Finite Group Databasebeta

← #10875899

Subgroups — #10875899

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
2#73693C211yesmaximal, characteristic
3C3×#1763C211yesmaximal, characteristic, complemented
4#71633C211yesmaximal, characteristic
5#70240C211yesmaximal, characteristic
6#70240C211yesmaximal, characteristic
7#70237C211yesmaximal, characteristic, complemented
8#70221C211yesmaximal, characteristic, complemented
9#69834C211yesmaximal, characteristic, complemented
10#69818C211yesmaximal, characteristic
11#69459C211yesmaximal, characteristic, complemented
12#68902C211yesmaximal, characteristic
13#68900C211yesmaximal, characteristic
14#68811C211yesmaximal, characteristic, complemented
15#68810C211yesmaximal, characteristic, complemented
16#27567—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#876—21no—
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30#4988C2211yescharacteristic
31#4988C2211yescharacteristic
32#4975—21no—
33#4975C2211yescharacteristic
34#4975C2211yescharacteristic
35#4972—21no—
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4940—21no—
41#4939C2211yescharacteristic
42C4×Dic3⋊C4—21no—
43C4×Dic3⋊C4C2211yescharacteristic
44#4928—21no—
45#4928C2211yescharacteristic
46#4676—21no—
47#4675—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61Q8×C4⋊C4—31no—
62#2100—31no—
63#2099—31no—
64#2071—31no—
65#2065—31no—
66#1936—31no—
67#1919—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1865—31no—
73#1803—31no—
74#1763D311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C4×Q8—22no—
80C3×C4×Q8—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88Q8×Dic3—22no—
89Q8×Dic3—22no—
90Q8×Dic3—22no—
91Q8×Dic3—41no—
92C2×#215—22no—
93C2×#215—22no—
94C2×#215—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#876—44no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#97—41no—
125C3×#95—21no—
126#835—22no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#821—22no—
133#821—41no—
134Dic3⋊C8—41no—
135#809—41no—
136#808—22no—
137C22×SD16—31no—
138C2×#126—61no—
139C2×C4×Q8—31no—
140C2×C4×Q8—31no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103C2×D311yescharacteristic
146C2×#103C2×D311yescharacteristic
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97C2×D311yescharacteristic
151#397—61no—
152#396—61no—
153#390—31no—
154#390—31no—
155#390—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#389—61no—
160#389—61no—
161#388—61no—
162#384—31no—
163#384—31no—
164#384—61no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—61no—
167C4×C4⋊C4C2×D311yescharacteristic
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178#336—31no—
179#336—31no—
180C23×D3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8#4112yes—
185C2×C3×Q8#4112yes—
186C2×C3×Q8#4112yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—42no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—42no—
223C3×C42—42no—
224#215—44no—
225(C2×D3)⋊C4—42no—
226(C2×D3)⋊C4—42no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—42no—
229(C2×D3)⋊C4—81no—
230(C2×D3)⋊C4—81no—
231C4⋊Dic3—24no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—42no—
238C4⋊Dic3#4112yes—
239Dic3⋊C4—22no—
240Dic3⋊C4—22no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—24no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—42no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×Q8C22×D311yescharacteristic
261C22×D4—31no—
262C2×SD16—62no—
263C2×SD16—62no—
264C2×SD16—62no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269#126—124no—
270C4×Q8—22no—
271C4×Q8—41no—
272C4×Q8—62no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×Q8—121no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#106—62no—
308C4⋊C8—121no—
309#103—22no—
310#103—22no—
311#103—62no—
312#102—62no—
313#102—121no—
314#102—121no—
315#97—41no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3C22×D411yescharacteristic
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8#12614yescomplemented
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3AGL(1,ℤ/8)12yes—
378C2×Dic3#13712yes—
379D12—41no—
380D12—41no—
381D12—41no—
382D12—41no—
383C3×C8—41no—
384C3×C8—41no—
385C3⋊C8—41no—
386C3⋊C8—41no—
387C24—61no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8#23512yes—
392C2×Q8#23512yes—
393C2×Q8#23512yes—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D411yescharacteristic
413C22×C4C23×D311yescharacteristic
414SD16—124no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—32no—
434C4⋊C4—42no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—64no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—241no—
448C4⋊C4—241no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C22⋊C4—122no—
454C22⋊C4—122no—
455C22⋊C4—241no—
456C22⋊C4—241no—
457C42—21no—
458C42—21no—
459C42—42no—
460C42—42no—
461C42—64no—
462C42—61no—
463C42—61no—
464C42—61no—
465C42—61no—
466C42—122no—
467C22×C3#52511yescharacteristic
468C22×C3#54611yescharacteristic
469C22×C3#54811yescharacteristic
470C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
471C22×C3C2×#13711yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57711yescharacteristic
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C3×C4—24no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4C2×#12611yescharacteristic
488C3×C4#53911yescharacteristic
489C3×C4#55211yescharacteristic
490C3×C4#55311yescharacteristic
491Dic3—22no—
492Dic3—22no—
493Dic3—41no—
494Dic3—41no—
495Dic3—42no—
496Dic3—81no—
497Dic3—81no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23C2×D3×D411yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8#87614yescomplemented
510D4—121no—
511D4—121no—
512D4—121no—
513D4—121no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—32no—
524C2×C4—32no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—61no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×#23511yescharacteristic
555C2×C4C2×#23511yescharacteristic
556C2×C4#100411yescharacteristic
557C2×C4D3×#4111yescharacteristic
558C2×C4#101011yescharacteristic
559C8—41no—
560C8—41no—
561C8—121no—
562C8—121no—
563C2×C3#248111yescharacteristic
564C2×C3#311211yescharacteristic
565C2×C3#312511yescharacteristic
566C2×C3#332011yescharacteristic
567C2×C3#333211yescharacteristic
568C2×C3#336811yescharacteristic
569C2×C3#336911yescharacteristic
570D3—161no—
571C22—241no—
572C22—241no—
573C22—241no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22—241no—
578C22#549011yescharacteristic
579C22#554711yescharacteristic
580C22#555211yescharacteristic
581C22D3×AGL(1,ℤ/8)11yescharacteristic
582C22D3×#13711yescharacteristic
583C22#577811yescharacteristic
584C22#577811yescharacteristic
585C4—24no—
586C4—41no—
587C4—41no—
588C4—62no—
589C4—62no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—121no—
594C4—121no—
595C4—122no—
596C4—241no—
597C4—241no—
598C4C2×#87611yescharacteristic
599C4#553111yescharacteristic
600C4#554011yescharacteristic
601C4#556411yescharacteristic
602C3#2756711yescharacteristic, complemented, Sylow
603C2—481no—
604C2#7350311yescharacteristic
605C2#7890811yescharacteristic
606C2#7896811yescharacteristic
607C2#7975211yescharacteristic
608C2#7978111yescharacteristic
609C2#7981511yescharacteristic
610C2#7985011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1763D3
hypocenterC3#27567
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. Q8×Dic3⋊C4C2semidirect
3. C3×#1763C2semidirect
7. #70237C2semidirect
8. #70221C2semidirect
9. #69834C2semidirect
11. #69459C2semidirect
14. #68811C2semidirect
15. #68810C2semidirect
74. #1763D3semidirect
333. C3×Q8#126semidirect
509. Q8#876semidirect
602. C3#27567semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3⋊C42
#736932
C3×#17632
#716332
#702402
#702372
#702212
#698342
#698182
#694592
#689022
#689002
#688112
#688102
#275673