Finite Group Databasebeta

← #10692250

Subgroups — #10692250

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2880C211yesmaximal, characteristic, complemented
2#84082C211yesmaximal, characteristic
3#72135C211yesmaximal, characteristic, complemented
4#69970C211yesmaximal, characteristic, complemented
5#69944C211yesmaximal, characteristic, complemented
6#67685C211yesmaximal, characteristic, complemented
7#67680C211yesmaximal, characteristic, complemented
8#9347—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#131—21no—
11C2×#949C411yescharacteristic, complemented
12C2×#949C411yescharacteristic, complemented
13C2×#930C2211yescharacteristic
14C3×#408—21no—
15C3×#408—22no—
16C3×#407—21no—
17C3×#407C2211yescharacteristic
18C2×C3×C22⋊C8—21no—
19C2×C3×C22⋊C8C2211yescharacteristic
20#5238—21no—
21#5238—21no—
22#5232C411yescharacteristic, complemented
23#5232C411yescharacteristic, complemented
24#5227—21no—
25C2×#835—21no—
26#5217—21no—
27C2×#832—21no—
28#5212C2211yescharacteristic
29#5207C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#4470—21no—
32#4470—21no—
33#4468—22no—
34#3006—31no—
35#2880D311yescharacteristic, complemented
36#1788—31no—
37#1519—31no—
38#1471—31no—
39#1305—31no—
40#1298—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—42no—
44C2×C3×#41—42no—
45C2×C3×#41C2×C411yescharacteristic
46C2×C3×#41C2×C411yescharacteristic
47C2×C3×#41D411yescharacteristic
48C2×C3×#41D411yescharacteristic
49C22×C3×Q8D411yescharacteristic
50C22×C3×D4—21no—
51C22×C3×D4D411yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—21no—
55C22×D12C2×C411yescharacteristic
56C3×#131—42no—
57C3×#131—42no—
58C2×C3×M16—22no—
59C2×C3×M16—21no—
60C22×C3×C8—22no—
61C22×C3×C8—21no—
62C2×#217—21no—
63#949—42no—
64#949—42no—
65C2×#215—21no—
66C2×#215—21no—
67C2×#213—21no—
68C2×#213—21no—
69#930—21no—
70#930—21no—
71#930—21no—
72#930—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3C2×C411yescharacteristic
76C2×#208C2×C411yescharacteristic
77C2×#208C2×C411yescharacteristic
78C22×C3⋊C8—21no—
79C3×C22⋊C8—22no—
80C3×C22⋊C8—22no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8D412yes—
84C3×C22⋊C8D411yescharacteristic
85C3×C22⋊C8D411yescharacteristic
86#835—21no—
87#835—21no—
88#835—42no—
89#832—21no—
90#832—21no—
91#832—42no—
92C3⋊#95—21no—
93#830—21no—
94#830—21no—
95#830—21no—
96#830—21no—
97C22×#41C2×D311yescharacteristic
98C2×AGL(1,ℤ/8)—31no—
99C2×AGL(1,ℤ/8)—31no—
100C2×#131—21no—
101C2×#121—31no—
102#449—61no—
103#449—61no—
104#448—61no—
105#447—61no—
106#417—31no—
107#417—31no—
108C2×#103—61no—
109C2×#102—61no—
110#408—21no—
111#408—22no—
112#407—21no—
113#407—31no—
114#407C2×D311yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—31no—
117C2×C22⋊C8C2×D311yescharacteristic
118#380—31no—
119#327—61no—
120#327—61no—
121#323—62no—
122C24×C3—21no—
123C24×C3#4111yescharacteristic
124C23×D3—21no—
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—41no—
130C3×#41—41no—
131C3×#41—41no—
132C3×#41—41no—
133C3×#41—44no—
134C3×#41—44no—
135C2×C3×Q8—21no—
136C2×C3×Q8—22no—
137C2×C3×Q8—41no—
138C2×C3×Q8C22⋊C411yescharacteristic
139C2×C3×Q8C22⋊C411yescharacteristic
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—41no—
147C2×C3×D4—44no—
148C2×C3×D4C22⋊C411yescharacteristic
149C2×C3×D4C22⋊C411yescharacteristic
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—44no—
158C22×C3×C4C22×C411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4#4111yescharacteristic
164C2×#67—41no—
165C2×#67—41no—
166C2×#67—81no—
167C2×#67—81no—
168C22×Dic3—21no—
169C22×Dic3—41no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—41no—
173C2×D12—41no—
174C3×M16—44no—
175C3×M16—41no—
176C3×M16—41no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—44no—
181C2×C3×C8—41no—
182C2×C3×C8—41no—
183#217—81no—
184#215—42no—
185#215—42no—
186#213—42no—
187#213—42no—
188(C2×D3)⋊C4—41no—
189(C2×D3)⋊C4—41no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—41no—
193C4⋊Dic3—41no—
194#208—41no—
195#208—41no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—41no—
199C2×C3⋊C8—41no—
200C2×C3⋊C8—41no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—42no—
204C2×#41—42no—
205C2×#41C4×D311yescharacteristic
206C2×#41C4×D311yescharacteristic
207C2×#41#6711yescharacteristic
208C2×#41#6711yescharacteristic
209C22×Q8D1211yescharacteristic
210C22×D4—21no—
211C22×D4—31no—
212C22×D4—61no—
213C22×D4D1211yescharacteristic
214C23×C4—21no—
215C23×C4C22×D311yescharacteristic
216AGL(1,ℤ/8)—122no—
217AGL(1,ℤ/8)—122no—
218C2×SD16—61no—
219C2×SD16—61no—
220C2×D8—61no—
221C2×D8—61no—
222#131—42no—
223#131—42no—
224C2×M16—22no—
225C2×M16—21no—
226C2×M16—31no—
227C2×M16—31no—
228C22×C8—22no—
229C22×C8—21no—
230C22×C8—61no—
231#121—61no—
232#121—61no—
233#121—61no—
234#121—61no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239#103—61no—
240#103—61no—
241#103—122no—
242#102—61no—
243#102—61no—
244#102—122no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—61no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8—61no—
253C22⋊C8#6712yes—
254C22⋊C8#6711yescharacteristic
255C22⋊C8#6711yescharacteristic
256#95—61no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—81no—
267C23×C3C4×D411yescharacteristic
268C23×C3#12111yescharacteristic
269C23×C3#12311yescharacteristic
270C22×D3—41no—
271C22×D3—42no—
272C22×D3—81no—
273C22×D3—81no—
274C3×Q8—21no—
275C3×Q8—21no—
276C3×Q8—21no—
277C3×Q8—42no—
278C3×Q8#9912yes—
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—21no—
282C3×D4—21no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—42no—
287C3×D4#9912yes—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4C2×C22⋊C411yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4#12011yescharacteristic
307C2×C3×C4#12111yescharacteristic
308#67—82no—
309#67—81no—
310#67—81no—
311C2×Dic3—41no—
312C2×Dic3—81no—
313C2×Dic3—81no—
314C2×Dic3—81no—
315C2×Dic3—81no—
316D12—41no—
317D12—41no—
318D12—81no—
319C3×C8—41no—
320C3×C8—42no—
321C3×C8—41no—
322C3⋊C8—41no—
323C3⋊C8—41no—
324C3⋊C8—81no—
325C24—21no—
326C24—61no—
327C24#23511yescharacteristic
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—44no—
337#41—44no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8(C2×D3)⋊C411yescharacteristic
342C2×Q8(C2×D3)⋊C411yescharacteristic
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—121no—
354C2×D4—121no—
355C2×D4—121no—
356C2×D4—121no—
357C2×D4—241no—
358C2×D4—241no—
359C2×D4(C2×D3)⋊C411yescharacteristic
360C2×D4(C2×D3)⋊C411yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—44no—
369C22×C4—61no—
370C22×C4—121no—
371C22×C4C2×C4×D311yescharacteristic
372C22×C4C2×D1211yescharacteristic
373C22×C4#23511yescharacteristic
374C22×C4C2×#6711yescharacteristic
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6711yescharacteristic
377SD16—122no—
378SD16—122no—
379D8—122no—
380D8—122no—
381M16—44no—
382M16—41no—
383M16—41no—
384M16—121no—
385M16—121no—
386C2×C8—21no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—44no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—61no—
393C2×C8—61no—
394C2×C8—121no—
395C2×C8—121no—
396C2×C8—121no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—121no—
400C4⋊C4—121no—
401C22⋊C4—121no—
402C22⋊C4—121no—
403C22⋊C4—241no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3#38611yescharacteristic
423C22×C3#47111yescharacteristic
424C22×C3#47211yescharacteristic
425C2×D3—81no—
426C2×D3—81no—
427C2×D3—81no—
428C2×D3—162no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—41no—
434C3×C4C2×#9911yescharacteristic
435C3×C4C2×#9911yescharacteristic
436Dic3—81no—
437Dic3—161no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—81no—
448C23—122no—
449C23—121no—
450C23—241no—
451C23—241no—
452C23C4×#6711yescharacteristic
453C23#92911yescharacteristic
454C23#93011yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—21no—
458Q8—42no—
459Q8#80612yes—
460D4—24no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—121no—
469D4—121no—
470D4—241no—
471D4—242no—
472D4—241no—
473D4—241no—
474D4#80612yes—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—121no—
492C2×C4—241no—
493C2×C4—241no—
494C2×C4—241no—
495C2×C4—241no—
496C2×C4C2×(C2×D3)⋊C411yescharacteristic
497C2×C4C4×#6711yescharacteristic
498C2×C4#93011yescharacteristic
499C2×C4#95311yescharacteristic
500C8—41no—
501C8—42no—
502C8—41no—
503C8—121no—
504C8—121no—
505C8—241no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—42no—
511C2×C3—81no—
512C2×C3#188111yescharacteristic
513C2×C3#189811yescharacteristic
514C2×C3#190711yescharacteristic
515D3—161no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—81no—
531C22—81no—
532C22—82no—
533C22—82no—
534C22—241no—
535C22—241no—
536C22—241no—
537C22—482no—
538C22#514111yescharacteristic
539C22#514211yescharacteristic
540C22#521311yescharacteristic
541C4—21no—
542C4—21no—
543C4—22no—
544C4—21no—
545C4—41no—
546C4—241no—
547C4—481no—
548C4C2×#80611yescharacteristic
549C4C2×#80611yescharacteristic
550C3#934711yescharacteristic, complemented, Sylow
551C2—21no—
552C2—21no—
553C2—41no—
554C2—41no—
555C2—42no—
556C2—81no—
557C2—481no—
558C2#7154311yescharacteristic
559C2#7201811yescharacteristic
560C2#7202011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2880C2
socleC22×C3#386
prime radical#84082C2
cosocleC23×C3×C4C23
hypercenter#2880D3
hypocenterC3#9347
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2880

Complements

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

NH ≅ G/Ntype
1. C3×#2880C2semidirect
3. #72135C2semidirect
4. #69970C2semidirect
5. #69944C2semidirect
6. #67685C2semidirect
7. #67680C2semidirect
11. C2×#949C4semidirect
12. C2×#949C4semidirect
22. #5232C4semidirect
23. #5232C4semidirect
35. #2880D3semidirect
550. C3#9347semidirect

Maximal subgroups

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

subgroupindex
C3×#28802
#840822
#721352
#699702
#699442
#676852
#676802
#93473