Finite Group Databasebeta

← #10944246

Subgroups — #10944246

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×C3×#465C211yesmaximal, characteristic, complemented
3C2×#5171C212yesmaximal, complemented
4#82820C211yesmaximal, characteristic, complemented
5#82709C211yesmaximal, characteristic
6#71970C211yesmaximal, characteristic, complemented
7#71697C211yesmaximal, characteristic, complemented
8#71682C211yesmaximal, characteristic, complemented
9#70577C212yesmaximal, complemented
10#70271C212yesmaximal, complemented
11#70203C212yesmaximal, complemented
12#32932—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C2×#946C2212yes—
19C2×#938C2212yes—
20C22×#208C2211yescharacteristic
21C2×#901—24no—
22C2×#901—28no—
23C22×ΓL(1,𝔽25)C2212yes—
24C22×D3×C8C2212yes—
25C3×#465—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×#835C2212yes—
30#5223—22no—
31C2×#832C2212yes—
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5171—28no—
35C2×#817C2212yes—
36#4991—21no—
37#4987C2212yes—
38#4987C2212yes—
39#4979—21no—
40#4960—21no—
41#4959—22no—
42#4947—21no—
43#4551C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C22×#131—31no—
47C2×#465—32no—
48C2×#465D311yescharacteristic, complemented
49#2616—31no—
50#2615—31no—
51#1895—32no—
52#1895—31no—
53#1894—32no—
54#1894—31no—
55#1807—32no—
56#1807—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67—22no—
62C22×#67D412yes—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—24no—
66C2×#235—24no—
67C22×D12D411yescharacteristic
68C22×C4×D3C2312yes—
69C22×Dic6D411yescharacteristic
70C2×C3×SD16—24no—
71C2×C3×SD16—22no—
72C22×C3×C8—24no—
73C22×C3×C8D414yescomplemented
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×#217—22no—
86C2×#217—21no—
87C2×#217—21no—
88#946—24no—
89#946—24no—
90#938—24no—
91#938—24no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2312yes—
97C2×Dic3⋊C4—22no—
98C2×#208—22no—
99C2×#208—24no—
100C22×C3⋊C8C2312yes—
101#901—232no—
102C2×ΓL(1,𝔽25)—24no—
103C2×ΓL(1,𝔽25)—24no—
104C2×ΓL(1,𝔽25)—24no—
105C2×D3×C8—24no—
106C2×D3×C8—24no—
107C2×D3×C8D418yescomplemented
108C3×#106—22no—
109C3×#103—22no—
110C3×#102—22no—
111#835—24no—
112#832—24no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117#817—24no—
118C22×#41—31no—
119C22×SD16C2×D311yescharacteristic
120C2×#131—68no—
121C2×#131—64no—
122C22×M16—31no—
123C22×M16—32no—
124C23×C8—32no—
125C23×C8C2×D311yescharacteristic
126C2×C22⋊Q8—32no—
127C2×C22⋊Q8C2×D311yescharacteristic
128C2×#121—32no—
129C2×#121C2×D311yescharacteristic
130#465—24no—
131#465—68no—
132C2×#106—32no—
133C2×#106C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103C2×D311yescharacteristic
136C2×#102—32no—
137C2×#102C2×D311yescharacteristic
138#395—61no—
139#394—62no—
140#394—61no—
141#393—61no—
142#392—61no—
143#392—62no—
144#380—31no—
145#380—31no—
146#380—32no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—21no—
152C24×C3C2×D411yescharacteristic
153C23×D3C2×D412yes—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—21no—
158C2×C3×D4—22no—
159C2×C3×D4—21no—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—42no—
163C2×C3×D4—82no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—24no—
174C2×#67—24no—
175C2×#67—24no—
176C2×#67—44no—
177C2×#67—82no—
178C2×#67—82no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3C2×D412yes—
182#235—28no—
183#235—216no—
184#235—28no—
185C2×D12—24no—
186C2×D12—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3C2×D414yes—
192C2×C4×D3#4114yes—
193C2×Dic6—22no—
194C2×Dic6—24no—
195C3×SD16—44no—
196C2×C3×C8—22no—
197C2×C3×C8—28no—
198C2×C3×C8C2×D414yes—
199C2×C3×C8C2×D412yes—
200C2×C3×C8C2×D412yes—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—42no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—42no—
208#217—44no—
209#217—81no—
210#217—81no—
211(C2×D3)⋊C4—44no—
212(C2×D3)⋊C4—82no—
213(C2×D3)⋊C4—82no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218Dic3⋊C4—44no—
219#208—28no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
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×Q8C22×D311yescharacteristic
231C22×D4—21no—
232C22×D4—32no—
233C22×D4—31no—
234C22×D4—62no—
235C22×D4C22×D311yescharacteristic
236C23×C4—32no—
237C23×C4C22×D311yescharacteristic
238C2×SD16—24no—
239C2×SD16—22no—
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×C8#6714yescomplemented
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—64no—
257C22⋊Q8—64no—
258#121—22no—
259#121—22no—
260#121—64no—
261#121—64no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276#106—22no—
277#106—64no—
278#103—22no—
279#103—64no—
280#102—22no—
281#102—64no—
282#95—62no—
283#95—62no—
284#95—61no—
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×Q8—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C3×D4—41no—
308C3×D4—41no—
309C3×D4—42no—
310C3×D4—84no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4#12112yes—
323C2×C3×C4C22×D412yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4112yes—
327#67—216no—
328#67—88no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3#12114yes—
338D12—28no—
339C4×D3—24no—
340C4×D3—24no—
341C4×D3—24no—
342C4×D3—24no—
343Dic6—28no—
344C3×C8—22no—
345C3×C8#12014yes—
346C3⋊C8—24no—
347C24—21no—
348C24—32no—
349C24D3×D411yescharacteristic
350#41—68no—
351#41—68no—
352#41—616no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—64no—
358C2×Q8—62no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—21no—
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—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—32no—
379C22×C4—34no—
380C22×C4—34no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4D3×D411yescharacteristic
388C22×C4#23712yes—
389C22×C4C2×#6712yes—
390C22×C4C23×D311yescharacteristic
391SD16—44no—
392M16—616no—
393M16—68no—
394C2×C8—22no—
395C2×C8—28no—
396C2×C8—34no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—616no—
400C2×C8D3×D414yes—
401C2×C8C2×#6712yes—
402C2×C8C2×#6712yes—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—42no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—124no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—242no—
418C22⋊C4—242no—
419C22⋊C4—241no—
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—82no—
430C22×C3—81no—
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—81no—
446C3×C4—81no—
447C3×C4C2×#12011yescharacteristic
448C3×C4#52511yescharacteristic
449C3×C4#52512yes—
450Dic3—24no—
451Dic3—82no—
452Dic3—82no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—34no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—62no—
464C23—62no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23#93812yes—
470C23C2×D3×D411yescharacteristic
471Q8—41no—
472Q8—41no—
473Q8—42no—
474Q8—68no—
475D4—41no—
476D4—41no—
477D4—42no—
478D4—68no—
479D4—616no—
480D4—84no—
481D4—248no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—34no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4#93812yes—
507C2×C4C2×D3×D412yes—
508C2×C4C2×#23711yescharacteristic
509C2×C4D3×#4112yes—
510C2×C4C22×#6711yescharacteristic
511C8—22no—
512C8—64no—
513C8#93714yes—
514C2×C3—42no—
515C2×C3—161no—
516C2×C3#284711yescharacteristic
517C2×C3#305712yes—
518C2×C3#305711yescharacteristic
519C2×C3#306012yes—
520C2×C3#306011yescharacteristic
521D3—44no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—64no—
528C22—64no—
529C22—64no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—122no—
539C22D3×#12112yes—
540C22#576212yes—
541C22#577011yescharacteristic
542C22#577011yescharacteristic
543C22C2×#93811yescharacteristic
544C4—22no—
545C4—64no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—242no—
550C4—242no—
551C4#560612yes—
552C4C2×#93711yescharacteristic
553C4#582811yescharacteristic
554C3#3293211yescharacteristic, complemented, Sylow
555C2—42no—
556C2—124no—
557C2—161no—
558C2#8065912yes—
559C2#8066012yes—
560C2#8272311yescharacteristic
561C2#8386311yescharacteristic
562C2#8391611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#465C2
socleC23×C3C22×D4
prime radical#82709C2
hypercenterC2×#465D3
hypocenterC3#32932
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. C22×#901C2semidirect
2. C2×C3×#465C2semidirect
3. C2×#5171C2semidirect
4. #82820C2semidirect
6. #71970C2semidirect
7. #71697C2semidirect
8. #71682C2semidirect
9. #70577C2semidirect
10. #70271C2semidirect
11. #70203C2semidirect
48. C2×#465D3semidirect
73. C22×C3×C8D4semidirect
107. C2×D3×C8D4semidirect
251. C22×C8#67semidirect
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×C3×#4652
C2×#51712
#828202
#827092
#719702
#716972
#716822
#705772
#702712
#702032
#329323