Finite Group Databasebeta

← #10686804

Subgroups — #10686804

Order 768.

1068 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×#5831C211yesmaximal, characteristic
2C3×#1789C211yesmaximal, characteristic, complemented
3#72138C211yesmaximal, characteristic, complemented
4#69980C211yesmaximal, characteristic, complemented
5#69968C211yesmaximal, characteristic, complemented
6C4×#806C211yesmaximal, characteristic, complemented
7#69445C211yesmaximal, characteristic, complemented
8#8366—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C22×D3×Q8C411yescharacteristic, complemented
11#5831—22no—
12#5831—22no—
13#5831—22no—
14#5831—22no—
15C2×Q8×Dic3C411yescharacteristic, complemented
16C2×#938—21no—
17C2×#938C411yescharacteristic, complemented
18C2×#936—21no—
19C2×#936C411yescharacteristic, complemented
20C2×C4×#67—21no—
21C2×C4×#67C2211yescharacteristic
22C2×#924—21no—
23C2×#924C2211yescharacteristic
24C2×C3×#104—22no—
25C3×#407C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27C2×#837—22no—
28C4×#217C2211yescharacteristic
29#5207C2211yescharacteristic
30C2×#806—21no—
31#4527—21no—
32#4469—21no—
33#4468—21no—
34#4467—22no—
35#4452—22no—
36C2×#548—31no—
37#1789—31no—
38#1789D311yescharacteristic, complemented
39C4×#99—31no—
40#1508—31no—
41#1508—31no—
42#1465—31no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C2×C3×#41D412yes—
47C2×C3×#41D412yes—
48C22×C3×Q8C2×C411yescharacteristic
49C22×C3×D4—21no—
50C22×C3×D4C2×C411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×#67—21no—
54C22×#67D411yescharacteristic
55C2×D3×Q8—22no—
56C2×D3×Q8—22no—
57C2×D3×Q8—44no—
58C22×C4×D3—21no—
59C22×C4×D3C2×C411yescharacteristic
60C22×Dic6—21no—
61C22×Dic6C2×C411yescharacteristic
62C2×C3×M16—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C42—21no—
65C2×#217—21no—
66C2×#217—21no—
67C2×#217—21no—
68C2×#217—21no—
69C2×#217—21no—
70C2×#217D411yescharacteristic
71Q8×Dic3—22no—
72Q8×Dic3—22no—
73#938—24no—
74#938—24no—
75#938—22no—
76#938—22no—
77#936—24no—
78#936—24no—
79#936—22no—
80#936—22no—
81C4×#67—24no—
82C4×#67—24no—
83C4×#67—21no—
84C4×#67—21no—
85C4×#67—21no—
86C4×#67—21no—
87C2×(C2×D3)⋊C4D411yescharacteristic
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2×C411yescharacteristic
90#924—24no—
91#924—24no—
92#924—21no—
93#924—21no—
94#924D412yes—
95#924D412yes—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4D411yescharacteristic
98C2×C4×Dic3—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3C2×C411yescharacteristic
101C2×#208—21no—
102C3×#104—44no—
103C3×C22⋊C8—21no—
104C3×C22⋊C8—21no—
105C3×#95—21no—
106#837—44no—
107C3⋊#95—21no—
108#830—21no—
109#830—21no—
110#806—24no—
111C22×#41C2×D311yescharacteristic
112C23×Q8—31no—
113#548—62no—
114#548—62no—
115#548—62no—
116#548—62no—
117C2×#124—31no—
118C2×#124—61no—
119C2×C22⋊Q8—31no—
120C2×C22⋊Q8—61no—
121C2×#121—31no—
122C2×#121—61no—
123C2×C4×Q8—31no—
124C2×C4×D4—31no—
125C2×C4×D4—61no—
126C2×#104—22no—
127C2×#104—62no—
128C2×#99—61no—
129#407—31no—
130#407C2×D311yescharacteristic
131C4×C22⋊C4—31no—
132C4×C22⋊C4C2×D311yescharacteristic
133#342—61no—
134#328—62no—
135#328—62no—
136#324—61no—
137#323—61no—
138C24×C3—21no—
139C24×C3C2×D411yescharacteristic
140C23×D3C22⋊C411yescharacteristic
141C3×#41—24no—
142C3×#41—24no—
143C3×#41—42no—
144C3×#41—42no—
145C3×#41—44no—
146C3×#41—44no—
147C2×C3×Q8—24no—
148C2×C3×Q8—21no—
149C2×C3×Q8C22⋊C412yes—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4—42no—
154C2×C3×D4—44no—
155C2×C3×D4C22⋊C412yes—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—42no—
160C22×C3×C4—44no—
161C22×C3×C4—41no—
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C2×#67—24no—
169C2×#67—21no—
170C2×#67—21no—
171C2×#67—22no—
172C2×#67—41no—
173C2×#67—42no—
174C2×#67—42no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—41no—
178C22×Dic3C22⋊C411yescharacteristic
179C22×Dic3C22⋊C411yescharacteristic
180C22×Dic3C22⋊C411yescharacteristic
181D3×Q8—44no—
182D3×Q8—48no—
183D3×Q8—44no—
184C2×C4×D3—24no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—44no—
188C2×C4×D3—42no—
189C2×Dic6—24no—
190C2×Dic6—21no—
191C2×Dic6—44no—
192C2×Dic6—42no—
193C2×Dic6C22⋊C412yes—
194C3×M16—42no—
195C2×C3×C8—41no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—41no—
198C3×C42—42no—
199#217—24no—
200#217—24no—
201#217—24no—
202#217—22no—
203#217—22no—
204#217—21no—
205#217—21no—
206#217—22no—
207#217—41no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210C4⋊Dic3—24no—
211C4⋊Dic3—21no—
212C4⋊Dic3C22⋊C412yes—
213Dic3⋊C4—22no—
214Dic3⋊C4—22no—
215Dic3⋊C4—21no—
216Dic3⋊C4—21no—
217C4×Dic3—24no—
218C4×Dic3—21no—
219C4×Dic3—21no—
220C4×Dic3—42no—
221#208—42no—
222C2×C3⋊C8—41no—
223C2×#41—21no—
224C2×#41—21no—
225C2×#41—44no—
226C2×#41D1212yes—
227C2×#41#6712yes—
228C22×Q8—31no—
229C22×Q8—61no—
230C22×Q8—62no—
231C22×Q8—62no—
232C22×Q8—124no—
233C22×Q8C4×D311yescharacteristic
234C22×D4—21no—
235C22×D4—31no—
236C22×D4—61no—
237C22×D4C4×D311yescharacteristic
238C23×C4—21no—
239C23×C4—31no—
240C23×C4—61no—
241C23×C4C22×D311yescharacteristic
242C2×M16—21no—
243C2×M16—61no—
244#124—62no—
245#124—64no—
246#124—64no—
247#124—62no—
248C22⋊Q8—32no—
249C22⋊Q8—32no—
250C22⋊Q8—64no—
251C22⋊Q8—64no—
252C22⋊Q8—61no—
253C22⋊Q8—61no—
254#121—62no—
255#121—64no—
256#121—64no—
257#121—62no—
258C4×Q8—62no—
259C4×Q8—62no—
260C4×D4—64no—
261C4×D4—64no—
262C4×D4—61no—
263C4×D4—61no—
264C4×D4—61no—
265C4×D4—61no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—31no—
272C2×C22⋊C4—31no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C42—21no—
279C2×C42—31no—
280C2×C42—61no—
281C2×C42—61no—
282#104—44no—
283#104—124no—
284#99—64no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—61no—
288C22⋊C8—61no—
289#95—21no—
290#95—61no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—22no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3C2×C22⋊C411yescharacteristic
301C23×C3#12011yescharacteristic
302C23×C3#12011yescharacteristic
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3#10412yescomplemented
308C3×Q8—24no—
309C3×Q8—22no—
310C3×Q8#9914yescomplemented
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—44no—
314C3×D4—44no—
315C3×D4—42no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4—42no—
327C2×C3×C4—81no—
328C2×C3×C4—81no—
329C2×C3×C4C2×C22⋊C411yescharacteristic
330C2×C3×C4C2×C22⋊C411yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12011yescharacteristic
333#67—42no—
334#67—42no—
335#67—48no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—42no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—81no—
350C2×Dic3—81no—
351C4×D3—42no—
352C4×D3—48no—
353C4×D3—42no—
354Dic6—48no—
355Dic6—42no—
356Dic6—42no—
357C3×C8—81no—
358C3⋊C8—81no—
359C24—21no—
360C24—31no—
361C24D3×D411yescharacteristic
362#41—24no—
363#41—24no—
364#41—42no—
365#41—42no—
366#41—44no—
367#41—44no—
368C2×Q8—24no—
369C2×Q8—21no—
370C2×Q8—32no—
371C2×Q8—64no—
372C2×Q8—61no—
373C2×Q8—122no—
374C2×Q8—124no—
375C2×Q8—128no—
376C2×Q8—124no—
377C2×Q8—124no—
378C2×Q8(C2×D3)⋊C412yes—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—62no—
385C2×D4—64no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—121no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4(C2×D3)⋊C412yes—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—41no—
401C22×C4—64no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—124no—
407C22×C4—122no—
408C22×C4—121no—
409C22×C4C2×C4×D311yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D411yescharacteristic
413C22×C4D3×D411yescharacteristic
414C22×C4C2×#6711yescharacteristic
415M16—42no—
416M16—122no—
417C2×C8—41no—
418C2×C8—121no—
419C4⋊C4—32no—
420C4⋊C4—64no—
421C4⋊C4—61no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—62no—
426C22⋊C4—22no—
427C22⋊C4—41no—
428C22⋊C4—64no—
429C22⋊C4—64no—
430C22⋊C4—64no—
431C22⋊C4—61no—
432C22⋊C4—62no—
433C22⋊C4—61no—
434C22⋊C4—62no—
435C22⋊C4—61no—
436C22⋊C4—61no—
437C22⋊C4—62no—
438C22⋊C4—121no—
439C42—42no—
440C42—64no—
441C42—61no—
442C42—61no—
443C42—122no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3#37911yescharacteristic
460C22×C3C2×#10411yescharacteristic
461C22×C3#42111yescharacteristic
462C2×D3—22no—
463C2×D3—42no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—42no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—24no—
470C3×C4—21no—
471C3×C4—82no—
472C3×C4C2×#9912yes—
473Dic3—41no—
474Dic3—41no—
475Dic3—42no—
476Dic3—41no—
477Dic3—42no—
478Dic3—82no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—32no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—61no—
490C23—61no—
491C23—121no—
492C23—121no—
493C23D3×C22⋊C411yescharacteristic
494C23#88211yescharacteristic
495C23#93711yescharacteristic
496Q8—24no—
497Q8—22no—
498Q8—122no—
499Q8—128no—
500Q8—122no—
501Q8#80614yescomplemented
502D4—22no—
503D4—22no—
504D4—44no—
505D4—44no—
506D4—42no—
507D4—122no—
508D4—122no—
509D4—128no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—61no—
522C2×C4—62no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—62no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—128no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—122no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—122no—
539C2×C4—241no—
540C2×C4—241no—
541C2×C4D3×C22⋊C411yescharacteristic
542C2×C4#88211yescharacteristic
543C2×C4C2×(C2×D3)⋊C411yescharacteristic
544C2×C4#93711yescharacteristic
545C8—81no—
546C8—241no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—82no—
552C2×C3#178111yescharacteristic
553C2×C3#178911yescharacteristic
554C2×C3#179611yescharacteristic
555D3—42no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—62no—
568C22—81no—
569C22—81no—
570C22—82no—
571C22—82no—
572C22—122no—
573C22—122no—
574C22—121no—
575C22—121no—
576C22D3×#10411yescharacteristic
577C22#482111yescharacteristic
578C22#495511yescharacteristic
579C4—21no—
580C4—21no—
581C4—24no—
582C4—21no—
583C4—82no—
584C4—121no—
585C4—122no—
586C4—121no—
587C4—121no—
588C4—122no—
589C4—242no—
590C4C2×#80612yes—
591C3#836611yescharacteristic, complemented, Sylow
592C2—21no—
593C2—21no—
594C2—41no—
595C2—41no—
596C2—82no—
597C2—122no—
598C2#6949811yescharacteristic
599C2#7038011yescharacteristic
600C2#7041511yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#379
prime radicalC2×#5831C2
cosocleC23×C3×C4C23
hypercenter#1789D3
hypocenterC3#8366
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. C3×#1789C2semidirect
3. #72138C2semidirect
4. #69980C2semidirect
5. #69968C2semidirect
6. C4×#806C2semidirect
7. #69445C2semidirect
10. C22×D3×Q8C4semidirect
15. C2×Q8×Dic3C4semidirect
17. C2×#938C4semidirect
19. C2×#936C4semidirect
38. #1789D3semidirect
307. C22×D3#104semidirect
310. C3×Q8#99semidirect
501. Q8#806semidirect
591. C3#8366semidirect

Maximal subgroups

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

subgroupindex
C2×#58312
C3×#17892
#721382
#699802
#699682
C4×#8062
#694452
#83663