Finite Group Databasebeta

← C2×#74405 #10963131

Subgroups — C2×#74405 #10963131

Order 768.

5566 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×Dic6C211yesmaximal, characteristic, complemented
2C22×#896C212yesmaximal, complemented
3C22×#67⋊C4C211yesmaximal, characteristic
4C2×C3×#385C211yesmaximal, characteristic, complemented
5C2×#5213C211yesmaximal, characteristic
6C2×#4979C212yesmaximal, complemented
7C2×#4958C211yesmaximal, characteristic
8C2×#4946C211yesmaximal, characteristic, complemented
9C2×#4942C211yesmaximal, characteristic, complemented
10#74405C2116yesmaximal, direct product factor
11C2×#4673C212yesmaximal, complemented
12C2×#4646C212yesmaximal, complemented
13C2×#2409—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C23×C4×D3C2211yescharacteristic
19C23×Dic6—28no—
20C23×Dic6—216no—
21C23×Dic6C2212yes—
22C23×Dic6C2212yes—
23C23×Dic6C2218yescomplemented
24C22×C3×C4⋊C4C2212yes—
25C22×C3×C22⋊C4C2211yescharacteristic
26C22×C3×C22⋊C4C2211yescharacteristic
27C22×#217C2211yescharacteristic
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2211yescharacteristic
30C22×(C2×D3)⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×Dic3⋊C4C2211yescharacteristic
34C22×C4×Dic3C2211yescharacteristic
35C2×#896—216no—
36C2×#896C22116yescomplemented
37C2×#67⋊C4—28no—
38C2×#67⋊C4C2218yes—
39C3×#385C2218yescomplemented
40C2×C3×#95C2212yes—
41#5213C2218yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45C2×C3⋊#95C2212yes—
46#4979C22116yescomplemented
47#4958C2218yes—
48#4946C2218yescomplemented
49#4942C2218yescomplemented
50#4673C22116yescomplemented
51#4646C22116yescomplemented
52C24×Q8—31no—
53C22×C22⋊Q8—32no—
54C22×C4×D4—31no—
55#2409—316no—
56C2×#395—32no—
57C2×#393—32no—
58C2×#391—32no—
59C2×#386—31no—
60C2×#386—31no—
61C2×#385—31no—
62C2×#385D311yescharacteristic, complemented
63C2×#380—31no—
64C25×C3C2311yescharacteristic
65C24×D3C2311yescharacteristic
66C23×C3×C4—24no—
67C23×C3×C4—28no—
68C23×C3×C4C2314yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2312yes—
72C23×C3×C4C2312yes—
73C22×#67—24no—
74C22×#67—24no—
75C22×#67Q818yescomplemented
76C22×#67C2314yes—
77C23×Dic3—24no—
78C23×Dic3—28no—
79C23×Dic3—28no—
80C23×Dic3D418yes—
81C23×Dic3C2314yes—
82C23×Dic3C2312yes—
83C23×Dic3C2311yescharacteristic
84C23×Dic3C2312yes—
85C23×Dic3C2314yes—
86C23×Dic3C2311yescharacteristic
87C23×Dic3C2311yescharacteristic
88C22×C4×D3—24no—
89C22×C4×D3—28no—
90C22×C4×D3C2314yes—
91C22×Dic6—2128no—
92C22×Dic6—28no—
93C22×Dic6—28no—
94C22×Dic6—28no—
95C22×Dic6—28no—
96C22×Dic6—28no—
97C22×Dic6—232no—
98C22×Dic6—264no—
99C22×Dic6D4116yescomplemented
100C22×Dic6C2318yes—
101C22×Dic6C2318yes—
102C2×C3×C4⋊C4—28no—
103C2×C3×C4⋊C4C2318yes—
104C2×C3×C22⋊C4—24no—
105C2×C3×C22⋊C4—24no—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C22⋊C4C2314yes—
108C2×#217—24no—
109C2×#217C2314yes—
110C2×(C2×D3)⋊C4—28no—
111C2×(C2×D3)⋊C4—24no—
112C2×(C2×D3)⋊C4—28no—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2318yes—
115C2×(C2×D3)⋊C4C2318yes—
116C2×Dic3⋊C4—24no—
117C2×Dic3⋊C4—28no—
118C2×Dic3⋊C4—28no—
119C2×Dic3⋊C4C2318yes—
120C2×Dic3⋊C4C2318yes—
121C2×Dic3⋊C4C2314yes—
122C2×C4×Dic3—24no—
123C2×C4×Dic3C2314yes—
124#896—264no—
125#67⋊C4—232no—
126C3×#95C2318yes—
127C3⋊#95C2318yes—
128C3⋊#95C2318yes—
129C3⋊#95C2318yes—
130C3⋊#95C2318yes—
131C23×Q8—32no—
132C23×Q8—32no—
133C23×Q8—38no—
134C23×Q8—68no—
135C23×Q8—616no—
136C23×D4—31no—
137C24×C4—31no—
138C24×C4—31no—
139C24×C4—31no—
140C24×C4C2×D311yescharacteristic
141C2×C22⋊Q8—316no—
142C2×C22⋊Q8—616no—
143C2×C4×D4—38no—
144C2×C4×D4—68no—
145C22×C4⋊C4—31no—
146C22×C4⋊C4—32no—
147C22×C4⋊C4—32no—
148C22×C4⋊C4C2×D312yes—
149C22×C22⋊C4—31no—
150C22×C22⋊C4—31no—
151C22×C22⋊C4—32no—
152C22×C22⋊C4—32no—
153C22×C22⋊C4C2×D311yescharacteristic
154C22×C22⋊C4C2×D311yescharacteristic
155C22×C42—31no—
156#395—316no—
157#393—316no—
158#391—316no—
159#386—38no—
160#386—38no—
161#385—38no—
162#385C2×D318yescomplemented
163#380—38no—
164C2×#95—32no—
165C2×#95—32no—
166C2×#95—32no—
167C2×#95—32no—
168C2×#95C2×D312yes—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3C2×D414yes—
172C24×C3C2×Q814yes—
173C24×C3#4114yes—
174C24×C3C2412yes—
175C24×C3C2411yescharacteristic
176C23×D3—24no—
177C23×D3—22no—
178C23×D3—22no—
179C23×D3—24no—
180C23×D3C2×Q814yescomplemented
181C23×D3C2412yes—
182C22×C3×C4—232no—
183C22×C3×C4—28no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—216no—
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×D418yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2414yes—
199C2×#67—232no—
200C2×#67—28no—
201C2×#67—28no—
202C2×#67C2×Q8116yescomplemented
203C22×Dic3—232no—
204C22×Dic3—22no—
205C22×Dic3—28no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—22no—
210C22×Dic3—232no—
211C22×Dic3—216no—
212C22×Dic3—216no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3C2×D4116yes—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×D418yes—
219C22×Dic3C2×D414yes—
220C22×Dic3C2×D418yes—
221C22×Dic3C2×Q814yescomplemented
222C22×Dic3C2414yes—
223C22×Dic3C2412yes—
224C22×Dic3C2414yes—
225C22×Dic3C2412yes—
226C22×Dic3C2412yes—
227C2×C4×D3—28no—
228C2×C4×D3—216no—
229C2×C4×D3—416no—
230C2×Dic6—264no—
231C2×Dic6—2256no—
232C2×Dic6—216no—
233C2×Dic6—264no—
234C2×Dic6—216no—
235C2×Dic6—216no—
236C2×Dic6—216no—
237C2×Dic6—216no—
238C2×Dic6C2×D4132yescomplemented
239C3×C4⋊C4—216no—
240C3×C22⋊C4—28no—
241C3×C22⋊C4—28no—
242#217—28no—
243(C2×D3)⋊C4—216no—
244(C2×D3)⋊C4—28no—
245(C2×D3)⋊C4—216no—
246Dic3⋊C4—28no—
247Dic3⋊C4—216no—
248Dic3⋊C4—216no—
249C4×Dic3—28no—
250C25—31no—
251C25C22×D311yescharacteristic
252C22×Q8—316no—
253C22×Q8—38no—
254C22×Q8—38no—
255C22×Q8—68no—
256C22×Q8—68no—
257C22×Q8—68no—
258C22×Q8—68no—
259C22×Q8—632no—
260C22×Q8—664no—
261C22×Q8—6128no—
262C22×Q8—68no—
263C22×D4—38no—
264C22×D4—34no—
265C22×D4—64no—
266C22×D4—64no—
267C23×C4—24no—
268C23×C4—28no—
269C23×C4—31no—
270C23×C4—38no—
271C23×C4—34no—
272C23×C4—31no—
273C23×C4—32no—
274C23×C4—31no—
275C23×C4—34no—
276C23×C4—32no—
277C23×C4—34no—
278C23×C4—64no—
279C23×C4—68no—
280C23×C4—68no—
281C23×C4—68no—
282C23×C4—64no—
283C23×C4C22×D314yes—
284C23×C4C22×D312yes—
285C23×C4C22×D311yescharacteristic
286C23×C4C22×D311yescharacteristic
287C23×C4C22×D312yes—
288C22⋊Q8—664no—
289C4×D4—632no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—34no—
292C2×C4⋊C4—38no—
293C2×C4⋊C4—38no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—68no—
296C2×C4⋊C4—68no—
297C2×C4⋊C4C22×D318yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—34no—
301C2×C22⋊C4—34no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—38no—
304C2×C22⋊C4—64no—
305C2×C22⋊C4—68no—
306C2×C22⋊C4—64no—
307C2×C22⋊C4—68no—
308C2×C22⋊C4C22×D314yes—
309C2×C22⋊C4C22×D314yes—
310C2×C42—34no—
311C2×C42—64no—
312#95—38no—
313#95—38no—
314#95—38no—
315#95—38no—
316#95C22×D318yes—
317C23×C3—28no—
318C23×C3—28no—
319C23×C3—216no—
320C23×C3—216no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3C22⋊Q8116yescomplemented
324C23×C3C22×D414yes—
325C23×C3C22×D412yes—
326C23×C3C22×D414yes—
327C23×C3C22×Q814yes—
328C23×C3C22×Q812yes—
329C23×C3C2×#4114yes—
330C23×C3C2×#4112yes—
331C23×C32−1+414yes—
332C23×C3C2511yescharacteristic
333C22×D3—28no—
334C22×D3—216no—
335C22×D3—24no—
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—24no—
339C22×D3—44no—
340C22×D3—44no—
341C22×D3—48no—
342C22×D3C22×Q814yescomplemented
343C2×C3×C4—232no—
344C2×C3×C4—216no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4—48no—
357C2×C3×C4C22×D418yes—
358#67—232no—
359C2×Dic3—232no—
360C2×Dic3—28no—
361C2×Dic3—216no—
362C2×Dic3—28no—
363C2×Dic3—22no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—232no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—48no—
373C2×Dic3#120116yes—
374C2×Dic3C22×D414yes—
375C2×Dic3C22×D418yes—
376C2×Dic3C22×D414yes—
377C2×Dic3C22×D418yes—
378C2×Dic3C22×Q814yescomplemented
379C4×D3—416no—
380Dic6—264no—
381Dic6—264no—
382C24—24no—
383C24—24no—
384C24—34no—
385C24—32no—
386C24—62no—
387C24—64no—
388C24—62no—
389C24—64no—
390C24#23514yes—
391C24D3×D414yes—
392C24D3×Q814yes—
393C24C23×D312yes—
394C24C23×D311yescharacteristic
395C2×Q8—332no—
396C2×Q8—616no—
397C2×Q8—616no—
398C2×Q8—616no—
399C2×Q8—616no—
400C2×Q8—664no—
401C2×Q8—6256no—
402C2×Q8—664no—
403C2×Q8—616no—
404C2×D4—316no—
405C2×D4—632no—
406C2×D4—68no—
407C2×D4—68no—
408C22×C4—232no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—34no—
421C22×C4—38no—
422C22×C4—32no—
423C22×C4—34no—
424C22×C4—38no—
425C22×C4—316no—
426C22×C4—34no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—34no—
431C22×C4—68no—
432C22×C4—616no—
433C22×C4—616no—
434C22×C4—632no—
435C22×C4—616no—
436C22×C4—632no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—68no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—1216no—
447C22×C4D3×D418yes—
448C22×C4C23×D312yes—
449C22×C4C23×D314yes—
450C22×C4C23×D312yes—
451C22×C4C23×D314yes—
452C4⋊C4—216no—
453C4⋊C4—68no—
454C4⋊C4—616no—
455C4⋊C4—616no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—68no—
459C22⋊C4—616no—
460C22⋊C4—68no—
461C22⋊C4—616no—
462C42—68no—
463C22×C3—24no—
464C22×C3—232no—
465C22×C3—24no—
466C22×C3—28no—
467C22×C3—28no—
468C22×C3C2×#12014yes—
469C22×C3C2×C22⋊Q814yes—
470C22×C3C2×C22⋊Q818yescomplemented
471C22×C3#53614yes—
472C22×C3#54818yes—
473C22×C3D4×Q818yes—
474C22×C3C23×D411yescharacteristic
475C22×C3C23×D412yes—
476C22×C3C23×Q811yescharacteristic
477C22×C3C22×#4111yescharacteristic
478C22×C3C2×2−1+412yes—
479C2×D3—24no—
480C2×D3—28no—
481C2×D3—42no—
482C2×D3—416no—
483C2×D3—42no—
484C2×D3—44no—
485C3×C4—28no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—44no—
489Dic3—24no—
490Dic3—28no—
491Dic3—24no—
492Dic3—44no—
493Dic3C2×#12018yes—
494C23—216no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—216no—
499C23—28no—
500C23—34no—
501C23—616no—
502C23—62no—
503C23—64no—
504C23—62no—
505C23—64no—
506C23—68no—
507C23—124no—
508C23—128no—
509C23—124no—
510C23#896116yescomplemented
511C23C2×#23514yes—
512C23C2×#23512yes—
513C23C2×D3×D414yes—
514C23C2×D3×D414yes—
515C23C2×D3×D412yes—
516C23C2×D3×Q814yes—
517C23C2×D3×Q812yes—
518C23#101014yes—
519C23C24×D311yescharacteristic
520Q8—664no—
521Q8—664no—
522D4—632no—
523C2×C4—232no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—216no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—34no—
535C2×C4—38no—
536C2×C4—316no—
537C2×C4—34no—
538C2×C4—38no—
539C2×C4—34no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—632no—
544C2×C4—632no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—68no—
551C2×C4—68no—
552C2×C4—616no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—1216no—
557C2×C4—128no—
558C2×C4C2×D3×D418yes—
559C2×C3—28no—
560C2×C3#240918yescomplemented
561C2×C3C22×#12011yescharacteristic
562C2×C3C22×C22⋊Q811yescharacteristic
563C2×C3C2×#53611yescharacteristic
564C2×C3C2×#54812yes—
565C2×C3C2×D4×Q812yes—
566D3—44no—
567C22—24no—
568C22—232no—
569C22—28no—
570C22—24no—
571C22—28no—
572C22—68no—
573C22—64no—
574C22—122no—
575C22—1216no—
576C22—122no—
577C22—124no—
578C22C2×#89614yes—
579C22C2×#89618yescomplemented
580C22#555218yes—
581C22D3×#12014yes—
582C22#563218yes—
583C22#564814yes—
584C22C22×#23511yescharacteristic
585C22C22×D3×D412yes—
586C22C22×D3×D411yescharacteristic
587C22C22×D3×Q811yescharacteristic
588C22C2×#101012yes—
589C4—28no—
590C4—38no—
591C4—42no—
592C4—42no—
593C4—44no—
594C4—64no—
595C4—68no—
596C4—64no—
597C4—124no—
598C3C2×#240911yescharacteristic, complemented, Sylow
599C2—28no—
600C2—124no—
601C2#7440518yesdirect product factor
602C2C22×#89611yescharacteristic
603C2C2×#555212yes—
604C2C2×D3×#12011yescharacteristic
605C2C2×#563212yes—
606C2C2×#564811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#385C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#385D3
hypocenterC3C2×#2409
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C24×Dic6C2semidirect
2. C22×#896C2semidirect
4. C2×C3×#385C2semidirect
6. C2×#4979C2semidirect
8. C2×#4946C2semidirect
9. C2×#4942C2semidirect
10. #74405C2direct (N × H)
11. C2×#4673C2semidirect
12. C2×#4646C2semidirect
23. C23×Dic6C22semidirect
36. C2×#896C22semidirect
39. C3×#385C22semidirect
46. #4979C22semidirect
48. #4946C22semidirect
49. #4942C22semidirect
50. #4673C22semidirect
51. #4646C22semidirect
62. C2×#385D3semidirect
75. C22×#67Q8semidirect
99. C22×Dic6D4semidirect
162. #385C2×D3semidirect
180. C23×D3C2×Q8semidirect
202. C2×#67C2×Q8semidirect
221. C22×Dic3C2×Q8semidirect
238. C2×Dic6C2×D4semidirect
323. C23×C3C22⋊Q8semidirect
342. C22×D3C22×Q8semidirect
378. C2×Dic3C22×Q8semidirect
470. C22×C3C2×C22⋊Q8semidirect
510. C23#896semidirect
560. C2×C3#2409semidirect
579. C22C2×#896semidirect
598. C3C2×#2409semidirect
601. C2#74405direct (N × H)

Maximal subgroups

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

subgroupindex
C24×Dic62
C22×#8962
C22×#67⋊C42
C2×C3×#3852
C2×#52132
C2×#49792
C2×#49582
C2×#49462
C2×#49422
#744052
C2×#46732
C2×#46462
C2×#24093