Finite Group Databasebeta

← #10702106

Subgroups — #10702106

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2C22×#830C212yesmaximal
3C22×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#4524C211yesmaximal, characteristic, complemented
5C2×#4524C212yesmaximal, complemented
6#11042—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C22⋊C4C411yescharacteristic
9C22×C3×C22⋊C4C411yescharacteristic
10C22×C3×C42C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C23×C3⋊C8C2212yes—
13C23×C3⋊C8C2212yes—
14C3×C4×C22⋊C4—22no—
15C3×C4×C22⋊C4—22no—
16C2×C3×#95C411yescharacteristic
17C2×C3×#95C411yescharacteristic
18C2×#830—24no—
19C2×#830—24no—
20C2×#830—24no—
21C2×C4⋊(C3⋊C8)—22no—
22C2×C4⋊(C3⋊C8)—24no—
23#4524—22no—
24#4524—24no—
25C22×C4⋊C8—31no—
26C22×C22⋊C8—32no—
27C2×C4×C22⋊C4D311yescharacteristic
28C2×#336—32no—
29C2×#336—31no—
30C25×C3C2×C411yescharacteristic
31C23×C3×C4—22no—
32C23×C3×C4—22no—
33C23×C3×C4—21no—
34C23×C3×C4—21no—
35C23×C3×C4—21no—
36C23×C3×C4—21no—
37C23×C3×C4—22no—
38C23×C3×C4—22no—
39C23×C3×C4C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2×C411yescharacteristic
43C23×C3×C4C2×C411yescharacteristic
44C23×C3×C4D412yes—
45C23×C3×C4Q812yes—
46C23×C3×C4C2311yescharacteristic
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4C814yescomplemented
53C2×C3×C42—22no—
54C2×C3×C42—22no—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×C3×C42—22no—
58C2×C3×C42—22no—
59C2×C3×C42D412yes—
60C2×C3×C42D412yes—
61C22×C3⋊C8—22no—
62C22×C3⋊C8—22no—
63C22×C3⋊C8—22no—
64C22×C3⋊C8—22no—
65C22×C3⋊C8—22no—
66C22×C3⋊C8—22no—
67C22×C3⋊C8—22no—
68C22×C3⋊C8—22no—
69C22×C3⋊C8—22no—
70C22×C3⋊C8—22no—
71C22×C3⋊C8—22no—
72C22×C3⋊C8—22no—
73C22×C3⋊C8D414yes—
74C22×C3⋊C8D414yes—
75C3×#95—22no—
76C3×#95—22no—
77#830—216no—
78C4⋊(C3⋊C8)—28no—
79C24×C4C2×D311yescharacteristic
80C23×C8—32no—
81C23×C8—32no—
82C22×C22⋊C4Dic311yescharacteristic
83C22×C22⋊C4Dic311yescharacteristic
84C22×C42C2×D311yescharacteristic
85C22×C42C2×D311yescharacteristic
86C2×C4⋊C8—64no—
87C2×C4⋊C8—62no—
88C2×C22⋊C8—64no—
89C2×C22⋊C8—64no—
90C2×C22⋊C8—64no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95Dic311yescharacteristic
94C2×#95Dic311yescharacteristic
95#336—64no—
96#336—62no—
97C24×C3—21no—
98C24×C3—21no—
99C24×C3—22no—
100C24×C3—22no—
101C24×C3—21no—
102C24×C3—21no—
103C24×C3—22no—
104C24×C3—22no—
105C24×C3C4⋊C412yes—
106C24×C3C2×C812yes—
107C24×C3M1612yes—
108C24×C3C22×C411yescharacteristic
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—41no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—41no—
141C22×C3×C4—42no—
142C22×C3×C4—41no—
143C22×C3×C4—42no—
144C22×C3×C4—41no—
145C22×C3×C4C22⋊C412yes—
146C22×C3×C4C22⋊C412yes—
147C22×C3×C4C4⋊C412yes—
148C22×C3×C4C2×C812yes—
149C22×C3×C4C2×C812yes—
150C22×C3×C4M1612yes—
151C22×C3×C4M1612yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×Q811yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C22×C3×C4#4111yescharacteristic
160C22×C3×C4#4112yes—
161C22×C3×C4#4112yes—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—28no—
164C3×C42—22no—
165C3×C42—24no—
166C3×C42—22no—
167C3×C42—42no—
168C3×C42—41no—
169C3×C42—41no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—24no—
172C2×C3⋊C8—24no—
173C2×C3⋊C8—24no—
174C2×C3⋊C8—24no—
175C2×C3⋊C8—24no—
176C2×C3⋊C8—42no—
177C2×C3⋊C8—42no—
178C2×C3⋊C8—42no—
179C2×C3⋊C8—42no—
180C2×C3⋊C8—42no—
181C2×C3⋊C8—42no—
182C2×C3⋊C8—42no—
183C2×C3⋊C8—42no—
184C25C2×Dic311yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4Dic612yes—
194C23×C4D1212yes—
195C23×C4C2×Dic311yescharacteristic
196C23×C4C2×Dic311yescharacteristic
197C23×C4C2×Dic311yescharacteristic
198C23×C4C2×Dic311yescharacteristic
199C23×C4C2×Dic311yescharacteristic
200C23×C4C22×D311yescharacteristic
201C22×C8—34no—
202C22×C8—34no—
203C22×C8—62no—
204C22×C8—62no—
205C22×C8—62no—
206C22×C8—62no—
207C22×C8—62no—
208C22×C8—62no—
209C22×C8—62no—
210C22×C8—62no—
211C22×C8—62no—
212C22×C8—62no—
213C22×C8—62no—
214C22×C8—62no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4C3⋊C814yescomplemented
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42#6712yes—
228C2×C42#6712yes—
229C4⋊C8—68no—
230C22⋊C8—616no—
231#95—22no—
232#95—22no—
233C23×C3—22no—
234C23×C3—24no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—24no—
243C23×C3—24no—
244C23×C3—24no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3C4⋊C814yescomplemented
256C23×C3C2×C22⋊C411yescharacteristic
257C23×C3C2×C4⋊C411yescharacteristic
258C23×C3#11711yescharacteristic
259C23×C3C4×D412yes—
260C23×C3C4×D412yes—
261C23×C3C22×C811yescharacteristic
262C23×C3C2×M1611yescharacteristic
263C23×C3C2×M1611yescharacteristic
264C23×C3C2×M1611yescharacteristic
265C23×C3#13112yes—
266C23×C3#13112yes—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4C22⋊C814yescomplemented
298C2×C3×C4#12012yes—
299C2×C3×C4#12011yescharacteristic
300C2×C3×C4#12112yes—
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12112yes—
304C2×C3×C4#12112yes—
305C2×C3×C4C22⋊Q812yes—
306C2×C3×C4C22⋊Q812yes—
307C2×C3×C4C22⋊Q811yescharacteristic
308C2×C3×C4C22⋊Q811yescharacteristic
309C2×C3×C4#12312yes—
310C2×C3×C4#12312yes—
311C2×C3×C4#12311yescharacteristic
312C2×C3×C4#12412yes—
313C2×C3×C4#12411yescharacteristic
314C2×C3×C4#12612yes—
315C2×C3×C4#12711yescharacteristic
316C3⋊C8—44no—
317C3⋊C8—44no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—21no—
326C24C2×C3⋊C812yes—
327C24#20812yes—
328C24C4⋊Dic312yes—
329C24C22×Dic311yescharacteristic
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4C2×C3⋊C812yes—
367C22×C4C2×C3⋊C812yes—
368C22×C4#20812yes—
369C22×C4#20812yes—
370C22×C4C4⋊Dic312yes—
371C22×C4#21712yes—
372C22×C4#21712yes—
373C22×C4C2×Dic611yescharacteristic
374C22×C4C2×D1211yescharacteristic
375C22×C4#23511yescharacteristic
376C22×C4#23511yescharacteristic
377C22×C4D3×D412yes—
378C22×C4D3×D412yes—
379C22×C4#23712yes—
380C22×C4#23712yes—
381C22×C4C2×#6711yescharacteristic
382C22×C4C2×#6711yescharacteristic
383C2×C8—64no—
384C2×C8—64no—
385C2×C8—64no—
386C2×C8—64no—
387C2×C8—64no—
388C2×C8—64no—
389C2×C8—122no—
390C2×C8—122no—
391C2×C8—122no—
392C2×C8—122no—
393C2×C8—122no—
394C2×C8—122no—
395C2×C8—122no—
396C2×C8—122no—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—41no—
403C42—42no—
404C42—41no—
405C22×C3—24no—
406C22×C3—28no—
407C22×C3—42no—
408C22×C3—44no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3#38011yescharacteristic
417C22×C3#38512yes—
418C22×C3#38611yescharacteristic
419C22×C3#38812yes—
420C22×C3#39011yescharacteristic
421C22×C3C2×C22⋊C811yescharacteristic
422C22×C3#40711yescharacteristic
423C22×C3#40811yescharacteristic
424C22×C3#40811yescharacteristic
425C22×C3C2×C4⋊C811yescharacteristic
426C22×C3#42311yescharacteristic
427C22×C3#42412yes—
428C22×C3#43111yescharacteristic
429C22×C3#43211yescharacteristic
430C22×C3#43312yes—
431C22×C3C8×D412yes—
432C22×C3C8×D412yes—
433C22×C3#43512yes—
434C22×C3#43512yes—
435C22×C3#43512yes—
436C22×C3#43512yes—
437C22×C3#43612yes—
438C22×C3#43612yes—
439C3×C4—24no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4#39211yescharacteristic
445C3×C4#39311yescharacteristic
446C3×C4#39412yes—
447C3×C4#39612yes—
448C3×C4#39711yescharacteristic
449C3×C4#39911yescharacteristic
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C4⋊(C3⋊C8)14yescomplemented
473C23C22×C3⋊C811yescharacteristic
474C23C2×#20811yescharacteristic
475C23C2×#20811yescharacteristic
476C23C2×#20811yescharacteristic
477C23C2×C4⋊Dic311yescharacteristic
478C23#92611yescharacteristic
479C23D4×Dic312yes—
480C23D4×Dic312yes—
481C23#94812yes—
482C23#94812yes—
483C23C2×#21711yescharacteristic
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4#83014yescomplemented
515C2×C4#87812yes—
516C2×C4#87812yes—
517C2×C4#87912yes—
518C2×C4#88212yes—
519C2×C4#88312yes—
520C2×C4#88412yes—
521C2×C4#88512yes—
522C2×C4#88612yes—
523C2×C4#92411yescharacteristic
524C2×C4#92411yescharacteristic
525C2×C4#93011yescharacteristic
526C2×C4#93011yescharacteristic
527C2×C4#93511yescharacteristic
528C2×C4#93611yescharacteristic
529C2×C4#93711yescharacteristic
530C2×C4#93812yes—
531C2×C4#93912yes—
532C2×C4#94011yescharacteristic
533C8—124no—
534C8—124no—
535C2×C3—44no—
536C2×C3#143911yescharacteristic
537C2×C3#179711yescharacteristic
538C2×C3#179811yescharacteristic
539C2×C3#185012yes—
540C2×C3#185112yes—
541C2×C3#186911yescharacteristic
542C2×C3#187011yescharacteristic
543C2×C3#192312yes—
544C2×C3#192412yes—
545C2×C3#195411yescharacteristic
546C2×C3#195511yescharacteristic
547C22—28no—
548C22—24no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22C2×C4⋊(C3⋊C8)11yescharacteristic
559C22#492411yescharacteristic
560C22#492511yescharacteristic
561C22#492611yescharacteristic
562C22#495012yes—
563C22#495112yes—
564C22#499912yes—
565C22#500012yes—
566C22D4×C3⋊C812yes—
567C22D4×C3⋊C812yes—
568C22#501112yes—
569C22#501112yes—
570C22#501112yes—
571C22#501112yes—
572C22#501212yes—
573C22#501212yes—
574C22C2×#83011yescharacteristic
575C22#520711yescharacteristic
576C22#521211yescharacteristic
577C22#522011yescharacteristic
578C22#522111yescharacteristic
579C22#523411yescharacteristic
580C22#523411yescharacteristic
581C4—24no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4#494711yescharacteristic
587C4#494812yes—
588C4#494911yescharacteristic
589C4#495311yescharacteristic
590C4#496011yescharacteristic
591C4#496112yes—
592C3#1104211yescharacteristic, complemented, Sylow
593C2—44no—
594C2#6944912yes—
595C2#6945012yes—
596C2#6945112yes—
597C2#6945212yes—
598C2#7153011yescharacteristic
599C2#7154611yescharacteristic
600C2#7158311yescharacteristic
601C2#7161011yescharacteristic
602C2#7161111yescharacteristic
603C2#7161411yescharacteristic
604C2#7161511yescharacteristic
subgroupG/H
center, FrattiniC23×C4C22×D3
commutatorC23×C3C22×C8
FittingC2×C3×C4×C22⋊C4C2
socleC24×C3C22×C4
prime radicalC25×C3C2×C4
cosocleC23×C3×C4C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#11042
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊(C3⋊C8)C2semidirect
4. C2×#4524C2semidirect
5. C2×#4524C2semidirect
52. C2×C3×C22⋊C4C8semidirect
220. C2×C22⋊C4C3⋊C8semidirect
255. C23×C3C4⋊C8semidirect
297. C2×C3×C4C22⋊C8semidirect
472. C23C4⋊(C3⋊C8)semidirect
514. C2×C4#830semidirect
592. C3#11042semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C42
C22×#8302
C22×C4⋊(C3⋊C8)2
C2×#45242
#110423