Finite Group Databasebeta

← #10845229

Subgroups — #10845229

Order 768.

776 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#491C211yesmaximal, characteristic
2#83229C211yesmaximal, characteristic
3C2×#5090C211yesmaximal, characteristic
4#83039C211yesmaximal, characteristic
5#71738C211yesmaximal, characteristic
6#70285C211yesmaximal, characteristic
7#70285C211yesmaximal, characteristic
8#70239C211yesmaximal, characteristic
9#70239C211yesmaximal, characteristic
10#70234C211yesmaximal, characteristic
11#70233C211yesmaximal, characteristic
12#70222C211yesmaximal, characteristic
13#70221C211yesmaximal, characteristic
14#69836C211yesmaximal, characteristic
15#69813C211yesmaximal, characteristic
16#25333—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C22×#216C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#491—24no—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5229C2211yescharacteristic
31#5229C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#5090—24no—
39#4984—21no—
40#4982—21no—
41#4982—21no—
42#4980—21no—
43#4980—21no—
44#4978—21no—
45#4978—21no—
46#4977C2211yescharacteristic
47#4975C2211yescharacteristic
48Dic3×C4⋊C4—21no—
49#4973C2211yescharacteristic
50#4973C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#491D311yescharacteristic
63C2×#462—31no—
64#2565—31no—
65#2365—31no—
66#2074—31no—
67#2072—31no—
68#2056—31no—
69#2056—31no—
70#1989—31no—
71#1940—31no—
72#1878—31no—
73#1878—31no—
74#1805—31no—
75#1803—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8D414yes—
84C3×#125—22no—
85C3×#125—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95Q8×Dic3—41no—
96Q8×Dic3—41no—
97C2×#216—22no—
98C2×#216—22no—
99C2×#216—22no—
100C2×#216—22no—
101C2×#216—41no—
102C2×#216—41no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#103—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#97—22no—
128#835—22no—
129#835—22no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143#816—22no—
144#816—22no—
145Dic3⋊C8—41no—
146Dic3⋊C8—41no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16—31no—
149C22×Q16—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#125C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155#491—24no—
156#462—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169#398—61no—
170#398—61no—
171#391—31no—
172#391—31no—
173#391—31no—
174#391—31no—
175#389—61no—
176#387—61no—
177#387—61no—
178#387—61no—
179#387—61no—
180#384—31no—
181#382—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8C2×D412yes—
204C2×C3×Q8C2×D412yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6#4112yes—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C42C2×D412yes—
241C3×C42#4112yes—
242#216—44no—
243#216—44no—
244C4⋊Dic3—21no—
245C4⋊Dic3—22no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—41no—
251C4⋊Dic3—42no—
252C4⋊Dic3—42no—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—42no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264C4×Dic3—42no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C22×Q8—31no—
272C22×Q8C22×D311yescharacteristic
273C22×Q8C22×D311yescharacteristic
274C2×Q16—62no—
275C2×Q16—62no—
276C2×Q16—62no—
277C2×Q16—62no—
278C2×Q16—121no—
279C2×Q16—121no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8#6714yes—
286#125—22no—
287#125—22no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—121no—
293C4×Q8—121no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317C4⋊C8—62no—
318C4⋊C8—121no—
319C4⋊C8—121no—
320#103—22no—
321#103—22no—
322#103—22no—
323#103—22no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#97—22no—
329#95—31no—
330#95—31no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3C22×D411yescharacteristic
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8—41no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4#12112yes—
368C2×C3×C4#12112yes—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C42−1+411yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—41no—
389C2×Dic3—41no—
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×Dic3#13712yes—
399Dic6—22no—
400Dic6—22no—
401Dic6—41no—
402Dic6—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—32no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8C2×#6712yes—
424C2×Q8C2×#6712yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4D3×D411yescharacteristic
438C22×C4D3×D411yescharacteristic
439C22×C4C23×D311yescharacteristic
440Q16—124no—
441Q16—124no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—32no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—62no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—121no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C42—61no—
476C42—61no—
477C42—61no—
478C42—61no—
479C42—122no—
480C42—122no—
481C42—122no—
482C42—122no—
483C42#23712yes—
484C42C2×#6712yes—
485C22×C3C2×#12111yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#54811yescharacteristic
488C22×C3C2×#13711yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57811yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#52911yescharacteristic
499C3×C4#54211yescharacteristic
500C3×C4#54611yescharacteristic
501C3×C4#54711yescharacteristic
502Dic3—22no—
503Dic3—42no—
504Dic3—41no—
505Dic3—41no—
506Dic3—41no—
507Dic3—42no—
508Dic3—81no—
509Dic3—81no—
510C23C2×D3×D411yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—62no—
520Q8—62no—
521Q8—121no—
522Q8—121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—32no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4#93912yes—
564C2×C4#93912yes—
565C2×C4C2×#23711yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4#100911yescharacteristic
569C2×C4#101011yescharacteristic
570C2×C4C22×#6711yescharacteristic
571C8—41no—
572C8—41no—
573C8—121no—
574C8—121no—
575C2×C3#242711yescharacteristic
576C2×C3#303711yescharacteristic
577C2×C3#304411yescharacteristic
578C2×C3#331211yescharacteristic
579C2×C3#331511yescharacteristic
580C2×C3#337411yescharacteristic
581C2×C3#337711yescharacteristic
582C22#567611yescharacteristic
583C22#567711yescharacteristic
584C22D3×#13711yescharacteristic
585C22#577711yescharacteristic
586C22#577911yescharacteristic
587C22#577911yescharacteristic
588C22C2×#93911yescharacteristic
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—62no—
594C4—81no—
595C4—81no—
596C4—122no—
597C4—121no—
598C4—121no—
599C4—121no—
600C4—122no—
601C4—241no—
602C4—241no—
603C4#568011yescharacteristic
604C4#568211yescharacteristic
605C4#582411yescharacteristic
606C4#582911yescharacteristic
607C3#2533311yescharacteristic, complemented, Sylow
608C2#8173311yescharacteristic
609C2#8173511yescharacteristic
610C2#8174911yescharacteristic
611C2#8175011yescharacteristic
612C2#8299611yescharacteristic
613C2#8396611yescharacteristic
614C2#8397011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#491C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#491D3
hypocenterC3#25333
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
607. C3#25333semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4912
#832292
C2×#50902
#830392
#717382
#702852
#702392
#702342
#702332
#702222
#702212
#698362
#698132
#253333