Finite Group Databasebeta

← #10844637

Subgroups — #10844637

Order 768.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2370C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic
3C2×#5069C211yesmaximal, characteristic, complemented
4#73802C211yesmaximal, characteristic
5#69635C211yesmaximal, characteristic, complemented
6#69634C211yesmaximal, characteristic, complemented
7#69628C211yesmaximal, characteristic
8#69628C211yesmaximal, characteristic
9#69488C211yesmaximal, characteristic
10#69488C211yesmaximal, characteristic
11#69486C211yesmaximal, characteristic, complemented
12#69485C211yesmaximal, characteristic, complemented
13#69474C211yesmaximal, characteristic
14#69465C211yesmaximal, characteristic, complemented
15#69464C211yesmaximal, characteristic, complemented
16#25289—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#875C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C3×#389—21no—
23C3×#389C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#387—21no—
26C3×#387—21no—
27C3×C4×C4⋊C4—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32#5069—24no—
33#5066—24no—
34#4987—21no—
35#4987—21no—
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56#4674—21no—
57#4674—21no—
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#491—31no—
63C2×#478—31no—
64#2547—31no—
65#2370D311yescharacteristic, complemented
66#2068—31no—
67#2066—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1927—31no—
76#1762—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×#125—22no—
82C3×#125D414yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#875—22no—
112#875—22no—
113#875—41no—
114#875—41no—
115C4×D12—22no—
116C4×D12—22no—
117#869—22no—
118#869—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#810—22no—
131#810—22no—
132#810—22no—
133#810—22no—
134#810—22no—
135#810—22no—
136#810—41no—
137#810—41no—
138#809—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C2×C4⋊Q8—31no—
149C2×#125C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×D4—31no—
153#491—64no—
154#478—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97—31no—
173#397—61no—
174#397—61no—
175#391—31no—
176#391—31no—
177#389—21no—
178#389C2×D311yescharacteristic
179#389C2×D311yescharacteristic
180#387—21no—
181#387—21no—
182#380—61no—
183#380—61no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C4⋊C4C2×D311yescharacteristic
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12#4112yes—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4C2×D412yes—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4#4112yes—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248C3×C42—81no—
249C3×C42—81no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—42no—
260C4⋊Dic3—42no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C22×Q8—31no—
274C22×Q8—31no—
275C22×D4—31no—
276C23×C4—61no—
277C22×C8—31no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8—31no—
281C4⋊Q8—62no—
282C4⋊Q8—62no—
283#125—22no—
284#125D1214yes—
285#124—62no—
286#124—62no—
287#124—121no—
288#124—121no—
289C4×D4—62no—
290C4×D4—62no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4#6711yescharacteristic
299C2×C4⋊C4#6711yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42C22×D311yescharacteristic
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#106—62no—
317C4⋊C8—62no—
318C4⋊C8—121no—
319C4⋊C8—121no—
320#103—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—121no—
327#103—121no—
328#102—62no—
329#102—62no—
330#102—62no—
331#102—62no—
332#97—62no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95#6711yescharacteristic
343#95#6711yescharacteristic
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4#12112yes—
381C2×C3×C4#12112yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397D12—22no—
398D12—22no—
399D12—41no—
400D12—41no—
401C4×D3—84no—
402Dic6—41no—
403Dic6—41no—
404Dic6—41no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C24—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C23×D311yescharacteristic
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—81no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4C2×D1212yes—
478C4⋊C4C2×D1212yes—
479C4⋊C4#23912yes—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—81no—
491C42—81no—
492C42C2×D1212yes—
493C42#23512yes—
494C22×C3C2×#12111yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3#54811yescharacteristic
497C22×C3#57511yescharacteristic
498C22×C3#57511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#54211yescharacteristic
520C3×C4#54811yescharacteristic
521C3×C4#55011yescharacteristic
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—62no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C22×#6711yescharacteristic
533Q8—121no—
534Q8—121no—
535Q8—121no—
536Q8—121no—
537Q8—121no—
538Q8—121no—
539Q8—121no—
540Q8—121no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—244no—
583C2×C4#93012yes—
584C2×C4#93012yes—
585C2×C4C22×D1211yescharacteristic
586C2×C4C2×#23511yescharacteristic
587C2×C4C2×#23911yescharacteristic
588C2×C4#100711yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4#100911yescharacteristic
591C8—121no—
592C8—121no—
593C8—121no—
594C8—121no—
595C2×C3#242611yescharacteristic
596C2×C3#304011yescharacteristic
597C2×C3#304611yescharacteristic
598C2×C3#330711yescharacteristic
599C2×C3#331511yescharacteristic
600C2×C3#337611yescharacteristic
601C2×C3#337711yescharacteristic
602D3—82no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22C2×#93011yescharacteristic
611C22#581711yescharacteristic
612C22#582211yescharacteristic
613C22#582211yescharacteristic
614C22#582911yescharacteristic
615C22#583611yescharacteristic
616C22#583811yescharacteristic
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#557611yescharacteristic
631C4#557711yescharacteristic
632C4#558011yescharacteristic
633C4#558211yescharacteristic
634C3#2528911yescharacteristic, complemented, Sylow
635C2—242no—
636C2#8249511yescharacteristic
637C2#8252711yescharacteristic
638C2#8257211yescharacteristic
639C2#8360011yescharacteristic
640C2#8361511yescharacteristic
641C2#8364511yescharacteristic
642C2#8370011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2370C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2370D3
hypocenterC3#25289
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C3×#2370C2semidirect
3. C2×#5069C2semidirect
5. #69635C2semidirect
6. #69634C2semidirect
11. #69486C2semidirect
12. #69485C2semidirect
14. #69465C2semidirect
15. #69464C2semidirect
65. #2370D3semidirect
634. C3#25289semidirect

Maximal subgroups

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

subgroupindex
C3×#23702
C2×#50662
C2×#50692
#738022
#696352
#696342
#696282
#694882
#694862
#694852
#694742
#694652
#694642
#252893