Finite Group Databasebeta

← #10944250

Subgroups — #10944250

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#901C211yesmaximal, characteristic, complemented
2C2×#5114C211yesmaximal, characteristic, complemented
3C2×#4864C212yesmaximal, complemented
4#74030C211yesmaximal, characteristic
5#73950C211yesmaximal, characteristic, complemented
6C3×#1807C211yesmaximal, characteristic, complemented
7#70577C212yesmaximal, complemented
8#69517C211yesmaximal, characteristic, complemented
9#69516C211yesmaximal, characteristic, complemented
10#68885C212yesmaximal, complemented
11#68884C212yesmaximal, complemented
12#32932—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×#930C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C22×#208C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#901—24no—
20C2×#901—28no—
21C22×ΓL(1,𝔽25)C2212yes—
22C22×D3×C8C2212yes—
23C2×#897C2212yes—
24C2×#895C2212yes—
25C2×C3×#106C2212yes—
26C3×#380C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5114—24no—
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#4987C2212yes—
33#4987C2212yes—
34C2×#810C2212yes—
35C2×#809C2212yes—
36#4961—22no—
37#4960—22no—
38#4864—28no—
39#4673—21no—
40#4672—21no—
41#4671—21no—
42#4656—21no—
43C3×#340C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#465—32no—
48C2×#465—31no—
49#2616—31no—
50#2615—31no—
51#1895—31no—
52#1895—32no—
53#1894—31no—
54#1894—32no—
55#1807—32no—
56#1807D311yescharacteristic, complemented
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C22×#67D412yes—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—24no—
63C2×#235—24no—
64C22×D12—22no—
65C22×D12D411yescharacteristic
66C22×D12C2311yescharacteristic
67C22×C4×D3C2312yes—
68C22×Dic6D411yescharacteristic
69C22×Dic6C2311yescharacteristic
70C22×C3×C8—24no—
71C22×C3×C8D414yescomplemented
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×#217—21no—
79#930—22no—
80#930—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89#924—22no—
90#924—22no—
91C2×Dic3⋊C4—21no—
92C2×#208—22no—
93C2×#208—24no—
94C22×C3⋊C8C2312yes—
95C2×#204—24no—
96C2×#204—22no—
97#901—232no—
98C2×ΓL(1,𝔽25)—24no—
99C2×ΓL(1,𝔽25)—24no—
100C2×ΓL(1,𝔽25)—24no—
101C2×D3×C8—24no—
102C2×D3×C8—24no—
103C2×D3×C8D418yescomplemented
104#897—24no—
105#897—24no—
106#895—24no—
107#895—24no—
108C3×#106—24no—
109C3×#95—21no—
110C3×#95—21no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#821—22no—
114#817—22no—
115#816—22no—
116#810—24no—
117#809—24no—
118C22×#41—31no—
119C22×SD16—31no—
120C2×#131—68no—
121C2×#131—64no—
122C22×M16—31no—
123C22×M16—32no—
124C23×C8—32no—
125C23×C8C2×D311yescharacteristic
126C2×C22⋊Q8—31no—
127C2×C22⋊Q8—32no—
128C2×#121—31no—
129C2×#121—32no—
130#465—68no—
131#465—64no—
132C2×#106—31no—
133C2×#106C2×D312yes—
134C2×#103—31no—
135C2×#103—32no—
136C2×#102—31no—
137C2×#102—32no—
138#395—61no—
139#394—61no—
140#394—62no—
141#393—61no—
142#392—61no—
143#392—62no—
144#380—32no—
145#380—32no—
146#380C2×D311yescharacteristic
147#380C2×D311yescharacteristic
148#340—32no—
149#340—32no—
150#340C2×D312yes—
151C24×C3C2×D411yescharacteristic
152C23×D3—21no—
153C23×D3C2×D412yes—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4#4112yes—
161C22×C3×C4C2411yescharacteristic
162C2×#67—24no—
163C2×#67—24no—
164C2×#67—24no—
165C2×#67—42no—
166C2×#67—82no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3C2×D412yes—
171#235—28no—
172#235—216no—
173#235—28no—
174C2×D12—24no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—22no—
179C2×D12—21no—
180C2×D12—44no—
181C2×D12—82no—
182C2×D12—82no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3C2×D414yes—
188C2×C4×D3#4114yes—
189C2×Dic6—22no—
190C2×Dic6—24no—
191C2×Dic6—22no—
192C2×Dic6—21no—
193C2×Dic6—22no—
194C2×Dic6—21no—
195C2×C3×C8—22no—
196C2×C3×C8—28no—
197C2×C3×C8C2×D412yes—
198C2×C3×C8C2×D412yes—
199C2×C3×C8C2×D414yes—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C22⋊C4—81no—
205C3×C22⋊C4—81no—
206#217—42no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—42no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—82no—
211(C2×D3)⋊C4—82no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—44no—
217Dic3⋊C4—42no—
218#208—28no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8C2×D414yes—
222#204—44no—
223ΓL(1,𝔽25)—216no—
224D3×C8—216no—
225C2×#41—64no—
226C2×#41—64no—
227C2×#41—62no—
228C2×#41—62no—
229C22×Q8—31no—
230C22×Q8—31no—
231C22×D4—32no—
232C22×D4—31no—
233C22×D4—31no—
234C22×D4—62no—
235C22×D4—61no—
236C23×C4—32no—
237C23×C4C22×D311yescharacteristic
238C2×SD16—62no—
239C2×SD16—64no—
240#131—632no—
241C2×M16—64no—
242C2×M16—64no—
243C2×M16—64no—
244C2×M16—62no—
245C2×M16—64no—
246C22×C8—24no—
247C22×C8—32no—
248C22×C8—38no—
249C22×C8—64no—
250C22×C8—64no—
251C22×C8D1214yescomplemented
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254C22⋊Q8—64no—
255C22⋊Q8—64no—
256C22⋊Q8—62no—
257C22⋊Q8—62no—
258#121—64no—
259#121—64no—
260#121—62no—
261#121—62no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4C22×D312yes—
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276#106—24no—
277#106—62no—
278#103—64no—
279#103—62no—
280#102—64no—
281#102—62no—
282#95—21no—
283#95—21no—
284#95—62no—
285#95—62no—
286C23×C3—22no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—22no—
290C23×C3#12112yes—
291C23×C3C22×D411yescharacteristic
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—42no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3#12114yes—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4#12112yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D412yes—
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4112yes—
319#67—216no—
320#67—84no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3#12114yes—
331D12—28no—
332D12—41no—
333D12—41no—
334D12—42no—
335D12—88no—
336C4×D3—24no—
337C4×D3—24no—
338C4×D3—24no—
339C4×D3—24no—
340Dic6—28no—
341Dic6—41no—
342Dic6—41no—
343Dic6—42no—
344C3×C8—22no—
345C3×C8#12014yes—
346C3⋊C8—24no—
347C24—32no—
348C24—61no—
349C24D3×D411yescharacteristic
350#41—68no—
351#41—68no—
352#41—616no—
353C2×Q8—64no—
354C2×Q8—62no—
355C2×Q8—61no—
356C2×Q8—62no—
357C2×Q8—61no—
358C2×Q8—62no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—64no—
362C2×D4—64no—
363C2×D4—62no—
364C2×D4—61no—
365C2×D4—62no—
366C2×D4—61no—
367C2×D4—62no—
368C2×D4—124no—
369C2×D4—122no—
370C2×D4—242no—
371C2×D4—242no—
372C2×D4—242no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—32no—
378C22×C4—34no—
379C22×C4—34no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4C2×D1212yes—
388C22×C4D3×D411yescharacteristic
389C22×C4#23912yes—
390C22×C4C23×D311yescharacteristic
391SD16—124no—
392M16—616no—
393M16—68no—
394C2×C8—22no—
395C2×C8—28no—
396C2×C8—34no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—616no—
400C2×C8C2×D1212yes—
401C2×C8C2×D1212yes—
402C2×C8D3×D414yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—124no—
412C4⋊C4—122no—
413C22⋊C4—81no—
414C22⋊C4—81no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—122no—
418C22⋊C4—122no—
419C22⋊C4—242no—
420C22⋊C4—242no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12112yes—
428C22×C3#57711yescharacteristic
429C22×C3#57711yescharacteristic
430C22×C3#57712yes—
431C2×D3—24no—
432C2×D3—24no—
433C2×D3—24no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—81no—
439C2×D3—82no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—82no—
443C3×C4—22no—
444C3×C4—82no—
445C3×C4—82no—
446C3×C4C2×#12011yescharacteristic
447C3×C4#52511yescharacteristic
448C3×C4#52512yes—
449Dic3—24no—
450Dic3—81no—
451Dic3—81no—
452Dic3—81no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—34no—
458C23—62no—
459C23—62no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—62no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23—122no—
468C23—122no—
469C23#89512yes—
470C23C2×D3×D411yescharacteristic
471Q8—68no—
472Q8—121no—
473Q8—121no—
474Q8—122no—
475D4—68no—
476D4—616no—
477D4—121no—
478D4—121no—
479D4—122no—
480D4—248no—
481D4—244no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—34no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4#89512yes—
507C2×C4C22×D1211yescharacteristic
508C2×C4C2×D3×D412yes—
509C2×C4C2×#23911yescharacteristic
510C2×C4D3×#4112yes—
511C8—22no—
512C8—64no—
513C8#88214yes—
514C2×C3—42no—
515C2×C3#284711yescharacteristic
516C2×C3#305712yes—
517C2×C3#305711yescharacteristic
518C2×C3#306011yescharacteristic
519C2×C3#306012yes—
520D3—44no—
521D3—161no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—64no—
528C22—64no—
529C22—64no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—122no—
534C22—241no—
535C22—242no—
536C22—242no—
537C22—241no—
538C22—241no—
539C22C2×#89511yescharacteristic
540C22D3×#12112yes—
541C22#576211yescharacteristic
542C22#576211yescharacteristic
543C22#577012yes—
544C4—22no—
545C4—64no—
546C4—82no—
547C4—82no—
548C4—241no—
549C4—241no—
550C4—241no—
551C4C2×#88211yescharacteristic
552C4#552411yescharacteristic
553C4#562712yes—
554C3#3293211yescharacteristic, complemented, Sylow
555C2—42no—
556C2—124no—
557C2—481no—
558C2#7404811yescharacteristic
559C2#7912011yescharacteristic
560C2#7918211yescharacteristic
561C2#8065912yes—
562C2#8066012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1807C2
socleC23×C3C22×D4
prime radical#74030C2
hypercenter#1807D3
hypocenterC3#32932
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C22×#901C2semidirect
2. C2×#5114C2semidirect
3. C2×#4864C2semidirect
5. #73950C2semidirect
6. C3×#1807C2semidirect
7. #70577C2semidirect
8. #69517C2semidirect
9. #69516C2semidirect
10. #68885C2semidirect
11. #68884C2semidirect
56. #1807D3semidirect
71. C22×C3×C8D4semidirect
103. C2×D3×C8D4semidirect
251. C22×C8D12semidirect
554. C3#32932semidirect

Maximal subgroups

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

subgroupindex
C22×#9012
C2×#51142
C2×#48642
#740302
#739502
C3×#18072
#705772
#695172
#695162
#688852
#688842
#329323