Finite Group Databasebeta

← #10875914

Subgroups — #10875914

Order 768.

808 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
1D4×Dic3⋊C4C211yesmaximal, characteristic, complemented
2#73695C211yesmaximal, characteristic, complemented
3C3×#1762C211yesmaximal, characteristic, complemented
4#71613C211yesmaximal, characteristic
5#70172C211yesmaximal, characteristic
6#70172C211yesmaximal, characteristic
7#70169C211yesmaximal, characteristic, complemented
8#70154C211yesmaximal, characteristic, complemented
9#69834C211yesmaximal, characteristic, complemented
10#69818C211yesmaximal, characteristic, complemented
11#69460C211yesmaximal, characteristic, complemented
12#68902C211yesmaximal, characteristic, complemented
13#68900C211yesmaximal, characteristic, complemented
14#68844C211yesmaximal, characteristic, complemented
15#68810C211yesmaximal, characteristic, complemented
16#27568—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C22×Dic3⋊C4—21no—
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#832C2211yescharacteristic
27#5210—21no—
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4988C2211yescharacteristic
32#4988C2211yescharacteristic
33#4977C2211yescharacteristic
34#4975C2211yescharacteristic
35#4972C2211yescharacteristic
36#4972C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4946—21no—
40#4942—21no—
41#4940—21no—
42#4940—21no—
43#4939C2211yescharacteristic
44C4×Dic3⋊C4C2211yescharacteristic
45#4928C2211yescharacteristic
46#4675—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#2513—31no—
61D4×C4⋊C4—31no—
62#2101—31no—
63#2099—31no—
64#2071—31no—
65#2065—31no—
66#1936—31no—
67#1920—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1864—31no—
73#1804—31no—
74#1762D311yescharacteristic, complemented
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C22×D12C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4—41no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92D4×Dic3—22no—
93D4×Dic3—22no—
94D4×Dic3—22no—
95D4×Dic3—41no—
96C2×#213—22no—
97C2×#213—22no—
98C2×#213—22no—
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×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—44no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C3×#102—22no—
121C3×#102—22no—
122C3×#97—41no—
123C3×#95—21no—
124#832—22no—
125C3⋊#95—21no—
126C3⋊#95—21no—
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×D8—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4C2×D311yescharacteristic
141C22×C4⋊C4—61no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×#97C2×D311yescharacteristic
151#397—61no—
152#397—61no—
153#390—31no—
154#390—31no—
155#390—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#388—61no—
160#388—61no—
161#385—61no—
162#385—61no—
163#384—31no—
164#382—31no—
165#380—61no—
166C4×C4⋊C4—31no—
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—
180C24×C3—21no—
181C23×D3—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4#4112yes—
186C2×C3×D4#4112yes—
187C2×C3×D4#4112yes—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4—81no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—24no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3—81no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C22⋊C4—42no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—81no—
232#217—24no—
233#217—42no—
234#213—44no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
240(C2×D3)⋊C4—81no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—22no—
248Dic3⋊C4—41no—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C22×D4—31no—
265C22×D4C22×D311yescharacteristic
266C23×C4—21no—
267C23×C4—61no—
268C23×C4—61no—
269C2×D8—62no—
270C2×D8—62no—
271C2×D8—62no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C4×D4—22no—
277C4×D4—41no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—121no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—124no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C42—21no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42C22×D311yescharacteristic
309#106—62no—
310C4⋊C8—121no—
311#102—22no—
312#102—22no—
313#102—62no—
314#102—62no—
315#102—121no—
316#102—121no—
317#97—41no—
318#95—21no—
319#95—31no—
320#95—31no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3C22×D411yescharacteristic
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4#12614yes—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—84no—
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×C3×C42−1+411yescharacteristic
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—48no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—84no—
389C2×Dic3AGL(1,ℤ/8)12yes—
390D12—41no—
391D12—41no—
392D12—41no—
393D12—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24—21no—
399C24—61no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4#23512yes—
410C2×D4#23512yes—
411C2×D4#23512yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—64no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—81no—
426C22×C4—122no—
427C22×C4—122no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C23×D311yescharacteristic
434D8—124no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—62no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—121no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C22⋊C4—42no—
467C22⋊C4—64no—
468C22⋊C4—122no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—241no—
472C22⋊C4—241no—
473C22⋊C4—241no—
474C22⋊C4—241no—
475C42—21no—
476C42—21no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—81no—
482C22×C3—24no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3#52511yescharacteristic
490C22×C3#54611yescharacteristic
491C22×C3#54811yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3#57511yescharacteristic
494C22×C3#57611yescharacteristic
495C22×C3#57711yescharacteristic
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4—81no—
508C3×C4C2×#12611yescharacteristic
509C3×C4#53911yescharacteristic
510C3×C4#55211yescharacteristic
511C3×C4#55311yescharacteristic
512Dic3—22no—
513Dic3—42no—
514Dic3—41no—
515Dic3—41no—
516Dic3—41no—
517Dic3—81no—
518Dic3—81no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23C2×D3×D411yescharacteristic
534D4—22no—
535D4—22no—
536D4—22no—
537D4—121no—
538D4—121no—
539D4—121no—
540D4—121no—
541D4#87614yes—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—32no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—84no—
571C2×C4—128no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—244no—
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×#23511yescharacteristic
584C2×C4#100411yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4#101011yescharacteristic
587C8—41no—
588C8—41no—
589C8—121no—
590C8—121no—
591C2×C3—44no—
592C2×C3#248111yescharacteristic
593C2×C3#311411yescharacteristic
594C2×C3#311911yescharacteristic
595C2×C3#330711yescharacteristic
596C2×C3#332511yescharacteristic
597C2×C3#337011yescharacteristic
598C2×C3#337311yescharacteristic
599D3—161no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#549011yescharacteristic
615C22#554711yescharacteristic
616C22#555211yescharacteristic
617C22D3×AGL(1,ℤ/8)11yescharacteristic
618C22#577411yescharacteristic
619C22#577511yescharacteristic
620C22#577711yescharacteristic
621C4—41no—
622C4—41no—
623C4—62no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—121no—
628C4—121no—
629C4—121no—
630C4—122no—
631C4—241no—
632C4—241no—
633C4C2×#87611yescharacteristic
634C4#553111yescharacteristic
635C4#554011yescharacteristic
636C4#556411yescharacteristic
637C3#2756811yescharacteristic, complemented, Sylow
638C2—44no—
639C2—481no—
640C2#7350311yescharacteristic
641C2#7889511yescharacteristic
642C2#7894111yescharacteristic
643C2#7975311yescharacteristic
644C2#7978211yescharacteristic
645C2#7985311yescharacteristic
646C2#7987311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1762D3
hypocenterC3#27568
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. D4×Dic3⋊C4C2semidirect
2. #73695C2semidirect
3. C3×#1762C2semidirect
7. #70169C2semidirect
8. #70154C2semidirect
9. #69834C2semidirect
10. #69818C2semidirect
11. #69460C2semidirect
12. #68902C2semidirect
13. #68900C2semidirect
14. #68844C2semidirect
15. #68810C2semidirect
74. #1762D3semidirect
637. C3#27568semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3⋊C42
#736952
C3×#17622
#716132
#701722
#701692
#701542
#698342
#698182
#694602
#689022
#689002
#688442
#688102
#275683