Finite Group Databasebeta

← #11233564

Subgroups — #11233564

Order 768.

1570 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2327C211yesmaximal, characteristic, complemented
2#83240C211yesmaximal, characteristic
3#83239C211yesmaximal, characteristic, complemented
4#82469C212yesmaximal
5#82465C212yesmaximal, complemented
6#76534C212yesmaximal
7#76521C212yesmaximal, complemented
8#75406C214yesmaximal, complemented
9#75217C214yesmaximal, complemented
10#75122C214yesmaximal, complemented
11#75045C214yesmaximal, complemented
12#74160C211yesmaximal, characteristic
13#74157C211yesmaximal, characteristic, complemented
14#73398C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic
16#45349—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2212yes—
19C2×#947C2212yes—
20C2×#944C2212yes—
21C2×#887C2212yes—
22C2×#875C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#391C2212yes—
26C3×#387C2214yes—
27C3×#387C2214yes—
28C3×#382C2211yescharacteristic
29C3×#376C2211yescharacteristic
30#5229C2212yes—
31#5229C2212yes—
32#4993C2214yes—
33#4992C2214yes—
34#4992C2212yes—
35#4992C2212yes—
36#4988C2212yes—
37#4987C2214yes—
38#4983C2212yes—
39#4983C2212yes—
40#4978C2214yes—
41#4976C2214yes—
42Dic3×C4⋊C4C2212yes—
43#4972C2212yes—
44#4972C2214yes—
45#4940C2214yes—
46#4940C2214yes—
47#4939C2211yescharacteristic
48#4935C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4933C2214yes—
52C4×Dic3⋊C4C2214yes—
53C4×Dic3⋊C4C2212yes—
54C4×Dic3⋊C4C2212yes—
55#4929C2211yescharacteristic
56#4676C2214yes—
57#4676C2214yes—
58#4670C2214yes—
59#4670C2214yes—
60#4669C2214yes—
61#4669C2214yes—
62#4668C2214yes—
63#4666C2212yes—
64#4664C2212yes—
65#4660C2212yes—
66#4660C2214yes—
67#4660C2212yes—
68#4659C2214yes—
69#4657C2212yes—
70#4657C2212yes—
71#4655C2214yes—
72#4653C2214yes—
73#4650C2212yes—
74#4649C2212yes—
75#4647C2214yes—
76#2772—34no—
77#2739—34no—
78#2596—34no—
79#2547—31no—
80#2546—31no—
81#2545—32no—
82#2522—34no—
83#2511—32no—
84#2448—32no—
85#2402—31no—
86#2365—32no—
87#2357—31no—
88#2351—31no—
89#2327D311yescharacteristic, complemented
90#2326—31no—
91C22×C3×Q8C2312yes—
92C22×D12C2311yescharacteristic
93C22×C4×D3C2312yes—
94C22×Dic6C2311yescharacteristic
95C3×C4⋊Q8—24no—
96C3×C4×Q8—44no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2314yes—
104C2×C3×C42C2311yescharacteristic
105#947—44no—
106#944—28no—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2312yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×C4⋊Dic3C2312yes—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2314yes—
119C2×Dic3⋊C4C2312yes—
120C2×Dic3⋊C4C2312yes—
121C2×Dic3⋊C4C2312yes—
122C2×Dic3⋊C4C2314yes—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2314yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2312yes—
127C2×Dic3⋊C4C2312yes—
128C2×C4×Dic3C2312yes—
129C2×C4×Dic3C2312yes—
130C2×C4×Dic3C2312yes—
131C2×C4×Dic3C2314yes—
132C2×C4×Dic3C2314yes—
133#887—44no—
134#875—24no—
135C4×D12—24no—
136C4×Dic6—24no—
137C3×#95C2312yes—
138C3×#95C2312yes—
139C3×#95C2314yes—
140C3×#95C2314yes—
141C3×#95C2312yes—
142C3×#95C2312yes—
143C3⋊#95C2314yes—
144C3⋊#95C2311yescharacteristic
145C3⋊#95C2311yescharacteristic
146C3⋊#95C2312yes—
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2311yescharacteristic
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2312yes—
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C2×C4⋊Q8—32no—
159C2×C4⋊Q8C2×D311yescharacteristic
160C2×#124—32no—
161C2×#124—31no—
162C2×C4×Q8—32no—
163C2×C4×Q8—31no—
164C2×C4×Q8C2×D312yes—
165C2×C4×D4—31no—
166#400—32no—
167#400—34no—
168#400—34no—
169#400—32no—
170#399—34no—
171#398—32no—
172#398—32no—
173#398—32no—
174#398—32no—
175#397—32no—
176#397—34no—
177#397—32no—
178#396—34no—
179#396—34no—
180#391—32no—
181#391—32no—
182#391—31no—
183#391C2×D312yes—
184#390—32no—
185#390—31no—
186#389—34no—
187#389—32no—
188#389—34no—
189#389—34no—
190#388—34no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#388—34no—
196#387—34no—
197#387—34no—
198#387—34no—
199#387—34no—
200#387C2×D314yes—
201#387C2×D314yes—
202#385—34no—
203#384—31no—
204#382—32no—
205#382C2×D311yescharacteristic
206#381—32no—
207#380—34no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—34no—
210C4×C4⋊C4—31no—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4—32no—
215C4×C22⋊C4—32no—
216#376C2×D311yescharacteristic
217C23×D3C2412yes—
218C2×C3×Q8—24no—
219C2×C3×Q8—24no—
220C2×C3×Q8—24no—
221C22×C3×C4C2412yes—
222C22×C3×C4C2414yes—
223C22×C3×C4C2412yes—
224C22×C3×C4C2414yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×Dic3C2412yes—
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C22×Dic3C2412yes—
235C2×D12—22no—
236C2×D12—22no—
237C2×D12—22no—
238C2×C4×D3—24no—
239C2×C4×D3—44no—
240C2×Dic6—22no—
241C2×Dic6—22no—
242C2×Dic6#4114yes—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—44no—
247C3×C42—22no—
248C3×C42—22no—
249C3×C42—44no—
250C3×C42#4114yes—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—28no—
254(C2×D3)⋊C4—44no—
255(C2×D3)⋊C4—44no—
256(C2×D3)⋊C4—44no—
257(C2×D3)⋊C4—44no—
258C4⋊Dic3—22no—
259C4⋊Dic3—22no—
260C4⋊Dic3—24no—
261C4⋊Dic3—44no—
262C4⋊Dic3—42no—
263Dic3⋊C4—24no—
264Dic3⋊C4—28no—
265Dic3⋊C4—28no—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268Dic3⋊C4—44no—
269Dic3⋊C4—42no—
270Dic3⋊C4—44no—
271Dic3⋊C4—42no—
272C4×Dic3—24no—
273C4×Dic3—24no—
274C4×Dic3—24no—
275C4×Dic3—44no—
276C4×Dic3—44no—
277C22×Q8—31no—
278C22×Q8C22×D312yes—
279C22×D4—31no—
280C23×C4—32no—
281C4⋊Q8—24no—
282C4⋊Q8—68no—
283#124—64no—
284#124—124no—
285C4×Q8—44no—
286C4×Q8—64no—
287C4×Q8—124no—
288C4×D4—64no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—34no—
293C2×C4⋊C4—34no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—34no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—34no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4C22×D312yes—
304C2×C4⋊C4C22×D312yes—
305C2×C4⋊C4C22×D312yes—
306C2×C4⋊C4C22×D314yes—
307C2×C22⋊C4—32no—
308C2×C22⋊C4—34no—
309C2×C22⋊C4—34no—
310C2×C22⋊C4—34no—
311C2×C22⋊C4—34no—
312C2×C22⋊C4—34no—
313C2×C22⋊C4—34no—
314C2×C42—32no—
315C2×C42—32no—
316C2×C42—32no—
317C2×C42—34no—
318C2×C42—34no—
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321C2×C42C22×D314yes—
322C2×C42C22×D311yescharacteristic
323#95—31no—
324#95—31no—
325#95—34no—
326#95—31no—
327#95—34no—
328#95—31no—
329#95—34no—
330#95—32no—
331#95—32no—
332#95—32no—
333#95—32no—
334#95—34no—
335#95—34no—
336#95—34no—
337#95—34no—
338#95C22×D312yes—
339#95C22×D312yes—
340#95C22×D312yes—
341#95C22×D312yes—
342#95C22×D314yes—
343#95C22×D314yes—
344C23×C3C2511yescharacteristic
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—24no—
349C22×D3—24no—
350C3×Q8—48no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4C2×#4112yes—
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2×#4112yes—
368C2×C3×C4C2×#4112yes—
369C2×C3×C4H(5,𝔽2)12yes—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—24no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—24no—
386C2×Dic3—22no—
387C2×Dic3C2×#4114yes—
388C2×Dic32−1+414yes—
389D12—44no—
390C4×D3—84no—
391Dic6—28no—
392C24—32no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—34no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—62no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—64no—
410C22×C4—124no—
411C22×C4C23×D312yes—
412C22×C4C23×D314yes—
413C22×C4C23×D312yes—
414C22×C4C23×D314yes—
415C22×C4C23×D311yescharacteristic
416C22×C4C23×D311yescharacteristic
417C22×C4C23×D311yescharacteristic
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—44no—
422C4⋊C4—68no—
423C4⋊C4—68no—
424C4⋊C4—64no—
425C4⋊C4—64no—
426C4⋊C4—64no—
427C4⋊C4—62no—
428C4⋊C4—64no—
429C4⋊C4—62no—
430C4⋊C4—124no—
431C4⋊C4—124no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C4⋊C4—122no—
435C4⋊C4—124no—
436C22⋊C4—68no—
437C22⋊C4—68no—
438C22⋊C4—64no—
439C22⋊C4—124no—
440C22⋊C4—124no—
441C22⋊C4—124no—
442C22⋊C4—124no—
443C42—22no—
444C42—22no—
445C42—44no—
446C42—64no—
447C42—64no—
448C42—64no—
449C42—124no—
450C42—124no—
451C42#23514yes—
452C22×C3C22×#4111yescharacteristic
453C22×C3C22×#4111yescharacteristic
454C22×C3C2×H(5,𝔽2)11yescharacteristic
455C22×C3C2×2−1+412yes—
456C22×C3#58512yes—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—44no—
461C2×D3—44no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—44no—
465C3×C4—42no—
466C3×C4—44no—
467C3×C4—42no—
468C3×C4#55314yes—
469Dic3—24no—
470Dic3—24no—
471Dic3—42no—
472Dic3—42no—
473Dic3—42no—
474Dic3—42no—
475Dic3—42no—
476C23—62no—
477C23—62no—
478C23—62no—
479C23—64no—
480C23—64no—
481C23C24×D311yescharacteristic
482Q8—48no—
483Q8—68no—
484D4—124no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—34no—
500C2×C4—34no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—64no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—244no—
519C2×C4C2×#23512yes—
520C2×C4C2×#23512yes—
521C2×C4C2×#23512yes—
522C2×C4#100412yes—
523C2×C4D3×#4112yes—
524C2×C3C2×#55311yescharacteristic
525C2×C3#350911yescharacteristic
526C2×C3#351011yescharacteristic
527C2×C3#351612yes—
528C2×C3#355412yes—
529D3—82no—
530C22—122no—
531C22—124no—
532C22—122no—
533C22—122no—
534C22—124no—
535C22C22×#23511yescharacteristic
536C22C2×#100411yescharacteristic
537C22C2×D3×#4111yescharacteristic
538C22D3×2−1+412yes—
539C22#596812yes—
540C4—22no—
541C4—22no—
542C4—44no—
543C4—42no—
544C4—44no—
545C4—42no—
546C4—64no—
547C4—64no—
548C4—122no—
549C4—122no—
550C4—122no—
551C4—122no—
552C4—122no—
553C4#555814yes—
554C3#4534911yescharacteristic, complemented, Sylow
555C2—242no—
556C2C2×#555811yescharacteristic
557C2#8587611yescharacteristic
558C2#8587711yescharacteristic
559C2#8635112yes—
560C2#8651212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2327C2
prime radicalC22×D12C23
hypercenter#2327D3
hypocenterC3#45349
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2327

Complements

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

NH ≅ G/Ntype
1. C3×#2327C2semidirect
3. #83239C2semidirect
5. #82465C2semidirect
7. #76521C2semidirect
8. #75406C2semidirect
9. #75217C2semidirect
10. #75122C2semidirect
11. #75045C2semidirect
13. #74157C2semidirect
14. #73398C2semidirect
89. #2327D3semidirect
554. C3#45349semidirect

Maximal subgroups

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

subgroupindex
C3×#23272
#832402
#832392
#824692
#824652
#765342
#765212
#754062
#752172
#751222
#750452
#741602
#741572
#733982
#733872
#453493