Finite Group Databasebeta

← #10691496

Subgroups — #10691496

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#452C211yesmaximal, characteristic
2#84075C211yesmaximal, characteristic
3#72138C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic, complemented
5#71688C211yesmaximal, characteristic, complemented
6#67679C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9239—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C22×C3×Q16C411yescharacteristic, complemented
11C22×C3×SD16C411yescharacteristic, complemented
12C2×C3×C22⋊Q8C2211yescharacteristic
13C2×#948—21no—
14C3×#452—21no—
15C3×#452—21no—
16C3×#452—21no—
17C3×#452—21no—
18C2×C3×#103C411yescharacteristic, complemented
19C2×C3×#102C411yescharacteristic, complemented
20C2×C3×C22⋊C8C2211yescharacteristic
21C2×#837—21no—
22C2×#837—21no—
23#5234—21no—
24#5234—22no—
25C4×#217C2211yescharacteristic
26#5207—21no—
27#5207C2211yescharacteristic
28#5207C2211yescharacteristic
29C2×#830—21no—
30C2×#830C2211yescharacteristic
31#4538—22no—
32#4528—21no—
33#4528—21no—
34C2×#452D311yescharacteristic
35#2880—31no—
36#1789—31no—
37#1508—31no—
38#1472—31no—
39#1293—31no—
40#1282—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—42no—
44C2×C3×#41—42no—
45C2×C3×#41D411yescharacteristic
46C2×C3×#41D411yescharacteristic
47C2×C3×#41D411yescharacteristic
48C2×C3×#41D411yescharacteristic
49C22×C3×Q8C2×C411yescharacteristic
50C22×C3×Q8C2×C411yescharacteristic
51C22×C3×D4—21no—
52C22×C3×D4C2×C411yescharacteristic
53C23×C3×C4—21no—
54C23×C3×C4C2311yescharacteristic
55C2×C3×Q16—21no—
56C2×C3×Q16—21no—
57C2×C3×Q16—42no—
58C2×C3×SD16—21no—
59C2×C3×SD16—21no—
60C2×C3×SD16—42no—
61C22×C3×C8C2×C411yescharacteristic
62C22×C3×C8C2×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—
76C2×#208—21no—
77C22×C3⋊C8—22no—
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#837—42no—
86#837—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
98C22×Q16Dic311yescharacteristic, complemented
99C22×SD16Dic311yescharacteristic, complemented
100C2×#131—61no—
101C2×C22⋊Q8C2×D311yescharacteristic
102#452—21no—
103#452—21no—
104#452—21no—
105#452—21no—
106C2×#104—61no—
107C2×#104—61no—
108C2×#103Dic311yescharacteristic, complemented
109C2×#102Dic311yescharacteristic, complemented
110#408—62no—
111#408—61no—
112#407—31no—
113#407—31no—
114#407—61no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—61no—
117C2×C22⋊C8C2×D311yescharacteristic
118C4×C22⋊C4—31no—
119#328—62no—
120#324—61no—
121#324—61no—
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—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—41no—
177C2×C3×C8—41no—
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—22no—
185#217—41no—
186C4×Dic3—41no—
187C4×Dic3—41no—
188#208—44no—
189#208—41no—
190#208—41no—
191#208—41no—
192#208—41no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—44no—
197C2×C3⋊C8—41no—
198C2×C3⋊C8—41no—
199C2×C3⋊C8—41no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—42no—
203C2×#41—42no—
204C2×#41#6711yescharacteristic
205C2×#41#6711yescharacteristic
206C2×#41#6711yescharacteristic
207C2×#41#6711yescharacteristic
208C22×Q8C2×Dic311yescharacteristic
209C22×Q8C2×Dic311yescharacteristic
210C22×D4—21no—
211C22×D4C2×Dic311yescharacteristic
212C23×C4—21no—
213C23×C4C22×D311yescharacteristic
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—42no—
217C2×SD16—21no—
218C2×SD16—21no—
219C2×SD16—42no—
220#131—122no—
221#131—122no—
222C2×M16—62no—
223C2×M16—61no—
224C2×M16—61no—
225C22×C8—62no—
226C22×C8—61no—
227C22×C8C2×Dic311yescharacteristic
228C22×C8C2×Dic311yescharacteristic
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C2×Dic311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—61no—
237C2×C42—61no—
238#104—122no—
239#104—122no—
240#103—21no—
241#103—21no—
242#102—21no—
243#102—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—31no—
247C22⋊C8—31no—
248C22⋊C8—32no—
249C22⋊C8—62no—
250C22⋊C8—62no—
251C22⋊C8—61no—
252C22⋊C8—61no—
253C22⋊C8—61no—
254C22⋊C8—61no—
255#95—61no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—81no—
266C23×C3C4×D411yescharacteristic
267C23×C3#12111yescharacteristic
268C23×C3#12311yescharacteristic
269C3×Q8—21no—
270C3×Q8—21no—
271C3×Q8—21no—
272C3×Q8—42no—
273C3×Q8—41no—
274C3×Q8—41no—
275C3×Q8—81no—
276C3×Q8#9912yes—
277C3×D4—24no—
278C3×D4—21no—
279C3×D4—21no—
280C3×D4—21no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—42no—
284C3×D4—42no—
285C3×D4#9912yes—
286C2×C3×C4—24no—
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—21no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
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—41no—
305C2×C3×C4—41no—
306C2×C3×C4—81no—
307C2×C3×C4C2×C22⋊C411yescharacteristic
308C2×C3×C4C4×D411yescharacteristic
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×Dic3—42no—
312C2×Dic3—81no—
313C2×Dic3—81no—
314C3×C8—41no—
315C3×C8—41no—
316C3⋊C8—41no—
317C3⋊C8—42no—
318C3⋊C8—41no—
319C3⋊C8—81no—
320C24—21no—
321C24#23711yescharacteristic
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—44no—
331#41—44no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8#21711yescharacteristic
340C2×Q8#21711yescharacteristic
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—44no—
349C2×D4#21711yescharacteristic
350C2×D4#21711yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—121no—
362C22×C4D3×D411yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4#23711yescharacteristic
365C22×C4C22×Dic311yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368Q16—42no—
369Q16—42no—
370SD16—42no—
371SD16—42no—
372M16—124no—
373M16—121no—
374M16—121no—
375M16—121no—
376M16—121no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—62no—
384C2×C8—124no—
385C2×C8—121no—
386C2×C8—121no—
387C2×C8—121no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—62no—
395C22⋊C4—121no—
396C42—121no—
397C42—121no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—82no—
415C22×C3—82no—
416C22×C3#38611yescharacteristic
417C22×C3#44311yescharacteristic
418C22×C3#44411yescharacteristic
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—41no—
424C3×C4—81no—
425C3×C4—81no—
426C3×C4C2×#9911yescharacteristic
427C3×C4C2×#9911yescharacteristic
428Dic3—82no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—81no—
439C23D4×Dic311yescharacteristic
440C23#93511yescharacteristic
441C23#93911yescharacteristic
442Q8—21no—
443Q8—21no—
444Q8—21no—
445Q8—42no—
446Q8—41no—
447Q8—41no—
448Q8—81no—
449Q8#83412yes—
450D4—24no—
451D4—21no—
452D4—21no—
453D4—21no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4#83412yes—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—122no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4D4×Dic311yescharacteristic
484C2×C4#93711yescharacteristic
485C2×C4#93911yescharacteristic
486C2×C4C2×#21711yescharacteristic
487C8—41no—
488C8—41no—
489C8—121no—
490C8—121no—
491C8—122no—
492C8—241no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—42no—
498C2×C3—81no—
499C2×C3#187611yescharacteristic
500C2×C3#189811yescharacteristic
501C2×C3#190211yescharacteristic
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22#519611yescharacteristic
521C22#519711yescharacteristic
522C22#522011yescharacteristic
523C4—21no—
524C4—21no—
525C4—22no—
526C4—21no—
527C4—41no—
528C4—81no—
529C4—81no—
530C4—242no—
531C4C2×#83411yescharacteristic
532C4C2×#83411yescharacteristic
533C3#923911yescharacteristic, complemented, Sylow
534C2—21no—
535C2—21no—
536C2—41no—
537C2—41no—
538C2—42no—
539C2—81no—
540C2#7167011yescharacteristic
541C2#7208711yescharacteristic
542C2#7210211yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#452C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenterC2×#452D3
hypocenterC3#9239
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#452

Complements

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

NH ≅ G/Ntype
4. #71691C2semidirect
5. #71688C2semidirect
6. #67679C2semidirect
7. #67677C2semidirect
10. C22×C3×Q16C4semidirect
11. C22×C3×SD16C4semidirect
18. C2×C3×#103C4semidirect
19. C2×C3×#102C4semidirect
98. C22×Q16Dic3semidirect
99. C22×SD16Dic3semidirect
108. C2×#103Dic3semidirect
109. C2×#102Dic3semidirect
533. C3#9239semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4522
#840752
#721382
#716912
#716882
#676792
#676772
#92393