Finite Group Databasebeta

← #10692238

Subgroups — #10692238

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3007C211yesmaximal, characteristic
2#84075C211yesmaximal, characteristic
3#72135C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic, complemented
5#71690C211yesmaximal, characteristic, complemented
6#67688C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9345—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#137C411yescharacteristic, complemented
11C2×C3×#137C411yescharacteristic, complemented
12C2×C3×C22⋊Q8C2211yescharacteristic
13C2×#948—21no—
14C3×#452—21no—
15C3×#452—21no—
16C3×#451—21no—
17C3×#450—21no—
18C3×#417C411yescharacteristic, complemented
19C3×#417C411yescharacteristic, complemented
20C3×#407C2211yescharacteristic
21#5234—21no—
22#5234—22no—
23C2×#835—21no—
24C2×#832—21no—
25#5212C2211yescharacteristic
26#5207—21no—
27#5207C2211yescharacteristic
28C2×#830—21no—
29C2×#830C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#4538—21no—
32#4538—21no—
33#4528—22no—
34#3007D311yescharacteristic
35#2880—31no—
36#1788—31no—
37#1520—31no—
38#1472—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×Q8C2×C411yescharacteristic
50C22×C3×Q8D411yescharacteristic
51C22×C3×D4—21no—
52C22×C3×D4D411yescharacteristic
53C23×C3×C4—21no—
54C23×C3×C4C2311yescharacteristic
55C3×#137—42no—
56C3×#137—42no—
57C2×C3×Q16—21no—
58C2×C3×Q16—21no—
59C2×C3×SD16—21no—
60C2×C3×SD16—21no—
61C2×C3×M16C2×C411yescharacteristic
62C2×C3×M16C2×C411yescharacteristic
63C3×C22⋊Q8—21no—
64C3×C22⋊Q8—21no—
65C3×C22⋊Q8—21no—
66C3×C22⋊Q8—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2×C411yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71#948—42no—
72#948—42no—
73C2×C4⋊Dic3—21no—
74C2×#208—22no—
75C2×#208—21no—
76C22×C3⋊C8—22no—
77C22×C3⋊C8—21no—
78C22×C3⋊C8—21no—
79C3×#103—21no—
80C3×#103—21no—
81C3×#102—21no—
82C3×#102—21no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8—21no—
85#835—42no—
86#832—42no—
87C3⋊#95—21no—
88#830—22no—
89#830—22no—
90#830—21no—
91#830—21no—
92#830—21no—
93#830—21no—
94#830D411yescharacteristic
95#830D411yescharacteristic
96#830D412yes—
97C22×#41C2×D311yescharacteristic
98C2×#137Dic311yescharacteristic, complemented
99C2×#137Dic311yescharacteristic, complemented
100C2×#131—61no—
101C2×C22⋊Q8C2×D311yescharacteristic
102#452—21no—
103#452—21no—
104#451—21no—
105#450—21no—
106#417Dic311yescharacteristic, complemented
107#417Dic311yescharacteristic, complemented
108C2×#103—61no—
109C2×#102—61no—
110#408—62no—
111#408—61no—
112#407—31no—
113#407—61no—
114#407C2×D311yescharacteristic
115C2×C22⋊C8—31no—
116C2×C22⋊C8—31no—
117C2×C22⋊C8—61no—
118#380—31no—
119#328—61no—
120#328—61no—
121#324—62no—
122C24×C3—21no—
123C24×C3#4111yescharacteristic
124C3×#41—22no—
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—41no—
129C3×#41—41no—
130C3×#41—41no—
131C3×#41—41no—
132C3×#41—44no—
133C3×#41—44no—
134C2×C3×Q8—21no—
135C2×C3×Q8—22no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—41no—
139C2×C3×Q8—41no—
140C2×C3×Q8—41no—
141C2×C3×Q8C22⋊C411yescharacteristic
142C2×C3×Q8C22⋊C411yescharacteristic
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—41no—
150C2×C3×D4—44no—
151C2×C3×D4C22⋊C411yescharacteristic
152C2×C3×D4C22⋊C411yescharacteristic
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—41no—
161C22×C3×C4—41no—
162C22×C3×C4—44no—
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4#4111yescharacteristic
169C22×Dic3—41no—
170C3×Q16—42no—
171C3×Q16—42no—
172C3×SD16—42no—
173C3×SD16—42no—
174C3×M16—41no—
175C3×M16—41no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—41no—
182C3×C22⋊C4—41no—
183C3×C22⋊C4—41no—
184#217—81no—
185C4⋊Dic3—41no—
186C4⋊Dic3—41no—
187#208—44no—
188#208—41no—
189#208—41no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—44no—
194C2×C3⋊C8—41no—
195C2×C3⋊C8—41no—
196C2×C3⋊C8—41no—
197C2×C3⋊C8—41no—
198C2×C3⋊C8—41no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—42no—
202C2×#41—42no—
203C2×#41C2×Dic311yescharacteristic
204C2×#41C2×Dic311yescharacteristic
205C2×#41#6711yescharacteristic
206C2×#41#6711yescharacteristic
207C22×Q8C2×Dic311yescharacteristic
208C22×Q8#6711yescharacteristic
209C22×D4—21no—
210C22×D4#6711yescharacteristic
211C23×C4—21no—
212C23×C4C22×D311yescharacteristic
213#137—42no—
214#137—42no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×SD16—21no—
218C2×SD16—21no—
219#131—122no—
220#131—122no—
221C2×M16—62no—
222C2×M16—61no—
223C2×M16C2×Dic311yescharacteristic
224C2×M16C2×Dic311yescharacteristic
225C22×C8—62no—
226C22×C8—61no—
227C22×C8—61no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4C2×Dic311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—61no—
237#103—21no—
238#103—21no—
239#103—122no—
240#102—21no—
241#102—21no—
242#102—122no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—32no—
246C22⋊C8—31no—
247C22⋊C8—31no—
248C22⋊C8—62no—
249C22⋊C8—62no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8—61no—
253C22⋊C8—61no—
254#95—61no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—81no—
265C23×C3C4×D411yescharacteristic
266C23×C3#12111yescharacteristic
267C23×C3#12311yescharacteristic
268C3×Q8—21no—
269C3×Q8—21no—
270C3×Q8—21no—
271C3×Q8—42no—
272C3×Q8—41no—
273C3×Q8—41no—
274C3×Q8—81no—
275C3×Q8#9912yes—
276C3×D4—24no—
277C3×D4—21no—
278C3×D4—21no—
279C3×D4—21no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—42no—
284C3×D4#9912yes—
285C2×C3×C4—24no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—81no—
305C2×C3×C4C2×C22⋊C411yescharacteristic
306C2×C3×C4C4×D411yescharacteristic
307C2×C3×C4#12011yescharacteristic
308C2×C3×C4#12111yescharacteristic
309C2×Dic3—81no—
310C2×Dic3—81no—
311C2×Dic3—81no—
312C3×C8—41no—
313C3×C8—41no—
314C3⋊C8—41no—
315C3⋊C8—42no—
316C3⋊C8—41no—
317C3⋊C8—81no—
318C24—21no—
319C24#23711yescharacteristic
320#41—22no—
321#41—22no—
322#41—22no—
323#41—22no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—44no—
329#41—44no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8#21711yescharacteristic
338C2×Q8#21711yescharacteristic
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4#21711yescharacteristic
348C2×D4#21711yescharacteristic
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—121no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4#23711yescharacteristic
363C22×C4C22×Dic311yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6711yescharacteristic
366Q16—42no—
367Q16—42no—
368SD16—42no—
369SD16—42no—
370M16—41no—
371M16—41no—
372M16—124no—
373M16—121no—
374M16—121no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—61no—
378C2×C8—62no—
379C2×C8—61no—
380C2×C8—124no—
381C2×C8—121no—
382C2×C8—121no—
383C2×C8—121no—
384C2×C8—121no—
385C2×C8—121no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—121no—
391C4⋊C4—121no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—241no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3#38611yescharacteristic
414C22×C3#47111yescharacteristic
415C22×C3#47311yescharacteristic
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—41no—
421C3×C4—81no—
422C3×C4—81no—
423C3×C4C2×#9911yescharacteristic
424C3×C4C2×#9911yescharacteristic
425Dic3—161no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—81no—
436C23D4×Dic311yescharacteristic
437C23#93511yescharacteristic
438C23#93911yescharacteristic
439Q8—21no—
440Q8—21no—
441Q8—21no—
442Q8—42no—
443Q8—41no—
444Q8—41no—
445Q8—81no—
446Q8#83412yes—
447D4—24no—
448D4—21no—
449D4—21no—
450D4—21no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4#83412yes—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—81no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4D4×Dic311yescharacteristic
480C2×C4#93711yescharacteristic
481C2×C4#93911yescharacteristic
482C2×C4C2×#21711yescharacteristic
483C8—41no—
484C8—41no—
485C8—121no—
486C8—122no—
487C8—121no—
488C8—241no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—42no—
494C2×C3—81no—
495C2×C3#188111yescharacteristic
496C2×C3#189811yescharacteristic
497C2×C3#190811yescharacteristic
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—81no—
513C22—81no—
514C22—82no—
515C22—82no—
516C22#520311yescharacteristic
517C22#520411yescharacteristic
518C22#522011yescharacteristic
519C4—21no—
520C4—21no—
521C4—22no—
522C4—21no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4—481no—
527C4C2×#83411yescharacteristic
528C4C2×#83411yescharacteristic
529C3#934511yescharacteristic, complemented, Sylow
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—42no—
535C2—81no—
536C2#7167011yescharacteristic
537C2#7210311yescharacteristic
538C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3007C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#3007D3
hypocenterC3#9345
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
4. #71691C2semidirect
5. #71690C2semidirect
6. #67688C2semidirect
7. #67684C2semidirect
10. C2×C3×#137C4semidirect
11. C2×C3×#137C4semidirect
18. C3×#417C4semidirect
19. C3×#417C4semidirect
98. C2×#137Dic3semidirect
99. C2×#137Dic3semidirect
106. #417Dic3semidirect
107. #417Dic3semidirect
529. C3#9345semidirect

Maximal subgroups

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

subgroupindex
C3×#30072
#840752
#721352
#716912
#716902
#676882
#676842
#93453