Finite Group Databasebeta

← #10858327

Subgroups — #10858327

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Q16C211yesmaximal, characteristic
2#82470C211yesmaximal, characteristic
3Q8×Dic3⋊C4C211yesmaximal, characteristic
4C2×#4793C211yesmaximal, characteristic
5#72065C211yesmaximal, characteristic
6#72065C211yesmaximal, characteristic
7#71628C211yesmaximal, characteristic
8#71628C211yesmaximal, characteristic
9#70619C211yesmaximal, characteristic
10#70617C211yesmaximal, characteristic
11#70249C211yesmaximal, characteristic
12#70248C211yesmaximal, characteristic
13#70236C211yesmaximal, characteristic
14#69784C211yesmaximal, characteristic
15#69770C211yesmaximal, characteristic
16#26381—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C3×C4×Q16—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4984C2211yescharacteristic
38#4982—21no—
39#4979—21no—
40#4975—21no—
41#4975C2211yescharacteristic
42#4972—21no—
43#4972C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4933—21no—
47C4×Dic3⋊C4—21no—
48C4×Dic3⋊C4C2211yescharacteristic
49#4929—21no—
50#4928—21no—
51#4928C2211yescharacteristic
52#4928C2211yescharacteristic
53#4793—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#477—31no—
63C2×C4×Q16D311yescharacteristic
64#2451—31no—
65Q8×C4⋊C4—31no—
66#2095—31no—
67#2083—31no—
68#2068—31no—
69#2057—31no—
70#1942—31no—
71#1916—31no—
72#1878—31no—
73#1878—31no—
74#1863—31no—
75#1863—31no—
76#1863—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×Dic6—21no—
80C2×C3×Q16—22no—
81C2×C3×Q16—22no—
82C2×C3×Q16Q814yescomplemented
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—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
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×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#888—22no—
122#888—41no—
123C3×#107—22no—
124C3×#103—22no—
125C3×#103D414yes—
126C3×C4×C8—22no—
127#835—22no—
128#835—22no—
129#835—22no—
130#835—41no—
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⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#818—22no—
142#818—22no—
143Dic3⋊C8—22no—
144#807—22no—
145#807—41no—
146C22×Q16C2×D311yescharacteristic
147C2×C4⋊Q8—31no—
148C2×C4×Q8—31no—
149C2×C4×Q8—31no—
150C2×C4×Q8C2×D311yescharacteristic
151C2×C4×Q8C2×D311yescharacteristic
152#477—64no—
153C4×Q16—24no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×C4×C8C2×D311yescharacteristic
167#398—61no—
168#395—61no—
169#391—31no—
170#391—31no—
171#391—61no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389—31no—
176#389—61no—
177#389—61no—
178#387—61no—
179#384—31no—
180#384—61no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#336—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
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×Q8C2×D412yes—
200C2×C3×Q8C2×Q812yes—
201C2×C3×Q8C2×Q812yes—
202C2×C3×Q8#4112yes—
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—42no—
219C2×Dic6—42no—
220C2×Dic6—81no—
221C3×Q16—28no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8C2×D412yes—
227C2×C3×C8C2×Q812yes—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4C2×D412yes—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—42no—
237C3×C42—42no—
238C4⋊Dic3—24no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3—81no—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—41no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254Dic3⋊C4—81no—
255Dic3⋊C4—81no—
256Dic3⋊C4Q1614yescomplemented
257C4×Dic3—24no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—42no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C22×Q8—61no—
270C22×Q8C22×D311yescharacteristic
271C22×Q8C22×D311yescharacteristic
272C2×Q16—22no—
273C2×Q16—22no—
274C2×Q16Dic614yescomplemented
275C22×C8—31no—
276C22×C8—31no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C4⋊Q8—62no—
280C4⋊Q8—121no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—41no—
284C4×Q8—41no—
285C4×Q8—62no—
286C4×Q8—62no—
287C4×Q8—62no—
288C4×Q8—121no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—31no—
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⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—22no—
316#107—62no—
317#107—62no—
318#107—62no—
319#107—121no—
320C4⋊C8—62no—
321#103—22no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—121no—
326#103#6714yes—
327C4×C8—22no—
328#95—31no—
329#95—31no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338C23×C3C22×D411yescharacteristic
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8C22⋊Q814yes—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
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×C4C22×Q811yescharacteristic
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—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
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—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3C2×Q1612yes—
394C2×Dic3C2×Q1612yes—
395Dic6—84no—
396C3×C8—22no—
397C3×C8—22no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—122no—
409C2×Q8—122no—
410C2×Q8—241no—
411C2×Q8C2×Dic612yes—
412C2×Q8C2×Dic612yes—
413C2×Q8#23512yes—
414C2×Q8C2×#6712yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430Q16—28no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8C2×Dic612yes—
442C2×C8C2×#6712yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—34no—
447C4⋊C4—32no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—64no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—121no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—241no—
463C4⋊C4—241no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4C2×#6712yes—
468C42—21no—
469C42—21no—
470C42—42no—
471C42—42no—
472C42—64no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—122no—
478C22×C3#52511yescharacteristic
479C22×C3D4211yescharacteristic
480C22×C3D4×Q811yescharacteristic
481C22×C3C22×Q1611yescharacteristic
482C22×C3#57511yescharacteristic
483C22×C3#57611yescharacteristic
484C22×C3#57811yescharacteristic
485C3×C4—24no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4C2×C22⋊Q811yescharacteristic
492C3×C4#53911yescharacteristic
493C3×C4#54711yescharacteristic
494C3×C4#55411yescharacteristic
495Dic3—22no—
496Dic3—22no—
497Dic3—41no—
498Dic3—42no—
499Dic3—41no—
500Dic3—81no—
501Dic3—81no—
502Dic3—81no—
503C23C2×D3×D411yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511Q8—244no—
512Q8#92414yes—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—32no—
524C2×C4—32no—
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—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—61no—
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×C4D3×Q1612yes—
555C2×C4C22×Dic611yescharacteristic
556C2×C4C2×#23511yescharacteristic
557C2×C4#100411yescharacteristic
558C2×C4D3×#4111yescharacteristic
559C2×C4#101011yescharacteristic
560C2×C4C22×#6711yescharacteristic
561C8—22no—
562C8—22no—
563C8—121no—
564C8—121no—
565C2×C3#244911yescharacteristic
566C2×C3#310411yescharacteristic
567C2×C3#312411yescharacteristic
568C2×C3#326811yescharacteristic
569C2×C3D4×Q1611yescharacteristic
570C2×C3Q8×Q1611yescharacteristic
571C2×C3#339011yescharacteristic
572C22D4×Dic611yescharacteristic
573C22#554711yescharacteristic
574C22C2×D3×Q1611yescharacteristic
575C22#576611yescharacteristic
576C22#576811yescharacteristic
577C22#576911yescharacteristic
578C22D4×#6711yescharacteristic
579C4—22no—
580C4—24no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—62no—
585C4—62no—
586C4—81no—
587C4—121no—
588C4—122no—
589C4—121no—
590C4—241no—
591C4—241no—
592C4—241no—
593C4#553911yescharacteristic
594C4#556411yescharacteristic
595C4C2×#92411yescharacteristic
596C4#582411yescharacteristic
597C3#2638111yescharacteristic, complemented, Sylow
598C2Q16×Dic611yescharacteristic
599C2#7970911yescharacteristic
600C2#7972211yescharacteristic
601C2#7974911yescharacteristic
602C2#8236011yescharacteristic
603C2Q16×#6711yescharacteristic
604C2#8392211yescharacteristic
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#26381
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
82. C2×C3×Q16Q8semidirect
256. Dic3⋊C4Q16semidirect
274. C2×Q16Dic6semidirect
597. C3#26381semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q162
#824702
Q8×Dic3⋊C42
C2×#47932
#720652
#716282
#706192
#706172
#702492
#702482
#702362
#697842
#697702
#263813