Finite Group Databasebeta

← #11235146

Subgroups — #11235146

Order 768.

2170 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5541C212yesmaximal, complemented
2C3×#2397C211yesmaximal, characteristic
3#82379C212yesmaximal, complemented
4#82376C212yesmaximal, complemented
5#82353C212yesmaximal, complemented
6#82351C212yesmaximal, complemented
7#82335C212yesmaximal, complemented
8#78069C214yesmaximal, complemented
9#78021C214yesmaximal, complemented
10#76724C214yesmaximal, complemented
11#76685C214yesmaximal, complemented
12#73390C211yesmaximal, characteristic, complemented
13#73388C211yesmaximal, characteristic, complemented
14#45432—31nomaximal, Sylow
15C2×C3×C4×D4C2212yes—
16C2×C3×C4×D4C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#935C2214yes—
20C2×#926C2212yes—
21C22×C4⋊Dic3C2212yes—
22C2×#924C2212yes—
23C2×#924C2212yes—
24C2×#924C2212yes—
25C2×#924C2212yes—
26#5541—216no—
27C2×#889C2214yes—
28C2×#879C2214yes—
29C2×#877C2214yes—
30C2×#870C2211yescharacteristic
31C2×#869C2211yescharacteristic
32C2×C4×Dic6C2212yes—
33C3×#399C2212yes—
34C3×#399C2212yes—
35C3×#392C2212yes—
36C3×#392C2212yes—
37C3×#376C2211yescharacteristic
38#5222C2214yes—
39#5221C2214yes—
40#5221C2214yes—
41#5220C2214yes—
42#5220C2214yes—
43#5212C2212yes—
44#5212C2212yes—
45#4982C2214yes—
46#4980C2214yes—
47#4976C2214yes—
48#4975C2214yes—
49#4972C2214yes—
50#4952C2214yes—
51#4951C2214yes—
52#4949C2214yes—
53#4949C2214yes—
54#4948C2214yes—
55#4948C2214yes—
56#4948C2214yes—
57#4945C2214yes—
58#4942C2214yes—
59Dic3×C22⋊C4C2214yes—
60#4936C2211yescharacteristic
61#4935C2211yescharacteristic
62C4×C4⋊Dic3C2212yes—
63#4929C2212yes—
64#4928C2212yes—
65#4655C2214yes—
66#4653C2214yes—
67#4653C2214yes—
68#4647C2214yes—
69C2×#552—32no—
70#2721—34no—
71#2705—34no—
72#2542—32no—
73#2538—32no—
74#2471—34no—
75#2441—34no—
76#2417—32no—
77#2415—32no—
78#2397D311yescharacteristic
79#2352—32no—
80#2330—31no—
81#2327—31no—
82C22×C3×D4C2311yescharacteristic
83C22×C3×D4C2312yes—
84C22×C3×D4C2312yes—
85C22×C3×D4C2311yescharacteristic
86C23×C3×C4C2312yes—
87C23×C3×C4C2312yes—
88C23×Dic3C2314yes—
89C22×Dic6C2312yes—
90C3×C4×D4—28no—
91C3×C4×D4—28no—
92C3×C4×D4—42no—
93C3×C4×D4—42no—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4C2312yes—
99C2×C3×C22⋊C4C2312yes—
100C2×C3×C22⋊C4C2312yes—
101C2×C3×C22⋊C4C2312yes—
102C2×C3×C22⋊C4C2312yes—
103C2×C3×C22⋊C4C2312yes—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×#217C2314yes—
108C2×#217C2314yes—
109C2×#217C2314yes—
110C2×#217C2314yes—
111C2×#217C2314yes—
112C2×#217C2314yes—
113C2×#217C2314yes—
114#935—216no—
115#926—28no—
116C2×C4⋊Dic3—28no—
117C2×C4⋊Dic3C2314yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2312yes—
123#924—28no—
124#924—28no—
125#924—44no—
126#924—44no—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×C4×Dic3C2314yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137#889—216no—
138#879—216no—
139#877—216no—
140#870D418yescomplemented
141#869—24no—
142C4×Dic6—28no—
143C3×#95C2312yes—
144C3×#95C2312yes—
145C3⋊#95C2314yes—
146C3⋊#95C2314yes—
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152#552—616no—
153C2×C4⋊Q8—31no—
154C2×#126—34no—
155C2×#125—34no—
156C2×#125—31no—
157C2×#123—34no—
158C2×C22⋊Q8—32no—
159C2×C22⋊Q8—32no—
160C2×C22⋊Q8—32no—
161C2×C22⋊Q8—32no—
162C2×C4×Q8—32no—
163C2×C4×D4C2×D312yes—
164C2×C4×D4C2×D312yes—
165C2×C4×D4C2×D311yescharacteristic
166C2×C4×D4C2×D311yescharacteristic
167C2×#117—32no—
168C2×#117—34no—
169C22×C4⋊C4—32no—
170#399—34no—
171#399—34no—
172#399C2×D312yes—
173#399C2×D312yes—
174#398—34no—
175#397—34no—
176#396—34no—
177#396—34no—
178#396—34no—
179#394—34no—
180#392C2×D312yes—
181#392C2×D312yes—
182#391—32no—
183#390—34no—
184#390—34no—
185#389—34no—
186#389—34no—
187#389—32no—
188#389—34no—
189#388—34no—
190#388—34no—
191#387—34no—
192#387—34no—
193#387—34no—
194#387—34no—
195#386—34no—
196#386—34no—
197#384—34no—
198#384—31no—
199#382—31no—
200#380—32no—
201#380—32no—
202#380—34no—
203C4×C4⋊C4—32no—
204C4×C22⋊C4—34no—
205#376C2×D311yescharacteristic
206C24×C3C2412yes—
207C24×C3C2412yes—
208C2×C3×D4—24no—
209C2×C3×D4—24no—
210C2×C3×D4—24no—
211C2×C3×D4—24no—
212C2×C3×D4—24no—
213C2×C3×D4—41no—
214C2×C3×D4—41no—
215C2×C3×D4—42no—
216C2×C3×D4—42no—
217C2×C3×D4—42no—
218C22×C3×C4—24no—
219C22×C3×C4—24no—
220C22×C3×C4—24no—
221C22×C3×C4—24no—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2412yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—28no—
232C22×Dic3—28no—
233C22×Dic3—44no—
234C22×Dic3C2414yes—
235C22×Dic3C2414yes—
236C22×Dic3C2414yes—
237C22×Dic3C2414yes—
238C2×Dic6—24no—
239C2×Dic6—24no—
240C2×Dic6—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C42—22no—
252C3×C42C2×D414yes—
253C3×C42#4114yes—
254#217—28no—
255#217—28no—
256#217—28no—
257#217—44no—
258#217—44no—
259#217—44no—
260#217—44no—
261C4⋊Dic3—24no—
262C4⋊Dic3—28no—
263C4⋊Dic3—24no—
264C4⋊Dic3—28no—
265C4⋊Dic3—44no—
266C4⋊Dic3C2×D418yes—
267Dic3⋊C4—28no—
268Dic3⋊C4—28no—
269Dic3⋊C4—28no—
270Dic3⋊C4—44no—
271Dic3⋊C4—44no—
272Dic3⋊C4—44no—
273Dic3⋊C4C2×D4116yes—
274C4×Dic3—28no—
275C4×Dic3—28no—
276C4×Dic3—44no—
277C22×Q8—32no—
278C22×D4C22×D312yes—
279C22×D4C22×D311yescharacteristic
280C22×D4C22×D312yes—
281C22×D4C22×D311yescharacteristic
282C23×C4—34no—
283C23×C4C22×D312yes—
284C23×C4C22×D312yes—
285C4⋊Q8—64no—
286#126—616no—
287#125—38no—
288#125—616no—
289#123—616no—
290C22⋊Q8—68no—
291C22⋊Q8—68no—
292C22⋊Q8—124no—
293C22⋊Q8—124no—
294C4×Q8—68no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—42no—
298C4×D4—42no—
299#117—616no—
300#117—68no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—34no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—68no—
315C2×C4⋊C4C22×D312yes—
316C2×C4⋊C4C22×D312yes—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—34no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4—34no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4C22×D312yes—
327C2×C22⋊C4C22×D312yes—
328C2×C22⋊C4C22×D312yes—
329C2×C22⋊C4C22×D312yes—
330C2×C22⋊C4C22×D312yes—
331C2×C22⋊C4C22×D312yes—
332C2×C42—34no—
333C2×C42—34no—
334C2×C42—34no—
335C2×C42C22×D311yescharacteristic
336C2×C42C22×D311yescharacteristic
337C2×C42C22×D311yescharacteristic
338#95—34no—
339#95—34no—
340#95—34no—
341#95—34no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95C22×D312yes—
346#95C22×D312yes—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C32−1+414yes—
361C23×C3C2511yescharacteristic
362C3×D4—48no—
363C3×D4—48no—
364C3×D4—44no—
365C3×D4—44no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—48no—
381C2×C3×C4—48no—
382C2×C3×C4C22×D414yes—
383C2×C3×C4C22×D412yes—
384C2×C3×C4C2×#4114yes—
385C2×C3×C4C2×#4112yes—
386C2×C3×C4H(5,𝔽2)12yes—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—24no—
396C2×Dic3—24no—
397C2×Dic3—24no—
398C2×Dic3—416no—
399C2×Dic3C22×D418yes—
400Dic6—48no—
401C24C23×D312yes—
402C24C23×D312yes—
403C2×Q8—64no—
404C2×Q8—64no—
405C2×Q8—64no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—34no—
426C22×C4—68no—
427C22×C4—68no—
428C22×C4—124no—
429C22×C4C23×D311yescharacteristic
430C22×C4C23×D311yescharacteristic
431C22×C4C23×D312yes—
432C22×C4C23×D312yes—
433C22×C4C23×D312yes—
434C22×C4C23×D312yes—
435C22×C4C23×D311yescharacteristic
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—38no—
439C4⋊C4—316no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C4⋊C4—68no—
445C4⋊C4—64no—
446C4⋊C4—68no—
447C4⋊C4—68no—
448C4⋊C4—64no—
449C4⋊C4—124no—
450C4⋊C4—124no—
451C4⋊C4—124no—
452C4⋊C4—124no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—68no—
460C22⋊C4—68no—
461C22⋊C4—68no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C22⋊C4—124no—
465C22⋊C4—124no—
466C42—22no—
467C42—68no—
468C42—68no—
469C42—124no—
470C42D3×D414yes—
471C42#23714yes—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3C23×D411yescharacteristic
487C22×C3C22×#4111yescharacteristic
488C22×C3C2×H(5,𝔽2)11yescharacteristic
489C22×C3C2×2−1+412yes—
490C22×C3#58512yes—
491C3×C4—24no—
492C3×C4—24no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4#55014yes—
498Dic3—28no—
499Dic3—44no—
500Dic3—44no—
501Dic3—44no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23#101014yes—
516C23C24×D311yescharacteristic
517Q8—128no—
518D4—48no—
519D4—48no—
520D4—44no—
521D4—44no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—38no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—1216no—
551C2×C4C2×D3×D414yes—
552C2×C4C2×D3×D412yes—
553C2×C4C2×#23714yes—
554C2×C4C2×#23712yes—
555C2×C4#100412yes—
556C2×C3—44no—
557C2×C3—82no—
558C2×C3C2×#55011yescharacteristic
559C2×C3#348912yes—
560C2×C3#349512yes—
561C2×C3#353811yescharacteristic
562C2×C3#353911yescharacteristic
563C22—24no—
564C22—24no—
565C22—24no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22C22×D3×D411yescharacteristic
578C22C22×#23711yescharacteristic
579C22C2×#100411yescharacteristic
580C22C2×#101012yes—
581C22#596412yes—
582C4—24no—
583C4—24no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—68no—
589C4—124no—
590C4—124no—
591C4—124no—
592C4#571814yes—
593C3#4543211yescharacteristic, complemented, Sylow
594C2—44no—
595C2—82no—
596C2#8602812yes—
597C2#8603111yescharacteristic
598C2#8604212yes—
599C2#8605311yescharacteristic
600C2C2×#571811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC3×#2397C2
hypercenter#2397D3
hypocenterC3#45432
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. C2×#5541C2semidirect
3. #82379C2semidirect
4. #82376C2semidirect
5. #82353C2semidirect
6. #82351C2semidirect
7. #82335C2semidirect
8. #78069C2semidirect
9. #78021C2semidirect
10. #76724C2semidirect
11. #76685C2semidirect
12. #73390C2semidirect
13. #73388C2semidirect
140. #870D4semidirect
593. C3#45432semidirect

Maximal subgroups

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

subgroupindex
C2×#55412
C3×#23972
#823792
#823762
#823532
#823512
#823352
#780692
#780212
#767242
#766852
#733902
#733882
#454323