Finite Group Databasebeta

← C2×#78030 #10964243

Subgroups — C2×#78030 #10964243

Order 768.

4222 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
1C22×#894C212yesmaximal, complemented
2C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C22×#883C212yesmaximal, complemented
4C22×#884C211yesmaximal, characteristic
5C2×C3×#399C211yesmaximal, characteristic, complemented
6C2×#4993C211yesmaximal, characteristic, complemented
7C2×#4975C212yesmaximal, complemented
8C2×#4949C211yesmaximal, characteristic, complemented
9C2×#4957C212yesmaximal, complemented
10#78030C2116yesmaximal, direct product factor
11C2×#4677C211yesmaximal, characteristic, complemented
12C2×#4666C211yesmaximal, characteristic, complemented
13C2×#2501—31nomaximal, Sylow
14C23×#67C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C23×D12C2211yescharacteristic
17C23×C4×D3C2212yes—
18C23×C4×D3C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×#217C2212yes—
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2212yes—
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×Dic3⋊C4C2211yescharacteristic
34C22×C4×Dic3C2212yes—
35C2×#894—216no—
36C2×#894C22116yescomplemented
37C2×D3×C4⋊C4—28no—
38C2×D3×C4⋊C4C2218yescomplemented
39C2×#884—28no—
40C2×#884C2218yes—
41C2×#883—216no—
42C2×#883C22116yescomplemented
43C3×#399C2218yescomplemented
44C2×C3×#95C2211yescharacteristic
45C2×C3⋊#95C2211yescharacteristic
46#4993C2218yescomplemented
47#4975C22116yescomplemented
48#4957C22116yescomplemented
49#4949C2218yescomplemented
50#4677C2218yescomplemented
51#4666C2218yescomplemented
52C22×#123—32no—
53C22×#123—32no—
54C22×#121—31no—
55C23×C4⋊C4—31no—
56#2501—316no—
57C2×#399—31no—
58C2×#399—31no—
59C2×#399—31no—
60C2×#399D311yescharacteristic, complemented
61C2×#390—32no—
62C2×#384—32no—
63C2×C4×C22⋊C4—31no—
64C25×C3C2311yescharacteristic
65C24×D3C2311yescharacteristic
66C24×D3C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2312yes—
71C22×#67—28no—
72C22×#67—24no—
73C22×#67—24no—
74C22×#67—28no—
75C22×#67C2314yes—
76C22×#67C2314yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2312yes—
80C23×Dic3C2311yescharacteristic
81C22×D12—24no—
82C22×D12—28no—
83C22×D12C2314yes—
84C22×C4×D3—28no—
85C22×C4×D3—28no—
86C22×C4×D3—28no—
87C22×C4×D3—28no—
88C22×C4×D3D418yes—
89C22×C4×D3C2318yes—
90C22×C4×D3C2314yes—
91C2×C3×C4⋊C4—24no—
92C2×C3×C4⋊C4—28no—
93C2×C3×C4⋊C4C2318yes—
94C2×C3×C4⋊C4C2314yes—
95C2×C3×C22⋊C4—28no—
96C2×C3×C22⋊C4—24no—
97C2×C3×C22⋊C4C2314yes—
98C2×C3×C22⋊C4C2318yes—
99C2×#217—28no—
100C2×#217C2318yes—
101C2×(C2×D3)⋊C4—24no—
102C2×(C2×D3)⋊C4—28no—
103C2×(C2×D3)⋊C4—28no—
104C2×(C2×D3)⋊C4D4116yescomplemented
105C2×(C2×D3)⋊C4C2318yes—
106C2×(C2×D3)⋊C4C2314yes—
107C2×(C2×D3)⋊C4C2318yes—
108C2×(C2×D3)⋊C4C2318yes—
109C2×C4⋊Dic3—28no—
110C2×C4⋊Dic3—24no—
111C2×C4⋊Dic3C2314yes—
112C2×C4⋊Dic3C2318yes—
113C2×Dic3⋊C4—28no—
114C2×Dic3⋊C4—24no—
115C2×Dic3⋊C4—28no—
116C2×Dic3⋊C4—28no—
117C2×Dic3⋊C4C2318yes—
118C2×Dic3⋊C4C2318yes—
119C2×Dic3⋊C4C2314yes—
120C2×Dic3⋊C4C2318yes—
121C2×C4×Dic3—28no—
122C2×C4×Dic3C2318yes—
123#894—264no—
124D3×C4⋊C4—232no—
125#884—232no—
126#883—264no—
127C3×#95C2314yes—
128C3⋊#95C2314yes—
129C23×D4—31no—
130C23×D4—31no—
131C23×D4—31no—
132C24×C4—31no—
133C24×C4—32no—
134C2×#123—316no—
135C2×#123—316no—
136C2×#123—616no—
137C2×#123—616no—
138C2×#121—38no—
139C2×#121—68no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4—32no—
142C22×C4⋊C4—32no—
143C22×C4⋊C4—32no—
144C22×C4⋊C4—32no—
145C22×C4⋊C4—31no—
146C22×C4⋊C4—38no—
147C22×C4⋊C4—68no—
148C22×C4⋊C4C2×D311yescharacteristic
149C22×C4⋊C4C2×D312yes—
150C22×C22⋊C4—32no—
151C22×C22⋊C4—32no—
152C22×C22⋊C4—31no—
153C22×C22⋊C4—32no—
154C22×C22⋊C4—32no—
155C22×C22⋊C4C2×D311yescharacteristic
156C22×C22⋊C4C2×D312yes—
157C22×C42—32no—
158#399—38no—
159#399—38no—
160#399—38no—
161#399C2×D318yescomplemented
162#390—316no—
163#384—316no—
164C4×C22⋊C4—38no—
165C2×#95—31no—
166C2×#95C2×D311yescharacteristic
167C24×C3—24no—
168C24×C3—22no—
169C24×C3—24no—
170C24×C3—22no—
171C24×C3—22no—
172C24×C3C2412yes—
173C24×C3C2411yescharacteristic
174C23×D3—24no—
175C23×D3—22no—
176C23×D3—22no—
177C23×D3—24no—
178C23×D3—22no—
179C23×D3—24no—
180C23×D3—22no—
181C23×D3—22no—
182C23×D3C2×D418yes—
183C23×D3C2×D414yes—
184C23×D3C2412yes—
185C23×D3C2412yes—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4C2×D418yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2414yes—
200C2×#67—216no—
201C2×#67—28no—
202C2×#67—28no—
203C2×#67—216no—
204C2×#67—416no—
205C2×#67—416no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—22no—
212C22×Dic3—24no—
213C22×Dic3C2×D418yes—
214C22×Dic3C2×D414yes—
215C22×Dic3#4118yes—
216C22×Dic3C2412yes—
217C22×Dic3C2414yes—
218C22×Dic3C2414yes—
219C22×Dic3C2412yes—
220C2×D12—28no—
221C2×D12—216no—
222C2×D12—416no—
223C2×C4×D3—232no—
224C2×C4×D3—216no—
225C2×C4×D3—216no—
226C2×C4×D3—216no—
227C2×C4×D3—216no—
228C2×C4×D3—432no—
229C2×C4×D3C2×D4116yes—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—216no—
232C3×C22⋊C4—216no—
233C3×C22⋊C4—28no—
234#217—216no—
235(C2×D3)⋊C4—28no—
236(C2×D3)⋊C4—216no—
237(C2×D3)⋊C4—216no—
238(C2×D3)⋊C4C2×D4132yescomplemented
239C4⋊Dic3—216no—
240C4⋊Dic3—28no—
241Dic3⋊C4—216no—
242Dic3⋊C4—28no—
243Dic3⋊C4—216no—
244Dic3⋊C4—216no—
245C4×Dic3—216no—
246C25—31no—
247C25—31no—
248C25C22×D311yescharacteristic
249C22×D4—34no—
250C22×D4—34no—
251C22×D4—34no—
252C22×D4—64no—
253C22×D4—68no—
254C22×D4—64no—
255C22×D4—68no—
256C22×D4—64no—
257C22×D4—68no—
258C23×C4—31no—
259C23×C4—32no—
260C23×C4—38no—
261C23×C4—34no—
262C23×C4—32no—
263C23×C4—31no—
264C23×C4—38no—
265C23×C4—68no—
266C23×C4—68no—
267C23×C4—68no—
268C23×C4—68no—
269C23×C4C22×D312yes—
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D312yes—
272C23×C4C22×D311yescharacteristic
273#123—664no—
274#123—664no—
275#121—632no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—38no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—38no—
282C2×C4⋊C4—38no—
283C2×C4⋊C4—38no—
284C2×C4⋊C4—632no—
285C2×C4⋊C4—68no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4—68no—
288C2×C4⋊C4—68no—
289C2×C4⋊C4—64no—
290C2×C4⋊C4—68no—
291C2×C4⋊C4C22×D314yes—
292C2×C4⋊C4C22×D318yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—316no—
296C2×C22⋊C4—38no—
297C2×C22⋊C4—38no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—68no—
302C2×C22⋊C4—68no—
303C2×C22⋊C4—64no—
304C2×C22⋊C4—68no—
305C2×C22⋊C4C22×D314yes—
306C2×C22⋊C4C22×D318yes—
307C2×C42—38no—
308C2×C42—68no—
309#95—34no—
310#95C22×D314yes—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—44no—
319C23×C3—48no—
320C23×C3—48no—
321C23×C3C22×D412yes—
322C23×C3C22×D414yes—
323C23×C3C2×#4114yes—
324C23×C32−1+412yes—
325C23×C32−1+412yes—
326C23×C3C2511yescharacteristic
327C22×D3—28no—
328C22×D3—28no—
329C22×D3—216no—
330C22×D3—216no—
331C22×D3—24no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—24no—
335C22×D3—22no—
336C22×D3—24no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—44no—
340C22×D3—44no—
341C22×D3—44no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3#127116yes—
345C22×D3C22×D418yes—
346C22×D3C22×D414yes—
347C2×C3×C4—28no—
348C2×C3×C4—216no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—48no—
358C2×C3×C4—44no—
359C2×C3×C4C22×D418yes—
360C2×C3×C4C22×D414yes—
361#67—416no—
362#67—416no—
363C2×Dic3—216no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—48no—
372C2×Dic3—44no—
373C2×Dic3#123116yescomplemented
374C2×Dic3C22×D418yes—
375C2×Dic3C22×D414yes—
376C2×Dic3C2×#4118yes—
377D12—416no—
378C4×D3—232no—
379C4×D3—432no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—22no—
385C24—38no—
386C24—34no—
387C24—32no—
388C24—32no—
389C24—62no—
390C24—64no—
391C24—62no—
392C24—64no—
393C24—62no—
394C24—62no—
395C24—64no—
396C24—62no—
397C24C23×D312yes—
398C24C23×D311yescharacteristic
399C2×D4—68no—
400C2×D4—616no—
401C2×D4—68no—
402C2×D4—616no—
403C2×D4—68no—
404C2×D4—616no—
405C2×D4—1216no—
406C2×D4—1216no—
407C2×D4—1216no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—34no—
417C22×C4—38no—
418C22×C4—32no—
419C22×C4—38no—
420C22×C4—34no—
421C22×C4—316no—
422C22×C4—34no—
423C22×C4—32no—
424C22×C4—632no—
425C22×C4—616no—
426C22×C4—616no—
427C22×C4—616no—
428C22×C4—616no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—1232no—
437C22×C4D3×D418yes—
438C22×C4D3×D414yes—
439C22×C4C23×D314yes—
440C22×C4C23×D312yes—
441C22×C4C23×D312yes—
442C22×C4C23×D314yes—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—616no—
446C4⋊C4—68no—
447C4⋊C4—616no—
448C4⋊C4—616no—
449C4⋊C4—68no—
450C4⋊C4—616no—
451C22⋊C4—28no—
452C22⋊C4—216no—
453C22⋊C4—332no—
454C22⋊C4—616no—
455C22⋊C4—616no—
456C22⋊C4—68no—
457C22⋊C4—616no—
458C42—616no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—44no—
463C22×C3—44no—
464C22×C3—88no—
465C22×C3C2×#12314yes—
466C22×C3C2×#12714yes—
467C22×C3#53614yes—
468C22×C3#54718yes—
469C22×C3#54718yes—
470C22×C3C23×D411yescharacteristic
471C22×C3C23×D412yes—
472C22×C3C22×#4112yes—
473C22×C3C2×2−1+411yescharacteristic
474C22×C3C2×2−1+411yescharacteristic
475C2×D3—24no—
476C2×D3—24no—
477C2×D3—28no—
478C2×D3—28no—
479C2×D3—232no—
480C2×D3—42no—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—44no—
484C2×D3—44no—
485C2×D3—88no—
486C2×D3C2×#12718yes—
487C3×C4—24no—
488C3×C4—28no—
489C3×C4—44no—
490C3×C4—42no—
491Dic3—28no—
492Dic3—44no—
493Dic3—42no—
494Dic3C2×#12318yescomplemented
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—38no—
501C23—34no—
502C23—316no—
503C23—44no—
504C23—48no—
505C23—44no—
506C23—44no—
507C23—48no—
508C23—68no—
509C23—616no—
510C23—62no—
511C23—64no—
512C23—62no—
513C23—68no—
514C23—616no—
515C23—64no—
516C23—62no—
517C23—62no—
518C23—64no—
519C23—62no—
520C23—124no—
521C23—128no—
522C23—124no—
523C23—124no—
524C23—128no—
525C23C2×D3×D412yes—
526C23C2×D3×D414yes—
527C23#100712yes—
528C23D3×#4114yes—
529C23#101012yes—
530C23C24×D311yescharacteristic
531D4—1216no—
532D4—1216no—
533D4—1216no—
534C2×C4—28no—
535C2×C4—216no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—316no—
545C2×C4—34no—
546C2×C4—38no—
547C2×C4—38no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—632no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—616no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—1232no—
560C2×C4—128no—
561C2×C4—124no—
562C2×C4C2×D3×D418yes—
563C2×C4C2×D3×D414yes—
564C2×C3—82no—
565C2×C3#250118yescomplemented
566C2×C3C22×#12311yescharacteristic
567C2×C3C22×#12711yescharacteristic
568C2×C3C2×#53611yescharacteristic
569C2×C3C2×#54712yes—
570C2×C3C2×#54712yes—
571D3—28no—
572D3—82no—
573C22—38no—
574C22—42no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—632no—
580C22—64no—
581C22—68no—
582C22—64no—
583C22—68no—
584C22—88no—
585C22—122no—
586C22—122no—
587C22—124no—
588C22—122no—
589C22—124no—
590C22—248no—
591C22#562814yes—
592C22#563118yes—
593C22D3×#12314yes—
594C22#565518yes—
595C22D3×#12714yes—
596C22C22×D3×D411yescharacteristic
597C22C22×D3×D412yes—
598C22C2×#100711yescharacteristic
599C22C2×D3×#4112yes—
600C22C2×#101011yescharacteristic
601C4—24no—
602C4—28no—
603C4—38no—
604C4—44no—
605C4—42no—
606C4—68no—
607C4—124no—
608C4—122no—
609C3C2×#250111yescharacteristic, complemented, Sylow
610C2—68no—
611C2—82no—
612C2—242no—
613C2#7803018yesdirect product factor
614C2C2×#562811yescharacteristic
615C2C2×#563112yes—
616C2C2×D3×#12311yescharacteristic
617C2C2×#565512yes—
618C2C2×D3×#12711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#399C2
socleC24×C3C24
prime radicalC22×#884C2
hypercenterC2×#399D3
hypocenterC3C2×#2501
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#894C2semidirect
2. C22×D3×C4⋊C4C2semidirect
3. C22×#883C2semidirect
5. C2×C3×#399C2semidirect
6. C2×#4993C2semidirect
7. C2×#4975C2semidirect
8. C2×#4949C2semidirect
9. C2×#4957C2semidirect
10. #78030C2direct (N × H)
11. C2×#4677C2semidirect
12. C2×#4666C2semidirect
36. C2×#894C22semidirect
38. C2×D3×C4⋊C4C22semidirect
42. C2×#883C22semidirect
43. C3×#399C22semidirect
46. #4993C22semidirect
47. #4975C22semidirect
48. #4957C22semidirect
49. #4949C22semidirect
50. #4677C22semidirect
51. #4666C22semidirect
60. C2×#399D3semidirect
104. C2×(C2×D3)⋊C4D4semidirect
161. #399C2×D3semidirect
238. (C2×D3)⋊C4C2×D4semidirect
373. C2×Dic3#123semidirect
494. Dic3C2×#123semidirect
565. C2×C3#2501semidirect
609. C3C2×#2501semidirect
613. C2#78030direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8942
C22×D3×C4⋊C42
C22×#8832
C22×#8842
C2×C3×#3992
C2×#49932
C2×#49752
C2×#49492
C2×#49572
#780302
C2×#46772
C2×#46662
C2×#25013