Finite Group Databasebeta

← #10702902

Subgroups — #10702902

Order 768.

890 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5132C211yesmaximal, characteristic
2#83762C211yesmaximal, characteristic
3#82070C211yesmaximal, characteristic
4C3×#1728C211yesmaximal, characteristic, complemented
5C2×#4559C211yesmaximal, characteristic, complemented
6C2×#4556C211yesmaximal, characteristic, complemented
7#71527C211yesmaximal, characteristic, complemented
8#11175—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5213C414yes—
17C2×#824—21no—
18#5132—22no—
19#5132—22no—
20C2×#823—21no—
21#5129—22no—
22#5126—22no—
23#5110—22no—
24#4939—21no—
25C4×(C2×D3)⋊C4—22no—
26#4929—21no—
27C3×#336—22no—
28#4559—22no—
29#4556—22no—
30#4524—22no—
31C2×#413—31no—
32#2879—31no—
33#2297—31no—
34C2×#344—31no—
35C2×#344—31no—
36#1728—31no—
37#1728D311yescharacteristic, complemented
38C23×C3×C4C2×C412yes—
39C23×C3×C4C2311yescharacteristic
40C22×#67C2×C412yes—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235—21no—
53C22×D12D411yescharacteristic
54C22×C4×D3D412yes—
55C22×Dic6D411yescharacteristic
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16D412yes—
68C22×C3×C8—21no—
69C22×C3×C8D412yes—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×#217C2×C412yes—
77C2×(C2×D3)⋊C4—24no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82C2×C4⋊Dic3—21no—
83C2×C4×Dic3—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208D412yes—
96C22×C3⋊C8—21no—
97C22×C3⋊C8D412yes—
98C3⋊#95—24no—
99#824—24no—
100#823—24no—
101(C2×D3)⋊C8—28no—
102C22×#41—31no—
103C22×M16—31no—
104C22×M16—31no—
105C22×M16C2×D311yescharacteristic
106C22×M16C2×D311yescharacteristic
107C2×#117—31no—
108C22×C42C2×D311yescharacteristic
109#413—62no—
110#413—62no—
111C2×#101—61no—
112C2×#100—61no—
113#408—62no—
114#408—62no—
115#407—62no—
116#391—61no—
117#390—61no—
118#386—34no—
119C4×C22⋊C4—62no—
120#344—62no—
121#344—62no—
122#336—22no—
123#336—62no—
124C24×C3C22×C411yescharacteristic
125C23×D3C22⋊C412yes—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—44no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C2×#67—22no—
145C2×#67—22no—
146C2×#67—22no—
147C2×#67—22no—
148C2×#67—22no—
149C2×#67—22no—
150C22×Dic3—22no—
151C22×Dic3C22⋊C412yes—
152#235—41no—
153#235—41no—
154#235—41no—
155#235—41no—
156#235—41no—
157#235—41no—
158#235—41no—
159#235—41no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—41no—
189C3×M16—41no—
190C3×M16—41no—
191C3×M16—41no—
192C3×M16—41no—
193C3×M16—41no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—41no—
199C2×C3×C8—41no—
200C3×C42—41no—
201C3×C42—41no—
202C3×C42—41no—
203C3×C42—41no—
204#217—22no—
205#217—22no—
206(C2×D3)⋊C4—48no—
207C4⋊Dic3—41no—
208C4⋊Dic3—41no—
209C4×Dic3—41no—
210C4×Dic3—41no—
211#208—22no—
212#208—22no—
213#208—41no—
214#208—41no—
215#208—41no—
216#208—41no—
217#208—41no—
218#208—41no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—41no—
224C2×C3⋊C8—41no—
225C2×#41—61no—
226C2×#41—61no—
227C2×#41—61no—
228C2×#41—61no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C2×#41—61no—
236C2×#41—61no—
237C22×Q8—31no—
238C22×D4—32no—
239C22×D4—31no—
240C23×C4—32no—
241C23×C4C4×D312yes—
242C23×C4C22×D311yescharacteristic
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—32no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16—61no—
263C2×M16—61no—
264C2×M16—61no—
265C2×M16—61no—
266C2×M16D1212yes—
267C22×C8—21no—
268C22×C8—32no—
269C22×C8—61no—
270C22×C8#6712yes—
271#117—61no—
272#117—61no—
273#117—61no—
274#117—61no—
275C2×C4⋊C4—61no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—64no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—61no—
285#101—64no—
286#100—64no—
287C22⋊C8—68no—
288#95—64no—
289C23×C3—42no—
290C23×C3C2×C22⋊C411yescharacteristic
291C23×C3#11711yescharacteristic
292C23×C3#11711yescharacteristic
293C23×C3C4×D411yescharacteristic
294C23×C3C4×D411yescharacteristic
295C23×C3C4×D411yescharacteristic
296C23×C3C4×D411yescharacteristic
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—44no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12311yescharacteristic
319C2×C3×C4#12311yescharacteristic
320C2×C3×C4#12311yescharacteristic
321C2×C3×C4#12311yescharacteristic
322C2×C3×C4#12311yescharacteristic
323C2×C3×C4#12311yescharacteristic
324C2×C3×C4#12311yescharacteristic
325C2×C3×C4#12311yescharacteristic
326C2×C3×C4#12311yescharacteristic
327C2×C3×C4#12311yescharacteristic
328C2×C3×C4#12411yescharacteristic
329C2×C3×C4#12411yescharacteristic
330C2×C3×C4#12411yescharacteristic
331C2×C3×C4#12411yescharacteristic
332C2×C3×C4#12611yescharacteristic
333C2×C3×C4#12611yescharacteristic
334C2×C3×C4#12611yescharacteristic
335C2×C3×C4#12611yescharacteristic
336#67—42no—
337#67—42no—
338#67—42no—
339#67—42no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—44no—
344C2×Dic3—42no—
345D12—41no—
346D12—41no—
347D12—41no—
348D12—41no—
349C4×D3—42no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—42no—
353Dic6—41no—
354Dic6—41no—
355Dic6—41no—
356Dic6—41no—
357C3×C8—42no—
358C3×C8—42no—
359C3⋊C8—42no—
360C3⋊C8—42no—
361C24—32no—
362C24C2×C4×D311yescharacteristic
363#41—121no—
364#41—121no—
365#41—121no—
366#41—121no—
367#41—121no—
368#41—121no—
369#41—121no—
370#41—121no—
371#41—121no—
372#41—121no—
373#41—121no—
374#41—121no—
375#41—121no—
376#41—121no—
377#41—121no—
378#41—121no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—32no—
401C22×C4—44no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4C2×D1211yescharacteristic
410C22×C4#23511yescharacteristic
411C22×C4#23511yescharacteristic
412C22×C4#23511yescharacteristic
413C22×C4#23511yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4#23711yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4#23911yescharacteristic
421C22×C4#23911yescharacteristic
422C22×C4C2×#6711yescharacteristic
423M16—22no—
424M16—22no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—62no—
432M16—62no—
433M16—121no—
434M16—121no—
435M16—121no—
436M16—121no—
437M16—121no—
438M16—121no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—121no—
450C2×C8—121no—
451C4⋊C4—121no—
452C4⋊C4—121no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—128no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—121no—
461C42—121no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3#38111yescharacteristic
470C22×C3#38611yescharacteristic
471C22×C3#38611yescharacteristic
472C22×C3#38811yescharacteristic
473C22×C3#38811yescharacteristic
474C22×C3#38811yescharacteristic
475C22×C3#38811yescharacteristic
476C2×D3—44no—
477C2×D3—44no—
478C2×D3—42no—
479C2×D3—82no—
480C3×C4—44no—
481C3×C4#39411yescharacteristic
482C3×C4#39411yescharacteristic
483C3×C4#39411yescharacteristic
484C3×C4#39411yescharacteristic
485C3×C4#39711yescharacteristic
486C3×C4#39711yescharacteristic
487C3×C4#39711yescharacteristic
488C3×C4#39711yescharacteristic
489Dic3—42no—
490Dic3—82no—
491C23—42no—
492C23—62no—
493C23—62no—
494C23—62no—
495C23—124no—
496C23#87211yescharacteristic
497C23C4×D1211yescharacteristic
498C23D3×C22⋊C411yescharacteristic
499C23#67⋊C411yescharacteristic
500C23#89211yescharacteristic
501C23#89311yescharacteristic
502C23C4×#6711yescharacteristic
503Q8—121no—
504Q8—121no—
505Q8—121no—
506Q8—121no—
507D4—122no—
508D4—122no—
509D4—122no—
510D4—122no—
511D4—121no—
512D4—121no—
513D4—121no—
514D4—121no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—124no—
530C2×C4—122no—
531C2×C4#87511yescharacteristic
532C2×C4#87611yescharacteristic
533C2×C4#88211yescharacteristic
534C2×C4#88311yescharacteristic
535C2×C4#88311yescharacteristic
536C2×C4#88311yescharacteristic
537C2×C4#88411yescharacteristic
538C2×C4#88411yescharacteristic
539C2×C4#88411yescharacteristic
540C2×C4#88511yescharacteristic
541C2×C4#88511yescharacteristic
542C2×C4#88611yescharacteristic
543C2×C4#89411yescharacteristic
544C2×C4#89411yescharacteristic
545C2×C4#89411yescharacteristic
546C2×C4#89411yescharacteristic
547C2×C4#89411yescharacteristic
548C2×C4#89511yescharacteristic
549C2×C4#89511yescharacteristic
550C2×C4#89811yescharacteristic
551C2×C4#89811yescharacteristic
552C2×C4#89811yescharacteristic
553C2×C4#92911yescharacteristic
554C2×C4#93011yescharacteristic
555C2×C4#93711yescharacteristic
556C2×C4#93811yescharacteristic
557C2×C4#93911yescharacteristic
558C2×C4#94711yescharacteristic
559C8—42no—
560C8—42no—
561C8—122no—
562C8—122no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3#144011yescharacteristic
570C2×C3#189112yes—
571D3—82no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—124no—
580C22—122no—
581C22—124no—
582C22—242no—
583C22#466411yescharacteristic
584C22#466511yescharacteristic
585C22#466911yescharacteristic
586C22#467011yescharacteristic
587C22#494011yescharacteristic
588C22#495611yescharacteristic
589C22#499011yescharacteristic
590C4—44no—
591C4—122no—
592C4—242no—
593C4#467211yescharacteristic
594C4#467411yescharacteristic
595C4#467511yescharacteristic
596C4#467511yescharacteristic
597C4#495911yescharacteristic
598C4#496111yescharacteristic
599C4#499111yescharacteristic
600C4#499211yescharacteristic
601C3#1117511yescharacteristic, complemented, Sylow
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—242no—
609C2#6942911yescharacteristic
610C2#6999212yes—
subgroupG/H
centerC2×C4#937
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1728C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1728D3
hypocenterC3#11175
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1728

Complements

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

NH ≅ G/Ntype
4. C3×#1728C2semidirect
5. C2×#4559C2semidirect
6. C2×#4556C2semidirect
7. #71527C2semidirect
37. #1728D3semidirect
601. C3#11175semidirect

Maximal subgroups

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

subgroupindex
C2×#51322
#837622
#820702
C3×#17282
C2×#45592
C2×#45562
#715272
#111753