Finite Group Databasebeta

← #10690310

Subgroups — #10690310

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2325C211yesmaximal, characteristic
2C2×#5040C211yesmaximal, characteristic
3C2×#4537C211yesmaximal, characteristic
4#69525C211yesmaximal, characteristic, complemented
5#69424C211yesmaximal, characteristic, complemented
6#67624C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9039—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×C3×C4×D4—21no—
12C22×#214—21no—
13C3×#388—21no—
14C3×#387—21no—
15C3×#386—21no—
16C3×#386—21no—
17C3×#385—21no—
18C3×#380C2211yescharacteristic
19C2×C3×#95C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5040—22no—
26#5040C414yescomplemented
27C2×#810—21no—
28C2×#810—21no—
29C2×#810—21no—
30#4537—22no—
31C2×#448—31no—
32#2325D311yescharacteristic
33#1787—31no—
34C2×#327—31no—
35#1446—31no—
36#1294—31no—
37#1280—31no—
38C22×C3×D4—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—21no—
51C22×Dic6—42no—
52C22×Dic6—42no—
53C22×Dic6C2×C412yescomplemented
54C3×#121—21no—
55C3×#121—21no—
56C3×#121C2×C412yes—
57C3×#121D412yes—
58C3×C4×D4—42no—
59C3×C4×D4—42no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C42—21no—
70C2×#214—24no—
71C2×#214—42no—
72C2×#214—42no—
73C22×C3⋊C8—21no—
74C22×C3⋊C8—21no—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95—21no—
78C3×#95—21no—
79C3×#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84#830—21no—
85#830—21no—
86#830—21no—
87#830C2×C412yescomplemented
88#810—24no—
89#810—42no—
90#810—42no—
91C23×Q8—31no—
92C22×SD16—61no—
93C2×#121C2×D311yescharacteristic
94C2×C4×D4—21no—
95#448—34no—
96#448—62no—
97C2×#103—61no—
98C2×#103—61no—
99C2×#103—61no—
100C2×C22⋊C8—31no—
101C2×C22⋊C8—31no—
102#388—21no—
103#387—21no—
104#386—21no—
105#386—21no—
106#385—21no—
107#380C2×D311yescharacteristic
108C2×#95—31no—
109C2×#95—61no—
110C2×#95—61no—
111C2×#95C2×D311yescharacteristic
112#327—62no—
113C24×C3—21no—
114C24×C3C2×D411yescharacteristic
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4C22⋊C412yes—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—42no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—81no—
141C22×C3×C4—81no—
142C22×C3×C4SD1612yescomplemented
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—41no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—44no—
170C2×Dic6—44no—
171C2×Dic6—44no—
172C2×Dic6—44no—
173C2×Dic6—41no—
174C2×Dic6—41no—
175C2×Dic6—41no—
176C2×Dic6—41no—
177C2×Dic6—41no—
178C2×Dic6—41no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—41no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4C22⋊C412yes—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—81no—
190C3×C22⋊C4—81no—
191C3×C42—42no—
192#214—48no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—41no—
195C2×C3⋊C8—41no—
196C2×C3⋊C8—41no—
197C2×C3⋊C8—41no—
198C2×C3⋊C8—41no—
199C22×Q8—32no—
200C22×Q8—61no—
201C22×Q8—61no—
202C22×Q8—61no—
203C22×Q8—61no—
204C22×Q8—61no—
205C22×Q8—122no—
206C22×Q8—122no—
207C22×D4—21no—
208C22×D4#6711yescharacteristic
209C23×C4—21no—
210C23×C4—61no—
211C23×C4—61no—
212C23×C4#6711yescharacteristic
213C23×C4#6711yescharacteristic
214C23×C4C22×D311yescharacteristic
215C2×SD16—64no—
216C2×SD16—122no—
217C2×SD16—122no—
218C22×C8—61no—
219C22×C8—61no—
220#121—21no—
221#121—21no—
222#121C4×D312yes—
223#121D1212yes—
224C4×D4—42no—
225C4×D4—42no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4#6711yescharacteristic
228C2×C4⋊C4#6711yescharacteristic
229C2×C4⋊C4#6711yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42—21no—
236#103—64no—
237#103—122no—
238#103—122no—
239C22⋊C8—32no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8—61no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—61no—
249#95—61no—
250#95—122no—
251#95—122no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—22no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3C4×D411yescharacteristic
264C23×C3#12011yescharacteristic
265C23×C3#12111yescharacteristic
266C23×C3C2×SD1611yescharacteristic
267C23×C3#13511yescharacteristic
268C23×C3AGL(1,ℤ/8)11yescharacteristic
269C23×C3#13711yescharacteristic
270C3×D4—42no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—81no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—81no—
302C2×C3×C4—84no—
303C2×C3×C4—81no—
304C2×C3×C4C2×C22⋊C411yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4#12111yescharacteristic
307C2×C3×C4#12311yescharacteristic
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—81no—
319C2×Dic3—81no—
320C2×Dic3—81no—
321C2×Dic3—81no—
322Dic6—24no—
323Dic6—24no—
324Dic6—42no—
325Dic6—42no—
326Dic6—42no—
327Dic6—42no—
328Dic6—42no—
329Dic6—42no—
330C3⋊C8—42no—
331C3⋊C8—81no—
332C24—21no—
333C24C2×#6711yescharacteristic
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—121no—
341C2×Q8—121no—
342C2×Q8—121no—
343C2×Q8—121no—
344C2×Q8—124no—
345C2×Q8—124no—
346C2×Q8—124no—
347C2×Q8—124no—
348C2×Q8—121no—
349C2×Q8—121no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4(C2×D3)⋊C412yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4—121no—
386C22×C4—121no—
387C22×C4—121no—
388C22×C4—121no—
389C22×C4—121no—
390C22×C4—121no—
391C22×C4#21412yescomplemented
392C22×C4C2×C4×D311yescharacteristic
393C22×C4C2×D1211yescharacteristic
394C22×C4#23511yescharacteristic
395C22×C4#23511yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398SD16—128no—
399C2×C8—62no—
400C2×C8—121no—
401C2×C8—121no—
402C2×C8—121no—
403C2×C8—121no—
404C2×C8—121no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4(C2×D3)⋊C412yes—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—42no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—81no—
416C22⋊C4—81no—
417C42—42no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3#38611yescharacteristic
438C22×C3C4×SD1611yescharacteristic
439C22×C3#44011yescharacteristic
440C22×C3#45011yescharacteristic
441C22×C3#45211yescharacteristic
442C22×C3#45511yescharacteristic
443C22×C3#45611yescharacteristic
444C22×C3B(4,𝔽2)11yescharacteristic
445C22×C3B(4,𝔽2)11yescharacteristic
446C22×C3#46111yescharacteristic
447C22×C3#46311yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4—81no—
457Dic3—44no—
458Dic3—82no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23C4×#6711yescharacteristic
471C23#93011yescharacteristic
472C23#93211yescharacteristic
473C23C2×#21411yescharacteristic
474C23#95011yescharacteristic
475C23#95111yescharacteristic
476C23#95311yescharacteristic
477Q8—64no—
478Q8—64no—
479Q8—122no—
480Q8—122no—
481Q8—122no—
482Q8—122no—
483Q8—122no—
484Q8—122no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—81no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—64no—
517C2×C4—81no—
518C2×C4—84no—
519C2×C4—81no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—241no—
530C2×C4—241no—
531C2×C4—241no—
532C2×C4—241no—
533C2×C4C2×(C2×D3)⋊C411yescharacteristic
534C2×C4C4×#6711yescharacteristic
535C2×C4#92911yescharacteristic
536C2×C4#93011yescharacteristic
537C8—122no—
538C8—241no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—82no—
544C2×C3#160111yescharacteristic
545C2×C3#160311yescharacteristic
546C2×C3#160411yescharacteristic
547C2×C3#160611yescharacteristic
548C2×C3#187711yescharacteristic
549C2×C3#188211yescharacteristic
550C2×C3#188911yescharacteristic
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22#501611yescharacteristic
571C22C4×#21411yescharacteristic
572C22#501811yescharacteristic
573C22#501911yescharacteristic
574C22#521311yescharacteristic
575C22#521811yescharacteristic
576C22#523911yescharacteristic
577C22#524211yescharacteristic
578C22#524411yescharacteristic
579C22#524411yescharacteristic
580C22#524611yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—124no—
591C4—242no—
592C3#903911yescharacteristic, complemented, Sylow
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—82no—
598C2#7161211yescharacteristic
599C2#7165411yescharacteristic
600C2#7166611yescharacteristic
601C2#7184611yescharacteristic
602C2#7188011yescharacteristic
603C2#7188511yescharacteristic
604C2#7188911yescharacteristic
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2325C2
socleC23×C3#120
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2325D3
hypocenterC3#9039
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2325

Complements

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

NH ≅ G/Ntype
4. #69525C2semidirect
5. #69424C2semidirect
6. #67624C2semidirect
7. #67613C2semidirect
26. #5040C4semidirect
53. C22×Dic6C2×C4semidirect
87. #830C2×C4semidirect
142. C22×C3×C4SD16semidirect
391. C22×C4#214semidirect
592. C3#9039semidirect

Maximal subgroups

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

subgroupindex
C3×#23252
C2×#50402
C2×#45372
#695252
#694242
#676242
#676132
#90393