Finite Group Databasebeta

← #10944248

Subgroups — #10944248

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×#907C211yesmaximal, characteristic, complemented
2#82709C211yesmaximal, characteristic
3C2×#5041C211yesmaximal, characteristic, complemented
4C2×#4778C212yesmaximal, complemented
5#73950C211yesmaximal, characteristic, complemented
6C3×#1895C211yesmaximal, characteristic, complemented
7#71989C211yesmaximal, characteristic, complemented
8#70203C212yesmaximal, complemented
9#69511C211yesmaximal, characteristic, complemented
10#68884C212yesmaximal, complemented
11#68868C212yesmaximal, complemented
12#32932—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×#938C2212yes—
17C22×#214C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C2×#907—24no—
21C2×#907—28no—
22C22×ΓL(1,𝔽25)C2212yes—
23C22×D3×C8C2212yes—
24C2×#897C2212yes—
25C2×C3×#102C2212yes—
26C3×#380C2211yescharacteristic
27#5223—22no—
28C2×#832C2211yescharacteristic
29#5212C2211yescharacteristic
30C2×#816C2212yes—
31#5041—24no—
32#4987C2212yes—
33#4987C2212yes—
34C2×#810C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#808C2212yes—
37#4961—22no—
38#4960—21no—
39#4947—21no—
40#4778—28no—
41#4672—21no—
42#4656—21no—
43C3×#340C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#465—32no—
48C2×#465—31no—
49#2616—31no—
50#2615—31no—
51#1895—32no—
52#1895D311yescharacteristic, complemented
53#1894—32no—
54#1894—31no—
55#1807—31no—
56#1807—32no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C22×#67D412yes—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—24no—
65C2×#235—24no—
66C22×D12D411yescharacteristic
67C22×C4×D3C2312yes—
68C22×Dic6D411yescharacteristic
69C22×Dic6C2311yescharacteristic
70C2×C3×M16—22no—
71C2×C3×M16—24no—
72C22×C3×C8C2312yes—
73C3×#121—22no—
74C3×#121—22no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×#217—21no—
80C2×#217—22no—
81C2×#217—21no—
82#938—24no—
83#938—24no—
84C2×#214—24no—
85C2×#214—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2311yescharacteristic
92#924—22no—
93#924—22no—
94C2×Dic3⋊C4—21no—
95C22×C3⋊C8—24no—
96C22×C3⋊C8D414yescomplemented
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99#907—232no—
100C2×ΓL(1,𝔽25)—24no—
101C2×ΓL(1,𝔽25)—24no—
102C2×ΓL(1,𝔽25)—24no—
103C2×D3×C8—24no—
104C2×D3×C8—24no—
105C2×D3×C8D418yescomplemented
106#897—24no—
107#897—24no—
108C3×#102—24no—
109C3×#95—21no—
110#832—22no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114#816—24no—
115#810—22no—
116#808—24no—
117#808—22no—
118C22×#41—31no—
119C22×SD16—31no—
120C2×#131—64no—
121C2×#131—68no—
122C22×M16—32no—
123C22×M16C2×D311yescharacteristic
124C23×C8—31no—
125C23×C8—32no—
126C2×C22⋊Q8—31no—
127C2×C22⋊Q8—32no—
128C2×#121—32no—
129C2×#121C2×D311yescharacteristic
130#465—68no—
131#465—64no—
132C2×#106—32no—
133C2×#106—31no—
134C2×#103—32no—
135C2×#103—31no—
136C2×#102—31no—
137C2×#102C2×D312yes—
138#395—61no—
139#394—61no—
140#394—62no—
141#393—61no—
142#392—61no—
143#392—62no—
144#380—31no—
145#380—32no—
146#380—32no—
147#380C2×D311yescharacteristic
148#340—32no—
149#340—32no—
150#340C2×D312yes—
151C24×C3—21no—
152C24×C3C2×D411yescharacteristic
153C23×D3C2×D412yes—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—42no—
159C2×C3×D4—82no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C2×#67—24no—
169C2×#67—24no—
170C2×#67—24no—
171C2×#67—44no—
172C2×#67—82no—
173C2×#67—82no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3C2×D412yes—
178#235—28no—
179#235—216no—
180#235—28no—
181C2×D12—24no—
182C2×D12—22no—
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—21no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—21no—
195C3×M16—28no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8C2×D414yes—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—81no—
205#217—42no—
206#217—44no—
207#217—81no—
208#214—44no—
209(C2×D3)⋊C4—44no—
210(C2×D3)⋊C4—82no—
211(C2×D3)⋊C4—82no—
212C4⋊Dic3—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—22no—
216C4⋊Dic3—44no—
217Dic3⋊C4—42no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—28no—
220C2×C3⋊C8C2×D412yes—
221C2×C3⋊C8C2×D412yes—
222C2×C3⋊C8C2×D414yes—
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—21no—
232C22×D4—32no—
233C22×D4—31no—
234C22×D4—62no—
235C22×D4C22×D311yescharacteristic
236C23×C4—32no—
237C23×C4C22×D311yescharacteristic
238C2×SD16—64no—
239C2×SD16—62no—
240#131—632no—
241C2×M16—22no—
242C2×M16—24no—
243C2×M16—64no—
244C2×M16—64no—
245C2×M16—64no—
246C22×C8—31no—
247C22×C8—34no—
248C22×C8—31no—
249C22×C8—38no—
250C22×C8—64no—
251C22×C8—64no—
252C22×C8—64no—
253C22×C8C22×D312yes—
254C22⋊Q8—64no—
255C22⋊Q8—64no—
256C22⋊Q8—62no—
257C22⋊Q8—62no—
258#121—22no—
259#121—22no—
260#121—64no—
261#121—64no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4C22×D311yescharacteristic
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—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276#106—64no—
277#106—62no—
278#103—62no—
279#103—64no—
280#102—24no—
281#102—62no—
282#95—21no—
283#95—62no—
284#95—62no—
285#95—61no—
286C23×C3—22no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—22no—
290C23×C3—42no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—41no—
295C23×C3#12112yes—
296C23×C3C22×D411yescharacteristic
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3#12114yes—
304C3×D4—41no—
305C3×D4—41no—
306C3×D4—42no—
307C3×D4—84no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4#12112yes—
318C2×C3×C4C22×D412yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4112yes—
322#67—216no—
323#67—88no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—41no—
334C2×Dic3#12114yes—
335D12—28no—
336C4×D3—24no—
337C4×D3—24no—
338C4×D3—24no—
339C4×D3—24no—
340Dic6—28no—
341Dic6—41no—
342Dic6—41no—
343Dic6—42no—
344C3×C8—24no—
345C3⋊C8—22no—
346C3⋊C8#12014yes—
347C24—21no—
348C24—32no—
349C24D3×D411yescharacteristic
350#41—68no—
351#41—68no—
352#41—616no—
353C2×Q8—64no—
354C2×Q8—62no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—64no—
365C2×D4—64no—
366C2×D4—64no—
367C2×D4—64no—
368C2×D4—62no—
369C2×D4—82no—
370C2×D4—124no—
371C2×D4—242no—
372C2×D4—242no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
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—62no—
386C22×C4—61no—
387C22×C4#23512yes—
388C22×C4D3×D411yescharacteristic
389C22×C4D3×D412yes—
390C22×C4C23×D311yescharacteristic
391SD16—124no—
392M16—28no—
393M16—616no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—32no—
397C2×C8—34no—
398C2×C8—32no—
399C2×C8—616no—
400C2×C8—68no—
401C2×C8—62no—
402C2×C8D3×D414yes—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—124no—
412C4⋊C4—122no—
413C22⋊C4—42no—
414C22⋊C4—81no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—122no—
418C22⋊C4—242no—
419C22⋊C4—242no—
420C22⋊C4—241no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—81no—
427C22×C3—82no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—82no—
431C22×C3C2×#12111yescharacteristic
432C22×C3C2×#12112yes—
433C22×C3#57712yes—
434C22×C3#57711yescharacteristic
435C22×C3#57711yescharacteristic
436C2×D3—24no—
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C3×C4—22no—
444C3×C4—81no—
445C3×C4—82no—
446C3×C4C2×#12011yescharacteristic
447C3×C4#52511yescharacteristic
448C3×C4#52512yes—
449Dic3—24no—
450Dic3—81no—
451Dic3—82no—
452Dic3—81no—
453C23—21no—
454C23—22no—
455C23—21no—
456C23—22no—
457C23—34no—
458C23—41no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—62no—
464C23—62no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23#88412yes—
470C23C2×D3×D411yescharacteristic
471Q8—68no—
472Q8—121no—
473Q8—122no—
474Q8—121no—
475D4—41no—
476D4—42no—
477D4—41no—
478D4—68no—
479D4—616no—
480D4—84no—
481D4—248no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—34no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4#88412yes—
507C2×C4C2×#23511yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×D3×D412yes—
510C2×C4D3×#4112yes—
511C8—24no—
512C8—34no—
513C8—62no—
514C2×C3—42no—
515C2×C3—161no—
516C2×C3#284711yescharacteristic
517C2×C3#305711yescharacteristic
518C2×C3#305712yes—
519C2×C3#306011yescharacteristic
520C2×C3#306012yes—
521D3—44no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—64no—
528C22—64no—
529C22—64no—
530C22—81no—
531C22—81no—
532C22—82no—
533C22—81no—
534C22—82no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—122no—
539C22C2×#88411yescharacteristic
540C22D3×#12112yes—
541C22#577511yescharacteristic
542C22#577512yes—
543C22#577811yescharacteristic
544C4—22no—
545C4—64no—
546C4—81no—
547C4—82no—
548C4—241no—
549C4—242no—
550C4—241no—
551C4#554712yes—
552C4D3×#12011yescharacteristic
553C4#565611yescharacteristic
554C3#3293211yescharacteristic, complemented, Sylow
555C2—42no—
556C2—124no—
557C2—161no—
558C2#7493511yescharacteristic
559C2#7889011yescharacteristic
560C2#7893111yescharacteristic
561C2#8080712yes—
562C2#8086612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1895C2
socleC23×C3C22×D4
prime radical#82709C2
hypercenter#1895D3
hypocenterC3#32932
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
1. C22×#907C2semidirect
3. C2×#5041C2semidirect
4. C2×#4778C2semidirect
5. #73950C2semidirect
6. C3×#1895C2semidirect
7. #71989C2semidirect
8. #70203C2semidirect
9. #69511C2semidirect
10. #68884C2semidirect
11. #68868C2semidirect
52. #1895D3semidirect
96. C22×C3⋊C8D4semidirect
105. C2×D3×C8D4semidirect
554. C3#32932semidirect

Maximal subgroups

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

subgroupindex
C22×#9072
#827092
C2×#50412
C2×#47782
#739502
C3×#18952
#719892
#702032
#695112
#688842
#688682
#329323