Finite Group Databasebeta

← #10703025

Subgroups — #10703025

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5139C211yesmaximal, characteristic
2#83768C211yesmaximal, characteristic
3#82228C211yesmaximal, characteristic
4C2×C3×#344C211yesmaximal, characteristic, complemented
5C2×#4555C211yesmaximal, characteristic, complemented
6C2×#4555C211yesmaximal, characteristic, complemented
7#71593C211yesmaximal, characteristic, complemented
8#11193—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16#5139—22no—
17#5139—22no—
18C2×#825—22no—
19#5133C414yescomplemented
20#5131—21no—
21#5120—21no—
22#5113—22no—
23#4988—21no—
24#4973—21no—
25#4958—22no—
26C4×(C2×D3)⋊C4—22no—
27C3×#344—22no—
28#4555—22no—
29#4555—22no—
30#4527—22no—
31C2×#421—31no—
32#2896—31no—
33#2297—31no—
34C2×#344D311yescharacteristic, complemented
35#1740—31no—
36C2×#339—31no—
37C2×#339—31no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C23×Dic3—21no—
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×#235C2×C412yes—
52C2×#235D412yes—
53C22×D12D411yescharacteristic
54C22×C4×D3—21no—
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×M16C2×C412yes—
67C2×C3×M16D412yes—
68C22×C3×C8—21no—
69C22×C3×C8—21no—
70C3×#117—21no—
71C3×#117—21no—
72C3×#117—21no—
73C3×#117—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42—21no—
77C2×#217—21no—
78C2×#217—21no—
79C2×(C2×D3)⋊C4—24no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926—21no—
87#926D412yes—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2×C412yes—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3—21no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3D412yes—
98C3⋊#95—24no—
99#825—28no—
100#821—24no—
101#816—24no—
102C22×#41—31no—
103C22×M16C2×D311yescharacteristic
104C22×M16C2×D311yescharacteristic
105C2×#117—31no—
106C2×#117—31no—
107C2×#117C2×D311yescharacteristic
108C22×C42—31no—
109#421—62no—
110#421—62no—
111C2×#104—62no—
112#419—61no—
113#418—61no—
114#417—34no—
115#416—62no—
116#391—61no—
117#390—61no—
118#386—62no—
119C4×C22⋊C4—62no—
120#344—22no—
121#342—62no—
122#339—62no—
123#339—62no—
124C24×C3C2×D411yescharacteristic
125C23×D3—21no—
126C22×C3×C4—22no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C2×#67—22no—
142C2×#67—22no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—41no—
147C22×Dic3—21no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—44no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12C22⋊C412yes—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—41no—
179C2×C4×D3—41no—
180C2×C4×D3—41no—
181C2×C4×D3—41no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6C22⋊C412yes—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—41no—
193C3×M16—41no—
194C3×M16—41no—
195C3×M16—41no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—41no—
199C2×C3×C8—41no—
200C2×C3×C8—41no—
201C2×C3×C8—41no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C42—41no—
207C3×C42—41no—
208#217—22no—
209#217—41no—
210#217—41no—
211#217—41no—
212(C2×D3)⋊C4—48no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—41no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3—41no—
221C4×Dic3—41no—
222C4×Dic3—41no—
223C4×Dic3—41no—
224C4×Dic3—41no—
225C2×#41—32no—
226C2×#41—32no—
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—31no—
239C22×D4—61no—
240C23×C4—61no—
241C23×C4—61no—
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×M16C4×D312yes—
254C2×M16#6712yes—
255C22×C8—21no—
256C22×C8—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—32no—
262#117—61no—
263#117—61no—
264#117—61no—
265#117—61no—
266#117—61no—
267#117—61no—
268#117—61no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4D1212yes—
276C2×C42—21no—
277C2×C42—32no—
278C2×C42—61no—
279C2×C42—61no—
280C2×C42—61no—
281C2×C42—61no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42—61no—
285#104—68no—
286#103—64no—
287#102—64no—
288#95—64no—
289C23×C3—42no—
290C23×C3C4×D411yescharacteristic
291C23×C3#12011yescharacteristic
292C23×C3#12111yescharacteristic
293C23×C3#12111yescharacteristic
294C23×C3#12111yescharacteristic
295C23×C3#12311yescharacteristic
296C23×C3#12411yescharacteristic
297C22×D3—22no—
298C22×D3—44no—
299C22×D3—41no—
300C22×D3—41no—
301C2×C3×C4—44no—
302C2×C3×C4—42no—
303C2×C3×C4C2×C22⋊C411yescharacteristic
304C2×C3×C4#11711yescharacteristic
305C2×C3×C4#11711yescharacteristic
306C2×C3×C4C4×D411yescharacteristic
307C2×C3×C4C4×D411yescharacteristic
308C2×C3×C4C4×D411yescharacteristic
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#12311yescharacteristic
316C2×C3×C4#12311yescharacteristic
317C2×C3×C4#12311yescharacteristic
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#12411yescharacteristic
325C2×C3×C4#12411yescharacteristic
326C2×C3×C4#12411yescharacteristic
327C2×C3×C4#12611yescharacteristic
328C2×C3×C4#12611yescharacteristic
329C2×C3×C4#12611yescharacteristic
330C2×C3×C4#12611yescharacteristic
331#67—42no—
332#67—42no—
333#67—42no—
334#67—42no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—42no—
338C2×Dic3—44no—
339C2×Dic3—44no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—82no—
346C2×Dic3—82no—
347D12—22no—
348D12—22no—
349D12—41no—
350D12—41no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—42no—
355Dic6—22no—
356Dic6—22no—
357Dic6—41no—
358Dic6—41no—
359C3×C8—42no—
360C3×C8—42no—
361C24—61no—
362C24C2×D1211yescharacteristic
363#41—62no—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41—62no—
369#41—62no—
370#41—62no—
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—32no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×D4—32no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4—121no—
396C2×D4—121no—
397C22×C4—22no—
398C22×C4—61no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—124no—
404C22×C4—121no—
405C22×C4—121no—
406C22×C4—121no—
407C22×C4—121no—
408C22×C4—121no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4C2×C4×D311yescharacteristic
412C22×C4#23511yescharacteristic
413C22×C4#23511yescharacteristic
414C22×C4#23511yescharacteristic
415C22×C4#23511yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4#23711yescharacteristic
421C22×C4#23711yescharacteristic
422C22×C4#23911yescharacteristic
423C22×C4#23911yescharacteristic
424C22×C4C2×#6711yescharacteristic
425M16—22no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—121no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—62no—
448C22⋊C4—128no—
449C22⋊C4—121no—
450C22⋊C4—121no—
451C22⋊C4—121no—
452C42—41no—
453C42—41no—
454C42—62no—
455C42—62no—
456C42—62no—
457C42—121no—
458C42—121no—
459C42—121no—
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#38611yescharacteristic
470C22×C3#38811yescharacteristic
471C22×C3#38811yescharacteristic
472C22×C3#39411yescharacteristic
473C22×C3#39411yescharacteristic
474C22×C3#39411yescharacteristic
475C22×C3#39711yescharacteristic
476C2×D3—44no—
477C2×D3—42no—
478C2×D3—82no—
479C2×D3—82no—
480C3×C4—82no—
481C3×C4#38111yescharacteristic
482C3×C4#38611yescharacteristic
483C3×C4#38811yescharacteristic
484C3×C4#38811yescharacteristic
485C3×C4#39411yescharacteristic
486C3×C4#39711yescharacteristic
487C3×C4#39711yescharacteristic
488C3×C4#39711yescharacteristic
489Dic3—44no—
490Dic3—42no—
491Dic3—82no—
492Dic3—82no—
493C23—42no—
494C23—62no—
495C23—124no—
496C23—121no—
497C23—121no—
498C23C4×D1211yescharacteristic
499C23#87511yescharacteristic
500C23#88211yescharacteristic
501C23#88611yescharacteristic
502C23#89511yescharacteristic
503C23#89511yescharacteristic
504C23#93011yescharacteristic
505Q8—62no—
506Q8—62no—
507Q8—121no—
508Q8—121no—
509D4—62no—
510D4—62no—
511D4—122no—
512D4—122no—
513D4—122no—
514D4—122no—
515D4—121no—
516D4—121no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—122no—
522C2×C4—124no—
523C2×C4—124no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—242no—
534C2×C4—242no—
535C2×C4#87211yescharacteristic
536C2×C4#87611yescharacteristic
537C2×C4D3×C22⋊C411yescharacteristic
538C2×C4#67⋊C411yescharacteristic
539C2×C4#88311yescharacteristic
540C2×C4#88311yescharacteristic
541C2×C4#88311yescharacteristic
542C2×C4#88411yescharacteristic
543C2×C4#88411yescharacteristic
544C2×C4#88411yescharacteristic
545C2×C4#88511yescharacteristic
546C2×C4#88511yescharacteristic
547C2×C4#89211yescharacteristic
548C2×C4#89311yescharacteristic
549C2×C4#89411yescharacteristic
550C2×C4#89411yescharacteristic
551C2×C4#89411yescharacteristic
552C2×C4#89411yescharacteristic
553C2×C4#89411yescharacteristic
554C2×C4#89811yescharacteristic
555C2×C4#89811yescharacteristic
556C2×C4#89811yescharacteristic
557C2×C4C4×#6711yescharacteristic
558C2×C4#92911yescharacteristic
559C2×C4#93711yescharacteristic
560C2×C4#93811yescharacteristic
561C2×C4#93911yescharacteristic
562C2×C4#94711yescharacteristic
563C8—42no—
564C8—42no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3#144011yescharacteristic
572C2×C3#190612yes—
573D3—82no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—122no—
582C22—124no—
583C22—242no—
584C22—242no—
585C22#466511yescharacteristic
586C22#467011yescharacteristic
587C22#467211yescharacteristic
588C22#467411yescharacteristic
589C22#494011yescharacteristic
590C22#496111yescharacteristic
591C22#499111yescharacteristic
592C4—82no—
593C4—124no—
594C4—122no—
595C4—242no—
596C4—242no—
597C4#466411yescharacteristic
598C4#466911yescharacteristic
599C4#467511yescharacteristic
600C4#467511yescharacteristic
601C4#495611yescharacteristic
602C4#495911yescharacteristic
603C4#499011yescharacteristic
604C4#499211yescharacteristic
605C3#1119311yescharacteristic, complemented, Sylow
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—242no—
613C2#6903912yes—
614C2#6942911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#344C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#344D3
hypocenterC3#11193
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×C3×#344C2semidirect
5. C2×#4555C2semidirect
6. C2×#4555C2semidirect
7. #71593C2semidirect
19. #5133C4semidirect
34. C2×#344D3semidirect
605. C3#11193semidirect

Maximal subgroups

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

subgroupindex
C2×#51392
#837682
#822282
C2×C3×#3442
C2×#45552
#715932
#111933