Finite Group Databasebeta

← #10698460

Subgroups — #10698460

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82710C211yesmaximal, characteristic
2C3×#1874C211yesmaximal, characteristic, complemented
3C2×#4537C211yesmaximal, characteristic, complemented
4#71570C211yesmaximal, characteristic, complemented
5#71556C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#10395—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#937—21no—
12C22×(C2×D3)⋊C4—21no—
13C22×(C2×D3)⋊C4C2211yescharacteristic
14C2×C3×#102—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16#5223—21no—
17#5220—21no—
18#5212C2211yescharacteristic
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22#4992—21no—
23C2×#808—21no—
24#4961—21no—
25#4961—21no—
26C3×#340—21no—
27#4537—22no—
28#4529—21no—
29#4468—22no—
30#2617—31no—
31#1874D311yescharacteristic, complemented
32#1853—31no—
33C2×#327—31no—
34C2×#323—31no—
35#1439—31no—
36#1294—31no—
37C24×D3D411yescharacteristic
38C22×C3×D4—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67—21no—
44C23×Dic3—21no—
45C23×Dic3D411yescharacteristic
46C22×C3×C8—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—42no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59#937—42no—
60#937—42no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—42no—
70C2×(C2×D3)⋊C4—42no—
71C2×(C2×D3)⋊C4—42no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4⋊Dic3D411yescharacteristic
74C22×C3⋊C8—21no—
75C3×#102—42no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—42no—
84#830—21no—
85#830D412yescomplemented
86#808—42no—
87C2×#121C2×D311yescharacteristic
88C2×#120—61no—
89C22×C4⋊C4C2×D311yescharacteristic
90C22×C22⋊C4—31no—
91C22×C22⋊C4—61no—
92C2×#106—61no—
93C2×#102—21no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8C2×D311yescharacteristic
96#397—61no—
97#394—61no—
98#394—61no—
99#392—61no—
100#386—61no—
101#380—31no—
102C2×#95—31no—
103C2×#95—61no—
104#340—21no—
105#340—61no—
106#327—62no—
107#323—62no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C23×D3—21no—
111C23×D3—21no—
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—21no—
116C23×D3—41no—
117C23×D3—41no—
118C23×D3—41no—
119C23×D3—41no—
120C23×D3SD1612yes—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C2×#67—42no—
148C2×#67—42no—
149C2×#67—81no—
150C2×#67—81no—
151C2×#67—81no—
152C2×#67—81no—
153C22×Dic3—22no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3—41no—
165C22×Dic3—41no—
166C22×Dic3—41no—
167C2×C3×C8—41no—
168C2×C3×C8—41no—
169C2×C3×C8—41no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178#217—41no—
179#217—41no—
180#217—81no—
181#217—81no—
182(C2×D3)⋊C4—42no—
183(C2×D3)⋊C4—42no—
184(C2×D3)⋊C4—48no—
185(C2×D3)⋊C4—41no—
186(C2×D3)⋊C4—41no—
187(C2×D3)⋊C4—41no—
188(C2×D3)⋊C4—41no—
189C4⋊Dic3—41no—
190C4⋊Dic3—41no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—41no—
193C2×C3⋊C8—41no—
194C25—31no—
195C22×D4—21no—
196C22×D4—61no—
197C22×D4—61no—
198C22×D4#6711yescharacteristic
199C23×C4—21no—
200C23×C4—31no—
201C23×C4—61no—
202C23×C4C22×D311yescharacteristic
203C22×C8—21no—
204C22×C8—61no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#120—122no—
210#120—122no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4#6711yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—61no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—61no—
223C2×C22⋊C4—61no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—122no—
230C2×C22⋊C4—122no—
231C2×C22⋊C4—122no—
232#106—122no—
233#102—42no—
234C22⋊C8—21no—
235C22⋊C8—21no—
236C22⋊C8—32no—
237C22⋊C8—61no—
238#95—61no—
239#95—61no—
240#95—61no—
241#95—61no—
242#95—61no—
243#95—122no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3#12011yescharacteristic
256C23×C3#12111yescharacteristic
257C23×C3#12311yescharacteristic
258C23×C3C2×SD1611yescharacteristic
259C23×C3#13511yescharacteristic
260C23×C3AGL(1,ℤ/8)11yescharacteristic
261C23×C3AGL(1,ℤ/8)11yescharacteristic
262C22×D3—22no—
263C22×D3—22no—
264C22×D3—22no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—41no—
279C22×D3—81no—
280C22×D3—81no—
281C22×D3—81no—
282C22×D3—81no—
283C22×D3—81no—
284C22×D3—81no—
285C3×D4—42no—
286C3×D4—81no—
287C3×D4—81no—
288C3×D4—81no—
289C3×D4—81no—
290C3×D4—81no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—81no—
306C2×C3×C4—81no—
307C2×C3×C4—81no—
308C2×C3×C4—81no—
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12311yescharacteristic
312C2×C3×C4#12411yescharacteristic
313#67—84no—
314#67—84no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—81no—
329C2×Dic3—81no—
330C2×Dic3—81no—
331C2×Dic3—81no—
332C2×Dic3—81no—
333C3×C8—81no—
334C3⋊C8—42no—
335C24—21no—
336C24—32no—
337C24—61no—
338C24—61no—
339C24—61no—
340C24—61no—
341C24—61no—
342C24—61no—
343C24—121no—
344C24—121no—
345C24—121no—
346C24—121no—
347C24D3×D411yescharacteristic
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—122no—
359C2×D4—122no—
360C2×D4—241no—
361C2×D4—241no—
362C2×D4—241no—
363C2×D4—241no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—62no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—121no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4—121no—
386C22×C4—121no—
387C22×C4—121no—
388C22×C4#23511yescharacteristic
389C22×C4#23511yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D411yescharacteristic
392C22×C4#23711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—62no—
398C2×C8—121no—
399C2×C8—121no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—122no—
411C22⋊C4—122no—
412C22⋊C4—128no—
413C22⋊C4—121no—
414C22⋊C4—121no—
415C22⋊C4—121no—
416C22⋊C4—121no—
417C22⋊C4—121no—
418C22⋊C4—121no—
419C22⋊C4—241no—
420C22⋊C4—241no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3#39411yescharacteristic
441C22×C3#44911yescharacteristic
442C22×C3#45011yescharacteristic
443C22×C3#45311yescharacteristic
444C22×C3#45511yescharacteristic
445C22×C3B(4,𝔽2)11yescharacteristic
446C22×C3#45811yescharacteristic
447C22×C3#46511yescharacteristic
448C22×C3#46911yescharacteristic
449C22×C3#48111yescharacteristic
450C22×C3#48211yescharacteristic
451C2×D3—44no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C2×D3—82no—
456C2×D3—82no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—82no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—82no—
467C3×C4—81no—
468Dic3—82no—
469Dic3—81no—
470Dic3—81no—
471Dic3—81no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—122no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—241no—
501C23—241no—
502C23—241no—
503C23—241no—
504C23—241no—
505C23—241no—
506C23#88311yescharacteristic
507C23#88411yescharacteristic
508C23#91111yescharacteristic
509C23#91211yescharacteristic
510C23D3×SD1611yescharacteristic
511C23#91411yescharacteristic
512C23#93711yescharacteristic
513D4—42no—
514D4—81no—
515D4—81no—
516D4—81no—
517D4—81no—
518D4—81no—
519D4—244no—
520D4—244no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—241no—
553C2×C4—241no—
554C2×C4—241no—
555C2×C4—241no—
556C2×C4—241no—
557C2×C4#88411yescharacteristic
558C2×C4#88511yescharacteristic
559C2×C4#92911yescharacteristic
560C2×C4#93911yescharacteristic
561C8—81no—
562C8—122no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—161no—
568C2×C3#160011yescharacteristic
569C2×C3#160111yescharacteristic
570C2×C3#161211yescharacteristic
571C2×C3#161311yescharacteristic
572C2×C3#203411yescharacteristic
573C2×C3#203711yescharacteristic
574C2×C3#204411yescharacteristic
575D3—82no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—124no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—242no—
600C22—242no—
601C22—242no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22#477711yescharacteristic
607C22#477811yescharacteristic
608C22#478211yescharacteristic
609C22#478411yescharacteristic
610C22#495911yescharacteristic
611C22#503211yescharacteristic
612C22#505311yescharacteristic
613C22#515811yescharacteristic
614C22#516911yescharacteristic
615C22#519811yescharacteristic
616C22#520211yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—82no—
622C4—81no—
623C4—242no—
624C4—241no—
625C4—241no—
626C4—241no—
627C3#1039511yescharacteristic, complemented, Sylow
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—161no—
633C2—242no—
634C2#6959911yescharacteristic
635C2#7020711yescharacteristic
636C2#7030611yescharacteristic
637C2#7096911yescharacteristic
638C2#7097211yescharacteristic
639C2#7116511yescharacteristic
640C2#7116811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1874C2
socleC23×C3#120
prime radical#82710C2
hypercenter#1874D3
hypocenterC3#10395
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C3×#1874C2semidirect
3. C2×#4537C2semidirect
4. #71570C2semidirect
5. #71556C2semidirect
6. C2×#4468C2semidirect
7. #67675C2semidirect
31. #1874D3semidirect
85. #830D4semidirect
627. C3#10395semidirect

Maximal subgroups

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

subgroupindex
#827102
C3×#18742
C2×#45372
#715702
#715562
C2×#44682
#676752
#103953