Finite Group Databasebeta

← #10694324

Subgroups — #10694324

Order 768.

764 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82700C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic, complemented
3C2×#4537C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic, complemented
5#69908C211yesmaximal, characteristic, complemented
6#69401C211yesmaximal, characteristic, complemented
7C2×#4451C211yesmaximal, characteristic, complemented
8#9715—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#940C411yescharacteristic, complemented
11C2×#939—21no—
12C2×#939—21no—
13C2×#930C2211yescharacteristic
14C22×C4×Dic3C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5221C411yescharacteristic, complemented
18C2×C3⋊#95—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×Dic3⋊C8—21no—
23#4988C411yescharacteristic, complemented
24#4975C411yescharacteristic, complemented
25#4957—21no—
26#4957—21no—
27C3×#327—22no—
28#4550—21no—
29#4537—22no—
30#4470—22no—
31#4451—22no—
32#2856—31no—
33#1797—31no—
34C2×#327—31no—
35C2×#327—31no—
36C2×#327—31no—
37C2×#327D311yescharacteristic, complemented
38#1434—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×#67—21no—
45C22×#67—21no—
46C23×Dic3—21no—
47C23×Dic3D411yescharacteristic
48C23×Dic3D411yescharacteristic
49C22×D12C2×C411yescharacteristic
50C22×C3×C8—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×#217—21no—
58C2×#217—21no—
59#940—42no—
60#939—42no—
61#939—42no—
62#939—42no—
63#939—42no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×C4⋊Dic3C2×C411yescharacteristic
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—42no—
75C2×C4×Dic3C2×C411yescharacteristic
76C2×C4×Dic3C2×C411yescharacteristic
77C2×C4×Dic3D411yescharacteristic
78C2×C4×Dic3D411yescharacteristic
79C22×C3⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×#95—21no—
83C3×#95D412yescomplemented
84C3⋊#95—21no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87C3⋊#95D412yescomplemented
88#830—21no—
89#830—21no—
90Dic3⋊C8—42no—
91C2×#127—31no—
92C2×#121—31no—
93C2×#121—61no—
94C2×#121—61no—
95C2×#121C2×D311yescharacteristic
96C22×C42—31no—
97C2×C4⋊C8—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#390—31no—
101#390—31no—
102#390—61no—
103#390—61no—
104#384—31no—
105C2×#95—31no—
106C2×#95—61no—
107C2×#95—61no—
108C2×#95C2×D311yescharacteristic
109#336—61no—
110#327—22no—
111#327—62no—
112#327—62no—
113#327—62no—
114C24×C3—21no—
115C24×C3C2×D411yescharacteristic
116C23×D3—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4C22×C411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×#67—42no—
144C2×#67—42no—
145C2×#67—42no—
146C2×#67—42no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C2×#67—41no—
151C2×#67—41no—
152C2×#67—41no—
153C2×#67—42no—
154C2×#67—42no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3—41no—
165C22×Dic3—41no—
166C22×Dic3—41no—
167C22×Dic3—41no—
168C22×Dic3—41no—
169C22×Dic3C22⋊C411yescharacteristic
170C22×Dic3C22⋊C411yescharacteristic
171C22×Dic3C22⋊C411yescharacteristic
172C22×Dic3C22⋊C411yescharacteristic
173C22×Dic3D812yes—
174C22×Dic3SD1612yes—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—41no—
178C2×D12—41no—
179C2×C3×C8—41no—
180C2×C3×C8—41no—
181C2×C3×C8—41no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186#217—42no—
187#217—42no—
188(C2×D3)⋊C4—42no—
189(C2×D3)⋊C4—41no—
190(C2×D3)⋊C4—41no—
191(C2×D3)⋊C4—42no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194Dic3⋊C4—42no—
195Dic3⋊C4—42no—
196C4×Dic3—22no—
197C4×Dic3—22no—
198C4×Dic3—41no—
199C4×Dic3—41no—
200C2×C3⋊C8—41no—
201C2×C3⋊C8—41no—
202C2×C3⋊C8—41no—
203C22×D4—21no—
204C22×D4—31no—
205C22×D4—61no—
206C22×D4—61no—
207C22×D4—61no—
208C22×D4C4×D311yescharacteristic
209C23×C4—21no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—61no—
213C23×C4C22×D311yescharacteristic
214C22×C8—21no—
215C22×C8—61no—
216#127—122no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221#121—61no—
222#121—61no—
223#121—61no—
224#121—61no—
225#121—122no—
226#121—122no—
227#121—122no—
228#121—122no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4C4×D311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C42—31no—
240C2×C42—31no—
241C2×C42—31no—
242C2×C42—31no—
243C2×C42—122no—
244C4⋊C8—122no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—61no—
248C22⋊C8—61no—
249#95—21no—
250#95—32no—
251#95—61no—
252#95—122no—
253#95—122no—
254#95#6712yescomplemented
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3C2×C22⋊C411yescharacteristic
267C23×C3#12111yescharacteristic
268C23×C3#12111yescharacteristic
269C23×C3C2×D811yescharacteristic
270C23×C3C2×SD1611yescharacteristic
271C23×C3AGL(1,ℤ/8)11yescharacteristic
272C23×C3AGL(1,ℤ/8)11yescharacteristic
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—41no—
279C22×D3—41no—
280C3×D4—81no—
281C3×D4—81no—
282C3×D4—81no—
283C3×D4—81no—
284C3×D4—81no—
285C3×D4—81no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—81no—
301C2×C3×C4—81no—
302C2×C3×C4—81no—
303C2×C3×C4—81no—
304C2×C3×C4C4×D411yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4#12411yescharacteristic
307C2×C3×C4#12711yescharacteristic
308#67—84no—
309#67—84no—
310#67—81no—
311#67—81no—
312#67—81no—
313#67—81no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—42no—
333C2×Dic3—81no—
334C2×Dic3—81no—
335C2×Dic3—81no—
336C2×Dic3—81no—
337C2×Dic3#9912yes—
338C2×Dic3#10212yes—
339C2×Dic3#10212yes—
340D12—81no—
341D12—81no—
342C3×C8—81no—
343C3⋊C8—81no—
344C24—21no—
345C24—61no—
346C24D3×D411yescharacteristic
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—121no—
360C2×D4—121no—
361C2×D4—122no—
362C2×D4—122no—
363C2×D4—122no—
364C2×D4—122no—
365C2×D4—122no—
366C2×D4—122no—
367C2×D4—121no—
368C2×D4—121no—
369C2×D4—121no—
370C2×D4—121no—
371C2×D4—121no—
372C2×D4—121no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—31no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—62no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—62no—
394C22×C4—121no—
395C22×C4—121no—
396C22×C4—121no—
397C22×C4—121no—
398C22×C4—121no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4C2×C4×D311yescharacteristic
404C22×C4#23511yescharacteristic
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4#23711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—121no—
413C2×C8—121no—
414C2×C8—121no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—122no—
420C4⋊C4—122no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—122no—
424C22⋊C4—122no—
425C22⋊C4—122no—
426C22⋊C4—122no—
427C22⋊C4—121no—
428C22⋊C4—121no—
429C42—62no—
430C42—62no—
431C42—121no—
432C42—121no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3#39011yescharacteristic
453C22×C3C2×#9911yescharacteristic
454C22×C3#41011yescharacteristic
455C22×C3C2×#10211yescharacteristic
456C22×C3#41811yescharacteristic
457C22×C3#42111yescharacteristic
458C22×C3#42111yescharacteristic
459C22×C3#45911yescharacteristic
460C22×C3#45911yescharacteristic
461C22×C3#46011yescharacteristic
462C22×C3#46011yescharacteristic
463C2×D3—81no—
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—82no—
475C3×C4—81no—
476Dic3—22no—
477Dic3—22no—
478Dic3—22no—
479Dic3—41no—
480Dic3—82no—
481Dic3—81no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—121no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23D3×C22⋊C411yescharacteristic
501C23#88411yescharacteristic
502C23D3×D811yescharacteristic
503C23#91111yescharacteristic
504C23D3×SD1611yescharacteristic
505C23#91411yescharacteristic
506C23#93911yescharacteristic
507D4—81no—
508D4—81no—
509D4—81no—
510D4—81no—
511D4—81no—
512D4—81no—
513D4—241no—
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—32no—
526C2×C4—32no—
527C2×C4—32no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—122no—
561C2×C4—241no—
562C2×C4—241no—
563C2×C4—241no—
564C2×C4—241no—
565C2×C4#67⋊C411yescharacteristic
566C2×C4#88511yescharacteristic
567C2×C4C4×#6711yescharacteristic
568C2×C4#94011yescharacteristic
569C8—81no—
570C8—241no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—161no—
576C2×C3#151011yescharacteristic
577C2×C3#151411yescharacteristic
578C2×C3#151711yescharacteristic
579C2×C3#151911yescharacteristic
580C2×C3#196311yescharacteristic
581C2×C3#196411yescharacteristic
582C2×C3#198211yescharacteristic
583D3—161no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22#474011yescharacteristic
611C22D3×#9911yescharacteristic
612C22#476211yescharacteristic
613C22D3×#10211yescharacteristic
614C22#477011yescharacteristic
615C22#478611yescharacteristic
616C22#482111yescharacteristic
617C22#482111yescharacteristic
618C22#495711yescharacteristic
619C22#515011yescharacteristic
620C22#516311yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—62no—
626C4—62no—
627C4—62no—
628C4—82no—
629C4—81no—
630C4—121no—
631C4—242no—
632C4—241no—
633C3#971511yescharacteristic, complemented, Sylow
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—161no—
639C2—481no—
640C2#6828111yescharacteristic
641C2#6828211yescharacteristic
642C2#6830411yescharacteristic
643C2#6830711yescharacteristic
644C2#7003011yescharacteristic
645C2#7016111yescharacteristic
646C2#7035711yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radical#82700C2
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9715
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×C3×#327C2semidirect
3. C2×#4537C2semidirect
4. C2×#4470C2semidirect
5. #69908C2semidirect
6. #69401C2semidirect
7. C2×#4451C2semidirect
10. C2×#940C4semidirect
17. #5221C4semidirect
23. #4988C4semidirect
24. #4975C4semidirect
37. C2×#327D3semidirect
83. C3×#95D4semidirect
87. C3⋊#95D4semidirect
254. #95#67semidirect
633. C3#9715semidirect

Maximal subgroups

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

subgroupindex
#827002
C2×C3×#3272
C2×#45372
C2×#44702
#699082
#694012
C2×#44512
#97153