Finite Group Databasebeta

← #10698464

Subgroups — #10698464

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#74031C211yesmaximal, characteristic
2C3×#1853C211yesmaximal, characteristic, complemented
3#71570C211yesmaximal, characteristic, complemented
4#71560C211yesmaximal, characteristic, complemented
5C2×#4470C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#10395—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C2×#882—21no—
14C2×C3×#106—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20#4991—21no—
21#4991—21no—
22C2×#809—21no—
23#4958—21no—
24#4674—21no—
25#4671—21no—
26C3×#340—21no—
27#4529—21no—
28#4470—22no—
29#4468—22no—
30#2617—31no—
31#1874—31no—
32#1853D311yescharacteristic, complemented
33C2×#327—31no—
34C2×#323—31no—
35#1439—31no—
36#1294—31no—
37C24×D3D411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C23×Dic3D411yescharacteristic
43C22×D12—21no—
44C22×D12—21no—
45C22×D12D411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—42no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
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—42no—
69C2×(C2×D3)⋊C4—42no—
70C2×(C2×D3)⋊C4—42no—
71C2×C4⋊Dic3D411yescharacteristic
72C22×C3⋊C8—21no—
73#882—42no—
74#882—42no—
75C3×#106—42no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yescomplemented
78C3×#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—42no—
84#830—21no—
85#830—21no—
86#809—42no—
87C2×#121—31no—
88C2×#120—61no—
89C22×C4⋊C4C2×D311yescharacteristic
90C22×C22⋊C4—31no—
91C22×C22⋊C4—61no—
92C2×#106—21no—
93C2×#102—61no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8C2×D311yescharacteristic
96#397—61no—
97#394—61no—
98#394—61no—
99#392—61no—
100#386—61no—
101#380C2×D311yescharacteristic
102C2×#95—31no—
103C2×#95—61no—
104#340—21no—
105#340—61no—
106#327—62no—
107#323—62no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
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—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C22×Dic3—22no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—41no—
157C2×D12—21no—
158C2×D12—21no—
159C2×D12—41no—
160C2×D12—41no—
161C2×D12—42no—
162C2×D12—42no—
163C2×D12—81no—
164C2×D12—81no—
165C2×D12—81no—
166C2×D12—81no—
167C2×C3×C8—22no—
168C2×C3×C8—41no—
169C2×C3×C8—41no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178(C2×D3)⋊C4—42no—
179(C2×D3)⋊C4—42no—
180(C2×D3)⋊C4—48no—
181(C2×D3)⋊C4—41no—
182(C2×D3)⋊C4—41no—
183(C2×D3)⋊C4—41no—
184(C2×D3)⋊C4—41no—
185(C2×D3)⋊C4—41no—
186(C2×D3)⋊C4—41no—
187(C2×D3)⋊C4—81no—
188(C2×D3)⋊C4—81no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C2×C3⋊C8—41no—
192C2×C3⋊C8—41no—
193C2×C3⋊C8—41no—
194C25—31no—
195C22×D4—31no—
196C22×D4—61no—
197C22×D4—61no—
198C22×D4—61no—
199C23×C4—21no—
200C23×C4—31no—
201C23×C4—61no—
202C23×C4C22×D311yescharacteristic
203C22×C8—21no—
204C22×C8—61no—
205#121—61no—
206#121—61no—
207#121—61no—
208#121—61no—
209#120—122no—
210#120—122no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4#6711yescharacteristic
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—42no—
233#102—122no—
234C22⋊C8—21no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8D1212yescomplemented
238#95—21no—
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#12011yescharacteristic
249C23×C3#12111yescharacteristic
250C23×C3#12311yescharacteristic
251C23×C3C2×SD1611yescharacteristic
252C23×C3#13511yescharacteristic
253C23×C3AGL(1,ℤ/8)11yescharacteristic
254C23×C3AGL(1,ℤ/8)11yescharacteristic
255C22×D3—22no—
256C22×D3—22no—
257C22×D3—22no—
258C22×D3—42no—
259C22×D3—42no—
260C22×D3—42no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—41no—
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—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—81no—
303C2×C3×C4—81no—
304C2×C3×C4—81no—
305C2×C3×C4—81no—
306C2×C3×C4#12111yescharacteristic
307C2×C3×C4#12111yescharacteristic
308C2×C3×C4#12311yescharacteristic
309C2×C3×C4#12411yescharacteristic
310#67—81no—
311#67—81no—
312#67—81no—
313#67—81no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—81no—
325C2×Dic3—81no—
326C2×Dic3—81no—
327C2×Dic3—81no—
328C2×Dic3—81no—
329D12—42no—
330D12—81no—
331D12—84no—
332D12—84no—
333C3×C8—42no—
334C3⋊C8—81no—
335C24—32no—
336C24—61no—
337C24—61no—
338C24—61no—
339C24—61no—
340C24—61no—
341C24—61no—
342C24—61no—
343C24—121no—
344C24—121no—
345C24—121no—
346C24—121no—
347C24D3×D411yescharacteristic
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—121no—
351C2×D4—121no—
352C2×D4—122no—
353C2×D4—122no—
354C2×D4—121no—
355C2×D4—121no—
356C2×D4—121no—
357C2×D4—121no—
358C2×D4—121no—
359C2×D4—121no—
360C2×D4—241no—
361C2×D4—241no—
362C2×D4—241no—
363C2×D4—241no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—62no—
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×C4C2×D1211yescharacteristic
389C22×C4#23511yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4#23911yescharacteristic
392C22×C4#23911yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—121no—
398C2×C8—121no—
399C2×C8—121no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—61no—
407C4⋊C4—61no—
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#39411yescharacteristic
434C22×C3#44911yescharacteristic
435C22×C3#45011yescharacteristic
436C22×C3#45311yescharacteristic
437C22×C3#45511yescharacteristic
438C22×C3B(4,𝔽2)11yescharacteristic
439C22×C3#45811yescharacteristic
440C22×C3#46511yescharacteristic
441C22×C3#46911yescharacteristic
442C22×C3#48111yescharacteristic
443C22×C3#48211yescharacteristic
444C2×D3—44no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—81no—
451C2×D3—81no—
452C2×D3—81no—
453C2×D3—81no—
454C2×D3—82no—
455C2×D3—81no—
456C2×D3—81no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—81no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—82no—
467C3×C4—81no—
468C3×C4—81no—
469Dic3—82no—
470Dic3—81no—
471Dic3—81no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—62no—
477C23—62no—
478C23—62no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23—122no—
483C23—122no—
484C23—122no—
485C23—121no—
486C23—121no—
487C23—121no—
488C23—121no—
489C23—121no—
490C23—121no—
491C23—121no—
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#89411yescharacteristic
507C23#89511yescharacteristic
508C23D3×SD1611yescharacteristic
509C23#91411yescharacteristic
510C23#91411yescharacteristic
511C23#91611yescharacteristic
512C23#93711yescharacteristic
513D4—122no—
514D4—241no—
515D4—244no—
516D4—244no—
517D4—241no—
518D4—241no—
519D4—241no—
520D4—241no—
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—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
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#89411yescharacteristic
558C2×C4#89511yescharacteristic
559C2×C4#93011yescharacteristic
560C2×C4#94711yescharacteristic
561C8—42no—
562C8—241no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3#160011yescharacteristic
568C2×C3#160111yescharacteristic
569C2×C3#161211yescharacteristic
570C2×C3#161311yescharacteristic
571C2×C3#203411yescharacteristic
572C2×C3#203711yescharacteristic
573C2×C3#204411yescharacteristic
574D3—82no—
575D3—161no—
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—124no—
589C22—122no—
590C22—122no—
591C22—122no—
592C22—242no—
593C22—242no—
594C22—242no—
595C22—241no—
596C22—241no—
597C22—241no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22#486311yescharacteristic
607C22#486411yescharacteristic
608C22#486511yescharacteristic
609C22#486611yescharacteristic
610C22#499111yescharacteristic
611C22#503211yescharacteristic
612C22#505311yescharacteristic
613C22#516911yescharacteristic
614C22#517211yescharacteristic
615C22#519811yescharacteristic
616C22#520211yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—81no—
622C4—81no—
623C4—82no—
624C4—242no—
625C4—241no—
626C4—241no—
627C3#1039511yescharacteristic, complemented, Sylow
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—242no—
633C2—481no—
634C2#6959611yescharacteristic
635C2#7058111yescharacteristic
636C2#7067511yescharacteristic
637C2#7097211yescharacteristic
638C2#7097711yescharacteristic
639C2#7116811yescharacteristic
640C2#7117311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1853C2
socleC23×C3#120
prime radical#74031C2
hypercenter#1853D3
hypocenterC3#10395
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C3×#1853C2semidirect
3. #71570C2semidirect
4. #71560C2semidirect
5. C2×#4470C2semidirect
6. C2×#4468C2semidirect
7. #67624C2semidirect
32. #1853D3semidirect
77. C3×C22⋊C8D4semidirect
237. C22⋊C8D12semidirect
627. C3#10395semidirect

Maximal subgroups

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

subgroupindex
#740312
C3×#18532
#715702
#715602
C2×#44702
C2×#44682
#676242
#103953