Finite Group Databasebeta

← #11025117

Subgroups — #11025117

Order 768.

2012 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
1C2×#5518C211yesmaximal, characteristic, complemented
2C22×#870C211yesmaximal, characteristic, complemented
3C3×#2290C211yesmaximal, characteristic
4#82353C212yesmaximal
5#82265C211yesmaximal, characteristic, complemented
6#82076C211yesmaximal, characteristic, complemented
7#75702C214yesmaximal, complemented
8#75165C214yesmaximal, complemented
9#74736C214yesmaximal
10#74417C214yesmaximal, complemented
11#73676C212yesmaximal, complemented
12#73505C212yesmaximal, complemented
13#73465C212yesmaximal, complemented
14#73407C211yesmaximal, characteristic, complemented
15#73390C211yesmaximal, characteristic, complemented
16#37705—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C2×#924C2212yes—
22C2×#924C2212yes—
23C22×Dic3⋊C4C2214yes—
24#5518—28no—
25C2×#889C2214yes—
26C2×#879C2214yes—
27C2×#870—28no—
28C2×#870C2212yes—
29C2×#870C2211yescharacteristic
30C2×#870C2214yes—
31C2×#870C2211yescharacteristic
32C2×#869C2211yescharacteristic
33C2×C4×Dic6C2212yes—
34C3×#388C2212yes—
35C3×#387C2212yes—
36C3×#386C2212yes—
37C3×#385C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C4×C22⋊C4C2211yescharacteristic
40C3×C4×C22⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42#5220C2214yes—
43#5212C2212yes—
44#5212C2214yes—
45#4983C2214yes—
46#4982C2214yes—
47#4982C2214yes—
48#4976C2214yes—
49#4952C2214yes—
50#4948C2214yes—
51#4948C2214yes—
52#4946C2214yes—
53#4944C2214yes—
54#4942C2214yes—
55Dic3×C22⋊C4C2214yes—
56#4935C2211yescharacteristic
57C4×C4⋊Dic3C2212yes—
58#4929C2212yes—
59#4929C2212yes—
60#4929C2212yes—
61#4928C2212yes—
62#4928C2214yes—
63#4928C2212yes—
64#4659C2214yes—
65#4659C2214yes—
66#4658C2214yes—
67#4658C2214yes—
68#4655C2214yes—
69#4655C2214yes—
70#4652C2214yes—
71#4650C2214yes—
72#4647C2214yes—
73#4647C2214yes—
74C2×#541—31no—
75C22×#125—31no—
76#2857—31no—
77#2725—34no—
78#2719—34no—
79#2642—31no—
80#2629—31no—
81#2489—32no—
82#2481—34no—
83#2453—32no—
84#2442—34no—
85#2417—32no—
86#2360—32no—
87#2330—31no—
88#2290D311yescharacteristic
89C22×C3×D4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2312yes—
92C23×C3×C4C2311yescharacteristic
93C23×Dic3C2314yes—
94C22×Dic6C2312yes—
95C3×C4×D4—24no—
96C3×#117—24no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4C2312yes—
101C2×C3×C22⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4C2312yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C42—24no—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×#217C2314yes—
113C2×#217C2314yes—
114C2×#217C2314yes—
115C2×C4⋊Dic3—28no—
116C2×C4⋊Dic3C2314yes—
117C2×C4⋊Dic3C2314yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2312yes—
122#924—28no—
123#924—28no—
124C2×Dic3⋊C4—216no—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
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—
136#889—216no—
137#879—216no—
138#870—24no—
139#870—28no—
140#870—216no—
141#870D418yescomplemented
142#869—24no—
143C4×Dic6—28no—
144C3×#95C2312yes—
145C3×#95C2312yes—
146C3×#95C2312yes—
147C3×#95C2312yes—
148C3×#95C2312yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159#541—68no—
160C2×C4⋊Q8—31no—
161C2×#126—34no—
162C2×#125—34no—
163C2×#125—31no—
164C2×#125—31no—
165C2×#125—32no—
166C2×#125—34no—
167C2×#125—68no—
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8—32no—
170C2×C4×Q8—32no—
171C2×C4×D4C2×D311yescharacteristic
172C2×#117C2×D311yescharacteristic
173C22×C4⋊C4—32no—
174C22×C4⋊C4—34no—
175C22×C42C2×D311yescharacteristic
176#400—34no—
177#400—34no—
178#400—34no—
179#400—34no—
180#398—34no—
181#398—34no—
182#398—34no—
183#397—34no—
184#396—34no—
185#396—34no—
186#391—32no—
187#391—32no—
188#391—32no—
189#389—34no—
190#389—32no—
191#389—34no—
192#389—32no—
193#389—34no—
194#389—34no—
195#388—34no—
196#388C2×D312yes—
197#387—34no—
198#387—34no—
199#387—34no—
200#387C2×D312yes—
201#386—34no—
202#386C2×D312yes—
203#385—34no—
204#385C2×D312yes—
205#384—31no—
206#382—34no—
207#380—34no—
208#380—32no—
209#380—34no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4C2×D311yescharacteristic
212C4×C22⋊C4—34no—
213C4×C22⋊C4C2×D311yescharacteristic
214C4×C22⋊C4C2×D311yescharacteristic
215#376C2×D311yescharacteristic
216C24×C3C2411yescharacteristic
217C24×C3C2411yescharacteristic
218C2×C3×D4—22no—
219C2×C3×D4—24no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—42no—
228C22×C3×C4#4114yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2412yes—
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2411yescharacteristic
238C22×Dic3—28no—
239C22×Dic3—28no—
240C22×Dic3—44no—
241C22×Dic3C2414yes—
242C22×Dic3C2414yes—
243C22×Dic3C2414yes—
244C22×Dic3C2414yes—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—42no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—42no—
256C3×C42—22no—
257C3×C42—24no—
258C3×C42—22no—
259C3×C42—22no—
260C3×C42—22no—
261C3×C42C2×D414yes—
262#217—28no—
263#217—28no—
264#217—44no—
265C4⋊Dic3—28no—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268C4⋊Dic3—28no—
269C4⋊Dic3—44no—
270C4⋊Dic3C2×D418yes—
271Dic3⋊C4—28no—
272Dic3⋊C4—28no—
273Dic3⋊C4—28no—
274Dic3⋊C4—28no—
275Dic3⋊C4—28no—
276Dic3⋊C4—28no—
277Dic3⋊C4—44no—
278Dic3⋊C4—44no—
279Dic3⋊C4C2×D4116yes—
280C4×Dic3—28no—
281C4×Dic3—44no—
282C22×Q8—32no—
283C22×D4C22×D311yescharacteristic
284C23×C4—34no—
285C23×C4C22×D311yescharacteristic
286C23×C4C22×D312yes—
287C23×C4C22×D311yescharacteristic
288C4⋊Q8—64no—
289#126—616no—
290#125—38no—
291#125—68no—
292#125—616no—
293#125—616no—
294#125—64no—
295C22⋊Q8—68no—
296C22⋊Q8—68no—
297C4×Q8—68no—
298C4×D4—24no—
299#117—24no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—34no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—616no—
316C2×C4⋊C4—68no—
317C2×C4⋊C4C22×D312yes—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D312yes—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4C22×D312yes—
324C2×C22⋊C4C22×D311yescharacteristic
325C2×C22⋊C4C22×D311yescharacteristic
326C2×C22⋊C4C22×D312yes—
327C2×C22⋊C4C22×D312yes—
328C2×C42—24no—
329C2×C42—34no—
330C2×C42—34no—
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D312yes—
335C2×C42C22×D311yescharacteristic
336C2×C42C22×D311yescharacteristic
337#95—34no—
338#95—34no—
339#95—34no—
340#95—34no—
341#95—34no—
342#95—34no—
343#95—34no—
344#95—34no—
345#95—34no—
346#95—34no—
347#95C22×D312yes—
348#95C22×D312yes—
349#95C22×D312yes—
350#95C22×D312yes—
351#95C22×D312yes—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3C2×#4112yes—
361C23×C32−1+414yes—
362C23×C3C2511yescharacteristic
363C3×D4—44no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—28no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—48no—
386C2×C3×C4—44no—
387C2×C3×C4C22×D414yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C2×#4112yes—
390C2×C3×C4C2×#4112yes—
391C2×C3×C42−1+412yes—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—24no—
396C2×Dic3—24no—
397C2×Dic3—24no—
398C2×Dic3—24no—
399C2×Dic3—24no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3—416no—
404C2×Dic3C22×D418yes—
405Dic6—48no—
406C24C23×D311yescharacteristic
407C24C23×D311yescharacteristic
408C2×Q8—64no—
409C2×Q8—64no—
410C2×Q8—64no—
411C2×D4—22no—
412C2×D4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—34no—
421C22×C4—34no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—42no—
425C22×C4—68no—
426C22×C4—68no—
427C22×C4—124no—
428C22×C4#23514yes—
429C22×C4C23×D311yescharacteristic
430C22×C4C23×D312yes—
431C22×C4C23×D312yes—
432C22×C4C23×D311yescharacteristic
433C22×C4C23×D311yescharacteristic
434C22×C4C23×D312yes—
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—38no—
441C4⋊C4—316no—
442C4⋊C4—42no—
443C4⋊C4—68no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—68no—
447C4⋊C4—68no—
448C4⋊C4—68no—
449C4⋊C4—64no—
450C4⋊C4—68no—
451C4⋊C4—64no—
452C4⋊C4—68no—
453C4⋊C4—124no—
454C4⋊C4—124no—
455C4⋊C4—124no—
456C22⋊C4—22no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—24no—
460C22⋊C4—42no—
461C22⋊C4—68no—
462C22⋊C4—68no—
463C22⋊C4—124no—
464C42—22no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—68no—
470C42—124no—
471C42D3×D414yes—
472C22×C3—22no—
473C22×C3—24no—
474C22×C3—22no—
475C22×C3—24no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—42no—
479C22×C3—41no—
480C22×C3—42no—
481C22×C3#54114yescomplemented
482C22×C3C23×D411yescharacteristic
483C22×C3C22×#4111yescharacteristic
484C22×C3C2×2−1+412yes—
485C22×C3C2×2−1+411yescharacteristic
486C22×C3#58512yes—
487C3×C4—22no—
488C3×C4—24no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—41no—
494C3×C4—42no—
495C3×C4#54814yes—
496Dic3—28no—
497Dic3—44no—
498Dic3—44no—
499Dic3—44no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—22no—
508C23C2×#23512yes—
509C23#101014yes—
510C23C24×D311yescharacteristic
511Q8—128no—
512D4—44no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—38no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—64no—
538C2×C4—64no—
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—1216no—
549C2×C4C2×#23512yes—
550C2×C4C2×#23512yes—
551C2×C4C2×D3×D414yes—
552C2×C4C2×D3×D412yes—
553C2×C4#101012yes—
554C2×C3—24no—
555C2×C3—81no—
556C2×C3C2×#54111yescharacteristic
557C2×C3C2×#54811yescharacteristic
558C2×C3#348912yes—
559C2×C3#351211yescharacteristic
560C2×C3#356312yes—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—22no—
565C22—42no—
566C22—41no—
567C22—42no—
568C22—41no—
569C22—41no—
570C22#551814yescomplemented
571C22C22×#23511yescharacteristic
572C22C22×D3×D411yescharacteristic
573C22C2×#101012yes—
574C22C2×#101011yescharacteristic
575C22#596412yes—
576C4—24no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—68no—
585C4—124no—
586C4—124no—
587C4—124no—
588C4#555214yes—
589C3#3770511yescharacteristic, complemented, Sylow
590C2—24no—
591C2—81no—
592C2C2×#551811yescharacteristic
593C2C2×#555211yescharacteristic
594C2#8585111yescharacteristic
595C2#8604212yes—
596C2#8614212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2290C2
prime radicalC22×C3×D4C23
hypercenter#2290D3
hypocenterC3#37705
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2290

Complements

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

NH ≅ G/Ntype
1. C2×#5518C2semidirect
2. C22×#870C2semidirect
5. #82265C2semidirect
6. #82076C2semidirect
7. #75702C2semidirect
8. #75165C2semidirect
10. #74417C2semidirect
11. #73676C2semidirect
12. #73505C2semidirect
13. #73465C2semidirect
14. #73407C2semidirect
15. #73390C2semidirect
141. #870D4semidirect
481. C22×C3#541semidirect
570. C22#5518semidirect
589. C3#37705semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
C22×#8702
C3×#22902
#823532
#822652
#820762
#757022
#751652
#747362
#744172
#736762
#735052
#734652
#734072
#733902
#377053