Finite Group Databasebeta

← #10732630

Subgroups — #10732630

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic, complemented
2C2×#4988C211yesmaximal, characteristic
3C2×#4988C211yesmaximal, characteristic
4C2×#4939C211yesmaximal, characteristic
5C3×C4⋊#95C211yesmaximal, characteristic, complemented
6#71563C211yesmaximal, characteristic, complemented
7C4×C3⋊#95C211yesmaximal, characteristic, complemented
8#69428C214yesmaximal, complemented
9#69427C214yesmaximal, complemented
10#15229—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C2214yes—
16C22×(C2×D3)⋊C4C2214yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C2×#890—24no—
25C2×#890—24no—
26C2×#890—24no—
27C2×C3×#95C2214yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2214yes—
31#4988—24no—
32#4988—24no—
33#4939—24no—
34C22×#125—31no—
35C2×#390—31no—
36C2×#390—31no—
37C2×#390—31no—
38#1439—34no—
39#1439—34no—
40C4⋊#95—31no—
41C4⋊#95D311yescharacteristic, complemented
42C4×#95—31no—
43C24×D3C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2311yescharacteristic
49C23×Dic3C2311yescharacteristic
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C22×D12—22no—
53C22×D12—22no—
54C22×D12—22no—
55C22×D12—22no—
56C22×D12—22no—
57C22×D12—22no—
58C22×D12Q814yes—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D414yes—
65C2×C3×C42—22no—
66C2×C3×C42—22no—
67C2×C3×C42—22no—
68C2×(C2×D3)⋊C4—28no—
69C2×(C2×D3)⋊C4—28no—
70C2×(C2×D3)⋊C4—28no—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4—28no—
74C2×(C2×D3)⋊C4—44no—
75C2×(C2×D3)⋊C4—44no—
76C2×(C2×D3)⋊C4—44no—
77C2×(C2×D3)⋊C4—44no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3D414yes—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3D414yes—
93#890—232no—
94C3×#95—44no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—44no—
98C23×D4—31no—
99C2×#125—64no—
100C2×#125—64no—
101C2×#125—64no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—32no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4C2×D311yescharacteristic
106C22×C4⋊C4C2×D311yescharacteristic
107C22×C22⋊C4—34no—
108C22×C22⋊C4—34no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4—32no—
111C22×C42—31no—
112C22×C42—31no—
113C22×C42C2×D311yescharacteristic
114#390—64no—
115#390—64no—
116#390—64no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—34no—
120C2×#95C2×D314yes—
121C24×C3C2411yescharacteristic
122C23×D3—22no—
123C23×D3—22no—
124C23×D3—22no—
125C23×D3—24no—
126C23×D3—24no—
127C23×D3—22no—
128C23×D3—22no—
129C23×D3—22no—
130C23×D3—24no—
131C23×D3—24no—
132C23×D3C2×Q814yes—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4C2×D414yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×Q812yes—
153C22×C3×C4#4112yes—
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×C3×C4#4114yes—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—24no—
170C22×Dic3—22no—
171C22×Dic3—24no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C22×Dic3C2×D414yes—
178C22×Dic3#4114yes—
179C2×D12—28no—
180C2×D12—28no—
181C2×D12—28no—
182C2×D12—44no—
183C2×D12—44no—
184C2×D12—44no—
185C2×D12—44no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—44no—
188C3×C42—44no—
189(C2×D3)⋊C4—416no—
190(C2×D3)⋊C4—416no—
191(C2×D3)⋊C4—48no—
192(C2×D3)⋊C4—48no—
193C4⋊Dic3—216no—
194C4⋊Dic3—28no—
195Dic3⋊C4—216no—
196C4×Dic3—28no—
197C4×Dic3—44no—
198C25—32no—
199C22×D4—34no—
200C22×D4—62no—
201C22×D4—62no—
202C22×D4—62no—
203C22×D4—62no—
204C22×D4—62no—
205C22×D4—62no—
206C23×C4—32no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—32no—
210C23×C4C22×D312yes—
211C23×C4C22×D312yes—
212C23×C4C22×D312yes—
213C23×C4C22×D311yescharacteristic
214#125—632no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—34no—
221C2×C4⋊C4—64no—
222C2×C4⋊C4—64no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—64no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4D1214yes—
230C2×C22⋊C4—68no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—64no—
233C2×C22⋊C4—68no—
234C2×C22⋊C4—68no—
235C2×C22⋊C4—68no—
236C2×C22⋊C4—124no—
237C2×C22⋊C4—124no—
238C2×C22⋊C4—124no—
239C2×C22⋊C4—124no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—34no—
244C2×C42—62no—
245C2×C42—62no—
246C2×C42—62no—
247C2×C42—62no—
248C2×C42—62no—
249#95—44no—
250#95—64no—
251#95—64no—
252#95—124no—
253C23×C3C22×D411yescharacteristic
254C23×C3C22×D411yescharacteristic
255C23×C3C22×D411yescharacteristic
256C23×C3C22×Q811yescharacteristic
257C23×C3C2×#4111yescharacteristic
258C23×C3C2×#4111yescharacteristic
259C23×C3C2×#4111yescharacteristic
260C23×C3C2×#4112yes—
261C23×C3C2×#4112yes—
262C23×C3H(5,𝔽2)12yes—
263C23×C3H(5,𝔽2)12yes—
264C22×D3—24no—
265C22×D3—24no—
266C22×D3—24no—
267C22×D3—28no—
268C22×D3—28no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—44no—
274C22×D3—44no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—44no—
280C22×D3—44no—
281C22×D3—44no—
282C22×D3—44no—
283C22×D3—44no—
284C22×D3—44no—
285C22×D3—44no—
286C22×D3—44no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—28no—
291C2×C3×C4—24no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—44no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—44no—
303C2×C3×C4—44no—
304C2×C3×C4#12014yes—
305C2×C3×C4C22⋊Q814yes—
306C2×C3×C4C22⋊Q814yes—
307C2×C3×C4C22⋊Q814yes—
308C2×C3×C4#12414yes—
309C2×C3×C4#12414yes—
310C2×C3×C4#12414yes—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—28no—
321C2×Dic3—24no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—44no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333D12—416no—
334C24—34no—
335C24—62no—
336C24—62no—
337C24—62no—
338C24—62no—
339C24—62no—
340C24—62no—
341C24—64no—
342C24—64no—
343C24—64no—
344C24—64no—
345C24C23×D311yescharacteristic
346C2×D4—68no—
347C2×D4—68no—
348C2×D4—68no—
349C2×D4—124no—
350C2×D4—124no—
351C2×D4—124no—
352C2×D4—124no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—34no—
371C22×C4—34no—
372C22×C4—62no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—64no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—64no—
389C22×C4—62no—
390C22×C4C2×D1214yes—
391C22×C4C2×D1212yes—
392C22×C4#23512yes—
393C22×C4#23512yes—
394C22×C4D3×D412yes—
395C22×C4D3×D412yes—
396C22×C4#23712yes—
397C22×C4D3×Q812yes—
398C22×C4#23914yes—
399C4⋊C4—28no—
400C4⋊C4—44no—
401C4⋊C4—616no—
402C4⋊C4—616no—
403C4⋊C4—68no—
404C22⋊C4—1216no—
405C22⋊C4—1216no—
406C22⋊C4—128no—
407C22⋊C4—128no—
408C42—44no—
409C42—68no—
410C42—124no—
411C22×C3C2×#12011yescharacteristic
412C22×C3C2×C22⋊Q811yescharacteristic
413C22×C3C2×C22⋊Q811yescharacteristic
414C22×C3C2×C22⋊Q811yescharacteristic
415C22×C3C2×#12411yescharacteristic
416C22×C3C2×#12411yescharacteristic
417C22×C3C2×#12411yescharacteristic
418C22×C3#52912yes—
419C22×C3#52912yes—
420C22×C3#53211yescharacteristic
421C22×C3#53211yescharacteristic
422C22×C3#53511yescharacteristic
423C22×C3#53511yescharacteristic
424C22×C3#54012yes—
425C22×C3#54012yes—
426C22×C3#54612yes—
427C22×C3#54612yes—
428C22×C3#55012yes—
429C22×C3#55012yes—
430C22×C3#55312yes—
431C22×C3#55312yes—
432C22×C3#55412yes—
433C22×C3#55412yes—
434C2×D3—44no—
435C2×D3—44no—
436C2×D3—44no—
437C2×D3—48no—
438C2×D3—48no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—82no—
442C2×D3—82no—
443C2×D3—84no—
444C2×D3—84no—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4—82no—
448C3×C4#39318yes—
449Dic3—42no—
450Dic3—42no—
451Dic3—44no—
452Dic3—44no—
453C23—68no—
454C23—68no—
455C23—64no—
456C23—64no—
457C23—64no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—122no—
466C23—124no—
467C23—124no—
468C23—124no—
469C23—124no—
470C23—124no—
471C23—124no—
472C23—124no—
473C23—124no—
474C23—124no—
475C23—124no—
476C23C22×D1211yescharacteristic
477C23C2×#23511yescharacteristic
478C23C2×#23511yescharacteristic
479C23C2×D3×D411yescharacteristic
480C23C2×D3×D411yescharacteristic
481C23C2×#23711yescharacteristic
482C23#100412yes—
483C23C2×D3×Q811yescharacteristic
484C23C2×#23912yes—
485C23D3×#4112yes—
486C23#100912yes—
487D4—1216no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—68no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—124no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4#87514yes—
528C2×C4#88214yes—
529C2×C4#88514yes—
530C2×C4#88514yes—
531C2×C4#89614yes—
532C2×C4#89614yes—
533C2×C4#89714yes—
534C2×C3C2×#39311yescharacteristic
535C2×C3#243111yescharacteristic
536C2×C3#243111yescharacteristic
537C2×C3#245412yes—
538C2×C3#245412yes—
539C2×C3#249012yes—
540C2×C3#249012yes—
541C2×C3#256611yescharacteristic
542C2×C3#256611yescharacteristic
543C2×C3#264111yescharacteristic
544C2×C3#286111yescharacteristic
545D3—84no—
546C22—124no—
547C22—124no—
548C22—124no—
549C22—128no—
550C22—128no—
551C22—242no—
552C22—244no—
553C22—242no—
554C22—242no—
555C22—242no—
556C22—244no—
557C22C2×#87511yescharacteristic
558C22C2×#88211yescharacteristic
559C22C2×#88511yescharacteristic
560C22C2×#88511yescharacteristic
561C22C2×#89611yescharacteristic
562C22C2×#89611yescharacteristic
563C22C2×#89711yescharacteristic
564C22#551412yes—
565C22#553012yes—
566C22#555412yes—
567C22#556212yes—
568C22#557712yes—
569C22#558012yes—
570C22#559912yes—
571C22#561912yes—
572C22#562412yes—
573C22#563012yes—
574C22#566811yescharacteristic
575C22#567411yescharacteristic
576C22#569012yes—
577C22#571511yescharacteristic
578C22#572511yescharacteristic
579C22#573012yes—
580C4—44no—
581C4—44no—
582C4—82no—
583C4—122no—
584C4—124no—
585C4—122no—
586C4—124no—
587C4#467318yes—
588C3#1522911yescharacteristic, complemented, Sylow
589C2—244no—
590C2C2×#467311yescharacteristic
591C2#7378712yes—
592C2#7391412yes—
593C2#7418411yescharacteristic
594C2#7433911yescharacteristic
595C2#7601512yes—
596C2#7601612yes—
597C2#7631311yescharacteristic
598C2#7637411yescharacteristic
599C2#7660611yescharacteristic
600C2#7662611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C4⋊#95C2
prime radicalC23×D12C22
hypercenterC4⋊#95D3
hypocenterC3#15229
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
1. C22×#890C2semidirect
5. C3×C4⋊#95C2semidirect
6. #71563C2semidirect
7. C4×C3⋊#95C2semidirect
8. #69428C2semidirect
9. #69427C2semidirect
41. C4⋊#95D3semidirect
588. C3#15229semidirect

Maximal subgroups

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

subgroupindex
C22×#8902
C2×#49882
C2×#49392
C3×C4⋊#952
#715632
C4×C3⋊#952
#694282
#694272
#152293