Finite Group Databasebeta

← #10855293

Subgroups — #10855293

Order 768.

792 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×C4×Q16C211yesmaximal, characteristic
2C2×#5182C211yesmaximal, characteristic
3#82468C211yesmaximal, characteristic
4#82461C211yesmaximal, characteristic
5#72065C211yesmaximal, characteristic
6#71632C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic
8#70645C211yesmaximal, characteristic
9#70621C211yesmaximal, characteristic
10#70284C211yesmaximal, characteristic
11#70239C211yesmaximal, characteristic
12#70237C211yesmaximal, characteristic
13#70234C211yesmaximal, characteristic
14#69784C211yesmaximal, characteristic
15#69767C211yesmaximal, characteristic
16#26167—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#944C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C3×C4×Q16—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5182—24no—
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4985—21no—
38#4983—21no—
39#4977—21no—
40#4977C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972—21no—
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4933—21no—
48#4933—21no—
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928—21no—
53#4928C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#462—31no—
63C2×C4×Q16D311yescharacteristic
64#2446—31no—
65#2351—31no—
66#2097—31no—
67#2086—31no—
68#2074—31no—
69#2043—31no—
70#1933—31no—
71#1916—31no—
72#1878—31no—
73#1878—31no—
74#1878—31no—
75#1865—31no—
76#1865—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16D414yes—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—41no—
88C3×C4×Q8—41no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96Q8×Dic3—22no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99Q8×Dic3—41no—
100#944—22no—
101#944—41no—
102C2×#216—22no—
103C2×#216—22no—
104C2×#216—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C3×#107—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×C4×C8—22no—
127#835—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—22no—
146#810—41no—
147#808—41no—
148C22×Q16—31no—
149C22×Q16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×Q8C2×D311yescharacteristic
155#462—64no—
156C4×Q16—24no—
157C2×#107C2×D311yescharacteristic
158C2×#106—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×C8C2×D311yescharacteristic
169#400—61no—
170#398—61no—
171#391—31no—
172#391—31no—
173#391—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389—61no—
178#389—61no—
179#387—61no—
180#387—61no—
181#382—31no—
182#382—61no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#336—31no—
190#336—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×Q8C2×D412yes—
202C2×C3×Q8C2×D412yes—
203C2×C3×Q8#4112yes—
204C2×C3×Q8#4112yes—
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—21no—
226C3×Q16—28no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C2×D412yes—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—42no—
242C3×C42—42no—
243#216—44no—
244C4⋊Dic3—24no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—42no—
251C4⋊Dic3#4112yes—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259C4×Dic3—24no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
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—22no—
275C2×Q16—22no—
276C2×Q16—62no—
277C2×Q16—62no—
278C2×Q16—62no—
279C2×Q16#6714yes—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C4⋊Q8—62no—
285C4⋊Q8—121no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—41no—
289C4×Q8—41no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—121no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
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—21no—
311C2×C42—21no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—22no—
318#106—121no—
319C4⋊C8—62no—
320#103—22no—
321#103—22no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—121no—
328C4×C8—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—22no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8#12314yes—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
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—41no—
368C2×C3×C4#13712yes—
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—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—41no—
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#13512yes—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402Dic6—41no—
403C3×C8—22no—
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—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8#23512yes—
422C2×Q8#23512yes—
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—28no—
441Q16—124no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8C2×#6712yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—62no—
462C4⋊C4—64no—
463C4⋊C4—62no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C42—21no—
476C42—21no—
477C42—42no—
478C42—42no—
479C42—64no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42—61no—
484C42—122no—
485C22×C3#52511yescharacteristic
486C22×C3D4211yescharacteristic
487C22×C3#54811yescharacteristic
488C22×C3C2×#13511yescharacteristic
489C22×C3C2×#13711yescharacteristic
490C22×C3#57811yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—24no—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4C2×#12311yescharacteristic
499C3×C4#53911yescharacteristic
500C3×C4#54711yescharacteristic
501C3×C4#55311yescharacteristic
502Dic3—22no—
503Dic3—41no—
504Dic3—42no—
505Dic3—41no—
506Dic3—41no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23C2×D3×D411yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—22no—
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518Q8—121no—
519Q8—121no—
520Q8—121no—
521Q8—121no—
522Q8#92914yes—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
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—121no—
564C2×C4#91812yes—
565C2×C4C2×#23511yescharacteristic
566C2×C4C2×#23511yescharacteristic
567C2×C4#100411yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4#101011yescharacteristic
570C2×C4C22×#6711yescharacteristic
571C8—22no—
572C8—41no—
573C8—121no—
574C8—121no—
575C2×C3#244311yescharacteristic
576C2×C3#310711yescharacteristic
577C2×C3#312711yescharacteristic
578C2×C3#327011yescharacteristic
579C2×C3#328511yescharacteristic
580C2×C3#336411yescharacteristic
581C2×C3#337411yescharacteristic
582C22#554711yescharacteristic
583C22#555211yescharacteristic
584C22C2×#91811yescharacteristic
585C22#576611yescharacteristic
586C22D3×#13511yescharacteristic
587C22#577111yescharacteristic
588C22D4×#6711yescharacteristic
589C4—22no—
590C4—24no—
591C4—42no—
592C4—41no—
593C4—62no—
594C4—81no—
595C4—81no—
596C4—121no—
597C4—121no—
598C4—122no—
599C4—121no—
600C4—241no—
601C4—241no—
602C4—241no—
603C4#554011yescharacteristic
604C4#556411yescharacteristic
605C4C2×#92911yescharacteristic
606C4#582411yescharacteristic
607C3#2616711yescharacteristic, complemented, Sylow
608C2#7970011yescharacteristic
609C2#7970211yescharacteristic
610C2#7972411yescharacteristic
611C2#7974711yescharacteristic
612C2#8235611yescharacteristic
613C2#8388811yescharacteristic
614C2#8392011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×C4×Q16D3
hypocenterC3#26167
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
607. C3#26167semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q162
C2×#51822
#824682
#824612
#720652
#716322
#716292
#706452
#706212
#702842
#702392
#702372
#702342
#697842
#697672
#261673