Finite Group Databasebeta

← #10795137

Subgroups — #10795137

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5546C211yesmaximal, characteristic
2#73445C212yesmaximal, complemented
3C3×C4×#99C211yesmaximal, characteristic, complemented
4C4×#834C211yesmaximal, characteristic, complemented
5#70028C212yesmaximal, complemented
6#70027C212yesmaximal
7C4×#806C212yesmaximal, complemented
8#68959C212yesmaximal, complemented
9#68953C212yesmaximal, complemented
10#20725—31nomaximal, Sylow
11C22×#237C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×D4×Dic3C2211yescharacteristic
14C2×C4×#67C411yescharacteristic, complemented
15C2×C4×#67C2211yescharacteristic
16C22×Dic3⋊C4C411yescharacteristic, complemented
17C22×Dic3⋊C4C2211yescharacteristic
18#5546—24no—
19#5546—24no—
20C2×D3×C4⋊C4C411yescharacteristic, complemented
21C2×#887C411yescharacteristic, complemented
22C2×#67⋊C4C411yescharacteristic, complemented
23C2×#67⋊C4C2211yescharacteristic
24C2×#877C411yescharacteristic, complemented
25C2×#877C2211yescharacteristic
26C2×#872C411yescharacteristic, complemented
27C2×C4×Dic6C411yescharacteristic, complemented
28C2×C3×#99C2212yes—
29C3×C4×C22⋊C4C2212yes—
30C2×#834C2212yes—
31C4×#217C2212yes—
32C2×#806C2214yes—
33#4943C2214yes—
34#4942C2214yes—
35#4933—24no—
36#4652—24no—
37C3×#342C2212yes—
38#4527C2214yes—
39#4526C2212yes—
40C2×#520—31no—
41#2313—32no—
42#1820—32no—
43#1799—32no—
44#1799—32no—
45#1782—32no—
46C4×#99—32no—
47C4×#99—31no—
48C4×#99D311yescharacteristic, complemented
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4C2×C411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×#67C2×C411yescharacteristic
53C22×#67C2311yescharacteristic
54C23×Dic3C2×C411yescharacteristic
55C23×Dic3C2×C411yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C2×#237—24no—
59C2×#237—24no—
60C2×#237—24no—
61C22×C4×D3C2×C411yescharacteristic
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3C2×C411yescharacteristic
64C22×Dic6C2×C411yescharacteristic
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C2×C3×C4⋊C4C2×C411yescharacteristic
68C2×C3×C22⋊C4C2×C411yescharacteristic
69C2×C3×C22⋊C4C2314yes—
70C2×C3×C22⋊C4C2311yescharacteristic
71C2×C3×C42—22no—
72C2×C3×C42C2×C411yescharacteristic
73C2×#217C2×C411yescharacteristic
74C2×#217C2314yes—
75C2×#217C2311yescharacteristic
76D4×Dic3—24no—
77C4×#67—22no—
78C4×#67—22no—
79C4×#67—22no—
80C4×#67—22no—
81C2×(C2×D3)⋊C4C2×C411yescharacteristic
82C2×(C2×D3)⋊C4C2×C411yescharacteristic
83C2×C4⋊Dic3C2×C411yescharacteristic
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4C2×C411yescharacteristic
88C2×Dic3⋊C4C2×C411yescharacteristic
89C2×Dic3⋊C4C2×C411yescharacteristic
90C2×Dic3⋊C4C2×C411yescharacteristic
91C2×Dic3⋊C4C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C411yescharacteristic
93C2×Dic3⋊C4C2×C411yescharacteristic
94C2×Dic3⋊C4C2×C411yescharacteristic
95C2×Dic3⋊C4D414yes—
96C2×Dic3⋊C4D414yes—
97C2×C4×Dic3—22no—
98C2×C4×Dic3C2×C411yescharacteristic
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103D3×C4⋊C4—24no—
104#887—24no—
105#67⋊C4—24no—
106#67⋊C4—24no—
107#877—24no—
108#877—24no—
109#872—22no—
110#872—22no—
111C4×Dic6—22no—
112C4×Dic6—22no—
113C3×#99—24no—
114C3×#95—22no—
115#834—24no—
116C3⋊#95—22no—
117C3⋊#95—24no—
118C3⋊#95—28no—
119#806—28no—
120C22×#41—31no—
121#520—64no—
122#520—64no—
123C2×C4×Q8—31no—
124C2×C4×Q8—31no—
125C2×C4×D4—31no—
126C2×C4×D4—31no—
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×C4×D4—31no—
130C2×C4×D4C2×D311yescharacteristic
131C2×#117—31no—
132C2×#117—31no—
133C2×#117—31no—
134C22×C4⋊C4—31no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—31no—
137C2×#99—32no—
138C2×#99—34no—
139C2×#99C2×D312yes—
140#387—64no—
141#387—64no—
142#381—34no—
143#380—34no—
144C4×C22⋊C4—32no—
145C4×C22⋊C4C2×D312yes—
146#342—32no—
147#342—34no—
148#342C2×D312yes—
149C24×C3C22×C411yescharacteristic
150C24×C3C2411yescharacteristic
151C23×D3C22×C411yescharacteristic
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4#4114yes—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—24no—
158C22×C3×C4—81no—
159C22×C3×C4—81no—
160C22×C3×C4C22×C411yescharacteristic
161C22×C3×C4C22×C411yescharacteristic
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C22×C411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C2×#67—22no—
168C2×#67—22no—
169C2×#67—22no—
170C2×#67—22no—
171C2×#67—22no—
172C2×#67#4114yes—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—24no—
180C22×Dic3—81no—
181C22×Dic3—81no—
182C22×Dic3C22×C411yescharacteristic
183C22×Dic3C22×C411yescharacteristic
184C22×Dic3C22×C411yescharacteristic
185C22×Dic3C22×C411yescharacteristic
186C22×Dic3C22×C411yescharacteristic
187C22×Dic3C22×C411yescharacteristic
188C22×Dic3C22×C411yescharacteristic
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D414yes—
191#237—232no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—81no—
198C2×C4×D3—81no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C3×C4⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—82no—
210#217—22no—
211#217—22no—
212#217—24no—
213#217—24no—
214(C2×D3)⋊C4—21no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—22no—
217C4⋊Dic3—22no—
218Dic3⋊C4—21no—
219Dic3⋊C4—21no—
220Dic3⋊C4—22no—
221Dic3⋊C4—22no—
222Dic3⋊C4—22no—
223Dic3⋊C4—84no—
224Dic3⋊C4C22⋊C412yes—
225Dic3⋊C4C22⋊C412yes—
226Dic3⋊C4C22⋊C412yes—
227Dic3⋊C4C22⋊C412yes—
228Dic3⋊C4C22⋊C414yes—
229Dic3⋊C4C22⋊C414yes—
230C4×Dic3—22no—
231C4×Dic3—22no—
232C4×Dic3—22no—
233C4×Dic3—22no—
234C4×Dic3—22no—
235C4×Dic3—82no—
236C2×#41—64no—
237C2×#41—64no—
238C2×#41—64no—
239C22×Q8—31no—
240C22×D4—31no—
241C22×D4—31no—
242C22×D4C22×D311yescharacteristic
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4C4×D311yescharacteristic
251C23×C4C22×D311yescharacteristic
252C4×Q8—62no—
253C4×Q8—62no—
254C4×Q8—64no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—64no—
258C4×D4—64no—
259C4×D4—62no—
260C4×D4—62no—
261C4×D4—64no—
262C4×D4—62no—
263C4×D4—62no—
264#117—62no—
265#117—62no—
266#117—64no—
267#117—64no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—34no—
273C2×C4⋊C4—34no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4C4×D311yescharacteristic
284C2×C22⋊C4—31no—
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—31no—
288C2×C22⋊C4—34no—
289C2×C22⋊C4C4×D311yescharacteristic
290C2×C22⋊C4C22×D314yes—
291C2×C22⋊C4C22×D311yescharacteristic
292C2×C42—22no—
293C2×C42—31no—
294C2×C42—31no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—62no—
299C2×C42C4×D311yescharacteristic
300#99—24no—
301#99—68no—
302#99—64no—
303#95—22no—
304#95—64no—
305#95—68no—
306#95—62no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3#11712yes—
314C23×C3C23×C411yescharacteristic
315C23×C3C22×D412yes—
316C23×C3C2×#4112yes—
317C23×C3C2×#4112yes—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3#11712yes—
323C3×D4—28no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—44no—
335C2×C3×C4—44no—
336C2×C3×C4—44no—
337C2×C3×C4—81no—
338C2×C3×C4—81no—
339C2×C3×C4C2×C22⋊C412yes—
340C2×C3×C4C2×C22⋊C411yescharacteristic
341C2×C3×C4C2×C22⋊C411yescharacteristic
342C2×C3×C4#11712yes—
343#67—28no—
344#67—28no—
345C2×Dic3—28no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—28no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—44no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—44no—
361C2×Dic3—44no—
362C2×Dic3—44no—
363C2×Dic3—82no—
364C2×Dic3C2×C22⋊C412yes—
365C2×Dic3C2×C22⋊C412yes—
366C2×Dic3C2×C22⋊C412yes—
367C2×Dic3C2×C22⋊C412yes—
368C2×Dic3#11712yes—
369C4×D3—28no—
370C4×D3—82no—
371C4×D3—82no—
372Dic6—28no—
373C24—31no—
374C24C2×C4×D311yescharacteristic
375C24C23×D311yescharacteristic
376#41—632no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—34no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4#23514yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—34no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—34no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—64no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—241no—
416C22×C4—241no—
417C22×C4—241no—
418C22×C4—241no—
419C22×C4C2×C4×D311yescharacteristic
420C22×C4C2×C4×D311yescharacteristic
421C22×C4C2×C4×D311yescharacteristic
422C22×C4C2×C4×D311yescharacteristic
423C22×C4C2×C4×D311yescharacteristic
424C22×C4D3×D412yes—
425C22×C4D3×D412yes—
426C4⋊C4—22no—
427C4⋊C4—32no—
428C4⋊C4—32no—
429C4⋊C4—34no—
430C4⋊C4—32no—
431C4⋊C4—32no—
432C4⋊C4—34no—
433C4⋊C4—62no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—244no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—62no—
445C22⋊C4—62no—
446C22⋊C4—62no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—64no—
450C22⋊C4—64no—
451C42—21no—
452C42—21no—
453C42—62no—
454C42—62no—
455C42—62no—
456C42—62no—
457C42—62no—
458C42—82no—
459C42—242no—
460C22×C3—22no—
461C22×C3—21no—
462C22×C3—22no—
463C22×C3—21no—
464C22×C3—22no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3C22×C22⋊C411yescharacteristic
472C22×C3C2×#11711yescharacteristic
473C22×C3C2×#11711yescharacteristic
474C22×C3#52514yes—
475C2×D3—22no—
476C2×D3—42no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—42no—
480C3×C4—22no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—22no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—84no—
487Dic3—22no—
488Dic3—22no—
489Dic3—22no—
490Dic3—22no—
491Dic3—22no—
492Dic3—41no—
493Dic3—41no—
494Dic3—84no—
495C23—21no—
496C23—21no—
497C23—32no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—61no—
503C23—61no—
504C23—121no—
505C23—121no—
506C23#87212yes—
507C23C22×C4×D311yescharacteristic
508C23C2×#23512yes—
509C23C2×D3×D412yes—
510C23D3×#4112yes—
511Q8—68no—
512D4—28no—
513D4—68no—
514D4—68no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—32no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—68no—
534C2×C4—68no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—68no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—124no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—124no—
552C2×C4—124no—
553C2×C4—124no—
554C2×C4—242no—
555C2×C4—242no—
556C2×C4—242no—
557C2×C4#87212yes—
558C2×C4D3×C22⋊C411yescharacteristic
559C2×C4D3×C22⋊C411yescharacteristic
560C2×C4D3×C22⋊C412yes—
561C2×C3—24no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—42no—
565C2×C3#228011yescharacteristic
566C2×C3#288311yescharacteristic
567C2×C3#288411yescharacteristic
568D3—42no—
569C22—22no—
570C22—21no—
571C22—22no—
572C22—21no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—62no—
581C22—122no—
582C22—122no—
583C22—121no—
584C22—121no—
585C22C2×#87211yescharacteristic
586C22C2×D3×C22⋊C411yescharacteristic
587C22D3×#11711yescharacteristic
588C22#554714yes—
589C4—22no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—62no—
596C4—62no—
597C4—62no—
598C4—62no—
599C4—62no—
600C4—84no—
601C4—121no—
602C4—121no—
603C4—244no—
604C3#2072511yescharacteristic, complemented, Sylow
605C2—24no—
606C2—21no—
607C2—21no—
608C2—42no—
609C2—122no—
610C2#7356111yescharacteristic
611C2#7868811yescharacteristic
612C2#7868911yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×C4×#99C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#237C22
cosocleC24×C3C24
hypercenterC4×#99D3
hypocenterC3#20725
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #73445C2semidirect
3. C3×C4×#99C2semidirect
4. C4×#834C2semidirect
5. #70028C2semidirect
7. C4×#806C2semidirect
8. #68959C2semidirect
9. #68953C2semidirect
14. C2×C4×#67C4semidirect
16. C22×Dic3⋊C4C4semidirect
20. C2×D3×C4⋊C4C4semidirect
21. C2×#887C4semidirect
22. C2×#67⋊C4C4semidirect
24. C2×#877C4semidirect
26. C2×#872C4semidirect
27. C2×C4×Dic6C4semidirect
48. C4×#99D3semidirect
604. C3#20725semidirect

Maximal subgroups

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

subgroupindex
C2×#55462
#734452
C3×C4×#992
C4×#8342
#700282
#700272
C4×#8062
#689592
#689532
#207253