Finite Group Databasebeta

← #10945146

Subgroups — #10945146

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C42×Q8C211yesmaximal, characteristic, complemented
2C2×#5028C211yesmaximal, characteristic
3C2×#5022C211yesmaximal, characteristic, complemented
4#73315C211yesmaximal, characteristic
5#71633C211yesmaximal, characteristic
6#71633C211yesmaximal, characteristic
7#71632C211yesmaximal, characteristic, complemented
8#71632C211yesmaximal, characteristic, complemented
9#69491C211yesmaximal, characteristic, complemented
10#69488C211yesmaximal, characteristic
11#69465C211yesmaximal, characteristic, complemented
12#69464C211yesmaximal, characteristic, complemented
13#69463C211yesmaximal, characteristic
14#69462C211yesmaximal, characteristic
15#69392C211yesmaximal, characteristic, complemented
16#33029—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C3×C4×C4⋊C4—21no—
25C3×C4×C4⋊C4—21no—
26C3×C4×C4⋊C4—21no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C43—21no—
31C3×C43C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5028—24no—
35#5022—24no—
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4940—21no—
45#4940—21no—
46#4940—21no—
47#4940—21no—
48#4939C2211yescharacteristic
49#4939C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#479—31no—
63C2×#464—31no—
64#2861—31no—
65C42×Q8D311yescharacteristic, complemented
66#1989—31no—
67#1983—31no—
68#1928—31no—
69#1927—31no—
70#1926—31no—
71#1925—31no—
72#1880—31no—
73#1880—31no—
74#1878—31no—
75#1878—31no—
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8D414yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—22no—
108C2×#215—22no—
109C2×#215—22no—
110C2×#215—22no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#875—22no—
126#875—22no—
127#870—22no—
128#870—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132#835—22no—
133#835—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#810—22no—
139#810—22no—
140#810—22no—
141#809—22no—
142#809—22no—
143#809—22no—
144#808—22no—
145#807—22no—
146C4⋊(C3⋊C8)—22no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#125—31no—
151C2×#124—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×Q8C2×D311yescharacteristic
155#479—64no—
156#464—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169#391—31no—
170#391—31no—
171#390—31no—
172#390—31no—
173#388—61no—
174#388—61no—
175#388—61no—
176#388—61no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4C2×D311yescharacteristic
181C4×C4⋊C4C2×D311yescharacteristic
182C4×C4⋊C4C2×D311yescharacteristic
183C43—21no—
184C43C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×D412yes—
198C2×C3×Q8#4112yes—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4C2×D412yes—
233C3×C4⋊C4#4112yes—
234C3×C4⋊C4#4112yes—
235C3×C42—24no—
236C3×C42—24no—
237C3×C42—24no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—41no—
245C3×C42—41no—
246C3×C42—42no—
247C3×C42—42no—
248C3×C42—42no—
249C3×C42C2×D412yes—
250C3×C42#4112yes—
251#216—44no—
252#215—44no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
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—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C22×D4—31no—
280C2×Q16—62no—
281C2×Q16—62no—
282C2×Q16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×SD16—62no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290#125—62no—
291#125—62no—
292#124—62no—
293#124—62no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8D1214yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
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×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42#6711yescharacteristic
323C2×C42#6711yescharacteristic
324C2×C42#6711yescharacteristic
325C2×C42#6711yescharacteristic
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C2×C42C22×D311yescharacteristic
329#107—62no—
330#106—62no—
331C4⋊C8—62no—
332C4⋊C8—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#103—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8#12414yes—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
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—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4#12112yes—
387C2×C3×C4#12112yes—
388C2×C3×C4#13512yes—
389C2×C3×C4#13712yes—
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405D12—41no—
406D12—41no—
407D12—41no—
408D12—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412Dic6—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C24—61no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8C2×D1212yes—
428C2×Q8#23512yes—
429C2×Q8#23512yes—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—124no—
451SD16—124no—
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—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4C2×D1212yes—
477C4⋊C4#23512yes—
478C4⋊C4#23512yes—
479C22⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—24no—
486C42—24no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42C2×D1212yes—
500C42#23512yes—
501C22×C3C2×#12111yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#52511yescharacteristic
504C22×C3C2×#13511yescharacteristic
505C22×C3C2×#13711yescharacteristic
506C22×C3#57511yescharacteristic
507C22×C3#57511yescharacteristic
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—22no—
520C3×C4—41no—
521C3×C4—42no—
522C3×C4—41no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4C2×#12411yescharacteristic
526C3×C4#52911yescharacteristic
527C3×C4#52911yescharacteristic
528C3×C4#53211yescharacteristic
529Dic3—81no—
530Dic3—81no—
531Dic3—81no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23C22×#6711yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—22no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546Q8—121no—
547Q8#87514yes—
548D4—121no—
549D4—121no—
550D4—121no—
551D4—121no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4#93012yes—
588C2×C4#93012yes—
589C2×C4#94212yes—
590C2×C4#95012yes—
591C2×C4C22×D1211yescharacteristic
592C2×C4C2×#23511yescharacteristic
593C2×C4C2×#23511yescharacteristic
594C2×C4C2×#23511yescharacteristic
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×#23511yescharacteristic
597C8—121no—
598C8—121no—
599C8—121no—
600C8—121no—
601C2×C3#285011yescharacteristic
602C2×C3#303511yescharacteristic
603C2×C3#303711yescharacteristic
604C2×C3#310211yescharacteristic
605C2×C3#310811yescharacteristic
606C2×C3#311111yescharacteristic
607C2×C3#311711yescharacteristic
608D3—161no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22C2×#93011yescharacteristic
617C22#579111yescharacteristic
618C22#579111yescharacteristic
619C22C2×#94211yescharacteristic
620C22C2×#95011yescharacteristic
621C22#582211yescharacteristic
622C22#582211yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4C2×#87511yescharacteristic
637C4#547711yescharacteristic
638C4#547911yescharacteristic
639C4#547911yescharacteristic
640C3#3302911yescharacteristic, complemented, Sylow
641C2—481no—
642C2#8207711yescharacteristic
643C2#8244111yescharacteristic
644C2#8244311yescharacteristic
645C2#8244511yescharacteristic
646C2#8252511yescharacteristic
647C2#8252811yescharacteristic
648C2#8253011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C42×Q8C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC42×Q8D3
hypocenterC3#33029
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C42×Q8C2semidirect
3. C2×#5022C2semidirect
7. #71632C2semidirect
8. #71632C2semidirect
9. #69491C2semidirect
11. #69465C2semidirect
12. #69464C2semidirect
15. #69392C2semidirect
65. C42×Q8D3semidirect
640. C3#33029semidirect

Maximal subgroups

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

subgroupindex
C3×C42×Q82
C2×#50282
C2×#50222
#733152
#716332
#716322
#694912
#694882
#694652
#694642
#694632
#694622
#693922
#330293