Finite Group Databasebeta

← #10799116

Subgroups — #10799116

Order 768.

1566 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#909C211yesmaximal, characteristic
2C2×C3×#428C211yesmaximal, characteristic, complemented
3C2×Dic3×M16C211yesmaximal, characteristic, complemented
4#82228C211yesmaximal, characteristic
5C2×#5004C211yesmaximal, characteristic, complemented
6C2×#4973C211yesmaximal, characteristic, complemented
7C2×#4966C211yesmaximal, characteristic
8#70629C212yesmaximal, complemented
9#70627C212yesmaximal, complemented
10#70575C212yesmaximal, complemented
11#70574C212yesmaximal, complemented
12#21237—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C4×Dic3C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C2×#909—24no—
21C2×#909—24no—
22C2×#909C418yescomplemented
23C22×Dic12C2212yes—
24C22×#204C2212yes—
25C3×#428—24no—
26C2×C3×#107C2212yes—
27C2×C3×#106C2212yes—
28C2×C3⋊#95—22no—
29Dic3×M16—24no—
30C2×#815C2212yes—
31C2×C8×Dic3C2212yes—
32#5004—24no—
33#4988C2211yescharacteristic
34#4988C2211yescharacteristic
35#4973—22no—
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38#4973C2212yes—
39#4973C2212yes—
40C2×#810C2212yes—
41C2×#810C2212yes—
42C2×#810C2212yes—
43#4966—24no—
44C2×#809C2212yes—
45#4958—22no—
46C4×(C2×D3)⋊C4—22no—
47C22×#137—31no—
48C2×#428D311yescharacteristic, complemented
49C2×#417—31no—
50C2×#419—31no—
51C2×C4×M16—31no—
52#2297—31no—
53C2×#391—31no—
54#1973—32no—
55#1971—32no—
56#1969—32no—
57#1968—32no—
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C23×Dic3—21no—
62C23×Dic3D412yes—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235C2×C414yes—
69C22×D12C2311yescharacteristic
70C22×C4×D3—21no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—21no—
76C22×Dic6—22no—
77C22×Dic6C2×C414yes—
78C22×Dic6C2312yes—
79C22×Dic6C2311yescharacteristic
80C2×C3×M16C2×C414yes—
81C2×C3×M16D414yes—
82C2×C3×M16D414yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3D412yes—
99C2×C4×Dic3C2311yescharacteristic
100C2×C4×Dic3C2311yescharacteristic
101C2×#208—22no—
102C2×#208—22no—
103C2×#208D414yes—
104C22×C3⋊C8C2312yes—
105#909—232no—
106C2×Dic12—24no—
107C2×Dic12—24no—
108C2×Dic12C2×C418yescomplemented
109C2×#204—24no—
110C2×#204—24no—
111C2×#204C2×C418yescomplemented
112C3×#107—24no—
113C3×#106—24no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—24no—
117C3⋊#95—24no—
118#815—24no—
119C8×Dic3—24no—
120#810—24no—
121#810—24no—
122#810—24no—
123#809—24no—
124C22×#41—31no—
125C23×Q8—31no—
126C2×#137—38no—
127C2×#137—64no—
128C2×#137—64no—
129C22×Q16—32no—
130C22×SD16—32no—
131C22×M16—31no—
132C22×M16C2×D311yescharacteristic
133C2×#117C2×D311yescharacteristic
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C42—31no—
136#428—24no—
137C2×#107C2×D312yes—
138C2×#106C2×D312yes—
139#419—64no—
140#417—64no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#102—32no—
145C4×M16—64no—
146C2×#97—32no—
147C2×C4×C8—32no—
148#391—32no—
149#391—31no—
150#391—31no—
151#391—32no—
152#391—62no—
153#390—31no—
154#390—31no—
155#386—62no—
156C4×C22⋊C4—62no—
157C2×#95—62no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—44no—
164C22×C3×C4—42no—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—24no—
174C2×#67—42no—
175C2×#67—42no—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—24no—
181C22×Dic3—42no—
182C22×Dic3—42no—
183C22×Dic3—42no—
184C22×Dic3C2×D412yes—
185C22×Dic3C2×D412yes—
186#235—28no—
187#235—28no—
188#235—44no—
189#235—44no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12C22×C412yes—
196C2×C4×D3—24no—
197C2×C4×D3—42no—
198C2×C4×D3—42no—
199C2×Dic6—28no—
200C2×Dic6—28no—
201C2×Dic6—22no—
202C2×Dic6—21no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—22no—
208C2×Dic6—21no—
209C2×Dic6—24no—
210C2×Dic6—22no—
211C2×Dic6—21no—
212C2×Dic6—44no—
213C2×Dic6—42no—
214C2×Dic6—44no—
215C2×Dic6—42no—
216C2×Dic6C22×C414yes—
217C2×Dic6C22×C412yes—
218C3×M16C22⋊C4116yescomplemented
219C2×C3×C8C22×C414yes—
220C2×C3×C8C2×D414yes—
221C2×C3×C8C2×D414yes—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C42—42no—
231(C2×D3)⋊C4—44no—
232(C2×D3)⋊C4—44no—
233C4×Dic3—24no—
234C4×Dic3—21no—
235C4×Dic3—21no—
236C4×Dic3—21no—
237C4×Dic3—21no—
238#208—28no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8C2×D414yes—
242Dic12—216no—
243#204—216no—
244C2×#41—34no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C22×Q8—34no—
251C22×Q8—32no—
252C22×Q8—31no—
253C22×Q8—61no—
254C22×Q8—62no—
255C22×Q8—62no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×D4—31no—
260C22×D4—61no—
261C23×C4—21no—
262C23×C4—32no—
263C23×C4—61no—
264C23×C4—61no—
265C23×C4C22×D311yescharacteristic
266#137—632no—
267C2×Q16—38no—
268C2×Q16—64no—
269C2×Q16—64no—
270C2×SD16—38no—
271C2×SD16—64no—
272C2×SD16—64no—
273C2×M16—34no—
274C2×M16—62no—
275C2×M16—62no—
276C2×M16C4×D314yes—
277C2×M16D1214yes—
278C2×M16#6714yes—
279C22×C8—32no—
280C22×C8C22×D312yes—
281#117—22no—
282#117—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4C22×D312yes—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—62no—
292C2×C42—21no—
293C2×C42—32no—
294C2×C42—31no—
295C2×C42—31no—
296C2×C42—62no—
297C2×C42—62no—
298#107—24no—
299#106—24no—
300#103—64no—
301#103—64no—
302#103—64no—
303#102—64no—
304#97—64no—
305C4×C8—64no—
306#95—64no—
307#95—64no—
308#95—62no—
309#95—62no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3C4×D412yes—
313C23×C3#12112yes—
314C23×C3#12712yes—
315C23×C3C22×D411yescharacteristic
316C22×D3—22no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—42no—
320C22×D3—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4—84no—
328C2×C3×C4C2×C22⋊C414yes—
329C2×C3×C4C4×D414yes—
330C2×C3×C4C4×D412yes—
331C2×C3×C4#12114yes—
332C2×C3×C4#12112yes—
333C2×C3×C4#12414yes—
334C2×C3×C4#12712yes—
335C2×C3×C4C23×C411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341#67—48no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—24no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—48no—
350C2×Dic3—42no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—41no—
354C2×Dic3—48no—
355C2×Dic3#13714yescomplemented
356D12—22no—
357D12—22no—
358D12—41no—
359D12—41no—
360C4×D3—48no—
361Dic6—24no—
362Dic6—22no—
363Dic6—24no—
364Dic6—22no—
365Dic6—42no—
366Dic6—41no—
367Dic6—42no—
368Dic6—41no—
369Dic6—48no—
370C3×C8C2×C22⋊C418yes—
371C3⋊C8—24no—
372C24—61no—
373C24D3×D411yescharacteristic
374#41—68no—
375#41—68no—
376#41—124no—
377#41—124no—
378C2×Q8—34no—
379C2×Q8—32no—
380C2×Q8—64no—
381C2×Q8—62no—
382C2×Q8—61no—
383C2×Q8—68no—
384C2×Q8—68no—
385C2×Q8—62no—
386C2×Q8—61no—
387C2×Q8—62no—
388C2×Q8—61no—
389C2×Q8—62no—
390C2×Q8—61no—
391C2×Q8—62no—
392C2×Q8—61no—
393C2×Q8—124no—
394C2×Q8—122no—
395C2×Q8—122no—
396C2×Q8—124no—
397C2×D4—32no—
398C2×D4—64no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—122no—
405C2×D4—122no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—64no—
414C22×C4—64no—
415C22×C4—64no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—64no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4C2×C4×D312yes—
425C22×C4C2×D1212yes—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D412yes—
428C22×C4#23912yes—
429C22×C4#23912yes—
430C22×C4C2×#6712yes—
431C22×C4C23×D311yescharacteristic
432Q16—616no—
433SD16—616no—
434M16—68no—
435M16(C2×D3)⋊C4116yescomplemented
436C2×C8—34no—
437C2×C8—62no—
438C2×C8—62no—
439C2×C8C2×C4×D314yes—
440C2×C8C2×D1214yes—
441C2×C8C2×#6714yes—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—124no—
451C22⋊C4—124no—
452C42—42no—
453C42—64no—
454C42—61no—
455C42—61no—
456C42—61no—
457C42—61no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—24no—
461C22×C3—24no—
462C22×C3#39014yes—
463C22×C3C2×C4×D411yescharacteristic
464C22×C3C2×#12111yescharacteristic
465C22×C3C2×#12711yescharacteristic
466C22×C3C2×#13712yes—
467C22×C3#57512yes—
468C2×D3—44no—
469C2×D3—42no—
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—82no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—82no—
476C3×C4#39014yes—
477C3×C4C22×C22⋊C411yescharacteristic
478C3×C4C2×C4×D411yescharacteristic
479C3×C4C2×#12111yescharacteristic
480C3×C4C2×#12411yescharacteristic
481Dic3—24no—
482Dic3—42no—
483Dic3—44no—
484Dic3—42no—
485C23—22no—
486C23—22no—
487C23—62no—
488C23—121no—
489C23—122no—
490C23—122no—
491C23—121no—
492C23#89312yes—
493C23#89512yes—
494C23#94012yes—
495C23C2×D3×D411yescharacteristic
496Q8—64no—
497Q8—62no—
498Q8—64no—
499Q8—62no—
500Q8—128no—
501Q8—122no—
502Q8—121no—
503Q8—122no—
504Q8—121no—
505D4—62no—
506D4—62no—
507D4—128no—
508D4—121no—
509D4—121no—
510C2×C4—34no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—84no—
525C2×C4—128no—
526C2×C4—128no—
527C2×C4—121no—
528C2×C4—122no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—128no—
532C2×C4#89312yes—
533C2×C4#89314yes—
534C2×C4#89512yes—
535C2×C4#89514yes—
536C2×C4C2×(C2×D3)⋊C414yes—
537C2×C4#94012yes—
538C2×C4#94714yes—
539C2×C4C22×C4×D311yescharacteristic
540C2×C4C22×D1211yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4C2×#23911yescharacteristic
543C2×C4C2×#23911yescharacteristic
544C2×C4C22×#6711yescharacteristic
545C8—64no—
546C8C2×(C2×D3)⋊C418yes—
547C2×C3—24no—
548C2×C3C2×#39011yescharacteristic
549C2×C3C4×#13712yes—
550C2×C3#305612yes—
551C2×C3#315211yescharacteristic
552C2×C3#315311yescharacteristic
553D3—82no—
554C22—22no—
555C22—24no—
556C22—24no—
557C22—22no—
558C22—124no—
559C22—122no—
560C22—241no—
561C22—242no—
562C22—241no—
563C22#498814yes—
564C22C2×#89311yescharacteristic
565C22C2×#89511yescharacteristic
566C22#577312yes—
567C22D3×#13712yes—
568C22C2×#94011yescharacteristic
569C4—64no—
570C4—81no—
571C4—82no—
572C4—81no—
573C4—122no—
574C4—122no—
575C4—124no—
576C4#498814yes—
577C4C2×#89311yescharacteristic
578C4C2×#89511yescharacteristic
579C4C22×(C2×D3)⋊C411yescharacteristic
580C4C2×#94711yescharacteristic
581C3#2123711yescharacteristic, complemented, Sylow
582C2—24no—
583C2—242no—
584C2#7920012yes—
585C2#7921212yes—
586C2C2×#498811yescharacteristic
587C2#8393811yescharacteristic
588C2#8397211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#428D3
hypocenterC3#21237
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C3×#428C2semidirect
3. C2×Dic3×M16C2semidirect
5. C2×#5004C2semidirect
6. C2×#4973C2semidirect
8. #70629C2semidirect
9. #70627C2semidirect
10. #70575C2semidirect
11. #70574C2semidirect
22. C2×#909C4semidirect
48. C2×#428D3semidirect
108. C2×Dic12C2×C4semidirect
111. C2×#204C2×C4semidirect
218. C3×M16C22⋊C4semidirect
355. C2×Dic3#137semidirect
435. M16(C2×D3)⋊C4semidirect
581. C3#21237semidirect

Maximal subgroups

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

subgroupindex
C22×#9092
C2×C3×#4282
C2×Dic3×M162
#822282
C2×#50042
C2×#49732
C2×#49662
#706292
#706272
#705752
#705742
#212373