Finite Group Databasebeta

← #10822737

Subgroups — #10822737

Order 768.

1610 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2C3×#2311C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5001C211yesmaximal, characteristic, complemented
5C2×#4998C211yesmaximal, characteristic, complemented
6C22×#809C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#69524C214yesmaximal, complemented
9#69523C214yesmaximal, complemented
10#23486—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15#5830—24no—
16#5830—24no—
17C2×#947—21no—
18C2×#940—21no—
19C2×#937—22no—
20C2×#930C2214yes—
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C3×#391—21no—
25C3×#390—21no—
26C3×#386—22no—
27C3×#380C2214yescomplemented
28#5234—24no—
29C2×#830C2212yescomplemented
30C2×#830C2212yescomplemented
31#5001—24no—
32#4998—24no—
33C2×#809—24no—
34C2×#809C2214yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#808C2212yescomplemented
38C2×#807C2212yescomplemented
39#4529C2214yescomplemented
40C2×#535—31no—
41C2×#427—31no—
42C2×#418—31no—
43C22×#102—31no—
44C2×#408—31no—
45#2311D311yescharacteristic, complemented
46#2034—34no—
47#2033—34no—
48#1736—31no—
49C24×D3—21no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C22×C3×Q8D411yescharacteristic
57C22×C3×D4D412yes—
58C22×C3×D4D411yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C22×#67—24no—
63C22×D12—22no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—21no—
68C22×D12—22no—
69C22×D12D414yes—
70C22×D12C2312yes—
71C3×#117—22no—
72C3×#117—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C42—21no—
81C2×#217—21no—
82#947—28no—
83#940—28no—
84#937—216no—
85#930—28no—
86#930—28no—
87C2×(C2×D3)⋊C4—24no—
88#926—22no—
89#926D414yes—
90C2×C4⋊Dic3C2312yes—
91C2×C4×Dic3—21no—
92C2×#208—22no—
93C2×#208—22no—
94C2×#208—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8D414yescomplemented
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C3×#95—24no—
102#830—24no—
103#830—24no—
104#809—28no—
105#809—24no—
106#809—24no—
107#808—24no—
108#807—24no—
109C22×#41C2×D311yescharacteristic
110C23×D4—31no—
111C22×M16—31no—
112C23×C8—31no—
113#535—64no—
114#535—64no—
115C2×#127—61no—
116C2×#124—61no—
117C2×#121—34no—
118C2×#120—62no—
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4C2×D311yescharacteristic
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#418—64no—
126C2×#102—32no—
127C2×#102—32no—
128C2×#102—34no—
129C2×#102—64no—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—21no—
134#390—21no—
135#386—22no—
136#380C2×D314yescomplemented
137#340—34no—
138C24×C3C2×D412yes—
139C24×C3C2×D411yescharacteristic
140C23×D3—22no—
141C23×D3—24no—
142C23×D3—42no—
143C23×D3—42no—
144C23×D3—82no—
145C23×D3—82no—
146C3×#41—44no—
147C3×#41—44no—
148C3×#41—44no—
149C3×#41—44no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—42no—
168C22×C3×C4—82no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4112yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—216no—
179C2×#67—84no—
180C2×#67—84no—
181C22×Dic3—22no—
182C2×D12—28no—
183C2×D12—28no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—24no—
189C2×D12—22no—
190C2×D12—44no—
191C2×D12—42no—
192C2×D12—42no—
193C2×D12—44no—
194C2×D12C2×D414yes—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C22⋊C4—84no—
202C3×C42—42no—
203#217—24no—
204(C2×D3)⋊C4—216no—
205C4⋊Dic3—22no—
206C4⋊Dic3C2×D414yes—
207C4×Dic3—24no—
208#208—44no—
209C2×C3⋊C8—28no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8C2×D412yes—
218C2×C3⋊C8C2×D412yes—
219C25—61no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C22×Q8#6711yescharacteristic
227C22×D4—34no—
228C22×D4—32no—
229C22×D4—64no—
230C22×D4—62no—
231C22×D4—62no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4—62no—
235C22×D4—61no—
236C22×D4#6712yes—
237C22×D4#6711yescharacteristic
238C23×C4—21no—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C2×M16—62no—
242C2×M16—62no—
243C2×M16—62no—
244C22×C8—34no—
245C22×C8—31no—
246C22×C8—31no—
247C22×C8—32no—
248C22×C8—62no—
249C22×C8—62no—
250#127—68no—
251#124—68no—
252#121—68no—
253#121—68no—
254#120—616no—
255#117—22no—
256#117—22no—
257#117—34no—
258#117—62no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4C22×D312yes—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—64no—
269C2×C42—21no—
270C2×C42—61no—
271#107—64no—
272#106—64no—
273#102—64no—
274#102—64no—
275#102—68no—
276C22⋊C8—64no—
277C22⋊C8—64no—
278#95—24no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3#12012yes—
287C23×C3#12112yes—
288C23×C3#12312yes—
289C23×C3C22×D411yescharacteristic
290C22×D3—24no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—44no—
294C22×D3—44no—
295C22×D3—48no—
296C22×D3—88no—
297C22×D3—88no—
298C22×D3—84no—
299C22×D3—84no—
300C3×Q8—44no—
301C3×D4—48no—
302C3×D4—44no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—44no—
314C2×C3×C4—42no—
315C2×C3×C4—84no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12114yes—
318C2×C3×C4#12114yes—
319C2×C3×C4#12112yes—
320C2×C3×C4#12314yes—
321C2×C3×C4#12312yes—
322C2×C3×C4#12414yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329#67—816no—
330C2×Dic3—24no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333D12—24no—
334D12—24no—
335D12—42no—
336D12—42no—
337D12—48no—
338C3⋊C8—22no—
339C3⋊C8—22no—
340C3⋊C8—42no—
341C24—62no—
342C24—64no—
343C24—122no—
344C24—122no—
345C24—242no—
346C24—242no—
347C24C2×#6712yes—
348C24C2×#6711yescharacteristic
349#41—44no—
350#41—44no—
351#41—44no—
352#41—44no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—34no—
363C2×D4—616no—
364C2×D4—68no—
365C2×D4—68no—
366C2×D4—64no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—122no—
375C2×D4—122no—
376C2×D4—244no—
377C2×D4—244no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—62no—
388C22×C4—82no—
389C22×C4#23512yes—
390C22×C4#23512yes—
391C22×C4D3×D412yes—
392C22×C4D3×D412yes—
393C22×C4D3×D412yes—
394C22×C4#23712yes—
395C22×C4C2×#6712yes—
396C22×C4C2×#6711yescharacteristic
397C22×C4C23×D311yescharacteristic
398M16—124no—
399C2×C8—32no—
400C2×C8—32no—
401C2×C8—68no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—34no—
414C4⋊C4—42no—
415C4⋊C4—62no—
416C22⋊C4—42no—
417C22⋊C4—616no—
418C22⋊C4—64no—
419C22⋊C4—84no—
420C42—42no—
421C42—64no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—44no—
427C22×C3—42no—
428C22×C3—44no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3#39414yes—
432C22×C3C2×#12011yescharacteristic
433C22×C3C2×#12111yescharacteristic
434C22×C3C2×#12311yescharacteristic
435C22×C3#57612yes—
436C22×C3#57712yes—
437C2×D3—44no—
438C2×D3—48no—
439C2×D3—816no—
440C2×D3—84no—
441C2×D3—82no—
442C2×D3—82no—
443C3×C4—42no—
444C3×C4—82no—
445C3×C4—82no—
446C3×C4#39414yes—
447C3×C4C2×#12111yescharacteristic
448C3×C4C2×#12111yescharacteristic
449C3×C4C2×#12311yescharacteristic
450C3×C4C2×#12411yescharacteristic
451Dic3—44no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—64no—
460C23—128no—
461C23—122no—
462C23—122no—
463C23—124no—
464C23—124no—
465C23—248no—
466C23—248no—
467C23—244no—
468C23—244no—
469C23#92912yes—
470C23#93712yes—
471C23#93912yes—
472C23C22×#6711yescharacteristic
473Q8—44no—
474D4—48no—
475D4—44no—
476D4—64no—
477D4—64no—
478D4—128no—
479D4—122no—
480D4—122no—
481D4—2416no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—64no—
495C2×C4—84no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4#88314yes—
499C2×C4#88414yes—
500C2×C4#88414yes—
501C2×C4#88514yes—
502C2×C4#92912yes—
503C2×C4#93712yes—
504C2×C4#93912yes—
505C2×C4C2×#23511yescharacteristic
506C2×C4C2×#23511yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×D3×D411yescharacteristic
510C2×C4C2×#23711yescharacteristic
511C8—62no—
512C8—62no—
513C8—122no—
514C2×C3—24no—
515C2×C3—82no—
516C2×C3C2×#39411yescharacteristic
517C2×C3#301212yes—
518C2×C3#305711yescharacteristic
519C2×C3#305811yescharacteristic
520C2×C3#309612yes—
521D3—84no—
522C22—24no—
523C22—22no—
524C22—22no—
525C22—24no—
526C22—44no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—124no—
532C22—128no—
533C22—2416no—
534C22—244no—
535C22—242no—
536C22—242no—
537C22#495914yes—
538C22C2×#92911yescharacteristic
539C22C2×#93711yescharacteristic
540C22C2×#93911yescharacteristic
541C22#583512yes—
542C22#583712yes—
543C4—42no—
544C4—82no—
545C4—82no—
546C4—124no—
547C4#495914yes—
548C4C2×#88311yescharacteristic
549C4C2×#88411yescharacteristic
550C4C2×#88411yescharacteristic
551C4C2×#88511yescharacteristic
552C3#2348611yescharacteristic, complemented, Sylow
553C2—24no—
554C2—82no—
555C2—244no—
556C2C2×#495911yescharacteristic
557C2#8256612yes—
558C2#8337811yescharacteristic
559C2#8339811yescharacteristic
560C2#8342412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC23×C3C22×D4
prime radicalC2×#5830C2
hypercenter#2311D3
hypocenterC3#23486
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C3×#2311C2semidirect
3. C2×#5234C2semidirect
4. C2×#5001C2semidirect
5. C2×#4998C2semidirect
6. C22×#809C2semidirect
7. #71558C2semidirect
8. #69524C2semidirect
9. #69523C2semidirect
27. C3×#380C22semidirect
29. C2×#830C22semidirect
30. C2×#830C22semidirect
37. C2×#808C22semidirect
38. C2×#807C22semidirect
39. #4529C22semidirect
45. #2311D3semidirect
97. C22×C3⋊C8D4semidirect
136. #380C2×D3semidirect
552. C3#23486semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C3×#23112
C2×#52342
C2×#50012
C2×#49982
C22×#8092
#715582
#695242
#695232
#234863