Finite Group Databasebeta

← #10796329

Subgroups — #10796329

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×M16C211yesmaximal, characteristic, complemented
2C2×#5524C211yesmaximal, characteristic
3C2×#5129C211yesmaximal, characteristic, complemented
4C2×#5110C211yesmaximal, characteristic, complemented
5C22×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C3×#1729C211yesmaximal, characteristic, complemented
7#71579C211yesmaximal, characteristic, complemented
8#68669C214yesmaximal, complemented
9#68668C214yesmaximal, complemented
10#20897—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C23×C3⋊C8C2211yescharacteristic
18C2×D3×M16—24no—
19C2×D3×M16—24no—
20C2×D3×M16—24no—
21C22×ΓL(1,𝔽25)C2212yes—
22C22×D3×C8C2212yes—
23#5524—24no—
24#5524—24no—
25C2×#897C411yescharacteristic
26C2×#897C411yescharacteristic
27C2×#895C411yescharacteristic
28C2×#895C411yescharacteristic
29C2×#886C412yes—
30C2×#882C412yes—
31C2×C4×D12C2212yes—
32C2×C4×D12C2212yes—
33#5129—24no—
34#5110—24no—
35C2×(C2×D3)⋊C8—24no—
36C2×(C2×D3)⋊C8C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38C2×(C2×D3)⋊C8C2214yescomplemented
39C2×(C2×D3)⋊C8C2214yescomplemented
40C3×#336C2214yescomplemented
41#4524C2214yescomplemented
42C23×M16—31no—
43C2×#525—31no—
44C2×#408—31no—
45C2×#408—31no—
46C22×C22⋊C8—31no—
47#1870—34no—
48#1869—34no—
49#1729—31no—
50#1729D311yescharacteristic, complemented
51C24×D3C2×C411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C22×#67C2×C412yes—
54C23×Dic3C2×C411yescharacteristic
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235—22no—
60C2×#235—22no—
61C22×D12C2×C411yescharacteristic
62C22×D12C2×C412yes—
63C22×D12C2×C412yes—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3C2×C411yescharacteristic
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3D414yes—
71C22×C4×D3D414yes—
72C22×C4×D3C2312yes—
73C22×C4×D3C2311yescharacteristic
74C22×C4×D3C2311yescharacteristic
75C22×Dic6C2×C411yescharacteristic
76C2×C3×M16—22no—
77C2×C3×M16—22no—
78C2×C3×M16D414yescomplemented
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4C2×C411yescharacteristic
87C2×C3×C4⋊C4C2×C411yescharacteristic
88C2×C3×C22⋊C4C2×C412yes—
89C2×C3×C42C2312yes—
90C2×(C2×D3)⋊C4C2×C412yes—
91C2×(C2×D3)⋊C4C2×C412yes—
92C2×(C2×D3)⋊C4C2×C412yes—
93C2×(C2×D3)⋊C4C2×C412yes—
94C2×C4⋊Dic3C2×C412yes—
95C2×C4⋊Dic3C2×C412yes—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208D414yescomplemented
99C22×C3⋊C8—22no—
100C22×C3⋊C8—22no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2312yes—
104D3×M16—232no—
105C2×ΓL(1,𝔽25)—24no—
106C2×ΓL(1,𝔽25)—24no—
107C2×ΓL(1,𝔽25)—24no—
108C2×D3×C8—24no—
109C2×D3×C8—24no—
110C2×D3×C8—24no—
111#897—24no—
112#897—24no—
113#895—24no—
114#895—24no—
115#886—28no—
116#882—28no—
117C4×D12—24no—
118C4×D12—24no—
119C4×D12—24no—
120C4×D12—24no—
121(C2×D3)⋊C8—24no—
122(C2×D3)⋊C8—24no—
123(C2×D3)⋊C8—28no—
124(C2×D3)⋊C8—28no—
125C22×#41—31no—
126C24×C4—31no—
127C22×M16—31no—
128C22×M16—32no—
129C22×M16—64no—
130C22×M16—64no—
131C22×M16—64no—
132C22×M16C2×D311yescharacteristic
133C23×C8—31no—
134C23×C8—32no—
135C23×C8C2×D311yescharacteristic
136#525—64no—
137#525—64no—
138C2×#123—32no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—31no—
141C2×#121—31no—
142C2×#121—31no—
143C2×#120—32no—
144C2×C4×D4—32no—
145C2×C4×D4—32no—
146C2×#117C2×D311yescharacteristic
147#408—64no—
148#408—64no—
149C2×C22⋊C8—32no—
150C2×C22⋊C8—32no—
151C2×C22⋊C8—34no—
152C2×C22⋊C8—34no—
153C2×C22⋊C8—64no—
154#336—34no—
155#336C2×D314yescomplemented
156C24×C3C22×C411yescharacteristic
157C23×D3—22no—
158C23×D3—22no—
159C23×D3—24no—
160C23×D3—42no—
161C23×D3C22⋊C414yes—
162C23×D3C22×C412yes—
163C23×D3C22×C411yescharacteristic
164C23×D3C22×C411yescharacteristic
165C22×C3×C4C22×C412yes—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—24no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3C22⋊C414yes—
181C22×Dic3C22×C412yes—
182C22×Dic3C22×C411yescharacteristic
183C22×Dic3C22×C411yescharacteristic
184#235—44no—
185#235—44no—
186#235—44no—
187#235—44no—
188C2×D12—24no—
189C2×D12—24no—
190C2×D12—28no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—44no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—28no—
200C2×C4×D3—28no—
201C2×C4×D3—28no—
202C2×C4×D3—28no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3C22⋊C414yes—
218C2×C4×D3C22⋊C414yes—
219C2×C4×D3C2×D412yes—
220C2×C4×D3C2×D412yes—
221C2×C4×D3C2×D412yes—
222C2×C4×D3C2×D412yes—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C3×M16—28no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—44no—
233C2×C3×C8C2×D414yes—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C22⋊C4—24no—
237C3×C42—22no—
238C3×C42—22no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—24no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—24no—
243C4⋊Dic3—24no—
244C4⋊Dic3—24no—
245#208—28no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—44no—
252C2×C3⋊C8C2×D414yes—
253ΓL(1,𝔽25)—216no—
254D3×C8—216no—
255C25—31no—
256C2×#41—62no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C22×Q8—31no—
263C22×D4—32no—
264C22×D4—32no—
265C22×D4—31no—
266C22×D4—32no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—34no—
271C23×C4—34no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—32no—
275C23×C4—62no—
276C23×C4—62no—
277C23×C4—62no—
278C23×C4—62no—
279C23×C4C22×D311yescharacteristic
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—34no—
283C2×M16—64no—
284C2×M16—64no—
285C2×M16—632no—
286C2×M16—62no—
287C2×M16—62no—
288C2×M16—64no—
289C2×M16D1214yescomplemented
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—32no—
294C22×C8—32no—
295C22×C8—64no—
296C22×C8—64no—
297C22×C8—64no—
298C22×C8—62no—
299C22×C8—62no—
300C22×C8—62no—
301C22×C8C22×D312yes—
302C22×C8C22×D312yes—
303#123—68no—
304C22⋊Q8—64no—
305C22⋊Q8—64no—
306#121—64no—
307#121—64no—
308#120—68no—
309C4×D4—64no—
310C4×D4—64no—
311C4×D4—64no—
312C4×D4—64no—
313#117—22no—
314#117—22no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4C4×D311yescharacteristic
318C2×C4⋊C4C4×D311yescharacteristic
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—32no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4C4×D312yes—
324C2×C42C22×D312yes—
325C22⋊C8—68no—
326C22⋊C8—64no—
327C22⋊C8—64no—
328C22⋊C8—68no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3C2×C22⋊C412yes—
332C23×C3C4×D412yes—
333C23×C3C4×D412yes—
334C23×C3C23×C411yescharacteristic
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—22no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—28no—
343C22×D3—28no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—24no—
348C22×D3—24no—
349C22×D3—44no—
350C22×D3—48no—
351C22×D3—48no—
352C22×D3C2×C22⋊C412yes—
353C22×D3C2×C22⋊C412yes—
354C22×D3#13114yes—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4C2×C22⋊C412yes—
360C2×C3×C4C4×D412yes—
361C2×C3×C4C4×D412yes—
362C2×C3×C4#12014yes—
363C2×C3×C4#12114yes—
364C2×C3×C4#12114yes—
365C2×C3×C4#12314yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—48no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—28no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3C2×C22⋊C412yes—
382C2×Dic3C2×C22⋊C412yes—
383D12—48no—
384D12—48no—
385D12—44no—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—22no—
391C4×D3—22no—
392C4×D3—22no—
393C4×D3—22no—
394C4×D3—28no—
395C4×D3—28no—
396C4×D3—42no—
397C4×D3—42no—
398C4×D3—42no—
399C4×D3—42no—
400Dic6—44no—
401C3×C8—24no—
402C3×C8—42no—
403C3⋊C8—24no—
404C3⋊C8—42no—
405C24—31no—
406C24—31no—
407C24—34no—
408C24—32no—
409C24—64no—
410C24—62no—
411C24—62no—
412C24—122no—
413C24C2×C4×D311yescharacteristic
414#41—124no—
415#41—124no—
416#41—124no—
417#41—124no—
418C2×Q8—62no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×D4—64no—
422C2×D4—64no—
423C2×D4—68no—
424C2×D4—64no—
425C2×D4—64no—
426C2×D4—64no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—124no—
431C22×C4—34no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—34no—
435C22×C4—34no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—68no—
446C22×C4—68no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—68no—
450C22×C4—68no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4C2×C4×D312yes—
466C22×C4C2×C4×D312yes—
467C22×C4C2×C4×D311yescharacteristic
468C22×C4C2×D1212yes—
469C22×C4#23512yes—
470C22×C4D3×D412yes—
471C22×C4D3×D412yes—
472C22×C4D3×D412yes—
473C22×C4#23912yes—
474C22×C4C23×D311yescharacteristic
475M16—28no—
476M16—616no—
477M16—68no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—34no—
484C2×C8—44no—
485C2×C8—616no—
486C2×C8—62no—
487C2×C8—62no—
488C2×C8—62no—
489C2×C8—62no—
490C2×C8—62no—
491C2×C8—124no—
492C2×C8C2×D1214yes—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C22⋊C4—24no—
498C22⋊C4—64no—
499C22⋊C4—64no—
500C22⋊C4—64no—
501C22⋊C4—64no—
502C42—22no—
503C42—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3#38614yes—
509C22×C3C22×C22⋊C411yescharacteristic
510C22×C3C2×C4×D411yescharacteristic
511C22×C3C2×C4×D411yescharacteristic
512C22×C3C2×#13112yes—
513C22×C3#56812yes—
514C2×D3—22no—
515C2×D3—24no—
516C2×D3—24no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—416no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—42no—
529C2×D3#40814yescomplemented
530C3×C4—42no—
531C3×C4—42no—
532C3×C4#38614yes—
533C3×C4C2×#12011yescharacteristic
534C3×C4C2×#12111yescharacteristic
535C3×C4C2×#12111yescharacteristic
536C3×C4C2×#12311yescharacteristic
537Dic3—22no—
538Dic3—22no—
539Dic3—42no—
540C23—22no—
541C23—22no—
542C23—32no—
543C23—34no—
544C23—32no—
545C23—68no—
546C23—61no—
547C23—62no—
548C23—61no—
549C23—68no—
550C23—61no—
551C23—62no—
552C23—62no—
553C23—61no—
554C23—62no—
555C23—62no—
556C23—64no—
557C23—64no—
558C23—62no—
559C23—124no—
560C23—128no—
561C23—128no—
562C23C4×D1212yes—
563C23D3×C22⋊C412yes—
564C23#89312yes—
565C23C22×C4×D311yescharacteristic
566Q8—124no—
567D4—128no—
568D4—128no—
569D4—128no—
570D4—124no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—68no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—68no—
583C2×C4—68no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4C4×D1212yes—
600C2×C4D3×C22⋊C412yes—
601C2×C4#88214yes—
602C2×C4#88414yes—
603C2×C4#89312yes—
604C2×C4#89414yes—
605C2×C4#89514yes—
606C2×C4C22×D1211yescharacteristic
607C2×C4C2×#23511yescharacteristic
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×D3×D411yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×#23911yescharacteristic
612C8—24no—
613C8—42no—
614C8—64no—
615C8—122no—
616C2×C3—24no—
617C2×C3C2×#38611yescharacteristic
618C2×C3C2×#40811yescharacteristic
619C2×C3#287911yescharacteristic
620C2×C3#293312yes—
621C2×C3#293312yes—
622D3—24no—
623D3—42no—
624D3—82no—
625C22—22no—
626C22—22no—
627C22—24no—
628C22—24no—
629C22—34no—
630C22—62no—
631C22—62no—
632C22—64no—
633C22—64no—
634C22—62no—
635C22—62no—
636C22—62no—
637C22—1216no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—124no—
644C22—124no—
645C22#466514yes—
646C22C2×C4×D1211yescharacteristic
647C22C2×D3×C22⋊C411yescharacteristic
648C22C2×#89311yescharacteristic
649C22D3×#13112yes—
650C22#575012yes—
651C4—42no—
652C4—42no—
653C4—62no—
654C4—62no—
655C4—122no—
656C4#466514yes—
657C4C2×#88211yescharacteristic
658C4C2×#88411yescharacteristic
659C4C2×#89411yescharacteristic
660C4C2×#89511yescharacteristic
661C3#2089711yescharacteristic, complemented, Sylow
662C2—24no—
663C2—64no—
664C2—122no—
665C2—242no—
666C2C2×#466511yescharacteristic
667C2#7855512yes—
668C2D3×#40811yescharacteristic
669C2#7866211yescharacteristic
670C2#7907512yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1729C2
prime radicalC2×#5524C2
cosocleC22×C3×C4C24
hypercenter#1729D3
hypocenterC3#20897
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
1. C22×D3×M16C2semidirect
3. C2×#5129C2semidirect
4. C2×#5110C2semidirect
5. C22×(C2×D3)⋊C8C2semidirect
6. C3×#1729C2semidirect
7. #71579C2semidirect
8. #68669C2semidirect
9. #68668C2semidirect
38. C2×(C2×D3)⋊C8C22semidirect
39. C2×(C2×D3)⋊C8C22semidirect
40. C3×#336C22semidirect
41. #4524C22semidirect
50. #1729D3semidirect
78. C2×C3×M16D4semidirect
98. C2×#208D4semidirect
155. #336C2×D3semidirect
289. C2×M16D12semidirect
529. C2×D3#408semidirect
661. C3#20897semidirect

Maximal subgroups

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

subgroupindex
C22×D3×M162
C2×#55242
C2×#51292
C2×#51102
C22×(C2×D3)⋊C82
C3×#17292
#715792
#686692
#686682
#208973