Finite Group Databasebeta

← #10716613

Subgroups — #10716613

Order 768.

2182 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
1C22×#872C211yesmaximal, characteristic
2C2×C4×Dic3⋊C4C212yesmaximal, complemented
3C2×#4667C211yesmaximal, characteristic
4C2×#4666C211yesmaximal, characteristic
5C3×C4×#95C211yesmaximal, characteristic, complemented
6#71537C212yesmaximal
7#71534C211yesmaximal, characteristic, complemented
8#69408C212yesmaximal
9Dic3⋊#95C212yesmaximal, complemented
10#69401C211yesmaximal, characteristic, complemented
11Dic3×#95C211yesmaximal, characteristic, complemented
12#13801—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C42C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C22×C4×Dic3C2212yes—
26C2×#872—28no—
27C2×#872C418yescomplemented
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2211yescharacteristic
38C4×Dic3⋊C4—28no—
39#4667C418yescomplemented
40#4666C418yescomplemented
41C22×#117—31no—
42C2×#380—31no—
43C2×C4×C4⋊C4—32no—
44C2×C4×C22⋊C4—31no—
45#1437—32no—
46#1437—32no—
47C4⋊#95—32no—
48#1434—31no—
49#1434—31no—
50C4×#95—31no—
51C4×#95D311yescharacteristic, complemented
52C24×D3C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×C4×D3—24no—
64C22×C4×D3—24no—
65C22×C4×D3C2×C414yes—
66C22×C4×D3C2×C414yes—
67C22×C4×D3C2×C414yes—
68C2×C3×C42—24no—
69C2×C3×C42—24no—
70C2×C3×C42—24no—
71C2×C3×C42—42no—
72C2×C3×C42C2×C414yes—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—42no—
78C2×(C2×D3)⋊C4—42no—
79C2×(C2×D3)⋊C4C2×C418yes—
80C2×(C2×D3)⋊C4C2×C414yes—
81C2×(C2×D3)⋊C4C2×C414yes—
82C2×(C2×D3)⋊C4C2×C418yes—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4C2×C418yes—
87C2×Dic3⋊C4C2×C414yes—
88C2×Dic3⋊C4C2×C414yes—
89C2×Dic3⋊C4D418yes—
90C2×C4×Dic3—24no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—42no—
96C2×C4×Dic3C2×C418yes—
97C2×C4×Dic3C2×C414yes—
98#872—232no—
99C3×#95—24no—
100C3×#95C2×C414yes—
101C3×#95C2×C414yes—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95C2×C414yes—
108C3⋊#95C2×C414yes—
109C24×C4—31no—
110C2×#117—38no—
111C2×#117—68no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—32no—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4—32no—
119C22×C42—31no—
120C22×C42—32no—
121C22×C42—32no—
122C22×C42C2×D312yes—
123C22×C42C2×D311yescharacteristic
124#380—38no—
125C4×C4⋊C4—68no—
126C4×C22⋊C4—38no—
127C2×#95—31no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95C2×D312yes—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D311yescharacteristic
137C24×C3C2411yescharacteristic
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—21no—
146C23×D3C22×C412yes—
147C23×D3C22×C412yes—
148C23×D3C22×C412yes—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4C22×C414yes—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C22×C412yes—
172C22×C3×C4C22×C412yes—
173C22×C3×C4C22×C412yes—
174C22×C3×C4C22×C414yes—
175C22×C3×C4C2×D414yes—
176C22×C3×C4#4114yes—
177C22×C3×C4#4114yes—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—21no—
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—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3C22×C414yes—
198C22×Dic3C22×C414yes—
199C22×Dic3C22×C412yes—
200C22×Dic3C22×C412yes—
201C22×Dic3C22×C412yes—
202C22×Dic3C22×C412yes—
203C22×Dic3C22×C412yes—
204C22×Dic3C2×D414yes—
205C22×Dic3C2×D414yes—
206C22×Dic3#4114yes—
207C2×C4×D3—28no—
208C2×C4×D3—28no—
209C2×C4×D3—216no—
210C2×C4×D3—216no—
211C2×C4×D3C42116yescomplemented
212C3×C42—28no—
213C3×C42—48no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—216no—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—48no—
218Dic3⋊C4—28no—
219Dic3⋊C4—216no—
220Dic3⋊C4C22⋊C4116yescomplemented
221C4×Dic3—28no—
222C4×Dic3—216no—
223C4×Dic3—48no—
224C25—31no—
225C23×C4—31no—
226C23×C4—31no—
227C23×C4—34no—
228C23×C4—34no—
229C23×C4—34no—
230C23×C4—32no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—64no—
234C23×C4—64no—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D312yes—
239C23×C4C22×D311yescharacteristic
240#117—632no—
241C2×C4⋊C4—34no—
242C2×C4⋊C4—38no—
243C2×C4⋊C4—34no—
244C2×C4⋊C4—38no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C22⋊C4—34no—
249C2×C22⋊C4—34no—
250C2×C22⋊C4—38no—
251C2×C22⋊C4—38no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—64no—
254C2×C22⋊C4—64no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—122no—
257C2×C22⋊C4—122no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—34no—
262C2×C42—38no—
263C2×C42—42no—
264C2×C42—64no—
265C2×C42—64no—
266C2×C42—64no—
267C2×C42—64no—
268C2×C42—64no—
269C2×C42—122no—
270C2×C42C4×D314yes—
271#95—24no—
272#95—34no—
273#95—34no—
274#95—64no—
275#95—64no—
276#95—64no—
277#95—122no—
278#95—122no—
279#95C4×D314yes—
280#95C4×D314yes—
281C23×C3C23×C411yescharacteristic
282C23×C3C23×C411yescharacteristic
283C23×C3C23×C411yescharacteristic
284C23×C3C22×D412yes—
285C23×C3C2×#4112yes—
286C23×C3C2×#4112yes—
287C23×C3C2×#4112yes—
288C23×C3H(5,𝔽2)11yescharacteristic
289C23×C3H(5,𝔽2)11yescharacteristic
290C23×C32−1+411yescharacteristic
291C23×C32−1+411yescharacteristic
292C22×D3—22no—
293C22×D3—24no—
294C22×D3—22no—
295C22×D3—24no—
296C22×D3—24no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—24no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3C2×C4214yes—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4C2×C4214yes—
342C2×C3×C4C2×C22⋊C414yes—
343C2×C3×C4#11718yescomplemented
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—22no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—24no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3C2×C4214yes—
368C2×Dic3C2×C22⋊C414yes—
369C2×Dic3C2×C22⋊C414yes—
370C2×Dic3#11714yes—
371C2×Dic3C4×D418yes—
372C2×Dic3C4×D418yes—
373C4×D3—232no—
374C24—32no—
375C24—32no—
376C24—32no—
377C24—62no—
378C24—62no—
379C24—62no—
380C24—61no—
381C24—61no—
382C24—62no—
383C24—61no—
384C24—61no—
385C24C23×D311yescharacteristic
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—34no—
409C22×C4—34no—
410C22×C4—32no—
411C22×C4—34no—
412C22×C4—316no—
413C22×C4—34no—
414C22×C4—34no—
415C22×C4—32no—
416C22×C4—68no—
417C22×C4—68no—
418C22×C4—616no—
419C22×C4—616no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4C2×C4×D314yes—
440C22×C4C2×C4×D312yes—
441C22×C4C2×C4×D312yes—
442C22×C4C2×C4×D312yes—
443C22×C4C2×C4×D312yes—
444C22×C4C2×C4×D314yes—
445C22×C4C2×C4×D312yes—
446C22×C4#23514yes—
447C22×C4#23514yes—
448C22×C4D3×D414yes—
449C4⋊C4—316no—
450C4⋊C4—68no—
451C4⋊C4—616no—
452C22⋊C4—68no—
453C22⋊C4—616no—
454C22⋊C4—68no—
455C22⋊C4—128no—
456C42—28no—
457C42—48no—
458C42—68no—
459C42—616no—
460C42—128no—
461C22×C3C22×C4211yescharacteristic
462C22×C3C22×C22⋊C411yescharacteristic
463C22×C3C2×#11712yes—
464C22×C3C2×#11711yescharacteristic
465C22×C3C2×C4×D412yes—
466C22×C3C2×C4×D412yes—
467C22×C3C4×#4111yescharacteristic
468C22×C3C4×#4111yescharacteristic
469C22×C3C4×#4112yes—
470C22×C3#51811yescharacteristic
471C22×C3#51911yescharacteristic
472C22×C3#52011yescharacteristic
473C22×C3#52011yescharacteristic
474C22×C3#52011yescharacteristic
475C22×C3#52011yescharacteristic
476C22×C3#54612yes—
477C22×C3#54712yes—
478C22×C3#54812yes—
479C22×C3#55012yes—
480C22×C3#55212yes—
481C22×C3#55312yes—
482C22×C3#55512yes—
483C22×C3#55612yes—
484C2×D3—24no—
485C2×D3—24no—
486C2×D3—24no—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—44no—
495C3×C4—24no—
496C3×C4—28no—
497C3×C4—24no—
498C3×C4—42no—
499C3×C4—44no—
500Dic3—24no—
501Dic3—42no—
502Dic3—44no—
503Dic3—44no—
504Dic3C4×C22⋊C418yes—
505C23—34no—
506C23—64no—
507C23—62no—
508C23—62no—
509C23—64no—
510C23—62no—
511C23—62no—
512C23—62no—
513C23—62no—
514C23—64no—
515C23—64no—
516C23—62no—
517C23—62no—
518C23—62no—
519C23—62no—
520C23—122no—
521C23—122no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23C22×C4×D311yescharacteristic
529C23C22×C4×D311yescharacteristic
530C23C22×C4×D311yescharacteristic
531C23C2×#23512yes—
532C23C2×#23512yes—
533C23C2×D3×D412yes—
534C23#100411yescharacteristic
535C23#100711yescharacteristic
536C23D3×#4112yes—
537C23#100911yescharacteristic
538C23#101011yescharacteristic
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—34no—
560C2×C4—34no—
561C2×C4—34no—
562C2×C4—38no—
563C2×C4—38no—
564C2×C4—34no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—632no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4C42×D314yes—
596C2×C4#87218yescomplemented
597C2×C4D3×C22⋊C414yes—
598C2×C3C2×C4×C22⋊C411yescharacteristic
599C2×C3C4×#11711yescharacteristic
600C2×C3#227411yescharacteristic
601C2×C3#233911yescharacteristic
602C2×C3#234111yescharacteristic
603C2×C3#234611yescharacteristic
604C2×C3#234711yescharacteristic
605C2×C3#235212yes—
606C2×C3#235412yes—
607C2×C3#236212yes—
608C2×C3#236312yes—
609D3—44no—
610C22—64no—
611C22—64no—
612C22—64no—
613C22—122no—
614C22—122no—
615C22—124no—
616C22—122no—
617C22—124no—
618C22—122no—
619C22—124no—
620C22—124no—
621C22C2×C42×D311yescharacteristic
622C22C2×#87212yes—
623C22C4×#23511yescharacteristic
624C22C4×#23511yescharacteristic
625C22C4×#23512yes—
626C22C2×D3×C22⋊C411yescharacteristic
627C22#549012yes—
628C22#550411yescharacteristic
629C22#550712yes—
630C22D3×#11711yescharacteristic
631C22#552311yescharacteristic
632C22#553112yes—
633C22#553212yes—
634C22#554012yes—
635C22C4×D3×D412yes—
636C22C4×D3×D412yes—
637C22#554511yescharacteristic
638C22#554611yescharacteristic
639C22#555212yes—
640C22#555312yes—
641C22#556812yes—
642C22#557211yescharacteristic
643C22#557411yescharacteristic
644C4—24no—
645C4—24no—
646C4—28no—
647C4—38no—
648C4—42no—
649C4—44no—
650C4—64no—
651C4—124no—
652C4—122no—
653C4—124no—
654C3#1380111yescharacteristic, complemented, Sylow
655C2—124no—
656C2C4×#87211yescharacteristic
657C2#7336811yescharacteristic
658C2#7339211yescharacteristic
659C2#7345411yescharacteristic
660C2#7346112yes—
661C2#7347012yes—
662C2C4×D3×C22⋊C411yescharacteristic
663C2#7363511yescharacteristic
664C2#7363912yes—
665C2#7364812yes—
666C2#7375011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC22×C3C22×C42
FittingC3×C4×#95C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenterC4×#95D3
hypocenterC3#13801
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic3⋊C4C2semidirect
5. C3×C4×#95C2semidirect
7. #71534C2semidirect
9. Dic3⋊#95C2semidirect
10. #69401C2semidirect
11. Dic3×#95C2semidirect
27. C2×#872C4semidirect
39. #4667C4semidirect
40. #4666C4semidirect
51. C4×#95D3semidirect
211. C2×C4×D3C42semidirect
220. Dic3⋊C4C22⋊C4semidirect
343. C2×C3×C4#117semidirect
596. C2×C4#872semidirect
654. C3#13801semidirect

Maximal subgroups

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

subgroupindex
C22×#8722
C2×C4×Dic3⋊C42
C2×#46672
C2×#46662
C3×C4×#952
#715372
#715342
#694082
Dic3⋊#952
#694012
Dic3×#952
#138013