Finite Group Databasebeta

← #10813615

Subgroups — #10813615

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic3⋊C4C211yesmaximal, characteristic
2C2×C4×#216C211yesmaximal, characteristic
3C2×#4790C211yesmaximal, characteristic
4C2×#4790C211yesmaximal, characteristic
5#75981C211yesmaximal, characteristic
6C3×#1863C211yesmaximal, characteristic
7#70642C211yesmaximal, characteristic
8#70606C211yesmaximal, characteristic
9#70236C211yesmaximal, characteristic
10#70221C211yesmaximal, characteristic
11#69619C211yesmaximal, characteristic
12#68904C211yesmaximal, characteristic
13#68859C211yesmaximal, characteristic
14#68683C211yesmaximal, characteristic
15#68651C211yesmaximal, characteristic
16C4⋊C4×Q16—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C4×#216—24no—
36#4984C2211yescharacteristic
37#4976—21no—
38#4975—21no—
39#4975—21no—
40#4975C2211yescharacteristic
41#4975C2211yescharacteristic
42#4972—21no—
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C4×Dic3⋊C4—21no—
49C4×Dic3⋊C4C2211yescharacteristic
50#4928—21no—
51#4928C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4790—24no—
54#4790C418yescomplemented
55#4649—21no—
56#4647—21no—
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×C4×Q16—31no—
64C2×C4×Q16—31no—
65C2×C4×Q16—31no—
66Q8×C4⋊C4—31no—
67Q8×C4⋊C4—31no—
68#1942—31no—
69#1942—31no—
70#1863—31no—
71#1863—31no—
72#1863—31no—
73#1863D311yescharacteristic
74#1845—31no—
75#1803—31no—
76#1803—31no—
77C8×C4⋊C4—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93Q8×Dic3—41no—
94Q8×Dic3C2×C414yes—
95C2×#216C2×C414yes—
96C2×#216D414yes—
97C2×#216Q814yes—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4D411yescharacteristic
114C2×Dic3⋊C4D411yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#887—22no—
125#887—22no—
126#887—41no—
127#887C2×C414yes—
128C4×Dic6—22no—
129C4×Dic6—41no—
130C3×#107—41no—
131C3×#103—22no—
132C3×#103C2×C414yes—
133C3×#95—21no—
134#835—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#818—41no—
141#816—22no—
142#816C2×C414yes—
143Dic3⋊C8—22no—
144C8×Dic3—22no—
145C8×Dic3C2×C414yes—
146#810—22no—
147#807—22no—
148#807—22no—
149#807C2×C414yes—
150C4×C3⋊C8—22no—
151C22×Q16—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155C2×C4×Q8—31no—
156C2×C4×Q8—31no—
157C2×C4×Q8C2×D311yescharacteristic
158C4×Q16—38no—
159C4×Q16—64no—
160C4×Q16—64no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#107C2×D311yescharacteristic
166C2×C4⋊C8—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103C2×D311yescharacteristic
172C2×#103C2×D311yescharacteristic
173C2×C4×C8—31no—
174C2×C4×C8—31no—
175C2×C4×C8—31no—
176#389—31no—
177#389—31no—
178#389—31no—
179#389—61no—
180#389—61no—
181#389—61no—
182#389—61no—
183#389C2×D311yescharacteristic
184#384—31no—
185#384—31no—
186#384—61no—
187#384—61no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4—61no—
190C4×C4⋊C4—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195#336—31no—
196#336C2×D311yescharacteristic
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C22×C412yes—
201C2×C3×Q8C2×D412yes—
202C2×C3×Q8C2×Q812yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6C22×C412yes—
222C2×Dic6C2×D412yes—
223C2×Dic6C2×Q812yes—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8C22×C412yes—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4C22×C412yes—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—42no—
241#216C4⋊C4116yescomplemented
242C4⋊Dic3—21no—
243C4⋊Dic3—24no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—42no—
249C4⋊Dic3—81no—
250C4⋊Dic3C22×C412yes—
251Dic3⋊C4—24no—
252Dic3⋊C4—22no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260Dic3⋊C4—81no—
261Dic3⋊C4Q1614yescomplemented
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—24no—
265C4×Dic3—24no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3C22×C412yes—
270C4×Dic3#4112yes—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8C22×C412yes—
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8C2×Q812yes—
277C22×Q8—31no—
278C22×Q8C22×D311yescharacteristic
279C2×Q16—34no—
280C2×Q16—34no—
281C2×Q16—34no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C4×Q8—22no—
287C4×Q8—34no—
288C4×Q8—34no—
289C4×Q8—41no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—121no—
296C4×Q8—121no—
297C4×Q8—121no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C42—21no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—34no—
330#107—41no—
331#107—62no—
332#107—62no—
333#107—121no—
334C4⋊C8—62no—
335#103—22no—
336#103—34no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103C4×D314yes—
341C4×C8—34no—
342C4×C8—62no—
343C4×C8—62no—
344#95—21no—
345#95—31no—
346#95—31no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8C2×C4⋊C414yes—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
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×C4C2×Q1612yes—
373C2×C3×C4C23×C411yescharacteristic
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×Q811yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C42−1+411yescharacteristic
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3C4×D412yes—
399C2×Dic3C4×D412yes—
400C2×Dic3C2×Q1612yes—
401C2×Dic3C2×Q1612yes—
402C2×Dic3#13512yes—
403Dic6—22no—
404Dic6—22no—
405Dic6—22no—
406Dic6C2×C4⋊C414yes—
407C3×C8—22no—
408C3×C8—41no—
409C3⋊C8—41no—
410C3⋊C8C2×C4⋊C414yes—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—32no—
415C2×Q8—32no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8C2×C4×D312yes—
421C2×Q8D3×D412yes—
422C2×Q8D3×Q812yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4C23×D311yescharacteristic
438Q16—316no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—32no—
445C2×C8—32no—
446C2×C8—32no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8C2×C4×D312yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—34no—
457C4⋊C4—32no—
458C4⋊C4—42no—
459C4⋊C4—62no—
460C4⋊C4—64no—
461C4⋊C4—64no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—81no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—241no—
474C4⋊C4—241no—
475C4⋊C4—241no—
476C4⋊C4—241no—
477C4⋊C4—241no—
478C4⋊C4C2×C4×D312yes—
479C42—21no—
480C42—21no—
481C42—32no—
482C42—32no—
483C42—42no—
484C42—64no—
485C42—64no—
486C42—61no—
487C42—61no—
488C42—122no—
489C42—122no—
490C42—122no—
491C22×C3C2×C4×D411yescharacteristic
492C22×C3D4211yescharacteristic
493C22×C3D4×Q811yescharacteristic
494C22×C3C22×Q1611yescharacteristic
495C22×C3C2×#13511yescharacteristic
496C22×C3#57611yescharacteristic
497C22×C3#57811yescharacteristic
498C3×C4—24no—
499C3×C4—22no—
500C3×C4—42no—
501C3×C4—41no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4C22×C4⋊C411yescharacteristic
505C3×C4#52011yescharacteristic
506C3×C4#54711yescharacteristic
507C3×C4#55411yescharacteristic
508Dic3—24no—
509Dic3—22no—
510Dic3—22no—
511Dic3—42no—
512Dic3—41no—
513Dic3—81no—
514Dic3—81no—
515Dic3C4×Q1614yescomplemented
516C23C2×D3×D411yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—22no—
520Q8—34no—
521Q8—62no—
522Q8—62no—
523Q8—62no—
524Q8D3×C4⋊C414yes—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—32no—
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—62no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4D3×Q1612yes—
567C2×C4C22×C4×D311yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4#100411yescharacteristic
570C2×C4C2×D3×Q811yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4#101011yescharacteristic
573C8—22no—
574C8—34no—
575C8—41no—
576C8—121no—
577C2×C3D4×C4⋊C411yescharacteristic
578C2×C3C2×C4×Q1611yescharacteristic
579C2×C3#295111yescharacteristic
580C2×C3D4×Q1611yescharacteristic
581C2×C3#328711yescharacteristic
582C2×C3Q8×Q1611yescharacteristic
583C2×C3#338311yescharacteristic
584C22C4×D3×D411yescharacteristic
585C22#559011yescharacteristic
586C22#563211yescharacteristic
587C22C2×D3×Q1611yescharacteristic
588C22D3×#13511yescharacteristic
589C22#577411yescharacteristic
590C22#577911yescharacteristic
591C4—22no—
592C4—24no—
593C4—34no—
594C4—41no—
595C4—42no—
596C4—62no—
597C4—64no—
598C4—62no—
599C4—81no—
600C4—81no—
601C4—121no—
602C4—122no—
603C4—241no—
604C4—241no—
605C4C2×D3×C4⋊C411yescharacteristic
606C4#554611yescharacteristic
607C4#563611yescharacteristic
608C4#565511yescharacteristic
609C3C4⋊C4×Q1611yescharacteristic, complemented, Sylow
610C2#7439311yescharacteristic
611C2C4×D3×Q1611yescharacteristic
612C2#7977811yescharacteristic
613C2#8013111yescharacteristic
614C2#8021111yescharacteristic
615C2#8110211yescharacteristic
616C2#8122211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socle, prime radicalC23×C3C22×D4
cosocleC22×C3×C4C24
hypercenter#1863D3
hypocenterC3C4⋊C4×Q16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
54. #4790C4semidirect
241. #216C4⋊C4semidirect
261. Dic3⋊C4Q16semidirect
515. Dic3C4×Q16semidirect
609. C3C4⋊C4×Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3⋊C42
C2×C4×#2162
C2×#47902
#759812
C3×#18632
#706422
#706062
#702362
#702212
#696192
#689042
#688592
#686832
#686512
C4⋊C4×Q163