Finite Group Databasebeta

← #10844677

Subgroups — #10844677

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#479C211yesmaximal, characteristic, complemented
2#83025C211yesmaximal, characteristic, complemented
3C2×#5067C211yesmaximal, characteristic
4#82476C211yesmaximal, characteristic
5#71631C211yesmaximal, characteristic, complemented
6#70639C211yesmaximal, characteristic, complemented
7#70623C211yesmaximal, characteristic, complemented
8#70585C211yesmaximal, characteristic, complemented
9#70569C211yesmaximal, characteristic, complemented
10#70469C211yesmaximal, characteristic, complemented
11#70457C211yesmaximal, characteristic
12#70245C211yesmaximal, characteristic, complemented
13#70245C211yesmaximal, characteristic, complemented
14#70230C211yesmaximal, characteristic, complemented
15#70230C211yesmaximal, characteristic, complemented
16#25291—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C2×#874C2211yescharacteristic
22C3×#479—24no—
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#5067—24no—
37#4991—21no—
38#4991—21no—
39#4988C2211yescharacteristic
40#4988C2211yescharacteristic
41#4984—21no—
42#4984C2211yescharacteristic
43#4984C2211yescharacteristic
44#4980—21no—
45#4980—21no—
46Dic3×C4⋊C4—21no—
47Dic3×C4⋊C4C2211yescharacteristic
48Dic3×C4⋊C4C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#809C2211yescharacteristic
53#4939—21no—
54#4939—21no—
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2211yescharacteristic
57C2×#803C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490—31no—
63C2×#479D311yescharacteristic, complemented
64#2548—31no—
65#2370—31no—
66#2067—31no—
67#2065—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1982—31no—
73#1940—31no—
74#1926—31no—
75#1925—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#125—22no—
85C3×#125D414yes—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96#947—22no—
97#947—22no—
98#947—41no—
99#947—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#874—22no—
122#874—22no—
123C3×#107—22no—
124C3×#106—22no—
125C3×C4⋊C8—22no—
126C3×#103—22no—
127C3×#103—22no—
128#835—22no—
129#835—22no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—22no—
138#821—22no—
139#821—41no—
140#821—41no—
141Dic3⋊C8—41no—
142Dic3⋊C8—41no—
143#809—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147#803—22no—
148C2×#127—31no—
149C2×#125C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×Q8C2×D311yescharacteristic
153#490—64no—
154#479—24no—
155C2×#107C2×D311yescharacteristic
156C2×#106C2×D311yescharacteristic
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103C2×D311yescharacteristic
163C2×#103C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97—31no—
173#394—61no—
174#394—61no—
175#390—31no—
176#390—31no—
177#390—61no—
178#390—61no—
179#389—31no—
180#389—31no—
181#389—61no—
182#387—61no—
183#387—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—61no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12—81no—
231C2×D12—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C42—42no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—41no—
262C4⋊Dic3—41no—
263C4⋊Dic3—42no—
264C4⋊Dic3—81no—
265C4⋊Dic3#4112yes—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—42no—
271C4×Dic3—81no—
272C4×Dic3—81no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C22×C8—31no—
285C22×C8—31no—
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288#127—62no—
289#127—62no—
290#125—22no—
291#125#6714yes—
292#124—62no—
293#124—62no—
294#124—121no—
295#124—121no—
296C4×Q8—22no—
297C4×Q8—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—21no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—22no—
326#106—22no—
327C4⋊C8—22no—
328C4⋊C8—121no—
329C4⋊C8—121no—
330#103—22no—
331#103—22no—
332#103—62no—
333#103—62no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—62no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411D12—41no—
412D12—41no—
413D12—41no—
414D12—41no—
415D12—41no—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—84no—
420D12—84no—
421C3×C8—41no—
422C3×C8—41no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C24—61no—
426C24—61no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8#23912yes—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—122no—
446C2×D4—122no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—241no—
450C2×D4—241no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—42no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—121no—
488C4⋊C4—121no—
489C4⋊C4—122no—
490C4⋊C4—241no—
491C4⋊C4C2×#6712yes—
492C4⋊C4C2×#6712yes—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C42—42no—
502C42—61no—
503C42—61no—
504C42—61no—
505C42—61no—
506C42—122no—
507C42—241no—
508C42—241no—
509C42#23712yes—
510C42C2×#6712yes—
511C22×C3C2×#12111yescharacteristic
512C22×C3#54611yescharacteristic
513C22×C3#54811yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—81no—
538C3×C4#52911yescharacteristic
539C3×C4#54211yescharacteristic
540C3×C4#54811yescharacteristic
541C3×C4#55011yescharacteristic
542Dic3—42no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567D4—121no—
568D4—121no—
569D4—121no—
570D4—121no—
571D4—121no—
572D4—121no—
573D4—121no—
574D4—121no—
575D4—244no—
576D4—244no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4#93912yes—
612C2×C4#93912yes—
613C2×C4C2×#23711yescharacteristic
614C2×C4C2×#23911yescharacteristic
615C2×C4#100711yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4#100911yescharacteristic
618C2×C4C22×#6711yescharacteristic
619C8—41no—
620C8—41no—
621C8—121no—
622C8—121no—
623C2×C3#242611yescharacteristic
624C2×C3#304011yescharacteristic
625C2×C3#304511yescharacteristic
626C2×C3#330911yescharacteristic
627C2×C3#331711yescharacteristic
628C2×C3#337011yescharacteristic
629C2×C3#337111yescharacteristic
630D3—161no—
631D3—161no—
632C22—241no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22#563011yescharacteristic
647C22#563311yescharacteristic
648C22#576911yescharacteristic
649C22#577011yescharacteristic
650C22#577711yescharacteristic
651C22#577711yescharacteristic
652C22C2×#93911yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—122no—
660C4—121no—
661C4—121no—
662C4—121no—
663C4—121no—
664C4—241no—
665C4#562411yescharacteristic
666C4#564311yescharacteristic
667C4#581711yescharacteristic
668C4#583411yescharacteristic
669C3#2529111yescharacteristic, complemented, Sylow
670C2—481no—
671C2—481no—
672C2#8121711yescharacteristic
673C2#8121811yescharacteristic
674C2#8124711yescharacteristic
675C2#8124911yescharacteristic
676C2#8275511yescharacteristic
677C2#8390111yescharacteristic
678C2#8396811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#479C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterC2×#479D3
hypocenterC3#25291
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
1. C2×C3×#479C2semidirect
2. #83025C2semidirect
5. #71631C2semidirect
6. #70639C2semidirect
7. #70623C2semidirect
8. #70585C2semidirect
9. #70569C2semidirect
10. #70469C2semidirect
12. #70245C2semidirect
13. #70245C2semidirect
14. #70230C2semidirect
15. #70230C2semidirect
63. C2×#479D3semidirect
669. C3#25291semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4792
#830252
C2×#50672
#824762
#716312
#706392
#706232
#705852
#705692
#704692
#704572
#702452
#702302
#252913