Finite Group Databasebeta

← #10720147

Subgroups — #10720147

Order 768.

1370 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4928C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic
3C2×C4×C4⋊Dic3C211yesmaximal, characteristic
4C3×C4×#95C211yesmaximal, characteristic
5#71535C211yesmaximal, characteristic
6#71535C211yesmaximal, characteristic
7#71535C211yesmaximal, characteristic
8#69421C212yesmaximal
9#69409C212yesmaximal
10#69406C212yesmaximal
11#69401C212yesmaximal
12#14132—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C4×C4⋊Dic3—24no—
37#4928—24no—
38#4928C418yescomplemented
39C2×#389—31no—
40C2×#389—31no—
41C2×C4×C4⋊C4—31no—
42#1445—32no—
43#1441—32no—
44#1436—32no—
45C4⋊#95—31no—
46C4⋊#95—31no—
47C4⋊#95—31no—
48#1434—32no—
49C4×#95D311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C2×C3×C42—22no—
62C2×C3×C42—22no—
63C2×C3×C42—22no—
64C2×C3×C42—22no—
65C2×C3×C42—22no—
66C2×C3×C42—22no—
67C2×C3×C42—22no—
68C2×C3×C42—22no—
69C2×C3×C42C2×C414yes—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3C2×C414yes—
79C2×C4⋊Dic3C2×C414yes—
80C2×C4⋊Dic3D414yes—
81C2×C4⋊Dic3Q814yes—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—42no—
85C2×Dic3⋊C4—42no—
86C2×Dic3⋊C4—42no—
87C2×Dic3⋊C4C2×C418yes—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3—42no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95Q814yes—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—42no—
99C3⋊#95—42no—
100C3⋊#95—42no—
101C3⋊#95—42no—
102C3⋊#95—42no—
103C3⋊#95C2×C418yes—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—32no—
110C22×C42—32no—
111C22×C42C2×D311yescharacteristic
112C22×C42C2×D311yescharacteristic
113C22×C42C2×D311yescharacteristic
114#389—38no—
115#389—64no—
116C4×C4⋊C4—64no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95C2×D311yescharacteristic
127C2×#95C2×D311yescharacteristic
128C2×#95C2×D311yescharacteristic
129C2×#95C2×D311yescharacteristic
130C24×C3C2411yescharacteristic
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×Q812yes—
170C22×C3×C4C2×Q812yes—
171C22×C3×C4C2×Q812yes—
172C22×C3×C4C2×Q812yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3C22×C414yes—
205C22×Dic3C22×C414yes—
206C22×Dic3C2×D414yes—
207C22×Dic3C2×Q814yes—
208C3×C42—28no—
209C3×C42—44no—
210C3×C42—44no—
211C4⋊Dic3—28no—
212C4⋊Dic3—28no—
213C4⋊Dic3—28no—
214C4⋊Dic3—28no—
215Dic3⋊C4—48no—
216Dic3⋊C4—48no—
217C4×Dic3—48no—
218C23×C4—32no—
219C23×C4—32no—
220C23×C4—32no—
221C23×C4—32no—
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D311yescharacteristic
229C2×C4⋊C4—34no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—34no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—38no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—122no—
245C2×C4⋊C4—122no—
246C2×C4⋊C4—122no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—64no—
256C2×C42—64no—
257C2×C42—122no—
258C2×C42C4×D314yes—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—38no—
263#95—64no—
264#95—64no—
265#95—64no—
266#95—122no—
267#95—122no—
268#95—122no—
269#95—122no—
270#95—122no—
271#95Dic614yes—
272C23×C3C23×C411yescharacteristic
273C23×C3C22×D411yescharacteristic
274C23×C3C22×Q811yescharacteristic
275C23×C3C22×Q811yescharacteristic
276C23×C3C2×#4111yescharacteristic
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4111yescharacteristic
279C23×C3C2×#4111yescharacteristic
280C23×C3C2×#4111yescharacteristic
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3H(5,𝔽2)11yescharacteristic
284C23×C3H(5,𝔽2)11yescharacteristic
285C23×C3H(5,𝔽2)11yescharacteristic
286C23×C32−1+411yescharacteristic
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4#11714yes—
329C2×C3×C4C4×D414yes—
330C2×C3×C4C4×Q814yes—
331C2×C3×C4C4×Q814yes—
332C2×C3×C4C22⋊Q814yes—
333C2×C3×C4#12314yes—
334C2×C3×C4#12514yes—
335C2×C3×C4#12614yes—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C24C23×D311yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—34no—
402C22×C4—34no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4C2×Dic612yes—
428C22×C4C2×Dic612yes—
429C22×C4C2×Dic612yes—
430C22×C4C2×C4×D312yes—
431C22×C4C2×C4×D312yes—
432C22×C4C2×C4×D312yes—
433C22×C4#23512yes—
434C22×C4#23512yes—
435C22×C4#23512yes—
436C22×C4#23512yes—
437C22×C4#23512yes—
438C22×C4D3×D412yes—
439C22×C4#23712yes—
440C22×C4#23712yes—
441C22×C4D3×Q812yes—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—128no—
447C4⋊C4—128no—
448C42—28no—
449C42—44no—
450C42—44no—
451C42—128no—
452C22×C3C2×#11711yescharacteristic
453C22×C3C2×C4×D411yescharacteristic
454C22×C3C2×C4×Q811yescharacteristic
455C22×C3C2×C4×Q811yescharacteristic
456C22×C3C4×#4111yescharacteristic
457C22×C3#51811yescharacteristic
458C22×C3#52011yescharacteristic
459C22×C3C2×C22⋊Q811yescharacteristic
460C22×C3C2×#12311yescharacteristic
461C22×C3C2×#12511yescharacteristic
462C22×C3C2×#12611yescharacteristic
463C22×C3#52911yescharacteristic
464C22×C3#52911yescharacteristic
465C22×C3#53311yescharacteristic
466C22×C3#53311yescharacteristic
467C22×C3#54011yescharacteristic
468C22×C3#54011yescharacteristic
469C22×C3#54211yescharacteristic
470C22×C3#54211yescharacteristic
471C22×C3#54611yescharacteristic
472C22×C3#54611yescharacteristic
473C22×C3D4×Q811yescharacteristic
474C22×C3#55011yescharacteristic
475C22×C3#55211yescharacteristic
476C22×C3#55211yescharacteristic
477C22×C3#55311yescharacteristic
478C22×C3#55311yescharacteristic
479C22×C3#55411yescharacteristic
480C22×C3#55411yescharacteristic
481C22×C3#55411yescharacteristic
482C22×C3#55411yescharacteristic
483C22×C3#55511yescharacteristic
484C22×C3#55511yescharacteristic
485C22×C3#55711yescharacteristic
486C22×C3Q8211yescharacteristic
487C3×C4—24no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4#38718yes—
494Dic3—44no—
495Dic3—44no—
496Dic3—44no—
497Dic3—44no—
498C23C22×Dic611yescharacteristic
499C23C22×C4×D311yescharacteristic
500C23C2×#23511yescharacteristic
501C23C2×#23511yescharacteristic
502C23C2×#23511yescharacteristic
503C23C2×#23511yescharacteristic
504C23C2×#23511yescharacteristic
505C23C2×D3×D411yescharacteristic
506C23C2×#23711yescharacteristic
507C23C2×#23711yescharacteristic
508C23C2×D3×Q811yescharacteristic
509C23#100911yescharacteristic
510C23#100911yescharacteristic
511C23#100911yescharacteristic
512C23#101011yescharacteristic
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4C4×Dic614yes—
583C2×C4#87214yes—
584C2×C4#87914yes—
585C2×C4#67⋊C414yes—
586C2×C4#88314yes—
587C2×C4#88714yes—
588C2×C4#88914yes—
589C2×C4#89614yes—
590C2×C3C2×#38711yescharacteristic
591C2×C3#229511yescharacteristic
592C2×C3#231911yescharacteristic
593C2×C3#236011yescharacteristic
594C2×C3#236311yescharacteristic
595C2×C3#236911yescharacteristic
596C2×C3#237011yescharacteristic
597C2×C3#242011yescharacteristic
598C2×C3#242311yescharacteristic
599C2×C3#245111yescharacteristic
600C2×C3#245411yescharacteristic
601C2×C3#249511yescharacteristic
602C2×C3#249711yescharacteristic
603C2×C3#250711yescharacteristic
604C2×C3#250811yescharacteristic
605C22C2×C4×Dic611yescharacteristic
606C22#547011yescharacteristic
607C22#547011yescharacteristic
608C22C2×#87211yescharacteristic
609C22C4×#23511yescharacteristic
610C22#547911yescharacteristic
611C22#547911yescharacteristic
612C22C2×#87911yescharacteristic
613C22C2×#67⋊C411yescharacteristic
614C22C2×#88311yescharacteristic
615C22C2×#88711yescharacteristic
616C22C2×#88911yescharacteristic
617C22C2×#89611yescharacteristic
618C22#552111yescharacteristic
619C22#552311yescharacteristic
620C22#553411yescharacteristic
621C22#553411yescharacteristic
622C22#553511yescharacteristic
623C22#553511yescharacteristic
624C22D4×Dic611yescharacteristic
625C22#554111yescharacteristic
626C22#554111yescharacteristic
627C22#554311yescharacteristic
628C22#554311yescharacteristic
629C22#555011yescharacteristic
630C22#555011yescharacteristic
631C22#555311yescharacteristic
632C22#556311yescharacteristic
633C22#556311yescharacteristic
634C22#556511yescharacteristic
635C22#556511yescharacteristic
636C22Q8×Dic611yescharacteristic
637C22#556711yescharacteristic
638C22#557911yescharacteristic
639C22#557911yescharacteristic
640C4—24no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—124no—
647C4—124no—
648C4—124no—
649C4—124no—
650C4#465218yes—
651C3#1413211yescharacteristic, complemented, Sylow
652C2#7330011yescharacteristic
653C2C2×#465211yescharacteristic
654C2#7341511yescharacteristic
655C2#7346511yescharacteristic
656C2#7346911yescharacteristic
657C2#7348411yescharacteristic
658C2#7351111yescharacteristic
659C2#7351311yescharacteristic
660C2#7358411yescharacteristic
661C2#7365611yescharacteristic
662C2#7365911yescharacteristic
663C2#7367411yescharacteristic
664C2#7367711yescharacteristic
665C2#7368411yescharacteristic
666C2#7368611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C4×#95C2
socle, prime radical, cosocleC24×C3C24
hypercenterC4×#95D3
hypocenterC3#14132
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
38. #4928C4semidirect
651. C3#14132semidirect

Maximal subgroups

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

subgroupindex
C2×#49282
C2×C4×C4⋊Dic32
C3×C4×#952
#715352
#694212
#694092
#694062
#694012
#141323