Finite Group Databasebeta

← #10844709

Subgroups — #10844709

Order 768.

782 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
1C2×#4795C211yesmaximal, characteristic
2C2×C8×Dic6C211yesmaximal, characteristic
3#76545C211yesmaximal, characteristic
4#76077C211yesmaximal, characteristic
5C3×#1933C211yesmaximal, characteristic
6#70258C211yesmaximal, characteristic
7#70248C211yesmaximal, characteristic
8#70225C211yesmaximal, characteristic
9#69775C211yesmaximal, characteristic
10#69681C211yesmaximal, characteristic
11#69635C211yesmaximal, characteristic
12#68864C211yesmaximal, characteristic
13#68850C211yesmaximal, characteristic
14#68795C211yesmaximal, characteristic
15#68781C211yesmaximal, characteristic
16#25294—31nomaximal, Sylow
17C2×#944C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C22×Dic12C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4983—21no—
37#4976—21no—
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4929C2211yescharacteristic
48#4929C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4795—24no—
51C8×Dic6—24no—
52#4660—21no—
53#4658—21no—
54#4657—21no—
55#4650—21no—
56#4646—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#462—31no—
63C2×C8×Q8—31no—
64#2582—31no—
65#2568—31no—
66#2068—31no—
67#2068—31no—
68#2056—31no—
69#2056—31no—
70#1969—31no—
71#1969—31no—
72#1956—31no—
73#1947—31no—
74#1933—31no—
75#1933D311yescharacteristic
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×Dic6D411yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88#944—22no—
89#944—22no—
90C2×#216—22no—
91C2×#216—22no—
92C2×#216—41no—
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4D411yescharacteristic
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×Dic12—22no—
111C2×Dic12—22no—
112C2×Dic12—41no—
113#887—22no—
114#887—22no—
115C4×Dic6—22no—
116C4×Dic6—22no—
117C3×#103—22no—
118C3×#103—22no—
119C3×C4×C8—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123#835—41no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#818—41no—
137#816—22no—
138#816—22no—
139Dic3⋊C8—22no—
140Dic3⋊C8D414yes—
141C8×Dic3—22no—
142C8×Dic3—22no—
143#810—22no—
144#810—22no—
145#810—41no—
146#807—41no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16—31no—
149C22×Q16—31no—
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153#462—64no—
154C8×Q8—64no—
155C2×#107—31no—
156C2×#107—31no—
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×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8C2×D311yescharacteristic
171#400—61no—
172#400—61no—
173#398—61no—
174#398—61no—
175#391—31no—
176#391—31no—
177#391—31no—
178#391—31no—
179#391—31no—
180#391—61no—
181#391—61no—
182#391C2×D311yescharacteristic
183#389—31no—
184#389—61no—
185#389C2×D311yescharacteristic
186#384—31no—
187#382—61no—
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6Q1614yescomplemented
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8C2×D412yes—
234C2×C3×C8#4112yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—81no—
242C3×C42—21no—
243C3×C42—21no—
244#216—44no—
245C4⋊Dic3—22no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—81no—
257Dic3⋊C4—81no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3C2×D412yes—
263C4×Dic3#4112yes—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D412yes—
270Dic12—44no—
271C22×Q8—31no—
272C22×Q8—31no—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C2×Q16—62no—
276C2×Q16—62no—
277C2×Q16—62no—
278C2×Q16—62no—
279C2×Q16—121no—
280C2×Q16—121no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
285C4⋊Q8—62no—
286C4⋊Q8—62no—
287C4×Q8—62no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—61no—
309C2×C42—61no—
310C2×C42C22×D311yescharacteristic
311#107—121no—
312#107—121no—
313C4⋊C8—34no—
314C4⋊C8—62no—
315C4⋊C8—62no—
316#103—22no—
317#103—22no—
318#103—62no—
319#103—62no—
320#103—62no—
321#103—62no—
322#103—121no—
323#103—121no—
324C4×C8—22no—
325C4×C8—62no—
326C4×C8—62no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—61no—
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—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—41no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
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×C4C2×Q1612yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C42−1+411yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3#12112yes—
391C2×Dic3#12112yes—
392C2×Dic3C2×Q1612yes—
393C2×Dic3C2×Q1612yes—
394C2×Dic3#13512yes—
395Dic6—28no—
396Dic6—22no—
397Dic6—22no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402Dic6—41no—
403Dic6—41no—
404C3×C8—22no—
405C3×C8—22no—
406C3⋊C8—22no—
407C3⋊C8—41no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—34no—
415C2×Q8—32no—
416C2×Q8—62no—
417C2×Q8—62no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4D3×D411yescharacteristic
442C22×C4D3×D411yescharacteristic
443C22×C4C23×D311yescharacteristic
444Q16—124no—
445Q16—124no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—32no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8#23512yes—
457C2×C8D3×D412yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—81no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—241no—
477C4⋊C4—241no—
478C42—21no—
479C42—21no—
480C42—32no—
481C42—32no—
482C42—61no—
483C42—61no—
484C42—122no—
485C42—122no—
486C22×C3C2×#12111yescharacteristic
487C22×C3#54811yescharacteristic
488C22×C3#55011yescharacteristic
489C22×C3C22×Q1611yescharacteristic
490C22×C3C2×#13511yescharacteristic
491C22×C3#57711yescharacteristic
492C22×C3#57811yescharacteristic
493C3×C4—22no—
494C3×C4—41no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4#52911yescharacteristic
500C3×C4#54211yescharacteristic
501C3×C4#54611yescharacteristic
502C3×C4#54811yescharacteristic
503Dic3—22no—
504Dic3—22no—
505Dic3—42no—
506Dic3—41no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510Dic3#46714yes—
511C23C2×D3×D411yescharacteristic
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—68no—
517Q8—62no—
518Q8—62no—
519Q8—121no—
520Q8—121no—
521Q8—121no—
522Q8—121no—
523Q8—121no—
524Q8—121no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
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—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
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—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4D3×Q1612yes—
567C2×C4C2×#23511yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4D3×#4111yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C2×C4#100911yescharacteristic
572C2×C4#101011yescharacteristic
573C8—22no—
574C8—22no—
575C8—62no—
576C8—121no—
577C2×C3#242611yescharacteristic
578C2×C3C2×#46711yescharacteristic
579C2×C3#306311yescharacteristic
580C2×C3#336311yescharacteristic
581C2×C3#336411yescharacteristic
582C2×C3#339411yescharacteristic
583C2×C3#339511yescharacteristic
584C22#555311yescharacteristic
585C22D3×#12111yescharacteristic
586C22#567711yescharacteristic
587C22#576211yescharacteristic
588C22C2×D3×Q1611yescharacteristic
589C22D3×#13511yescharacteristic
590C22#577111yescharacteristic
591C4—22no—
592C4—34no—
593C4—41no—
594C4—62no—
595C4—62no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—121no—
601C4—122no—
602C4—241no—
603C4—241no—
604C4—241no—
605C4#552811yescharacteristic
606C4#555211yescharacteristic
607C4#565411yescharacteristic
608C4#571011yescharacteristic
609C3#2529411yescharacteristic, complemented, Sylow
610C2#7549911yescharacteristic
611C2#7963911yescharacteristic
612C2#7971111yescharacteristic
613C2#8066211yescharacteristic
614C2D3×#46711yescharacteristic
615C2#8162411yescharacteristic
616C2#8163411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socle, prime radicalC23×C3C22×D4
hypercenter#1933D3
hypocenterC3#25294
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
227. C2×Dic6Q16semidirect
609. C3#25294semidirect

Maximal subgroups

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

subgroupindex
C2×#47952
C2×C8×Dic62
#765452
#760772
C3×#19332
#702582
#702482
#702252
#697752
#696812
#696352
#688642
#688502
#687952
#687812
#252943