Finite Group Databasebeta

← #10797155

Subgroups — #10797155

Order 768.

1654 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2C2×#5745C211yesmaximal, characteristic
3C2×C3×#419C211yesmaximal, characteristic, complemented
4C22×#836C211yesmaximal, characteristic, complemented
5#82213C211yesmaximal, characteristic
6#71998C211yesmaximal, characteristic, complemented
7#71587C211yesmaximal, characteristic, complemented
8#70286C218yesmaximal, complemented
9#21006—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×#216—21no—
15C2×#942—24no—
16C2×#942—22no—
17C2×#942C418yescomplemented
18C2×#942C2214yes—
19C22×#215—21no—
20C22×#208C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23#5745—24no—
24#5745—24no—
25C2×#907—22no—
26C2×D3×M16—22no—
27C2×#901C2214yes—
28C3×#419—22no—
29C3×#419—22no—
30C2×C3×#103C2214yescomplemented
31C2×#836—24no—
32C2×#836C2214yescomplemented
33#5004—21no—
34#5002—22no—
35#5001—21no—
36#4966C2214yes—
37#4552C2214yescomplemented
38#4530C2214yescomplemented
39C22×#137—31no—
40C2×#568—31no—
41#2896—31no—
42C2×#419D311yescharacteristic, complemented
43C22×#101—31no—
44#1908—38no—
45#1742—31no—
46#1738—31no—
47C22×C3×Q8—22no—
48C22×C3×Q8—21no—
49C22×C3×Q8—21no—
50C22×C3×Q8—21no—
51C22×C3×Q8—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8C2×C414yes—
56C22×C3×Q8C2312yes—
57C23×C3×C4—21no—
58C23×C3×C4C2311yescharacteristic
59C22×#67—21no—
60C2×#235—21no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—21no—
64C2×#235C2×C414yes—
65C2×#235D414yes—
66C2×#235D412yes—
67C2×#235C2312yes—
68C22×D12D411yescharacteristic
69C22×C4×D3—21no—
70C22×Dic6D411yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16D412yes—
74C2×C3×M16D412yes—
75C22×C3×C8C2312yes—
76C3×#117—21no—
77C3×#117—21no—
78C3×#117—21no—
79C3×#117—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—21no—
83C2×#216—28no—
84C2×#216—24no—
85#942—216no—
86#942—28no—
87#942C2×C4116yes—
88C2×#215—28no—
89C2×#215—24no—
90C2×#208—21no—
91C2×#208—22no—
92C2×#208—22no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208C2×C414yes—
98C2×#208D414yes—
99C2×#208C2312yes—
100C2×#208C2312yes—
101C2×#208C2312yes—
102C22×C3⋊C8—21no—
103C22×C3⋊C8—21no—
104C22×C3⋊C8—21no—
105#907—216no—
106D3×M16—216no—
107#901—28no—
108#901—28no—
109C2×ΓL(1,𝔽25)—24no—
110C2×D3×C8—24no—
111C3×#103—28no—
112#836—28no—
113#810—24no—
114#809—24no—
115C22×#41—31no—
116C23×Q8C2×D311yescharacteristic
117C2×#137—38no—
118C2×#137—34no—
119C2×#137—64no—
120C2×#137—62no—
121C22×Q16—61no—
122C22×SD16—61no—
123#568—64no—
124#568—64no—
125C2×#131—34no—
126C2×#131—62no—
127C22×M16—31no—
128C22×M16—31no—
129C22×M16—31no—
130C22×M16—62no—
131C22×M16C2×D311yescharacteristic
132C2×#117C2×D311yescharacteristic
133#419—22no—
134#419—22no—
135#419—61no—
136#418—61no—
137#417—34no—
138#416—62no—
139C2×#103C2×D314yescomplemented
140C2×#101—34no—
141C2×#101—64no—
142#343—34no—
143#341—34no—
144C24×C3#4111yescharacteristic
145C23×D3—21no—
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—24no—
150C2×C3×Q8—24no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×Q8—41no—
161C2×C3×Q8—41no—
162C2×C3×Q8—42no—
163C2×C3×Q8C22×C414yes—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—42no—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4112yes—
178C22×C3×C4#4111yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C2×#67—24no—
181C2×#67—22no—
182C2×#67—24no—
183C2×#67—22no—
184C22×Dic3—21no—
185#235—24no—
186#235—24no—
187#235—24no—
188#235—24no—
189#235—24no—
190#235—22no—
191#235—22no—
192#235C22⋊C418yes—
193#235C22×C414yes—
194#235C2×D414yes—
195C2×D12—22no—
196C2×D12—21no—
197C2×D12C22⋊C414yes—
198C2×D12C2×D412yes—
199C2×C4×D3—24no—
200C2×C4×D3—24no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×Dic6—22no—
204C2×Dic6—21no—
205C2×Dic6C22⋊C414yes—
206C2×Dic6C2×D412yes—
207C3×M16—24no—
208C3×M16—24no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8C2×D414yes—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C22⋊C4—41no—
215C3×C22⋊C4—41no—
216C3×C42—41no—
217C3×C42—41no—
218#216—216no—
219#215—216no—
220#208—24no—
221#208—24no—
222#208—22no—
223#208—22no—
224#208—22no—
225#208—22no—
226#208—42no—
227#208C22×C414yes—
228#208C2×D414yes—
229C2×C3⋊C8—24no—
230C2×C3⋊C8—24no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—42no—
235ΓL(1,𝔽25)—216no—
236D3×C8—216no—
237C2×#41—34no—
238C2×#41—34no—
239C2×#41—32no—
240C2×#41—32no—
241C2×#41—62no—
242C2×#41—62no—
243C2×#41—61no—
244C2×#41—61no—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—31no—
254C22×Q8C4×D314yes—
255C22×Q8C22×D312yes—
256C22×D4—31no—
257C22×D4—61no—
258C23×C4—21no—
259C23×C4—61no—
260C23×C4C22×D311yescharacteristic
261#137—316no—
262#137—616no—
263#137—68no—
264C2×Q16—68no—
265C2×Q16—64no—
266C2×SD16—68no—
267C2×SD16—64no—
268#131—68no—
269#131—68no—
270#131—616no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—34no—
274C2×M16—34no—
275C2×M16—32no—
276C2×M16—32no—
277C2×M16—32no—
278C2×M16—616no—
279C2×M16—62no—
280C2×M16—62no—
281C2×M16—61no—
282C2×M16—61no—
283C2×M16—61no—
284C2×M16—64no—
285C2×M16—62no—
286C2×M16—62no—
287C2×M16#6712yes—
288C2×M16#6712yes—
289C22×C8—61no—
290C22×C8—61no—
291C22×C8—64no—
292C22×C8—61no—
293C22×C8C22×D312yes—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298C2×C4⋊C4C22×D312yes—
299C2×C22⋊C4—21no—
300C2×C42—21no—
301#103—28no—
302#103—64no—
303#102—64no—
304#101—68no—
305C23×C3—42no—
306C23×C3C4×D412yes—
307C23×C3#12111yescharacteristic
308C23×C3#12111yescharacteristic
309C23×C3#12312yes—
310C23×C3C2×#4111yescharacteristic
311C22×D3—22no—
312C22×D3—24no—
313C22×D3—82no—
314C3×Q8—24no—
315C3×Q8—24no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—44no—
319C3×Q8—44no—
320C2×C3×C4—24no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—44no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4—82no—
327C2×C3×C4C2×C22⋊C412yes—
328C2×C3×C4C2×C22⋊C412yes—
329C2×C3×C4C4×D412yes—
330C2×C3×C4C4×D412yes—
331C2×C3×C4C4×D412yes—
332C2×C3×C4#12012yes—
333C2×C3×C4#12012yes—
334C2×C3×C4#12112yes—
335C2×C3×C4#12112yes—
336C2×C3×C4#12111yescharacteristic
337C2×C3×C4#12111yescharacteristic
338C2×C3×C4#12312yes—
339C2×C3×C4C23×C411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345#67—24no—
346#67—24no—
347#67—28no—
348C2×Dic3—22no—
349C2×Dic3—24no—
350D12—22no—
351D12—24no—
352D12C2×C22⋊C414yes—
353C4×D3—24no—
354C4×D3—24no—
355C4×D3—28no—
356Dic6—22no—
357Dic6—24no—
358Dic6C2×C22⋊C414yes—
359C3×C8—24no—
360C3⋊C8—24no—
361C3⋊C8—24no—
362C3⋊C8—42no—
363C24—61no—
364C24#23711yescharacteristic
365#41—38no—
366#41—34no—
367#41—34no—
368#41—64no—
369#41—64no—
370#41—64no—
371#41—64no—
372#41—64no—
373#41—62no—
374#41—62no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—34no—
386C2×Q8—32no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—62no—
395C2×Q8—61no—
396C2×Q8C2×C4×D314yes—
397C2×D4—34no—
398C2×D4—32no—
399C2×D4—64no—
400C2×D4—64no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—61no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—61no—
412C22×C4—64no—
413C22×C4—64no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4C2×C4×D312yes—
417C22×C4#23512yes—
418C22×C4D3×D412yes—
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D412yes—
422C22×C4#23711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C23×D311yescharacteristic
426Q16—616no—
427SD16—616no—
428M16—24no—
429M16—24no—
430M16—34no—
431M16—34no—
432M16—616no—
433M16—64no—
434M16—64no—
435M16—62no—
436M16—62no—
437M16—62no—
438M16—62no—
439M16—122no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—616no—
443C2×C8—64no—
444C2×C8—64no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—122no—
449C2×C8C2×#6714yes—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C42—41no—
455C42—41no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3#38612yes—
462C22×C3#38612yes—
463C22×C3C2×C4×D411yescharacteristic
464C22×C3C2×#12111yescharacteristic
465C22×C3C2×#12311yescharacteristic
466C2×D3—24no—
467C2×D3—44no—
468C2×D3—84no—
469C3×C4—44no—
470C3×C4—82no—
471C3×C4#38612yes—
472C3×C4#38612yes—
473C3×C4C22×C22⋊C411yescharacteristic
474C3×C4C2×C4×D411yescharacteristic
475C3×C4C2×#12011yescharacteristic
476C3×C4C2×#12111yescharacteristic
477Dic3—24no—
478C23—42no—
479C23—62no—
480C23—64no—
481C23—242no—
482C23#67⋊C412yes—
483C23#88312yes—
484C23#93811yescharacteristic
485C23#93811yescharacteristic
486C23C2×#23711yescharacteristic
487Q8—24no—
488Q8—24no—
489Q8—34no—
490Q8—44no—
491Q8—44no—
492Q8—42no—
493Q8—42no—
494Q8—64no—
495Q8—62no—
496D4—34no—
497D4—68no—
498D4—64no—
499D4—64no—
500D4—64no—
501D4—62no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—68no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—82no—
514C2×C4D3×C22⋊C412yes—
515C2×C4D3×C22⋊C412yes—
516C2×C4#67⋊C412yes—
517C2×C4#88312yes—
518C2×C4#88412yes—
519C2×C4#88412yes—
520C2×C4C4×#6712yes—
521C2×C4C4×#6712yes—
522C2×C4#93712yes—
523C2×C4#93712yes—
524C2×C4#93811yescharacteristic
525C2×C4#93811yescharacteristic
526C2×C4C22×C4×D311yescharacteristic
527C2×C4C2×#23511yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C2×C4C22×#6711yescharacteristic
532C8—24no—
533C8—64no—
534C8—64no—
535C8—122no—
536C2×C3—22no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—22no—
540C2×C3C2×#38611yescharacteristic
541C2×C3#306912yes—
542D3—44no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—64no—
549C22—124no—
550C22—244no—
551C22#495612yes—
552C22#495612yes—
553C22C2×#67⋊C411yescharacteristic
554C22C2×#88311yescharacteristic
555C22C2×#93811yescharacteristic
556C4—44no—
557C4—64no—
558C4—82no—
559C4#495612yes—
560C4#495612yes—
561C4C2×D3×C22⋊C411yescharacteristic
562C4C2×#88411yescharacteristic
563C4C2×C4×#6711yescharacteristic
564C4C2×#93711yescharacteristic
565C3#2100611yescharacteristic, complemented, Sylow
566C2—22no—
567C2—22no—
568C2—21no—
569C2—21no—
570C2—124no—
571C2C2×#495611yescharacteristic
572C2#8399812yes—
subgroupG/H
centerC22C2×#938
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC22×C3C2×#121
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#419D3
hypocenterC3#21006
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×C3×#419C2semidirect
4. C22×#836C2semidirect
6. #71998C2semidirect
7. #71587C2semidirect
8. #70286C2semidirect
17. C2×#942C4semidirect
30. C2×C3×#103C22semidirect
32. C2×#836C22semidirect
37. #4552C22semidirect
38. #4530C22semidirect
42. C2×#419D3semidirect
139. C2×#103C2×D3semidirect
565. C3#21006semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C2×#57452
C2×C3×#4192
C22×#8362
#822132
#719982
#715872
#702862
#210063