Finite Group Databasebeta

← #10836636

Subgroups — #10836636

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4877C211yesmaximal, characteristic, complemented
2C2×#4709C211yesmaximal, characteristic
3#76251C211yesmaximal, characteristic
4#76197C211yesmaximal, characteristic
5C3×#1944C211yesmaximal, characteristic, complemented
6#70637C211yesmaximal, characteristic
7#70582C211yesmaximal, characteristic
8#70563C211yesmaximal, characteristic, complemented
9#69817C211yesmaximal, characteristic, complemented
10#69618C211yesmaximal, characteristic, complemented
11#69614C211yesmaximal, characteristic, complemented
12#68904C211yesmaximal, characteristic, complemented
13#68891C211yesmaximal, characteristic
14#68824C211yesmaximal, characteristic
15#68801C211yesmaximal, characteristic, complemented
16#24719—31nomaximal, Sylow
17C2×#897C2211yescharacteristic
18C2×#890C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×#872C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4994—21no—
33#4990—21no—
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4983—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4972—21no—
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4929C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4877—24no—
49#4709—24no—
50#4676—21no—
51#4669—21no—
52#4657—21no—
53#4654—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×#433—31no—
64#2629—31no—
65#2323—31no—
66#2077—31no—
67#2076—31no—
68#2035—31no—
69#2035—31no—
70#1989—31no—
71#1948—31no—
72#1944D311yescharacteristic, complemented
73#1941—31no—
74#1935—31no—
75#1933—31no—
76#1803—31no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4D411yescharacteristic
105C2×Dic3⋊C4D411yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#897—22no—
112#897—22no—
113#890—22no—
114#890—22no—
115#887—22no—
116#887—22no—
117#872—22no—
118#872—22no—
119C3×#107—22no—
120C3×#106—22no—
121C3×#97—22no—
122C3×#95—21no—
123C3×#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131(C2×D3)⋊C8—22no—
132(C2×D3)⋊C8—22no—
133#816—22no—
134#816—22no—
135Dic3⋊C8—22no—
136Dic3⋊C8D414yes—
137#810—22no—
138#810—22no—
139#808—22no—
140#808—22no—
141C4×C3⋊C8—22no—
142C2×#125—31no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C2×#117—31no—
146#479—64no—
147#433—64no—
148C2×#107C2×D311yescharacteristic
149C2×#106—31no—
150C2×#106—31no—
151C2×#106C2×D311yescharacteristic
152C2×C4⋊C8—31no—
153C2×C4⋊C8—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#103—31no—
157C2×#103—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×#97C2×D311yescharacteristic
161C2×C4×C8—31no—
162#398—61no—
163#398—61no—
164#396—61no—
165#396—61no—
166#391—31no—
167#389—31no—
168#389—61no—
169#389C2×D311yescharacteristic
170#389C2×D311yescharacteristic
171#388—61no—
172#388—61no—
173#387—61no—
174#384—31no—
175#380—31no—
176#380—31no—
177#380—31no—
178C4×C4⋊C4—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340—31no—
183#340—31no—
184#340—31no—
185#340C2×D311yescharacteristic
186#340C2×D311yescharacteristic
187C23×D3C2×D411yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6#4112yes—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8C2×D412yes—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4—81no—
229C3×C4⋊C4#4112yes—
230C3×C42—21no—
231C3×C42—21no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—22no—
235(C2×D3)⋊C4—81no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238C4⋊Dic3—24no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—42no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—81no—
251Dic3⋊C4—81no—
252Dic3⋊C4—81no—
253C4×Dic3—42no—
254C4×Dic3—81no—
255C4×Dic3C2×D412yes—
256C4×Dic3#4112yes—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8C2×D412yes—
263C22×Q8—31no—
264C23×C4—31no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269#125—62no—
270#125—62no—
271C22⋊Q8—62no—
272C22⋊Q8—62no—
273C4×Q8—62no—
274C4×Q8—62no—
275#117—62no—
276#117—62no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—31no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42C22×D311yescharacteristic
305#107—22no—
306#106—22no—
307#106—62no—
308#106—62no—
309C4⋊C8—34no—
310C4⋊C8—62no—
311#103—62no—
312#103—62no—
313#103—62no—
314#103—62no—
315C22⋊C8—62no—
316C22⋊C8—62no—
317#97—22no—
318C4×C8—62no—
319#95—21no—
320#95—21no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3C22×D411yescharacteristic
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C2×C3×C42−1+411yescharacteristic
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3#12112yes—
380C2×Dic3#12112yes—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—41no—
385Dic6—22no—
386Dic6—22no—
387Dic6—41no—
388Dic6—41no—
389C3×C8—22no—
390C3×C8—41no—
391C3⋊C8—22no—
392C3⋊C8—41no—
393C24—31no—
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4C23×D311yescharacteristic
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—32no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8D3×D412yes—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—64no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—122no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4#23512yes—
457C22⋊C4—62no—
458C22⋊C4—62no—
459C22⋊C4—64no—
460C22⋊C4—241no—
461C22⋊C4—241no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—32no—
466C42—32no—
467C42—122no—
468C42—241no—
469C22×C3C2×#12111yescharacteristic
470C22×C3#52511yescharacteristic
471C22×C3#53611yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57711yescharacteristic
474C22×C3#57711yescharacteristic
475C22×C3#57811yescharacteristic
476C2×D3—41no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—41no—
482C2×D3—41no—
483C3×C4—42no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4#54711yescharacteristic
490C3×C4#54811yescharacteristic
491C3×C4#55211yescharacteristic
492C3×C4#55611yescharacteristic
493Dic3—22no—
494Dic3—22no—
495Dic3—42no—
496Dic3—41no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—61no—
506C23—61no—
507C23C2×D3×D411yescharacteristic
508Q8—62no—
509Q8—62no—
510Q8—121no—
511Q8—121no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—62no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4C2×#23511yescharacteristic
556C2×C4C2×D3×D411yescharacteristic
557C2×C4#100711yescharacteristic
558C2×C4#100711yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C8—22no—
562C8—41no—
563C8—62no—
564C8—121no—
565C2×C3#241511yescharacteristic
566C2×C3#306011yescharacteristic
567C2×C3#306311yescharacteristic
568C2×C3#312011yescharacteristic
569C2×C3#312611yescharacteristic
570C2×C3#324211yescharacteristic
571C2×C3#324411yescharacteristic
572D3—81no—
573C22—121no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22#554711yescharacteristic
581C22D3×#12111yescharacteristic
582C22#572411yescharacteristic
583C22#577511yescharacteristic
584C22#577511yescharacteristic
585C22#577811yescharacteristic
586C22#577911yescharacteristic
587C4—41no—
588C4—42no—
589C4—41no—
590C4—62no—
591C4—62no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—121no—
596C4—122no—
597C4—241no—
598C4—241no—
599C4—241no—
600C4#550711yescharacteristic
601C4#556811yescharacteristic
602C4#563311yescharacteristic
603C4#569311yescharacteristic
604C3#2471911yescharacteristic, complemented, Sylow
605C2—241no—
606C2#7599511yescharacteristic
607C2#7978611yescharacteristic
608C2#7978911yescharacteristic
609C2#8086611yescharacteristic
610C2#8090711yescharacteristic
611C2#8198711yescharacteristic
612C2#8198911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1944C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1944D3
hypocenterC3#24719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
1. C2×#4877C2semidirect
5. C3×#1944C2semidirect
8. #70563C2semidirect
9. #69817C2semidirect
10. #69618C2semidirect
11. #69614C2semidirect
12. #68904C2semidirect
15. #68801C2semidirect
72. #1944D3semidirect
604. C3#24719semidirect

Maximal subgroups

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

subgroupindex
C2×#48772
C2×#47092
#762512
#761972
C3×#19442
#706372
#705822
#705632
#698172
#696182
#696142
#689042
#688912
#688242
#688012
#247193