Finite Group Databasebeta

← #10691631

Subgroups — #10691631

Order 768.

966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5036C211yesmaximal, characteristic
2#74007C211yesmaximal, characteristic, complemented
3C3×#1519C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#69959C211yesmaximal, characteristic, complemented
6#67678C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9259—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×#213C412yescomplemented
12C2×#893—21no—
13C2×D3×C22⋊C4—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C2×#834—21no—
17C2×#834—21no—
18#5212C2211yescharacteristic
19C4×#217C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×#821—22no—
22#5036—21no—
23#5036—22no—
24#5036—21no—
25#4988—21no—
26C2×#809C412yescomplemented
27#4955—21no—
28#4665—22no—
29C3×#327—22no—
30C2×#447—31no—
31D4×C22⋊C4—31no—
32#1964—31no—
33#1786—31no—
34#1519D311yescharacteristic, complemented
35#1289—31no—
36#1282—31no—
37C24×D3—21no—
38C22×C3×D4—21no—
39C22×C3×D4C2×C411yescharacteristic, complemented
40C23×C3×C4C2311yescharacteristic
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—44no—
45C22×D12C2×C412yes—
46C22×D12D411yescharacteristic
47C22×D12D412yes—
48C22×D12D411yescharacteristic
49C22×C4×D3—22no—
50C22×C3×C8—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217D411yescharacteristic
61C2×#217D411yescharacteristic
62C2×#213—22no—
63C2×#213—22no—
64C2×#213—44no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3—21no—
69C2×C4×Dic3—21no—
70C22×C3⋊C8C2×C412yescomplemented
71#893—44no—
72D3×C22⋊C4—42no—
73D3×C22⋊C4—42no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×#95D412yes—
77#834—42no—
78#834—42no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#821—44no—
84#809—22no—
85#809—22no—
86C23×D4—31no—
87C22×D8—32no—
88C2×#121C2×D311yescharacteristic
89C2×C4×D4—61no—
90C22×C22⋊C4—61no—
91#447—61no—
92#447—62no—
93#447—61no—
94C2×#102—32no—
95C2×#102—62no—
96C2×#99—61no—
97C2×#99—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#390—61no—
101#386—62no—
102#380—31no—
103#380C2×D311yescharacteristic
104#379—61no—
105C4×C22⋊C4—31no—
106#327—22no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C23×D3—21no—
111C23×D3—22no—
112C23×D3—22no—
113C23×D3—44no—
114C23×D3—41no—
115C23×D3—41no—
116C23×D3—41no—
117C23×D3—41no—
118C23×D3—82no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—42no—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C2×D12—21no—
141C2×D12—24no—
142C2×D12—24no—
143C2×D12—24no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—21no—
147C2×D12—42no—
148C2×D12—42no—
149C2×D12—42no—
150C2×D12—44no—
151C2×D12—44no—
152C2×D12—44no—
153C2×D12C22⋊C412yes—
154C2×D12C22⋊C412yes—
155C2×C4×D3—44no—
156C2×C4×D3—44no—
157C2×C4×D3—82no—
158C2×C3×C8—42no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—42no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167#217—21no—
168#217—21no—
169#217—21no—
170#217—41no—
171#217—41no—
172#217D812yescomplemented
173#213—44no—
174#213—44no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—44no—
177(C2×D3)⋊C4—82no—
178C4⋊Dic3—42no—
179C4×Dic3—42no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—42no—
182C25—61no—
183C22×D4—21no—
184C22×D4—32no—
185C22×D4—31no—
186C22×D4—32no—
187C22×D4—31no—
188C22×D4—61no—
189C22×D4—61no—
190C22×D4—61no—
191C22×D4—124no—
192C22×D4C4×D311yescharacteristic, complemented
193C23×C4—62no—
194C23×C4C22×D311yescharacteristic
195C2×D8—62no—
196C2×D8—62no—
197C2×D8—124no—
198C22×C8—21no—
199C22×C8—32no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204C4×D4—124no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4C4×D311yescharacteristic, complemented
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—31no—
211C2×C22⋊C4—31no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—122no—
217C2×C22⋊C4—122no—
218C2×C42—61no—
219#102—62no—
220#102—62no—
221#102—124no—
222#99—122no—
223#99—122no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—61no—
229#95—61no—
230#95#6712yes—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—81no—
237C23×C3—81no—
238C23×C3C4×D411yescharacteristic
239C23×C3#12011yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3C2×D811yescharacteristic
242C23×C3#13511yescharacteristic
243C23×C3AGL(1,ℤ/8)11yescharacteristic
244C23×C3AGL(1,ℤ/8)11yescharacteristic
245C22×D3—24no—
246C22×D3—22no—
247C22×D3—22no—
248C22×D3—44no—
249C22×D3—44no—
250C22×D3—42no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—42no—
260C22×D3—42no—
261C22×D3—82no—
262C22×D3—82no—
263C22×D3—84no—
264C22×D3—84no—
265C22×D3—84no—
266C22×D3—84no—
267C3×D4—42no—
268C3×D4—81no—
269C3×D4—81no—
270C3×D4—81no—
271C3×D4—81no—
272C3×D4—81no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4C2×C22⋊C411yescharacteristic
286C2×C3×C4C4×D411yescharacteristic
287C2×C3×C4#12111yescharacteristic
288C2×C3×C4#12311yescharacteristic
289C2×Dic3—22no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298D12—24no—
299D12—22no—
300D12—22no—
301D12—22no—
302D12—44no—
303D12—44no—
304D12—42no—
305D12#9914yescomplemented
306C4×D3—88no—
307C3×C8—81no—
308C3⋊C8—42no—
309C24—21no—
310C24—61no—
311C24—61no—
312C24—62no—
313C24—62no—
314C24—124no—
315C24—121no—
316C24—121no—
317C24—121no—
318C24—121no—
319C24—242no—
320C24D3×D411yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—32no—
324C2×D4—32no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—64no—
333C2×D4—64no—
334C2×D4—64no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—62no—
338C2×D4—62no—
339C2×D4—122no—
340C2×D4—122no—
341C2×D4—122no—
342C2×D4—124no—
343C2×D4—124no—
344C2×D4—124no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—124no—
351C22×C4—124no—
352C22×C4—121no—
353C22×C4—242no—
354C22×C4C2×C4×D311yescharacteristic
355C22×C4#23511yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4#23711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360D8—124no—
361D8—124no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—62no—
365C2×C8—122no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—42no—
369C4⋊C4—122no—
370C22⋊C4—32no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4—61no—
378C22⋊C4—124no—
379C22⋊C4—124no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—242no—
383C42—122no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—161no—
398C22×C3—161no—
399C22×C3#38611yescharacteristic
400C22×C3C4×D811yescharacteristic
401C22×C3#44211yescharacteristic
402C22×C3#44711yescharacteristic
403C22×C3#44911yescharacteristic
404C22×C3#45911yescharacteristic
405C22×C3#46311yescharacteristic
406C2×D3—24no—
407C2×D3—44no—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—84no—
415C2×D3—84no—
416C2×D3—84no—
417C2×D3—84no—
418C2×D3—82no—
419C2×D3—82no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—82no—
424C3×C4—81no—
425C3×C4C2×#9912yes—
426Dic3—42no—
427Dic3—81no—
428Dic3—161no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—64no—
435C23—62no—
436C23—62no—
437C23—81no—
438C23—81no—
439C23—124no—
440C23—124no—
441C23—122no—
442C23—122no—
443C23—122no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23—121no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—242no—
453C23—242no—
454C23—244no—
455C23—244no—
456C23—244no—
457C23—244no—
458C23#67⋊C411yescharacteristic
459C23#88411yescharacteristic
460C23D3×D811yescharacteristic
461C23#91111yescharacteristic
462C23#91411yescharacteristic
463C23#91611yescharacteristic
464C23#93711yescharacteristic
465D4—34no—
466D4—42no—
467D4—64no—
468D4—62no—
469D4—62no—
470D4—62no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4—81no—
476D4—124no—
477D4—124no—
478D4—122no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—62no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—248no—
498C2×C4—241no—
499C2×C4—241no—
500C2×C4—241no—
501C2×C4D3×C22⋊C411yescharacteristic
502C2×C4#88311yescharacteristic
503C2×C4C4×#6711yescharacteristic
504C2×C4#93811yescharacteristic
505C8—81no—
506C8—122no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—161no—
514C2×C3#187911yescharacteristic
515C2×C3#189811yescharacteristic
516C2×C3#190411yescharacteristic
517D3—44no—
518D3—82no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—64no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—124no—
534C22—124no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—122no—
539C22—122no—
540C22—161no—
541C22—161no—
542C22—244no—
543C22—244no—
544C22—244no—
545C22—244no—
546C22—242no—
547C22—242no—
548C22#475611yescharacteristic
549C22#478511yescharacteristic
550C22#478611yescharacteristic
551C22#478711yescharacteristic
552C22#495611yescharacteristic
553C22#515611yescharacteristic
554C22#517211yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—82no—
559C4—81no—
560C4—122no—
561C4—241no—
562C4—481no—
563C4D3×#9912yes—
564C3#925911yescharacteristic, complemented, Sylow
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—124no—
572C2—161no—
573C2—242no—
574C2#7003611yescharacteristic
575C2#7020811yescharacteristic
576C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#386
prime radicalC2×#5036C2
cosocleC23×C3×C4C23
hypercenter#1519D3
hypocenterC3#9259
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #74007C2semidirect
3. C3×#1519C2semidirect
4. #71686C2semidirect
5. #69959C2semidirect
6. #67678C2semidirect
7. #67625C2semidirect
11. C22×#213C4semidirect
26. C2×#809C4semidirect
34. #1519D3semidirect
39. C22×C3×D4C2×C4semidirect
56. C2×C3×C4⋊C4C2×C4semidirect
70. C22×C3⋊C8C2×C4semidirect
172. #217D8semidirect
192. C22×D4C4×D3semidirect
207. C2×C4⋊C4C4×D3semidirect
305. D12#99semidirect
564. C3#9259semidirect

Maximal subgroups

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

subgroupindex
C2×#50362
#740072
C3×#15192
#716862
#699592
#676782
#676252
#92593