Finite Group Databasebeta

← #10943840

Subgroups — #10943840

Order 768.

3798 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×Q8C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic, complemented
3C2×#5186C214yesmaximal, complemented
4#82834C211yesmaximal, characteristic
5C2×#5050C211yesmaximal, characteristic, complemented
6#72132C211yesmaximal, characteristic, complemented
7#70266C214yesmaximal, complemented
8#69963C211yesmaximal, characteristic, complemented
9#69955C211yesmaximal, characteristic
10#32882—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×D3×Q8—24no—
13C22×D3×Q8—216no—
14C22×D3×Q8—22no—
15C22×D3×Q8—24no—
16C22×D3×Q8C2214yes—
17C23×C4×D3—21no—
18C23×C4×D3C2211yescharacteristic
19C23×Dic6—21no—
20C23×Dic6C2211yescharacteristic
21C22×C3×Q16C2212yescomplemented
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×#946C2212yes—
24C22×#216C2212yescomplemented
25C3×#451—24no—
26C2×C3×#103C2212yescomplemented
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×#835C2214yescomplemented
29#5212C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#5186—216no—
32C2×(C2×D3)⋊C8C2214yes—
33C2×#816C2214yescomplemented
34#5050—24no—
35#4989—22no—
36#4987C2212yes—
37C2×#810C2212yescomplemented
38#4955—21no—
39#4946—21no—
40C24×Q8—31no—
41C2×#451—34no—
42C2×#451—31no—
43C2×#451D311yescharacteristic, complemented
44#2848—31no—
45#1787—34no—
46#1787—31no—
47#1787—31no—
48#1779—31no—
49C24×D3D411yescharacteristic
50C22×C3×Q8—28no—
51C22×C3×Q8—22no—
52C22×C3×Q8—21no—
53C22×C3×Q8D414yes—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×Q8C2312yes—
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3D411yescharacteristic
60C2×D3×Q8—264no—
61C2×D3×Q8—216no—
62C2×D3×Q8—216no—
63C2×D3×Q8—264no—
64C2×D3×Q8—216no—
65C2×D3×Q8—28no—
66C2×D3×Q8—28no—
67C2×D3×Q8—28no—
68C2×D3×Q8—28no—
69C2×D3×Q8—28no—
70C2×D3×Q8D4116yes—
71C22×C4×D3—28no—
72C22×C4×D3—216no—
73C22×C4×D3—24no—
74C22×C4×D3—22no—
75C22×C4×D3—22no—
76C22×C4×D3—28no—
77C22×C4×D3D412yes—
78C22×C4×D3C2312yes—
79C22×Dic6—28no—
80C22×Dic6—216no—
81C22×Dic6—21no—
82C22×Dic6—22no—
83C22×Dic6—24no—
84C22×Dic6—28no—
85C22×Dic6D414yes—
86C22×Dic6C2312yes—
87C2×C3×Q16—28no—
88C2×C3×Q16—24no—
89C22×C3×C8C2312yes—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×#217—21no—
96#946—28no—
97C2×#216—28no—
98C2×#216—24no—
99C2×(C2×D3)⋊C4—22no—
100C2×(C2×D3)⋊C4—22no—
101C2×C4⋊Dic3C2312yes—
102C2×Dic3⋊C4—22no—
103C22×C3⋊C8C2312yes—
104C3×#103—24no—
105C3×C22⋊C8—22no—
106#835—28no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#830—22no—
110(C2×D3)⋊C8—28no—
111#816—28no—
112#810—24no—
113C23×Q8—34no—
114C23×Q8—31no—
115C23×Q8—61no—
116C23×Q8—62no—
117C23×Q8—616no—
118C23×Q8—64no—
119C23×Q8—64no—
120C23×Q8C2×D311yescharacteristic
121C24×C4—31no—
122C24×C4—61no—
123C22×Q16—32no—
124C22×Q16C2×D312yescomplemented
125C2×C22⋊Q8—32no—
126C2×C22⋊Q8C2×D311yescharacteristic
127#451—24no—
128#451—64no—
129#451—616no—
130C2×#103—34no—
131C2×#103—34no—
132C2×#103—32no—
133C2×#103C2×D312yescomplemented
134C2×C22⋊C8—31no—
135C2×C22⋊C8—34no—
136C2×C22⋊C8C2×D311yescharacteristic
137#385—61no—
138#385—62no—
139#380—31no—
140#380—32no—
141#379—61no—
142C24×C3C2×D411yescharacteristic
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—24no—
146C23×D3—24no—
147C23×D3Q1614yescomplemented
148C23×D3C2×D412yes—
149C2×C3×Q8—216no—
150C2×C3×Q8—28no—
151C2×C3×Q8—216no—
152C2×C3×Q8—24no—
153C2×C3×Q8—22no—
154C2×C3×Q8—24no—
155C2×C3×Q8—21no—
156C2×C3×Q8—24no—
157C2×C3×Q8—21no—
158C2×C3×Q8C2×D418yes—
159C2×C3×Q8C2×D414yes—
160C22×C3×C4—28no—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—28no—
171C22×Dic3—22no—
172C22×Dic3—24no—
173C22×Dic3—22no—
174C22×Dic3—24no—
175C22×Dic3—22no—
176C22×Dic3C2×D412yes—
177D3×Q8—2128no—
178D3×Q8—232no—
179D3×Q8—232no—
180D3×Q8—232no—
181D3×Q8—232no—
182C2×C4×D3—264no—
183C2×C4×D3—216no—
184C2×C4×D3—216no—
185C2×C4×D3—216no—
186C2×C4×D3—216no—
187C2×C4×D3—216no—
188C2×C4×D3—24no—
189C2×C4×D3—24no—
190C2×C4×D3—22no—
191C2×C4×D3—24no—
192C2×C4×D3—22no—
193C2×C4×D3—24no—
194C2×C4×D3C2×D418yes—
195C2×Dic6—264no—
196C2×Dic6—216no—
197C2×Dic6—216no—
198C2×Dic6—28no—
199C2×Dic6—216no—
200C2×Dic6—24no—
201C2×Dic6—24no—
202C2×Dic6—22no—
203C2×Dic6—216no—
204C2×Dic6—216no—
205C2×Dic6C2×D414yes—
206C2×Dic6C2×D418yes—
207C3×Q16—48no—
208C2×C3×C8—24no—
209C2×C3×C8—22no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—42no—
213C3×C22⋊C4—42no—
214#217—42no—
215#216—48no—
216(C2×D3)⋊C4—44no—
217(C2×D3)⋊C4—44no—
218C4⋊Dic3—24no—
219Dic3⋊C4—44no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—24no—
222C25—31no—
223C22×Q8—28no—
224C22×Q8—22no—
225C22×Q8—21no—
226C22×Q8—316no—
227C22×Q8—34no—
228C22×Q8—32no—
229C22×Q8—64no—
230C22×Q8—68no—
231C22×Q8—616no—
232C22×Q8—68no—
233C22×Q8—68no—
234C22×Q8—68no—
235C22×Q8—68no—
236C22×Q8—616no—
237C22×Q8—616no—
238C22×Q8—664no—
239C22×Q8—664no—
240C22×Q8—616no—
241C22×Q8—68no—
242C22×Q8—68no—
243C22×Q8—61no—
244C22×Q8—62no—
245C22×Q8#6714yes—
246C22×Q8C22×D311yescharacteristic
247C22×Q8C22×D312yes—
248C23×C4—21no—
249C23×C4—32no—
250C23×C4—31no—
251C23×C4—32no—
252C23×C4—68no—
253C23×C4—64no—
254C23×C4—616no—
255C23×C4—68no—
256C23×C4—61no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4C22×D311yescharacteristic
260C2×Q16—28no—
261C2×Q16—24no—
262C2×Q16—64no—
263C2×Q16—68no—
264C22×C8—32no—
265C22×C8C22×D312yes—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—68no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—61no—
277#103—24no—
278#103—64no—
279#103—68no—
280#103—68no—
281C22⋊C8—22no—
282C22⋊C8—68no—
283C22⋊C8—62no—
284#95—62no—
285#95—61no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3#12012yes—
289C23×C3C2×Q1612yes—
290C23×C3#13712yes—
291C23×C3C22×D411yescharacteristic
292C22×D3—28no—
293C22×D3—216no—
294C22×D3—28no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—28no—
300C22×D3—216no—
301C22×D3#12014yes—
302C22×D3C2×Q1614yes—
303C22×D3#13714yes—
304C3×Q8—216no—
305C3×Q8—24no—
306C3×Q8—28no—
307C3×Q8—42no—
308C3×Q8—42no—
309C3×Q8#12018yes—
310C2×C3×C4—216no—
311C2×C3×C4—24no—
312C2×C3×C4—28no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4#12012yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D414yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×Dic3—28no—
325C2×Dic3—216no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—28no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3#12014yes—
334C4×D3—216no—
335C4×D3—264no—
336C4×D3—28no—
337C4×D3—28no—
338Dic6—264no—
339Dic6—216no—
340Dic6—28no—
341Dic6—24no—
342Dic6#12018yes—
343C3×C8—42no—
344C3⋊C8—42no—
345C24—32no—
346C24—34no—
347C24—64no—
348C24—64no—
349C24—62no—
350C24—62no—
351C24D3×D411yescharacteristic
352C2×Q8—216no—
353C2×Q8—28no—
354C2×Q8—216no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—21no—
359C2×Q8—24no—
360C2×Q8—21no—
361C2×Q8—34no—
362C2×Q8—38no—
363C2×Q8—616no—
364C2×Q8—616no—
365C2×Q8—664no—
366C2×Q8—616no—
367C2×Q8—616no—
368C2×Q8—68no—
369C2×Q8—616no—
370C2×Q8—632no—
371C2×Q8—6128no—
372C2×Q8—632no—
373C2×Q8—632no—
374C2×Q8—632no—
375C2×Q8—64no—
376C2×Q8—64no—
377C2×Q8—62no—
378C2×Q8D3×D418yes—
379C2×Q8C2×#6714yes—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—32no—
387C22×C4—38no—
388C22×C4—64no—
389C22×C4—64no—
390C22×C4—616no—
391C22×C4—664no—
392C22×C4—616no—
393C22×C4—616no—
394C22×C4—616no—
395C22×C4—616no—
396C22×C4—64no—
397C22×C4—68no—
398C22×C4—62no—
399C22×C4—64no—
400C22×C4—62no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—64no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4D3×D411yescharacteristic
407C22×C4D3×D412yes—
408C22×C4C2×#6712yes—
409C22×C4C23×D311yescharacteristic
410Q16—48no—
411Q16—128no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—64no—
415C2×C8—62no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—42no—
419C4⋊C4—64no—
420C4⋊C4—124no—
421C22⋊C4—42no—
422C22⋊C4—124no—
423C22⋊C4—124no—
424C22⋊C4—122no—
425C22×C3—22no—
426C22×C3—24no—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3#45114yescomplemented
430C22×C3C2×#12011yescharacteristic
431C22×C3C2×#12012yes—
432C22×C3C22×Q1611yescharacteristic
433C22×C3C2×#13711yescharacteristic
434C22×C3C2×#13712yes—
435C2×D3—232no—
436C2×D3—24no—
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—24no—
441C2×D3—24no—
442C2×D3#45118yescomplemented
443C3×C4—28no—
444C3×C4—22no—
445C3×C4—81no—
446C3×C4—81no—
447C3×C4C2×#12012yes—
448C3×C4C2×#12012yes—
449Dic3—24no—
450Dic3—28no—
451Dic3—82no—
452C23—22no—
453C23—22no—
454C23—34no—
455C23—34no—
456C23—34no—
457C23—616no—
458C23—68no—
459C23—68no—
460C23—616no—
461C23—68no—
462C23—62no—
463C23—62no—
464C23—62no—
465C23—62no—
466C23D3×Q1612yes—
467C23#91812yes—
468C23#93712yes—
469C23C2×D3×D411yescharacteristic
470Q8—216no—
471Q8—24no—
472Q8—28no—
473Q8—38no—
474Q8—42no—
475Q8—42no—
476Q8—664no—
477Q8—616no—
478Q8—68no—
479Q8—64no—
480Q8#93718yes—
481C2×C4—216no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—34no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—664no—
493C2×C4—616no—
494C2×C4—68no—
495C2×C4—616no—
496C2×C4—62no—
497C2×C4—64no—
498C2×C4—62no—
499C2×C4—68no—
500C2×C4—68no—
501C2×C4—68no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4#93712yes—
506C2×C4C2×D3×D414yes—
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C22×#6711yescharacteristic
509C8—42no—
510C8—122no—
511C2×C3—24no—
512C2×C3#284611yescharacteristic
513C2×C3C2×#45112yes—
514C2×C3C2×#45111yescharacteristic
515C2×C3#300411yescharacteristic
516C2×C3#300412yes—
517D3—28no—
518C22—22no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—38no—
523C22—632no—
524C22—64no—
525C22—64no—
526C22—64no—
527C22—64no—
528C22—64no—
529C22—64no—
530C22#518614yescomplemented
531C22D3×#12012yes—
532C22C2×D3×Q1611yescharacteristic
533C22C2×#91811yescharacteristic
534C22D3×#13712yes—
535C22C2×#93711yescharacteristic
536C4—28no—
537C4—22no—
538C4—64no—
539C4—68no—
540C4—81no—
541C4—81no—
542C4—242no—
543C4D3×#12012yes—
544C4C2×#93712yes—
545C3#3288211yescharacteristic, complemented, Sylow
546C2—24no—
547C2—68no—
548C2D3×#45112yes—
549C2#8018312yes—
550C2#8261211yescharacteristic
551C2C2×#518611yescharacteristic
552C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenterC2×#451D3
hypocenterC3#32882
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
2. C2×C3×#451C2semidirect
3. C2×#5186C2semidirect
5. C2×#5050C2semidirect
6. #72132C2semidirect
7. #70266C2semidirect
8. #69963C2semidirect
21. C22×C3×Q16C22semidirect
24. C22×#216C22semidirect
26. C2×C3×#103C22semidirect
28. C2×#835C22semidirect
33. C2×#816C22semidirect
37. C2×#810C22semidirect
43. C2×#451D3semidirect
124. C22×Q16C2×D3semidirect
133. C2×#103C2×D3semidirect
147. C23×D3Q16semidirect
429. C22×C3#451semidirect
442. C2×D3#451semidirect
530. C22#5186semidirect
545. C3#32882semidirect

Maximal subgroups

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

subgroupindex
C23×D3×Q82
C2×C3×#4512
C2×#51862
#828342
C2×#50502
#721322
#702662
#699632
#699552
#328823