Finite Group Databasebeta

← #10799118

Subgroups — #10799118

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×#942C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic, complemented
3C2×#5225C211yesmaximal, characteristic, complemented
4C2×#4966C211yesmaximal, characteristic
5C2×#4964C211yesmaximal, characteristic, complemented
6#82070C211yesmaximal, characteristic
7C2×C4×#208C211yesmaximal, characteristic, complemented
8#69679C212yesmaximal, complemented
9#69678C212yesmaximal, complemented
10#69677C212yesmaximal, complemented
11#69676C212yesmaximal, complemented
12#21237—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#216C2212yes—
18C2×#942—24no—
19C2×#942—24no—
20C2×#942C418yescomplemented
21C22×#215C2212yes—
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C3×#391—22no—
26C3×#391C2212yes—
27C3×#391C2212yes—
28C2×C3×#95—22no—
29#5225—24no—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5213—22no—
33C2×#810C2212yes—
34#4966—24no—
35C2×#809C2212yes—
36#4964—24no—
37C2×#808C2212yes—
38C2×#807C2212yes—
39#4939C2211yescharacteristic
40#4939C2211yescharacteristic
41C4×(C2×D3)⋊C4—22no—
42#4929C2211yescharacteristic
43#4929C2211yescharacteristic
44C4×#208—24no—
45C2×#803C2212yes—
46C2×C4×C3⋊C8C2212yes—
47C22×#137—31no—
48C2×#428—31no—
49C2×#417—31no—
50C2×#419—31no—
51C2×C4×M16—31no—
52#2297—31no—
53C2×#391D311yescharacteristic, complemented
54#1973—32no—
55#1971—32no—
56#1969—32no—
57#1968—32no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8C2×C414yes—
65C22×C3×Q8C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4D412yes—
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235—22no—
76C2×#235C2×C414yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C42—22no—
85C2×C3×C42—22no—
86C2×C3×C42D412yes—
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#216—24no—
91C2×#216—24no—
92C2×#216C2×C418yescomplemented
93#942—232no—
94C2×#215—24no—
95C2×#215—24no—
96C2×#215C2×C418yescomplemented
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99#926—22no—
100#926—22no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4×Dic3—21no—
104C2×#208—22no—
105C2×#208—22no—
106C2×#208C2×C414yes—
107C2×#208D414yes—
108C2×#208D414yes—
109C2×#208D414yes—
110C22×C3⋊C8C2312yes—
111C22×C3⋊C8C2312yes—
112C3×#95—24no—
113C3×#95—24no—
114#835—24no—
115#835—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#810—24no—
119#809—24no—
120#808—24no—
121#807—24no—
122#803—24no—
123C4×C3⋊C8—24no—
124C22×#41—31no—
125C23×Q8C2×D311yescharacteristic
126C2×#137—38no—
127C2×#137—64no—
128C2×#137—64no—
129C22×Q16—32no—
130C22×SD16—32no—
131C22×M16—31no—
132C22×M16—31no—
133C2×#117—31no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C42C2×D311yescharacteristic
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
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—22no—
149#391—31no—
150#391—31no—
151#391C2×D312yes—
152#391C2×D312yes—
153#390—31no—
154#390—31no—
155#386—62no—
156C4×C22⋊C4—62no—
157C2×#95—22no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—28no—
163C2×C3×Q8—28no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—44no—
169C2×C3×Q8—42no—
170C2×C3×Q8—44no—
171C2×C3×Q8—42no—
172C2×C3×Q8C22×C414yes—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4—44no—
182C22×C3×C4—42no—
183C22×C3×C4C22×C412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4#4112yes—
191C22×C3×C4#4112yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—24no—
194C2×#67—42no—
195C2×#67—42no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199#235—28no—
200#235—28no—
201#235—44no—
202#235—44no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12C22×C412yes—
209C2×C4×D3—24no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6C22×C412yes—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C42—24no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226#217—42no—
227#216—216no—
228#215—216no—
229(C2×D3)⋊C4—44no—
230(C2×D3)⋊C4—44no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4×Dic3—42no—
236#208—28no—
237#208C22⋊C4116yescomplemented
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8C22×C414yes—
241C2×C3⋊C8C2×D414yes—
242C2×C3⋊C8C2×D414yes—
243C2×C3⋊C8C2×D414yes—
244C2×#41—34no—
245C2×#41—62no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C22×Q8—22no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—31no—
257C22×Q8C4×D314yes—
258C22×Q8C22×D312yes—
259C22×D4—31no—
260C22×D4—61no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—61no—
264C23×C4#6712yes—
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—34no—
275C2×M16—34no—
276C2×M16—34no—
277C2×M16—62no—
278C2×M16—62no—
279C22×C8—32no—
280C22×C8—32no—
281#117—62no—
282#117—62no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4C22×D312yes—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—62no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—61no—
295C2×C42#6712yes—
296C2×C42C22×D311yescharacteristic
297C2×C42C22×D311yescharacteristic
298#107—64no—
299#106—64no—
300#103—64no—
301#103—64no—
302#103—64no—
303#102—64no—
304#97—64no—
305C4×C8—64no—
306#95—24no—
307#95—24no—
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—
321C3×Q8—24no—
322C3×Q8—24no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—48no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—48no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—48no—
336C2×C3×C4—42no—
337C2×C3×C4—44no—
338C2×C3×C4—84no—
339C2×C3×C4C2×C22⋊C414yes—
340C2×C3×C4C4×D414yes—
341C2×C3×C4C4×D412yes—
342C2×C3×C4#12114yes—
343C2×C3×C4#12112yes—
344C2×C3×C4#12414yes—
345C2×C3×C4#12712yes—
346C2×C3×C4#13714yescomplemented
347C2×C3×C4C23×C411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353#67—48no—
354C2×Dic3—22no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361D12—22no—
362D12—22no—
363D12—41no—
364D12—41no—
365C4×D3—48no—
366Dic6—22no—
367Dic6—22no—
368Dic6—41no—
369Dic6—41no—
370C3⋊C8—24no—
371C3⋊C8C2×C22⋊C418yes—
372C24—61no—
373C24C2×#6711yescharacteristic
374#41—68no—
375#41—68no—
376#41—124no—
377#41—124no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—32no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8C2×C4×D314yes—
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—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—61no—
417C22×C4—64no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4C2×C4×D312yes—
423C22×C4#23512yes—
424C22×C4D3×D412yes—
425C22×C4D3×D412yes—
426C22×C4D3×D412yes—
427C22×C4#23712yes—
428C22×C4C2×#6712yes—
429C22×C4C2×#6712yes—
430C22×C4C2×#6711yescharacteristic
431C22×C4C23×D311yescharacteristic
432Q16—616no—
433SD16—616no—
434M16—316no—
435M16—68no—
436C2×C8—34no—
437C2×C8—34no—
438C2×C8—34no—
439C2×C8—34no—
440C2×C8—62no—
441C2×C8—62no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C22⋊C4—124no—
450C22⋊C4—122no—
451C22⋊C4—124no—
452C42—24no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—122no—
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—24no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4—82no—
477C3×C4#39014yes—
478C3×C4C22×C22⋊C411yescharacteristic
479C3×C4C2×C4×D411yescharacteristic
480C3×C4C2×#12111yescharacteristic
481C3×C4C2×#12411yescharacteristic
482Dic3—42no—
483Dic3—81no—
484Dic3—81no—
485C23—22no—
486C23—22no—
487C23—62no—
488C23—121no—
489C23—121no—
490C23—122no—
491C23—122no—
492C23C4×#6712yes—
493C23#93912yes—
494C23#94012yes—
495C23C22×#6711yescharacteristic
496Q8—24no—
497Q8—24no—
498Q8—48no—
499Q8—42no—
500Q8—42no—
501Q8—62no—
502Q8—62no—
503Q8—121no—
504Q8—121no—
505D4—62no—
506D4—62no—
507D4—128no—
508D4—121no—
509D4—121no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—62no—
523C2×C4—84no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—128no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4D3×C22⋊C414yes—
532C2×C4#67⋊C414yes—
533C2×C4#88414yes—
534C2×C4#88514yes—
535C2×C4C4×#6712yes—
536C2×C4#93912yes—
537C2×C4#94012yes—
538C2×C4#94214yescomplemented
539C2×C4C22×C4×D311yescharacteristic
540C2×C4C2×#23511yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C2×C4C2×D3×D411yescharacteristic
543C2×C4C2×D3×D411yescharacteristic
544C2×C4C2×#23711yescharacteristic
545C8—38no—
546C8—64no—
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—122no—
559C22—124no—
560C22—241no—
561C22—242no—
562C22—241no—
563C22#495714yes—
564C22C2×C4×#6711yescharacteristic
565C22C2×#93911yescharacteristic
566C22C2×#94011yescharacteristic
567C22C2×#94212yes—
568C22#582212yes—
569C4—24no—
570C4—42no—
571C4—44no—
572C4—82no—
573C4—122no—
574C4—241no—
575C4—241no—
576C4#495714yes—
577C4C2×D3×C22⋊C411yescharacteristic
578C4C2×#67⋊C411yescharacteristic
579C4C2×#88411yescharacteristic
580C4C2×#88511yescharacteristic
581C3#2123711yescharacteristic, complemented, Sylow
582C2—24no—
583C2—242no—
584C2C2×#495711yescharacteristic
585C2C4×#94212yes—
586C2#8273212yes—
587C2#8294311yescharacteristic
588C2#8320311yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#391C2
socleC23×C3C22×D4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#391D3
hypocenterC3#21237
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×C3×#391C2semidirect
3. C2×#5225C2semidirect
5. C2×#4964C2semidirect
7. C2×C4×#208C2semidirect
8. #69679C2semidirect
9. #69678C2semidirect
10. #69677C2semidirect
11. #69676C2semidirect
20. C2×#942C4semidirect
53. C2×#391D3semidirect
92. C2×#216C2×C4semidirect
96. C2×#215C2×C4semidirect
237. #208C22⋊C4semidirect
346. C2×C3×C4#137semidirect
538. C2×C4#942semidirect
581. C3#21237semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C2×C3×#3912
C2×#52252
C2×#49662
C2×#49642
#820702
C2×C4×#2082
#696792
#696782
#696772
#696762
#212373