Finite Group Databasebeta

← #10875908

Subgroups — #10875908

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
1C3×D4×C4⋊C4C211yesmaximal, characteristic, complemented
2#73694C211yesmaximal, characteristic, complemented
3#71613C211yesmaximal, characteristic
4#71613C211yesmaximal, characteristic
5#71613C211yesmaximal, characteristic
6#71609C211yesmaximal, characteristic, complemented
7#71607C211yesmaximal, characteristic, complemented
8#71605C211yesmaximal, characteristic, complemented
9#69629C211yesmaximal, characteristic, complemented
10#69617C211yesmaximal, characteristic, complemented
11#69613C211yesmaximal, characteristic, complemented
12#69602C211yesmaximal, characteristic, complemented
13#69585C211yesmaximal, characteristic, complemented
14#69571C211yesmaximal, characteristic, complemented
15#69483C211yesmaximal, characteristic, complemented
16#27568—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×C3×C4⋊C4—21no—
21C22×#213C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#385—21no—
25C3×#385—21no—
26C3×#384C2211yescharacteristic
27C3×#380—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#4992—21no—
33#4990—21no—
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4939C2211yescharacteristic
39#4939C2211yescharacteristic
40#4939C2211yescharacteristic
41#4936C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4675—21no—
47#4669—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61D4×C4⋊C4D311yescharacteristic, complemented
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#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C22×D12C2311yescharacteristic
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—41no—
85C3×C4×D4—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—44no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#213—22no—
103C2×#213—22no—
104C2×#213—22no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C3×#95—21no—
121C3×#95—21no—
122C3×#95D411yescharacteristic
123C3×#95D411yescharacteristic
124#832—22no—
125#832—22no—
126#832—22no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#809—22no—
132#809—41no—
133#809—41no—
134#808—22no—
135C4⋊(C3⋊C8)—41no—
136#803—41no—
137C22×D8—31no—
138C2×C4×D4C2×D311yescharacteristic
139C2×C4×D4C2×D311yescharacteristic
140C2×C4×D4C2×D311yescharacteristic
141C22×C4⋊C4—21no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97—31no—
151#397—61no—
152#397—61no—
153#390—31no—
154#390—31no—
155#390—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#385—21no—
162#385—21no—
163#384C2×D311yescharacteristic
164#382—31no—
165#380—21no—
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#340—31no—
175#340—31no—
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—24no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—81no—
199C22×C3×C4—81no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4#4112yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237#213—44no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241(C2×D3)⋊C4—81no—
242(C2×D3)⋊C4—81no—
243(C2×D3)⋊C4—81no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250Dic3⋊C4—81no—
251C4×Dic3—81no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—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—21no—
268C23×C4—21no—
269C2×D8—62no—
270C2×D8—62no—
271C2×D8—62no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—41no—
281C4×D4—41no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4#6711yescharacteristic
292C2×C4⋊C4#6711yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
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—61no—
306C2×C42C22×D311yescharacteristic
307C2×C42C22×D311yescharacteristic
308C2×C42C22×D311yescharacteristic
309#106—62no—
310C4⋊C8—121no—
311#102—62no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—121no—
316#102—121no—
317#97—121no—
318#95—21no—
319#95—21no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95#6711yescharacteristic
325#95#6711yescharacteristic
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—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—48no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—84no—
373C2×C3×C4—84no—
374C2×C3×C4AGL(1,ℤ/8)12yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C42−1+411yescharacteristic
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
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#23712yes—
411C2×D4#23912yes—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C2×#6711yescharacteristic
430C22×C4C2×#6711yescharacteristic
431C22×C4C2×#6711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433C22×C4C23×D311yescharacteristic
434D8—124no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—81no—
464C4⋊C4—241no—
465C4⋊C4#23512yes—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—241no—
472C22⋊C4—241no—
473C22⋊C4—241no—
474C22⋊C4—241no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—241no—
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—22no—
504C3×C4—42no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4C2×#12611yescharacteristic
513C3×C4#53911yescharacteristic
514C3×C4#55211yescharacteristic
515C3×C4#55311yescharacteristic
516Dic3—81no—
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—
533C23C22×#6711yescharacteristic
534D4—22no—
535D4—22no—
536D4—22no—
537D4—121no—
538D4—121no—
539D4—121no—
540D4—121no—
541D4#89814yes—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
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—48no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—121no—
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#93212yes—
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4#100411yescharacteristic
585C2×C4C2×#23911yescharacteristic
586C2×C4#100711yescharacteristic
587C8—121no—
588C8—121no—
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#579111yescharacteristic
615C22C2×#93211yescharacteristic
616C22#580611yescharacteristic
617C22#581711yescharacteristic
618C22#582211yescharacteristic
619C22#583511yescharacteristic
620C22#583711yescharacteristic
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—241no—
631C4—241no—
632C4—241no—
633C4C2×#89811yescharacteristic
634C4#551311yescharacteristic
635C4#551411yescharacteristic
636C4#551511yescharacteristic
637C3#2756811yescharacteristic, complemented, Sylow
638C2—44no—
639C2—481no—
640C2#8217511yescharacteristic
641C2#8231111yescharacteristic
642C2#8256811yescharacteristic
643C2#8348711yescharacteristic
644C2#8351211yescharacteristic
645C2#8356911yescharacteristic
646C2#8363911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×D4×C4⋊C4C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterD4×C4⋊C4D3
hypocenterC3#27568
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×D4×C4⋊C4C2semidirect
2. #73694C2semidirect
6. #71609C2semidirect
7. #71607C2semidirect
8. #71605C2semidirect
9. #69629C2semidirect
10. #69617C2semidirect
11. #69613C2semidirect
12. #69602C2semidirect
13. #69585C2semidirect
14. #69571C2semidirect
15. #69483C2semidirect
61. D4×C4⋊C4D3semidirect
637. C3#27568semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C4⋊C42
#736942
#716132
#716092
#716072
#716052
#696292
#696172
#696132
#696022
#695852
#695712
#694832
#275683