×D311yescharacteristic
144C22×C22⋊C4—32no—
145C22×C22⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4—32no—
148C22×C22⋊C4—32no—
149C22×C22⋊C4—32no—
150C22×C22⋊C4C2×D312yes—
151C22×C22⋊C4C2×D312yes—
152C22×C42—31no—
153C22×C42—31no—
154C22×C42—31no—
155C22×C42—32no—
156#394—38no—
157#394—38no—
158#394—38no—
159#394C2×D318yescomplemented
160#390—316no—
161#390—316no—
162#390—316no—
163C4×C4⋊C4—38no—
164C2×#95—31no—
165C2×#95C2×D311yescharacteristic
166C24×C3—24no—
167C24×C3—24no—
168C24×C3—24no—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3—24no—
173C24×C3C2414yes—
174C24×C3C2411yescharacteristic
175C23×D3—24no—
176C23×D3—24no—
177C23×D3—24no—
178C23×D3—24no—
179C23×D3—24no—
180C23×D3—24no—
181C23×D3—24no—
182C23×D3C2414yes—
183C2×C3×D4—28no—
184C2×C3×D4—48no—
185C2×C3×D4—416no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2412yes—
200C2×#67—216no—
201C2×#67—216no—
202C2×#67—216no—
203C2×#67—216no—
204C2×#67—216no—
205C2×#67—216no—
206C2×#67—432no—
207C2×#67—432no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3C2×D414yes—
216C22×Dic3C2×D414yes—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×D414yes—
219C22×Dic3#4114yes—
220C22×Dic3#4118yes—
221C22×Dic3C2414yes—
222C22×Dic3C2412yes—
223C22×Dic3C2412yes—
224C22×Dic3C2412yes—
225C22×Dic3C2412yes—
226C2×D12—28no—
227C2×D12—48no—
228C2×D12—416no—
229C2×Dic6—232no—
230C2×Dic6—28no—
231C2×Dic6—28no—
232C2×Dic6C2×D4116yescomplemented
233C3×C4⋊C4—28no—
234C3×C22⋊C4—216no—
235C3×C22⋊C4—48no—
236#217—216no—
237#217—48no—
238(C2×D3)⋊C4—216no—
239(C2×D3)⋊C4—216no—
240(C2×D3)⋊C4—48no—
241(C2×D3)⋊C4—48no—
242Dic3⋊C4—28no—
243Dic3⋊C4—28no—
244Dic3⋊C4—216no—
245C4×Dic3—216no—
246C4×Dic3—28no—
247C4×Dic3—28no—
248C4×Dic3C2×D4116yes—
249C25—32no—
250C25C22×D312yes—
251C22×Q8—38no—
252C22×Q8—34no—
253C22×Q8—64no—
254C22×Q8—64no—
255C22×D4—24no—
256C22×D4—38no—
257C22×D4—38no—
258C22×D4—34no—
259C22×D4—44no—
260C22×D4—68no—
261C22×D4—68no—
262C22×D4—68no—
263C22×D4—68no—
264C22×D4—68no—
265C22×D4—68no—
266C22×D4—64no—
267C22×D4—124no—
268C22×D4C22×D314yes—
269C23×C4—31no—
270C23×C4—31no—
271C23×C4—31no—
272C23×C4—31no—
273C23×C4—32no—
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D312yes—
276C23×C4C22×D311yescharacteristic
277C23×C4C22×D311yescharacteristic
278#127—632no—
279#124—664no—
280C4×Q8—632no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—34no—
283C2×C4⋊C4—34no—
284C2×C4⋊C4—38no—
285C2×C4⋊C4—64no—
286C2×C4⋊C4—68no—
287C2×C4⋊C4—64no—
288C2×C4⋊C4C22×D314yes—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—38no—
291C2×C22⋊C4—38no—
292C2×C22⋊C4—38no—
293C2×C22⋊C4—38no—
294C2×C22⋊C4—38no—
295C2×C22⋊C4—38no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—68no—
298C2×C22⋊C4—68no—
299C2×C22⋊C4—68no—
300C2×C22⋊C4—124no—
301C2×C22⋊C4—124no—
302C2×C22⋊C4—124no—
303C2×C22⋊C4C22×D318yes—
304C2×C22⋊C4C22×D318yes—
305C2×C42—38no—
306C2×C42—34no—
307C2×C42—34no—
308C2×C42—34no—
309C2×C42—38no—
310C2×C42—64no—
311C2×C42—64no—
312C2×C42—68no—
313#95—34no—
314#95C22×D314yes—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3—48no—
323C23×C3—48no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—48no—
327C23×C3—48no—
328C23×C3—48no—
329C23×C3C22×D412yes—
330C23×C3C22×D412yes—
331C23×C3C2×#4112yes—
332C23×C3C2×#4114yes—
333C23×C3H(5,𝔽2)14yes—
334C23×C3C2511yescharacteristic
335C22×D3—24no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—24no—
341C22×D3—24no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—48no—
345C22×D3—48no—
346C22×D3—48no—
347C22×D3—48no—
348C22×D3—48no—
349C3×D4—416no—
350C2×C3×C4—28no—
351C2×C3×C4—28no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—44no—
361C2×C3×C4—48no—
362C2×C3×C4C22×D414yes—
363C2×C3×C4C22×D414yes—
364#67—432no—
365#67—432no—
366C2×Dic3—28no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—44no—
375C2×Dic3—48no—
376C2×Dic3#121116yes—
377C2×Dic3#124116yescomplemented
378C2×Dic3C22×D414yes—
379C2×Dic3C22×D414yes—
380C2×Dic3C22×D414yes—
381C2×Dic3C22×D414yes—
382C2×Dic3C2×#4114yes—
383C2×Dic3C2×#4118yes—
384D12—416no—
385Dic6—232no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—34no—
394C24—64no—
395C24—64no—
396C24—64no—
397C24—64no—
398C24—64no—
399C24—64no—
400C24—64no—
401C24C23×D314yes—
402C24C23×D311yescharacteristic
403C2×Q8—316no—
404C2×Q8—68no—
405C2×Q8—68no—
406C2×Q8—632no—
407C2×D4—28no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—616no—
411C2×D4—616no—
412C2×D4—616no—
413C2×D4—616no—
414C2×D4—616no—
415C2×D4—616no—
416C2×D4—68no—
417C2×D4—128no—
418C2×D4—1216no—
419C2×D4—1232no—
420C2×D4—1232no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—34no—
430C22×C4—34no—
431C22×C4—34no—
432C22×C4—38no—
433C22×C4—34no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—34no—
437C22×C4—32no—
438C22×C4—34no—
439C22×C4—32no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4D3×D414yes—
448C22×C4D3×D414yes—
449C22×C4C23×D314yes—
450C22×C4C23×D312yes—
451C22×C4C23×D312yes—
452C22×C4C23×D312yes—
453C4⋊C4—28no—
454C4⋊C4—68no—
455C4⋊C4—616no—
456C4⋊C4—68no—
457C22⋊C4—216no—
458C22⋊C4—48no—
459C22⋊C4—616no—
460C22⋊C4—616no—
461C22⋊C4—616no—
462C22⋊C4—128no—
463C22⋊C4—128no—
464C22⋊C4—128no—
465C42—316no—
466C42—68no—
467C42—68no—
468C42—616no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—816no—
477C22×C3C2×#12114yes—
478C22×C3C2×#12414yes—
479C22×C3#53514yes—
480C22×C3#55018yes—
481C22×C3#55918yes—
482C22×C3C23×D411yescharacteristic
483C22×C3C23×D411yescharacteristic
484C22×C3C22×#4111yescharacteristic
485C22×C3C22×#4112yes—
486C22×C3C2×H(5,𝔽2)12yes—
487C2×D3—44no—
488C2×D3—44no—
489C2×D3—44no—
490C2×D3—44no—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—816no—
495C3×C4—24no—
496C3×C4—24no—
497C3×C4—42no—
498C3×C4—44no—
499Dic3—24no—
500Dic3—42no—
501Dic3—44no—
502Dic3C2×#12118yes—
503Dic3C2×#12418yescomplemented
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—64no—
519C23—64no—
520C23—64no—
521C23—64no—
522C23—64no—
523C23—64no—
524C23—64no—
525C23—128no—
526C23—128no—
527C23—128no—
528C23—128no—
529C23—128no—
530C23—128no—
531C23—128no—
532C23C2×D3×D412yes—
533C23C2×D3×D412yes—
534C23#100414yes—
535C23D3×#4112yes—
536C23D3×#4114yes—
537C23C24×D311yescharacteristic
538Q8—632no—
539D4—416no—
540D4—1216no—
541D4—1232no—
542D4—1232no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—316no—
554C2×C4—34no—
555C2×C4—34no—
556C2×C4—34no—
557C2×C4—38no—
558C2×C4—34no—
559C2×C4—316no—
560C2×C4—34no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—68no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—124no—
572C2×C4—128no—
573C2×C4C2×D3×D414yes—
574C2×C4C2×D3×D414yes—
575C2×C3—84no—
576C2×C3#254818yescomplemented
577C2×C3C22×#12111yescharacteristic
578C2×C3C22×#12411yescharacteristic
579C2×C3C2×#53511yescharacteristic
580C2×C3C2×#55012yes—
581C2×C3C2×#55912yes—
582D3—84no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—816no—
591C22—124no—
592C22—124no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—124no—
597C22—124no—
598C22—2416no—
599C22#559214yes—
600C22#559918yes—
601C22D3×#12114yes—
602C22#566418yes—
603C22D3×#12414yes—
604C22C22×D3×D411yescharacteristic
605C22C22×D3×D411yescharacteristic
606C22C2×#100412yes—
607C22C2×D3×#4112yes—
608C22C2×D3×#4111yescharacteristic
609C4—24no—
610C4—24no—
611C4—38no—
612C4—38no—
613C4—42no—
614C4—44no—
615C4—64no—
616C4—122no—
617C4—124no—
618C3C2×#254811yescharacteristic, complemented, Sylow
619C2—84no—
620C2—244no—
621C2#7694618yesdirect product factor
622C2C2×#559211yescharacteristic
623C2C2×#559912yes—
624C2C2×D3×#12111yescharacteristic
625C2C2×#566412yes—
626C2C2×D3×#12411yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#394C2
socleC24×C3C24
prime radicalC22×#940C2
hypercenterC2×#394D3
hypocenterC3C2×#2548
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×#887C2semidirect
3. C22×#885C2semidirect
4. C2×C3×#394C2semidirect
5. C2×#5222C2semidirect
6. C2×#4991C2semidirect
7. C2×#4957C2semidirect
8. C2×#4957C2semidirect
9. C2×#4957C2semidirect
10. #76946C2direct (N × H)
11. C2×#4672C2semidirect
12. C2×#4649C2semidirect
38. C2×#887C22semidirect
40. C2×#885C22semidirect
41. C3×#394C22semidirect
43. #5222C22semidirect
45. #4991C22semidirect
46. #4957C22semidirect
47. #4957C22semidirect
48. #4957C22semidirect
49. #4672C22semidirect
50. #4649C22semidirect
58. C2×#394D3semidirect
90. C22×Dic6D4semidirect
159. #394C2×D3semidirect
232. C2×Dic6C2×D4semidirect
377. C2×Dic3#124semidirect
503. Dic3C2×#124semidirect
576. C2×C3#2548semidirect
618. C3C2×#2548semidirect
621. C2#76946direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9402
C22×#8872
C22×#8852
C2×C3×#3942
C2×#52222
C2×#49912
C2×#49572
#769462
C2×#46722
C2×#46492
C2×#25483