Finite Group Databasebeta

← #10720149

Subgroups — #10720149

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×#4984C211yesmaximal, characteristic
2C2×Dic3×C4⋊C4C211yesmaximal, characteristic
3C2×#4928C211yesmaximal, characteristic
4C3×C4⋊#95C211yesmaximal, characteristic
5#71563C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic
7C4×C3⋊#95C211yesmaximal, characteristic
8#69423C212yesmaximal
9#69411C212yesmaximal
10#69406C212yesmaximal
11#69403C212yesmaximal
12#14132—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2211yescharacteristic
21C22×C4×Dic3C2212yes—
22C22×C4×Dic3C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4984C418yescomplemented
37Dic3×C4⋊C4—24no—
38#4928—24no—
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⋊#95D311yescharacteristic
48#1434—32no—
49C4×#95—31no—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×Dic3C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2×C414yes—
66C2×C3×C4⋊C4Q814yes—
67C2×C3×C42—22no—
68C2×C3×C42—22no—
69C2×C3×C42—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—42no—
75C2×C4⋊Dic3—42no—
76C2×C4⋊Dic3C2×C414yes—
77C2×C4⋊Dic3C2×C418yes—
78C2×C4⋊Dic3D414yes—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—42no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C2×C4×Dic3—42no—
90C2×C4×Dic3C2×C414yes—
91C3×#95—42no—
92C3×#95—42no—
93C3⋊#95—22no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—42no—
100C3⋊#95—42no—
101C3⋊#95—42no—
102C3⋊#95C2×C418yes—
103C3⋊#95Q814yes—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4—32no—
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C4⋊C4C2×D311yescharacteristic
110C22×C42—31no—
111C22×C42—31no—
112C22×C42—32no—
113C22×C42C2×D311yescharacteristic
114#389—38no—
115#389—64no—
116C4×C4⋊C4—64no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—31no—
121C2×#95—31no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95C2×D312yes—
129C2×#95C2×D312yes—
130C24×C3C2411yescharacteristic
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4C22×C414yes—
156C22×C3×C4C22×C412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×Q814yes—
159C22×C3×C4C2×Q812yes—
160C22×C3×C4C2×Q812yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3C22×C414yes—
201C22×Dic3C22×C412yes—
202C22×Dic3C22×C412yes—
203C22×Dic3C2×D414yes—
204C22×Dic3C2×Q812yes—
205C22×Dic3C2×Q812yes—
206C22×Dic3#4112yes—
207C22×Dic3#4112yes—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—28no—
210C3×C42—44no—
211C4⋊Dic3—28no—
212C4⋊Dic3—28no—
213C4⋊Dic3—48no—
214Dic3⋊C4—48no—
215C4×Dic3—28no—
216C4×Dic3—48no—
217C4×Dic3—44no—
218C23×C4—31no—
219C23×C4—32no—
220C23×C4—31no—
221C23×C4—32no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4C22×D312yes—
225C23×C4C22×D312yes—
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D311yescharacteristic
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—34no—
234C2×C4⋊C4—34no—
235C2×C4⋊C4—38no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—122no—
243C2×C4⋊C4—122no—
244C2×C4⋊C4—122no—
245C2×C4⋊C4Dic614yes—
246C2×C4⋊C4C4×D314yes—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—34no—
251C2×C42—62no—
252C2×C42—62no—
253C2×C42—64no—
254C2×C42—64no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42—62no—
258C2×C42—122no—
259#95—38no—
260#95—34no—
261#95—42no—
262#95—42no—
263#95—62no—
264#95—62no—
265#95—64no—
266#95—64no—
267#95—64no—
268#95—62no—
269#95—122no—
270#95—122no—
271#95—122no—
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—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
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—24no—
298C2×C3×C4—24no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4#11714yes—
316C2×C3×C4C4×D414yes—
317C2×C3×C4C4×Q814yes—
318C2×C3×C4C22⋊Q814yes—
319C2×C3×C4#12314yes—
320C2×C3×C4#12514yes—
321C2×C3×C4#12614yes—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—42no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3C4×Q814yes—
364C24C23×D311yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—34no—
393C22×C4—32no—
394C22×C4—34no—
395C22×C4—32no—
396C22×C4—32no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4C2×Dic614yes—
432C22×C4C2×Dic612yes—
433C22×C4C2×C4×D314yes—
434C22×C4C2×C4×D312yes—
435C22×C4#23512yes—
436C22×C4D3×D412yes—
437C22×C4#23712yes—
438C22×C4#23712yes—
439C22×C4D3×Q812yes—
440C22×C4#23912yes—
441C22×C4#23912yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—128no—
447C4⋊C4—128no—
448C42—44no—
449C42—68no—
450C42—128no—
451C42—124no—
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—44no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—44no—
491C3×C4#38718yes—
492Dic3—24no—
493Dic3—42no—
494Dic3—44no—
495Dic3—44no—
496Dic3—42no—
497Dic3—42no—
498C23C22×Dic611yescharacteristic
499C23C22×C4×D311yescharacteristic
500C23C2×#23511yescharacteristic
501C23C2×D3×D411yescharacteristic
502C23C2×#23711yescharacteristic
503C23C2×#23711yescharacteristic
504C23#100411yescharacteristic
505C23#100411yescharacteristic
506C23C2×D3×Q811yescharacteristic
507C23C2×#23911yescharacteristic
508C23C2×#23911yescharacteristic
509C23D3×#4111yescharacteristic
510C23D3×#4111yescharacteristic
511C23#100911yescharacteristic
512C23#101011yescharacteristic
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—34no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—62no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—122no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—122no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—121no—
583C2×C4C4×Dic614yes—
584C2×C4#87714yes—
585C2×C4#87814yes—
586C2×C4#88314yes—
587C2×C4#89014yes—
588C2×C4#89314yes—
589C2×C4#89814yes—
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
606C22C2×#87711yescharacteristic
607C22C2×#87811yescharacteristic
608C22C2×#88311yescharacteristic
609C22C2×#89011yescharacteristic
610C22C2×#89311yescharacteristic
611C22#551111yescharacteristic
612C22C2×#89811yescharacteristic
613C22#551411yescharacteristic
614C22#552911yescharacteristic
615C22#553211yescharacteristic
616C22#553911yescharacteristic
617C22#553911yescharacteristic
618C22#554511yescharacteristic
619C22#554611yescharacteristic
620C22Q8×Dic611yescharacteristic
621C22#557011yescharacteristic
622C22C4×D3×Q811yescharacteristic
623C22C4×#23911yescharacteristic
624C22#559911yescharacteristic
625C22#560311yescharacteristic
626C22#561211yescharacteristic
627C22#561911yescharacteristic
628C22#562211yescharacteristic
629C22#562411yescharacteristic
630C22#563011yescharacteristic
631C22#563211yescharacteristic
632C22#568011yescharacteristic
633C22#568211yescharacteristic
634C22#569411yescharacteristic
635C22#569511yescharacteristic
636C22#571911yescharacteristic
637C22#572811yescharacteristic
638C22#573011yescharacteristic
639C22#573411yescharacteristic
640C4—44no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—64no—
645C4—122no—
646C4—124no—
647C4—122no—
648C4—124no—
649C4—122no—
650C4#465418yes—
651C3#1413211yescharacteristic, complemented, Sylow
652C2C2×#465411yescharacteristic
653C2#7365711yescharacteristic
654C2#7366011yescharacteristic
655C2#7389111yescharacteristic
656C2#7389411yescharacteristic
657C2#7423511yescharacteristic
658C2#7436511yescharacteristic
659C2#7598511yescharacteristic
660C2#7603511yescharacteristic
661C2#7604011yescharacteristic
662C2#7620511yescharacteristic
663C2#7629511yescharacteristic
664C2#7632411yescharacteristic
665C2#7647711yescharacteristic
666C2#7650811yescharacteristic
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
36. #4984C4semidirect
651. C3#14132semidirect

Maximal subgroups

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

subgroupindex
C2×#49842
C2×Dic3×C4⋊C42
C2×#49282
C3×C4⋊#952
#715632
C4×C3⋊#952
#694232
#694112
#694062
#694032
#141323