Finite Group Databasebeta

← #10732590

Subgroups — #10732590

Order 768.

1678 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4988C211yesmaximal, characteristic
2C2×#4988C211yesmaximal, characteristic
3C2×#4977C211yesmaximal, characteristic, complemented
4C2×#4939C211yesmaximal, characteristic
5C3×C4⋊#95C211yesmaximal, characteristic, complemented
6#71563C211yesmaximal, characteristic, complemented
7C4×C3⋊#95C211yesmaximal, characteristic, complemented
8#69429C214yesmaximal, complemented
9#69429C214yesmaximal, complemented
10#15224—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⋊Dic3C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C2×C3×#95C2214yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2214yes—
28#4988—24no—
29#4988—24no—
30#4977—24no—
31#4939—24no—
32C2×#390—31no—
33C2×#390—31no—
34C2×#390—31no—
35C2×#382—31no—
36#1440—34no—
37#1440—34no—
38C4⋊#95—31no—
39C4⋊#95D311yescharacteristic, complemented
40C4×#95—31no—
41C24×D3C2312yes—
42C23×C3×C4C2312yes—
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×Dic3C2312yes—
47C23×Dic3C2311yescharacteristic
48C23×Dic3C2311yescharacteristic
49C23×Dic3C2312yes—
50C22×D12—22no—
51C22×D12—22no—
52C22×D12—22no—
53C22×D12—22no—
54C22×D12—22no—
55C22×D12—22no—
56C22×D12D414yes—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C42—22no—
64C2×C3×C42—22no—
65C2×C3×C42—22no—
66C2×(C2×D3)⋊C4—28no—
67C2×(C2×D3)⋊C4—28no—
68C2×(C2×D3)⋊C4—28no—
69C2×(C2×D3)⋊C4—24no—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—28no—
72C2×(C2×D3)⋊C4—44no—
73C2×(C2×D3)⋊C4—44no—
74C2×(C2×D3)⋊C4—44no—
75C2×(C2×D3)⋊C4—44no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3—22no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3D414yes—
85C3×#95—44no—
86C3⋊#95—24no—
87C3⋊#95—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—44no—
91C23×D4—31no—
92C22×C4⋊C4—31no—
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C4⋊C4C2×D311yescharacteristic
95C22×C22⋊C4—34no—
96C22×C22⋊C4—34no—
97C22×C22⋊C4—32no—
98C22×C22⋊C4—32no—
99C22×C42—31no—
100C22×C42—31no—
101C22×C42C2×D311yescharacteristic
102#390—64no—
103#390—64no—
104#390—64no—
105#382—64no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95—34no—
111C2×#95C2×D314yes—
112C24×C3C2411yescharacteristic
113C23×D3—22no—
114C23×D3—24no—
115C23×D3—24no—
116C23×D3—22no—
117C23×D3—22no—
118C23×D3—22no—
119C23×D3—22no—
120C23×D3—22no—
121C23×D3—24no—
122C23×D3—24no—
123C23×D3C2×D414yes—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4#4112yes—
143C22×C3×C4#4112yes—
144C22×C3×C4#4112yes—
145C22×C3×C4#4112yes—
146C22×C3×C4#4114yes—
147C22×C3×C4#4112yes—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—22no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—24no—
166C22×Dic3—24no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C22×Dic3#4114yes—
170C2×D12—28no—
171C2×D12—28no—
172C2×D12—28no—
173C2×D12—44no—
174C2×D12—44no—
175C2×D12—44no—
176C2×D12—44no—
177C3×C4⋊C4—44no—
178C3×C4⋊C4—44no—
179C3×C42—44no—
180(C2×D3)⋊C4—416no—
181(C2×D3)⋊C4—48no—
182(C2×D3)⋊C4—48no—
183(C2×D3)⋊C4—416no—
184C4⋊Dic3—44no—
185C4×Dic3—28no—
186C4×Dic3—44no—
187C25—32no—
188C22×D4—34no—
189C22×D4—62no—
190C22×D4—62no—
191C22×D4—62no—
192C22×D4—62no—
193C22×D4—62no—
194C22×D4—62no—
195C23×C4—31no—
196C23×C4—31no—
197C23×C4—32no—
198C23×C4—32no—
199C23×C4C22×D312yes—
200C23×C4C22×D312yes—
201C23×C4C22×D312yes—
202C23×C4C22×D311yescharacteristic
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—62no—
210C2×C4⋊C4—62no—
211C2×C4⋊C4—62no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—64no—
214C2×C22⋊C4—68no—
215C2×C22⋊C4—68no—
216C2×C22⋊C4—68no—
217C2×C22⋊C4—68no—
218C2×C22⋊C4—124no—
219C2×C22⋊C4—124no—
220C2×C22⋊C4—124no—
221C2×C22⋊C4—124no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—34no—
226C2×C42—62no—
227C2×C42—62no—
228C2×C42—62no—
229C2×C42—62no—
230C2×C42—62no—
231#95—44no—
232#95—64no—
233#95—64no—
234#95—64no—
235#95—64no—
236#95—124no—
237C23×C3C22×D411yescharacteristic
238C23×C3C22×D411yescharacteristic
239C23×C3C2×#4111yescharacteristic
240C23×C3C2×#4111yescharacteristic
241C23×C3C2×#4111yescharacteristic
242C23×C3C2×#4111yescharacteristic
243C23×C3C2×#4111yescharacteristic
244C23×C3C2×#4112yes—
245C23×C3C2×#4112yes—
246C23×C3H(5,𝔽2)12yes—
247C23×C3H(5,𝔽2)12yes—
248C22×D3—24no—
249C22×D3—24no—
250C22×D3—24no—
251C22×D3—28no—
252C22×D3—28no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—44no—
258C22×D3—44no—
259C22×D3—42no—
260C22×D3—42no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3—44no—
264C22×D3—44no—
265C22×D3—44no—
266C22×D3—44no—
267C22×D3—44no—
268C22×D3—44no—
269C22×D3—44no—
270C22×D3—44no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
286C2×C3×C4—44no—
287C2×C3×C4—44no—
288C2×C3×C4#12114yes—
289C2×C3×C4#12314yes—
290C2×C3×C4#12314yes—
291C2×C3×C4#12314yes—
292C2×C3×C4#12414yes—
293C2×C3×C4#12614yes—
294C2×C3×C4#12614yes—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—24no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—44no—
316C2×Dic3—44no—
317D12—416no—
318C24—34no—
319C24—62no—
320C24—62no—
321C24—62no—
322C24—62no—
323C24—62no—
324C24—64no—
325C24—64no—
326C24—62no—
327C24—64no—
328C24—64no—
329C24C23×D311yescharacteristic
330C2×D4—68no—
331C2×D4—68no—
332C2×D4—68no—
333C2×D4—124no—
334C2×D4—124no—
335C2×D4—124no—
336C2×D4—124no—
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—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—34no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—64no—
374C22×C4—64no—
375C22×C4#23512yes—
376C22×C4#23512yes—
377C22×C4#23512yes—
378C22×C4D3×D412yes—
379C22×C4D3×D412yes—
380C22×C4#23712yes—
381C22×C4#23914yes—
382C22×C4#23912yes—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—124no—
386C22⋊C4—1216no—
387C22⋊C4—1216no—
388C22⋊C4—128no—
389C22⋊C4—128no—
390C42—44no—
391C42—68no—
392C42—124no—
393C22×C3C2×#12111yescharacteristic
394C22×C3C2×#12311yescharacteristic
395C22×C3C2×#12311yescharacteristic
396C22×C3C2×#12311yescharacteristic
397C22×C3C2×#12411yescharacteristic
398C22×C3C2×#12611yescharacteristic
399C22×C3C2×#12611yescharacteristic
400C22×C3#52912yes—
401C22×C3#52912yes—
402C22×C3#53211yescharacteristic
403C22×C3#53211yescharacteristic
404C22×C3#53511yescharacteristic
405C22×C3#53511yescharacteristic
406C22×C3#54012yes—
407C22×C3#54012yes—
408C22×C3#54612yes—
409C22×C3#54612yes—
410C22×C3#55312yes—
411C22×C3#55312yes—
412C22×C3#55512yes—
413C22×C3#55512yes—
414C22×C3#55912yes—
415C22×C3#55912yes—
416C2×D3—44no—
417C2×D3—48no—
418C2×D3—48no—
419C2×D3—44no—
420C2×D3—44no—
421C2×D3—82no—
422C2×D3—82no—
423C2×D3—82no—
424C2×D3—82no—
425C2×D3—84no—
426C2×D3—84no—
427C3×C4—44no—
428C3×C4—82no—
429C3×C4—82no—
430C3×C4#39718yes—
431Dic3—44no—
432Dic3—42no—
433Dic3—42no—
434Dic3—82no—
435C23—64no—
436C23—64no—
437C23—64no—
438C23—68no—
439C23—68no—
440C23—122no—
441C23—122no—
442C23—124no—
443C23—122no—
444C23—122no—
445C23—122no—
446C23—124no—
447C23—122no—
448C23—122no—
449C23—122no—
450C23—124no—
451C23—124no—
452C23—124no—
453C23—124no—
454C23—124no—
455C23—124no—
456C23—124no—
457C23—124no—
458C23C2×#23511yescharacteristic
459C23C2×#23511yescharacteristic
460C23C2×#23511yescharacteristic
461C23C2×D3×D411yescharacteristic
462C23C2×D3×D411yescharacteristic
463C23C2×#23711yescharacteristic
464C23#100412yes—
465C23C2×#23911yescharacteristic
466C23C2×#23912yes—
467C23D3×#4112yes—
468C23#100912yes—
469D4—1216no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—64no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—124no—
508C2×C4—124no—
509C2×C4#87614yes—
510C2×C4#88314yes—
511C2×C4#88414yes—
512C2×C4#88514yes—
513C2×C4#89414yes—
514C2×C4#89414yes—
515C2×C4#89814yes—
516C2×C3C2×#39711yescharacteristic
517C2×C3#242012yes—
518C2×C3#242012yes—
519C2×C3#250011yescharacteristic
520C2×C3#250011yescharacteristic
521C2×C3#251312yes—
522C2×C3#251312yes—
523C2×C3#254411yescharacteristic
524C2×C3#254411yescharacteristic
525C2×C3#264111yescharacteristic
526C2×C3#286111yescharacteristic
527D3—84no—
528C22—124no—
529C22—124no—
530C22—128no—
531C22—128no—
532C22—124no—
533C22—244no—
534C22—242no—
535C22—242no—
536C22—242no—
537C22—242no—
538C22—244no—
539C22C2×#87611yescharacteristic
540C22C2×#88311yescharacteristic
541C22C2×#88411yescharacteristic
542C22C2×#88511yescharacteristic
543C22C2×#89411yescharacteristic
544C22C2×#89411yescharacteristic
545C22C2×#89811yescharacteristic
546C22#551412yes—
547C22#553212yes—
548C22#555912yes—
549C22#556212yes—
550C22#558012yes—
551C22#558512yes—
552C22#561212yes—
553C22#561912yes—
554C22#562412yes—
555C22#563012yes—
556C22#566811yescharacteristic
557C22#567411yescharacteristic
558C22#569512yes—
559C22#571511yescharacteristic
560C22#572511yescharacteristic
561C22#573412yes—
562C4—44no—
563C4—82no—
564C4—82no—
565C4—122no—
566C4—124no—
567C4—122no—
568C4—242no—
569C4#467518yes—
570C3#1522411yescharacteristic, complemented, Sylow
571C2—244no—
572C2C2×#467511yescharacteristic
573C2#7369512yes—
574C2#7389112yes—
575C2#7416611yescharacteristic
576C2#7436711yescharacteristic
577C2#7604812yes—
578C2#7621312yes—
579C2#7635011yescharacteristic
580C2#7637411yescharacteristic
581C2#7653311yescharacteristic
582C2#7660611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C4⋊#95C2
prime radicalC23×D12C22
hypercenterC4⋊#95D3
hypocenterC3#15224
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#4977C2semidirect
5. C3×C4⋊#95C2semidirect
6. #71563C2semidirect
7. C4×C3⋊#95C2semidirect
8. #69429C2semidirect
9. #69429C2semidirect
39. C4⋊#95D3semidirect
570. C3#15224semidirect

Maximal subgroups

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

subgroupindex
C2×#49882
C2×#49772
C2×#49392
C3×C4⋊#952
#715632
C4×C3⋊#952
#694292
#152243