Finite Group Databasebeta

← C2×#82661 #10964634

Subgroups — C2×#82661 #10964634

Order 768.

3158 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C22×#890C211yesmaximal, characteristic, complemented
3C2×#5222C212yesmaximal, complemented
4C2×#5221C211yesmaximal, characteristic, complemented
5C2×#5221C211yesmaximal, characteristic, complemented
6C2×#5221C212yesmaximal, complemented
7C2×#5212C211yesmaximal, characteristic, complemented
8C2×C4×#217C211yesmaximal, characteristic, complemented
9#82661C2116yesmaximal, direct product factor
10C2×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
11C2×#4953C212yesmaximal, complemented
12C2×#4945C212yesmaximal, complemented
13C2×#2490—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×C3×D4C2212yes—
16C24×C3×C4C2211yescharacteristic
17C2×C3×#121—28no—
18C2×C3×#121C2218yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2211yescharacteristic
22C22×#217C2211yescharacteristic
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×#217C2212yes—
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30C22×C4⋊Dic3C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×C4×Dic3C2211yescharacteristic
33C22×C4×Dic3C2211yescharacteristic
34C22×C4×Dic3C2212yes—
35C2×#890—28no—
36C2×#890C2218yescomplemented
37#5222C22116yescomplemented
38#5221C2218yescomplemented
39#5221C2218yescomplemented
40#5221C22116yescomplemented
41#5212C2218yescomplemented
42C4×#217C2218yescomplemented
43C2×C3⋊#95C2211yescharacteristic
44C2×C3⋊#95C2211yescharacteristic
45C2×C3⋊#95C2212yes—
46C2×C3⋊#95C2212yes—
47Dic3×C4⋊C4C2218yescomplemented
48#4953C22116yescomplemented
49#4945C22116yescomplemented
50C22×#125—31no—
51C22×#121D311yescharacteristic
52#2490—316no—
53C2×#397—32no—
54C2×#394—32no—
55C2×#390—32no—
56C2×#390—31no—
57C2×#390—31no—
58C2×#388—32no—
59C2×#380—31no—
60C2×C4×C4⋊C4—31no—
61C2×C4×C22⋊C4—31no—
62C25×C3C2312yes—
63C25×C3C2311yescharacteristic
64C22×C3×D4—28no—
65C22×C3×D4—28no—
66C22×C3×D4—24no—
67C22×C3×D4—24no—
68C22×C3×D4—24no—
69C22×C3×D4—44no—
70C22×C3×D4C2314yes—
71C22×C3×D4C2318yes—
72C23×C3×C4—24no—
73C23×C3×C4—24no—
74C23×C3×C4—24no—
75C23×C3×C4C2311yescharacteristic
76C23×C3×C4C2314yes—
77C23×C3×C4C2312yes—
78C23×C3×C4C2311yescharacteristic
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2312yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2312yes—
84C23×Dic3C2311yescharacteristic
85C3×#121—232no—
86C2×C3×C4⋊C4D418yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C22⋊C4—28no—
89C2×C3×C22⋊C4C2318yes—
90C2×#217—24no—
91C2×#217—24no—
92C2×#217—28no—
93C2×#217—44no—
94C2×#217—44no—
95C2×#217—44no—
96C2×#217—44no—
97C2×#217C2318yes—
98C2×#217C2314yes—
99C2×#217C2314yes—
100C2×#217C2318yes—
101C2×#217C2318yes—
102C2×#217C2318yes—
103C2×#217C2318yes—
104C2×C4⋊Dic3—28no—
105C2×C4⋊Dic3—24no—
106C2×C4⋊Dic3—24no—
107C2×C4⋊Dic3C2318yes—
108C2×C4⋊Dic3C2314yes—
109C2×C4⋊Dic3C2314yes—
110C2×Dic3⋊C4—28no—
111C2×Dic3⋊C4C2318yes—
112C2×C4×Dic3—24no—
113C2×C4×Dic3—24no—
114C2×C4×Dic3—44no—
115C2×C4×Dic3C2314yes—
116C2×C4×Dic3C2314yes—
117C2×C4×Dic3C2318yes—
118#890—232no—
119C3⋊#95C2314yes—
120C3⋊#95C2318yes—
121C3⋊#95C2314yes—
122C3⋊#95C2318yes—
123C23×D4C2×D311yescharacteristic
124C23×D4C2×D312yes—
125C24×C4C2×D311yescharacteristic
126C2×#125—38no—
127C2×#125—68no—
128C2×#121—28no—
129C2×#121C2×D318yes—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C22⋊C4—31no—
136C22×C22⋊C4—31no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4C2×D312yes—
143C22×C42—31no—
144C22×C42—31no—
145C22×C42—32no—
146#397—316no—
147#394—316no—
148#390—38no—
149#390—38no—
150#390—316no—
151#388—316no—
152#380—38no—
153C4×C4⋊C4—38no—
154C4×C22⋊C4—38no—
155C2×#95—32no—
156C2×#95—31no—
157C2×#95—32no—
158C2×#95—31no—
159C24×C3—22no—
160C24×C3—22no—
161C24×C3—22no—
162C24×C3—22no—
163C24×C3—22no—
164C24×C3—22no—
165C24×C3—24no—
166C24×C3—24no—
167C24×C3—24no—
168C24×C3—24no—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3#4114yes—
173C24×C3C2414yes—
174C24×C3C2412yes—
175C24×C3C2411yescharacteristic
176C2×C3×D4—216no—
177C2×C3×D4—216no—
178C2×C3×D4—28no—
179C2×C3×D4—28no—
180C2×C3×D4—28no—
181C2×C3×D4—48no—
182C2×C3×D4—432no—
183C2×C3×D4—416no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—28no—
190C22×C3×C4—28no—
191C22×C3×C4—28no—
192C22×C3×C4—416no—
193C22×C3×C4C2×D418yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4#4114yes—
196C22×C3×C4#4114yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2414yes—
199C22×C3×C4C2412yes—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—24no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—22no—
213C22×Dic3—24no—
214C22×Dic3—22no—
215C22×Dic3—24no—
216C22×Dic3—22no—
217C22×Dic3#4114yes—
218C22×Dic3C2412yes—
219C22×Dic3C2414yes—
220C22×Dic3C2412yes—
221C22×Dic3C2412yes—
222C22×Dic3C2414yes—
223C22×Dic3C2412yes—
224C3×C4⋊C4C2×D4116yes—
225C3×C22⋊C4—216no—
226#217—28no—
227#217—28no—
228#217—216no—
229#217—48no—
230#217—48no—
231#217—48no—
232#217—48no—
233C4⋊Dic3—216no—
234C4⋊Dic3—28no—
235C4⋊Dic3—28no—
236Dic3⋊C4—216no—
237C4×Dic3—28no—
238C4×Dic3—28no—
239C4×Dic3—48no—
240C25C22×D312yes—
241C25C22×D311yescharacteristic
242C22×D4—28no—
243C22×D4—24no—
244C22×D4—24no—
245C22×D4—28no—
246C22×D4—24no—
247C22×D4—44no—
248C22×D4C22×D314yes—
249C22×D4C22×D318yes—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—31no—
254C23×C4—32no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—32no—
258C23×C4—31no—
259C23×C4C22×D314yes—
260C23×C4C22×D311yescharacteristic
261C23×C4C22×D312yes—
262C23×C4C22×D311yescharacteristic
263#125—632no—
264#121—232no—
265C2×C4⋊C4—34no—
266C2×C4⋊C4—38no—
267C2×C4⋊C4—38no—
268C2×C4⋊C4—34no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—68no—
271C2×C4⋊C4—68no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4#6718yes—
274C2×C4⋊C4C22×D314yes—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—34no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—38no—
281C2×C22⋊C4—38no—
282C2×C22⋊C4—38no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—68no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—124no—
287C2×C22⋊C4—124no—
288C2×C22⋊C4—124no—
289C2×C22⋊C4—124no—
290C2×C22⋊C4C22×D318yes—
291C2×C42—34no—
292C2×C42—34no—
293C2×C42—38no—
294C2×C42—64no—
295C2×C42—64no—
296C2×C42—124no—
297#95—34no—
298#95—34no—
299#95—38no—
300#95—38no—
301C23×C3—28no—
302C23×C3—28no—
303C23×C3—28no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—44no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—48no—
322C23×C3—48no—
323C23×C3—48no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—48no—
327C23×C3—48no—
328C23×C3C22×D412yes—
329C23×C3C2×#4114yes—
330C23×C3C2×#4112yes—
331C23×C3C2×#4112yes—
332C23×C3C2×#4112yes—
333C23×C3C2×#4112yes—
334C23×C3H(5,𝔽2)12yes—
335C23×C3H(5,𝔽2)12yes—
336C23×C3C2511yescharacteristic
337C3×D4—432no—
338C3×D4—416no—
339C2×C3×C4—216no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—44no—
346C2×C3×C4—416no—
347C2×C3×C4#124116yes—
348C2×C3×C4C22×D418yes—
349C2×C3×C4C22×D414yes—
350C2×C3×C4C2×#4114yes—
351C2×C3×C4C2×#4114yes—
352C2×Dic3—28no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—22no—
370C2×Dic3—48no—
371C2×Dic3—44no—
372C2×Dic3—44no—
373C2×Dic3—48no—
374C2×Dic3—44no—
375C2×Dic3C2×#4114yes—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24#23714yes—
390C24C23×D314yes—
391C24C23×D312yes—
392C24C23×D311yescharacteristic
393C2×D4—216no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—216no—
397C2×D4—28no—
398C2×D4—48no—
399C2×D4—432no—
400C2×D4—416no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—34no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—34no—
416C22×C4—416no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—64no—
423C22×C4—64no—
424C22×C4—62no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—64no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—64no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4#23714yes—
435C22×C4#23714yes—
436C22×C4C2×#6718yes—
437C22×C4C2×#6714yes—
438C22×C4C23×D312yes—
439C22×C4C23×D314yes—
440C22×C4C23×D312yes—
441C4⋊C4—68no—
442C4⋊C4—616no—
443C4⋊C4—616no—
444C4⋊C4—68no—
445C4⋊C4C2×#67116yes—
446C22⋊C4—216no—
447C22⋊C4—68no—
448C22⋊C4—616no—
449C22⋊C4—68no—
450C22⋊C4—128no—
451C22⋊C4—128no—
452C22⋊C4—128no—
453C22⋊C4—128no—
454C42—68no—
455C42—68no—
456C42—128no—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—416no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—816no—
473C22×C3C2×#12414yes—
474C22×C3#52914yes—
475C22×C3#54014yes—
476C22×C3#54614yes—
477C22×C3#55014yes—
478C22×C3#55314yes—
479C22×C3#55314yes—
480C22×C3C23×D411yescharacteristic
481C22×C3C22×#4111yescharacteristic
482C22×C3C22×#4111yescharacteristic
483C22×C3C22×#4111yescharacteristic
484C22×C3C22×#4111yescharacteristic
485C22×C3C2×H(5,𝔽2)11yescharacteristic
486C22×C3C2×H(5,𝔽2)11yescharacteristic
487C3×C4—28no—
488C3×C4—42no—
489C3×C4C2×#12418yes—
490Dic3—24no—
491Dic3—44no—
492Dic3—42no—
493Dic3—42no—
494Dic3—44no—
495Dic3—42no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23C2×#23714yes—
524C23C2×#23712yes—
525C23C2×#23712yes—
526C23C2×#23712yes—
527C23#100412yes—
528C23D3×#4112yes—
529C23#100912yes—
530C23C22×#6712yes—
531C23C24×D311yescharacteristic
532D4—432no—
533D4—416no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—216no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—34no—
541C2×C4—44no—
542C2×C4—416no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—68no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4—128no—
564C2×C4—124no—
565C2×C4—128no—
566C2×C4#936116yes—
567C2×C4C2×#23714yes—
568C2×C4C2×#23714yes—
569C2×C4C22×#6718yes—
570C2×C4C22×#6714yes—
571C2×C3—44no—
572C2×C3—84no—
573C2×C3#249018yescomplemented
574C2×C3C22×#12411yescharacteristic
575C2×C3C2×#52911yescharacteristic
576C2×C3C2×#54011yescharacteristic
577C2×C3C2×#54611yescharacteristic
578C2×C3C2×#55011yescharacteristic
579C2×C3C2×#55311yescharacteristic
580C2×C3C2×#55311yescharacteristic
581C22—24no—
582C22—24no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—416no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—816no—
597C22#560014yes—
598C22#561414yes—
599C22#561814yes—
600C22#561914yes—
601C22C2×#93614yes—
602C22#580614yes—
603C22#583414yes—
604C22C22×#23711yescharacteristic
605C22C22×#23711yescharacteristic
606C22C22×#23711yescharacteristic
607C22C2×#100411yescharacteristic
608C22C2×D3×#4111yescharacteristic
609C22C2×#100911yescharacteristic
610C22C23×#6711yescharacteristic
611C4—28no—
612C4—42no—
613C4—64no—
614C4—124no—
615C4—122no—
616C4—122no—
617C4—122no—
618C4—124no—
619C4C2×#93618yes—
620C3C2×#249011yescharacteristic, complemented, Sylow
621C2—44no—
622C2—84no—
623C2#8266118yesdirect product factor
624C2C2×#560011yescharacteristic
625C2C2×#561911yescharacteristic
626C2C2×#561811yescharacteristic
627C2C2×#561411yescharacteristic
628C2C22×#93611yescharacteristic
629C2C2×#580611yescharacteristic
630C2C2×#583411yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
Fitting, prime radicalC22×C3×#121C2
socleC24×C3C24
hypercenterC22×#121D3
hypocenterC3C2×#2490
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
2. C22×#890C2semidirect
3. C2×#5222C2semidirect
4. C2×#5221C2semidirect
5. C2×#5221C2semidirect
6. C2×#5221C2semidirect
7. C2×#5212C2semidirect
8. C2×C4×#217C2semidirect
9. #82661C2direct (N × H)
10. C2×Dic3×C4⋊C4C2semidirect
11. C2×#4953C2semidirect
12. C2×#4945C2semidirect
36. C2×#890C22semidirect
37. #5222C22semidirect
38. #5221C22semidirect
39. #5221C22semidirect
40. #5221C22semidirect
41. #5212C22semidirect
42. C4×#217C22semidirect
47. Dic3×C4⋊C4C22semidirect
48. #4953C22semidirect
49. #4945C22semidirect
573. C2×C3#2490semidirect
620. C3C2×#2490semidirect
623. C2#82661direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×#8902
C2×#52222
C2×#52212
C2×#52122
C2×C4×#2172
#826612
C2×Dic3×C4⋊C42
C2×#49532
C2×#49452
C2×#24903