Finite Group Databasebeta

← #11235112

Subgroups — #11235112

Order 768.

1726 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2542C211yesmaximal, characteristic, complemented
2#83260C211yesmaximal, characteristic, complemented
3#83228C211yesmaximal, characteristic, complemented
4#82800C214yesmaximal, complemented
5#82783C214yesmaximal, complemented
6#82338C211yesmaximal, characteristic, complemented
7#82329C211yesmaximal, characteristic
8#78121C214yesmaximal, complemented
9#78034C214yesmaximal, complemented
10#76353C212yesmaximal
11#76341C212yesmaximal, complemented
12#74333C212yesmaximal, complemented
13#74325C212yesmaximal, complemented
14#73669C211yesmaximal, characteristic, complemented
15#73645C211yesmaximal, characteristic, complemented
16#45430—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2214yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×#940C2211yescharacteristic
21C2×#898C2214yes—
22C2×#885C2214yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C3×#399C2212yes—
26C3×#399C2212yes—
27C3×#390C2212yes—
28C3×#384C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30#5231C2212yes—
31#5231C2212yes—
32#5229C2214yes—
33#5221C2211yescharacteristic
34#5221C2212yes—
35#5213C2214yes—
36#5211C2212yes—
37C4×#217C2212yes—
38#4994C2214yes—
39#4992C2214yes—
40#4990C2214yes—
41#4990C2214yes—
42#4988C2212yes—
43#4988C2211yescharacteristic
44#4987C2214yes—
45#4984C2212yes—
46#4982C2212yes—
47#4980C2214yes—
48#4979C2212yes—
49#4978C2214yes—
50#4978C2214yes—
51#4977C2212yes—
52#4977C2211yescharacteristic
53Dic3×C4⋊C4C2212yes—
54Dic3×C4⋊C4C2212yes—
55Dic3×C4⋊C4C2214yes—
56#4973C2214yes—
57#4961C2214yes—
58#4959C2214yes—
59#4957C2214yes—
60#4956C2214yes—
61#4953C2214yes—
62#4952C2214yes—
63#4951C2214yes—
64#4951C2214yes—
65#4945C2214yes—
66#4939C2212yes—
67#4936C2211yescharacteristic
68C4×C4⋊Dic3C2212yes—
69C4×C4⋊Dic3C2211yescharacteristic
70#4932C2211yescharacteristic
71C4×Dic3⋊C4C2212yes—
72#4666C2212yes—
73#4664C2212yes—
74#4657C2212yes—
75#4656C2212yes—
76#2740—34no—
77#2700—34no—
78#2563—31no—
79#2560—34no—
80#2548—32no—
81#2542D311yescharacteristic, complemented
82#2521—34no—
83#2513—32no—
84#2512—32no—
85#2399—31no—
86#2357—31no—
87#2357—31no—
88#2355—32no—
89#2331—31no—
90#2328—31no—
91C22×C3×Q8C2312yes—
92C22×C3×D4C2311yescharacteristic
93C23×C3×C4C2312yes—
94C22×#67C2314yes—
95C22×D12C2311yescharacteristic
96C22×C4×D3C2312yes—
97C22×Dic6C2312yes—
98C3×C4⋊Q8—24no—
99C3×C22⋊Q8—48no—
100C3×C4×D4—24no—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C42C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×#217C2314yes—
113C2×#217C2314yes—
114C2×#217C2314yes—
115#940—24no—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2314yes—
127C2×Dic3⋊C4C2312yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2312yes—
131C2×C4×Dic3C2311yescharacteristic
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2314yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2311yescharacteristic
137#898—48no—
138#885—48no—
139C4×D12—24no—
140C4×Dic6—28no—
141C3×#95C2311yescharacteristic
142C3×#95C2311yescharacteristic
143C3⋊#95C2312yes—
144C3⋊#95C2312yes—
145C3⋊#95C2312yes—
146C3⋊#95C2312yes—
147C3⋊#95C2314yes—
148C3⋊#95C2312yes—
149C3⋊#95C2311yescharacteristic
150C3⋊#95C2311yescharacteristic
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C2×C4⋊Q8C2×D311yescharacteristic
159C2×#127—31no—
160C2×#126—34no—
161C2×#124—34no—
162C2×C22⋊Q8C2×D314yes—
163C2×C4×Q8—32no—
164C2×C4×D4—31no—
165C2×C4×D4C2×D311yescharacteristic
166#399C2×D312yes—
167#399C2×D312yes—
168#398—32no—
169#398—32no—
170#397—34no—
171#397—34no—
172#397—34no—
173#396—34no—
174#395—32no—
175#395—32no—
176#394—34no—
177#394—34no—
178#393—32no—
179#393—32no—
180#391—34no—
181#391—34no—
182#390—31no—
183#390—32no—
184#390—32no—
185#390—31no—
186#390—34no—
187#390—32no—
188#390C2×D312yes—
189#389—32no—
190#388—34no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#387—34no—
196#387—34no—
197#387—34no—
198#386—34no—
199#386—34no—
200#384C2×D312yes—
201#382—32no—
202#382—31no—
203#382—31no—
204#381—32no—
205#381—32no—
206#380—34no—
207C4×C4⋊C4—34no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—32no—
210C4×C4⋊C4—31no—
211C4×C4⋊C4—32no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4C2×D311yescharacteristic
214C4×C22⋊C4—32no—
215C4×C22⋊C4—32no—
216#376—31no—
217C24×C3C2412yes—
218C23×D3C2412yes—
219C2×C3×Q8—24no—
220C2×C3×Q8—44no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—22no—
224C22×C3×C4—24no—
225C22×C3×C4—44no—
226C22×C3×C4C2412yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C2×#67—28no—
234C2×#67—44no—
235C2×#67—44no—
236C22×Dic3C2412yes—
237C22×Dic3C2414yes—
238C22×Dic3C2412yes—
239C22×Dic3C2412yes—
240C22×Dic3C2412yes—
241C22×Dic3C2412yes—
242C2×D12—22no—
243C2×D12—22no—
244C2×D12—22no—
245C2×C4×D3—24no—
246C2×C4×D3—44no—
247C2×Dic6—24no—
248C2×Dic6—24no—
249C2×Dic6—42no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—28no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—44no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—28no—
258C3×C22⋊C4—44no—
259C3×C42—24no—
260C3×C42#4114yes—
261#217—44no—
262#217—44no—
263#217—44no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—28no—
266(C2×D3)⋊C4—44no—
267(C2×D3)⋊C4—44no—
268(C2×D3)⋊C4—44no—
269(C2×D3)⋊C4—44no—
270C4⋊Dic3—22no—
271C4⋊Dic3—24no—
272C4⋊Dic3—24no—
273C4⋊Dic3—44no—
274C4⋊Dic3—42no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—42no—
279C4×Dic3—24no—
280C4×Dic3—22no—
281C4×Dic3—44no—
282C4×Dic3—44no—
283C4×Dic3—44no—
284C4×Dic3#4114yes—
285C22×Q8—32no—
286C22×Q8C22×D312yes—
287C22×D4—34no—
288C22×D4—31no—
289C22×D4C22×D311yescharacteristic
290C23×C4—32no—
291C23×C4C22×D312yes—
292C4⋊Q8—24no—
293#127—64no—
294#126—128no—
295#124—128no—
296C22⋊Q8—48no—
297C4×Q8—68no—
298C4×D4—24no—
299C4×D4—64no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D312yes—
311C2×C4⋊C4C22×D314yes—
312C2×C4⋊C4C22×D312yes—
313C2×C4⋊C4C22×D312yes—
314C2×C4⋊C4C22×D314yes—
315C2×C22⋊C4—32no—
316C2×C22⋊C4—34no—
317C2×C22⋊C4—34no—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4C22×D312yes—
325C2×C22⋊C4C22×D314yes—
326C2×C22⋊C4C22×D314yes—
327C2×C42—31no—
328C2×C42—32no—
329C2×C42—31no—
330C2×C42—34no—
331C2×C42—34no—
332C2×C42—34no—
333C2×C42C22×D312yes—
334C2×C42C22×D311yescharacteristic
335#95—32no—
336#95—32no—
337#95—31no—
338#95—32no—
339#95—32no—
340#95—34no—
341#95—31no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95—34no—
346#95—32no—
347#95—32no—
348#95—34no—
349#95—34no—
350#95C22×D311yescharacteristic
351#95C22×D311yescharacteristic
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—24no—
355C23×C3—22no—
356C23×C3—24no—
357C23×C3C2511yescharacteristic
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—24no—
361C22×D3—22no—
362C22×D3—24no—
363C3×Q8—84no—
364C3×D4—44no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—84no—
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4H(5,𝔽2)14yes—
383C2×C3×C42−1+412yes—
384#67—88no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—24no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—24no—
396C2×Dic3—24no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3C2×#4114yes—
402D12—44no—
403C4×D3—84no—
404Dic6—48no—
405C24—32no—
406C24C23×D312yes—
407C2×Q8—24no—
408C2×Q8—44no—
409C2×Q8—64no—
410C2×Q8—64no—
411C2×Q8—122no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—68no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—124no—
420C2×D4—124no—
421C22×C4—24no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—44no—
429C22×C4—64no—
430C22×C4—124no—
431C22×C4C23×D312yes—
432C22×C4C23×D314yes—
433C22×C4C23×D312yes—
434C22×C4C23×D311yescharacteristic
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—28no—
443C4⋊C4—44no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—62no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—124no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C22⋊C4—24no—
454C22⋊C4—28no—
455C22⋊C4—44no—
456C22⋊C4—64no—
457C22⋊C4—68no—
458C22⋊C4—124no—
459C22⋊C4—124no—
460C22⋊C4—124no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C22⋊C4—124no—
465C42—24no—
466C42—34no—
467C42—62no—
468C42—64no—
469C42—124no—
470C42—124no—
471C42—124no—
472C42#23714yes—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3C22×#4111yescharacteristic
479C22×C3C22×#4111yescharacteristic
480C22×C3C2×H(5,𝔽2)12yes—
481C22×C3C2×2−1+411yescharacteristic
482C22×C3#58512yes—
483C2×D3—42no—
484C2×D3—44no—
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—44no—
488C3×C4—24no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—44no—
494C3×C4#55614yes—
495Dic3—24no—
496Dic3—42no—
497Dic3—42no—
498Dic3—42no—
499Dic3—44no—
500Dic3—42no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—62no—
507C23—64no—
508C23—62no—
509C23—62no—
510C23—64no—
511C23C24×D311yescharacteristic
512Q8—84no—
513Q8—128no—
514D4—44no—
515D4—124no—
516D4—248no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—34no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—64no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—84no—
547C2×C4—244no—
548C2×C4C2×#23712yes—
549C2×C4C2×#23712yes—
550C2×C4C2×#23712yes—
551C2×C4#100414yes—
552C2×C4#100712yes—
553C2×C4D3×#4112yes—
554C2×C3—82no—
555C2×C3C2×#55611yescharacteristic
556C2×C3#349412yes—
557C2×C3#350811yescharacteristic
558C2×C3#351011yescharacteristic
559C2×C3#355412yes—
560D3—82no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—122no—
567C22—124no—
568C22—122no—
569C22—122no—
570C22—124no—
571C22C22×#23711yescharacteristic
572C22C2×#100412yes—
573C22C2×#100711yescharacteristic
574C22C2×D3×#4111yescharacteristic
575C22#596812yes—
576C4—22no—
577C4—24no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—64no—
583C4—122no—
584C4—122no—
585C4—122no—
586C4—124no—
587C4—122no—
588C4#564114yes—
589C3#4543011yescharacteristic, complemented, Sylow
590C2—82no—
591C2—242no—
592C2C2×#564111yescharacteristic
593C2#8619011yescharacteristic
594C2#8634411yescharacteristic
595C2#8667812yes—
596C2#8676512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2542C2
prime radicalC2×#940C22
hypercenter#2542D3
hypocenterC3#45430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C3×#2542C2semidirect
2. #83260C2semidirect
3. #83228C2semidirect
4. #82800C2semidirect
5. #82783C2semidirect
6. #82338C2semidirect
8. #78121C2semidirect
9. #78034C2semidirect
11. #76341C2semidirect
12. #74333C2semidirect
13. #74325C2semidirect
14. #73669C2semidirect
15. #73645C2semidirect
81. #2542D3semidirect
589. C3#45430semidirect

Maximal subgroups

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

subgroupindex
C3×#25422
#832602
#832282
#828002
#827832
#823382
#823292
#781212
#780342
#763532
#763412
#743332
#743252
#736692
#736452
#454303