Finite Group Databasebeta

← #10796347

Subgroups — #10796347

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5791C211yesmaximal, characteristic
2C22×#907C211yesmaximal, characteristic, complemented
3C2×C3×#407C211yesmaximal, characteristic, complemented
4C22×#830C211yesmaximal, characteristic, complemented
5C2×#5129C211yesmaximal, characteristic, complemented
6#71980C211yesmaximal, characteristic, complemented
7#71980C211yesmaximal, characteristic, complemented
8#69958C214yesmaximal, complemented
9#69957C214yesmaximal, complemented
10#20899—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×#953C412yes—
16#5791—24no—
17#5791—24no—
18C2×#930C412yes—
19C2×#929C412yes—
20C2×C4×#67C2214yes—
21C2×#926C2211yescharacteristic
22C2×#924C412yes—
23C23×C3⋊C8C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C2×#907—24no—
26C2×#907—24no—
27C2×#907—24no—
28C22×ΓL(1,𝔽25)C2212yescomplemented
29C22×D3×C8C2212yescomplemented
30C3×#407—24no—
31C2×C3×C22⋊C8C2214yes—
32C2×#830—24no—
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5129—24no—
36C2×(C2×D3)⋊C8C2214yescomplemented
37#4550C2212yescomplemented
38#4550C2214yescomplemented
39#4550C2212yescomplemented
40C22×#131—31no—
41C2×#525—31no—
42C2×#408—31no—
43C2×#407D311yescharacteristic, complemented
44C22×C22⋊C8—31no—
45#1870—34no—
46#1869—34no—
47#1729—31no—
48#1729—31no—
49C25×C3C2×C411yescharacteristic
50C23×C3×C4—22no—
51C23×C3×C4—22no—
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4C2×C412yes—
57C23×C3×C4C2×C411yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C414yes—
61C22×#67C2×C412yes—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235D414yes—
67C2×#235D414yes—
68C22×D12C2×C411yescharacteristic
69C22×C4×D3C2312yes—
70C22×Dic6C2×C411yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16—22no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16D414yescomplemented
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79#953—28no—
80C2×#217C2×C414yes—
81C2×#217C2×C412yes—
82#930—28no—
83#929—28no—
84C4×#67—28no—
85C4×#67—28no—
86C2×(C2×D3)⋊C4C2×C414yes—
87#926—22no—
88#926—22no—
89C2×C4⋊Dic3C2×C412yes—
90#924—28no—
91C2×Dic3⋊C4C2×C414yes—
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8—22no—
94C22×C3⋊C8—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8—22no—
98C22×C3⋊C8D414yescomplemented
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#907—232no—
103C2×ΓL(1,𝔽25)—24no—
104C2×ΓL(1,𝔽25)—24no—
105C2×ΓL(1,𝔽25)—24no—
106C2×D3×C8—24no—
107C2×D3×C8—24no—
108C2×D3×C8—24no—
109C3×C22⋊C8—28no—
110#830—24no—
111#830—24no—
112(C2×D3)⋊C8—28no—
113C22×#41—31no—
114C24×C4C2×D311yescharacteristic
115C2×#131—64no—
116C2×#131—64no—
117C2×#131—64no—
118C22×M16—32no—
119C22×M16C2×D311yescharacteristic
120C22×M16C2×D311yescharacteristic
121C23×C8—31no—
122C23×C8—31no—
123C23×C8—32no—
124#525—64no—
125#525—64no—
126C2×#123—32no—
127C2×C22⋊Q8—32no—
128C2×#121—32no—
129C2×#120—32no—
130C2×C4×D4—34no—
131C2×#117—31no—
132#408—64no—
133#407—24no—
134C2×C22⋊C8—32no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8—34no—
137C2×C22⋊C8—64no—
138C2×C22⋊C8C2×D314yes—
139#336—32no—
140#336—34no—
141#336—32no—
142C24×C3—22no—
143C24×C3—24no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—42no—
147C24×C3C22×C412yes—
148C24×C3C22×C411yescharacteristic
149C23×D3C22×C412yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—44no—
163C22×C3×C4—44no—
164C22×C3×C4—44no—
165C22×C3×C4—44no—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—28no—
178C2×#67—28no—
179C2×#67—44no—
180C2×#67C22⋊C418yes—
181C22×Dic3C22×C414yes—
182C22×Dic3C22×C412yes—
183#235—28no—
184#235—28no—
185#235—28no—
186#235—28no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12C22⋊C414yes—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3C2×D414yes—
195C2×C4×D3C2×D414yes—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6C22⋊C414yes—
199C3×M16—28no—
200C3×M16—44no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8C2×D414yes—
207#217—24no—
208#217—28no—
209(C2×D3)⋊C4—28no—
210C4⋊Dic3—24no—
211Dic3⋊C4—28no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214C2×C3⋊C8—28no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—44no—
223C2×C3⋊C8C2×D412yes—
224C2×C3⋊C8C2×D412yes—
225ΓL(1,𝔽25)—216no—
226D3×C8—216no—
227C25C4×D311yescharacteristic
228C2×#41—34no—
229C2×#41—34no—
230C2×#41—62no—
231C2×#41—62no—
232C2×#41—62no—
233C2×#41—62no—
234C22×Q8—31no—
235C22×D4—32no—
236C22×D4—34no—
237C22×D4—31no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—32no—
245C23×C4C4×D312yes—
246C23×C4C4×D311yescharacteristic
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249#131—632no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16—22no—
255C2×M16—64no—
256C2×M16—64no—
257C2×M16—64no—
258C2×M16#6714yescomplemented
259C22×C8—34no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8—32no—
263C22×C8—64no—
264C22×C8—64no—
265C22×C8—62no—
266C22×C8—62no—
267C22×C8—64no—
268C22×C8—62no—
269C22×C8—62no—
270C22×C8—62no—
271C22×C8C22×D312yes—
272C22×C8C22×D312yes—
273#123—68no—
274C22⋊Q8—68no—
275#121—68no—
276#120—68no—
277C4×D4—68no—
278C4×D4—68no—
279#117—62no—
280#117—62no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—34no—
283C2×C22⋊C4—32no—
284C2×C22⋊C4—34no—
285C2×C22⋊C4—34no—
286C2×C42—32no—
287C22⋊C8—28no—
288C22⋊C8—68no—
289C22⋊C8—64no—
290C22⋊C8—64no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—28no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—48no—
301C23×C3—48no—
302C23×C3—44no—
303C23×C3—44no—
304C23×C3C2×C22⋊C412yes—
305C23×C3C4×D412yes—
306C23×C3C4×D412yes—
307C23×C3#13114yescomplemented
308C23×C3C23×C411yescharacteristic
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3C2×C22⋊C414yes—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—48no—
323C2×C3×C4C2×C22⋊C412yes—
324C2×C3×C4C4×D412yes—
325C2×C3×C4C4×D412yes—
326C2×C3×C4#12014yes—
327C2×C3×C4#12114yes—
328C2×C3×C4#12114yes—
329C2×C3×C4#12314yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336#67—216no—
337#67—416no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3C2×C22⋊C414yes—
344D12—28no—
345C4×D3—24no—
346C4×D3—24no—
347C4×D3—24no—
348C4×D3—24no—
349Dic6—28no—
350C3×C8—24no—
351C3×C8—42no—
352C3⋊C8—22no—
353C3⋊C8—22no—
354C3⋊C8—42no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—32no—
360C24—42no—
361C24C2×C4×D312yes—
362C24C2×C4×D311yescharacteristic
363#41—68no—
364#41—68no—
365#41—68no—
366#41—68no—
367C2×Q8—34no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×D4—38no—
371C2×D4—34no—
372C2×D4—64no—
373C2×D4—64no—
374C2×D4—68no—
375C2×D4—68no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—124no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—32no—
392C22×C4—34no—
393C22×C4—34no—
394C22×C4—34no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4C2×C4×D312yes—
404C22×C4C2×C4×D311yescharacteristic
405C22×C4#23512yes—
406C22×C4D3×D412yes—
407C22×C4D3×D412yes—
408C22×C4D3×D412yes—
409C22×C4#23712yes—
410C22×C4C2×#6712yes—
411C22×C4C23×D311yescharacteristic
412M16—28no—
413M16—44no—
414M16—616no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—32no—
421C2×C8—32no—
422C2×C8—616no—
423C2×C8—68no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—62no—
429C2×C8—62no—
430C2×C8—62no—
431C2×C8—124no—
432C2×C8C2×#6714yes—
433C4⋊C4—68no—
434C4⋊C4—64no—
435C22⋊C4—68no—
436C22⋊C4—68no—
437C22⋊C4—64no—
438C42—62no—
439C42—62no—
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—416no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3#38614yes—
453C22×C3C22×C22⋊C411yescharacteristic
454C22×C3C2×C4×D411yescharacteristic
455C22×C3C2×C4×D411yescharacteristic
456C22×C3C2×#13112yes—
457C22×C3#56812yes—
458C2×D3—28no—
459C2×D3—24no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—44no—
463C3×C4—42no—
464C3×C4#38614yes—
465C3×C4C2×#12011yescharacteristic
466C3×C4C2×#12111yescharacteristic
467C3×C4C2×#12111yescharacteristic
468C3×C4C2×#12311yescharacteristic
469Dic3—24no—
470Dic3—44no—
471C23—22no—
472C23—22no—
473C23—28no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—34no—
481C23—48no—
482C23—48no—
483C23—44no—
484C23—44no—
485C23—62no—
486C23—64no—
487C23—64no—
488C23—62no—
489C23D3×C22⋊C412yes—
490C23#67⋊C412yes—
491C23#90714yescomplemented
492C23C4×#6712yes—
493C23C22×C4×D311yescharacteristic
494Q8—68no—
495D4—616no—
496D4—68no—
497D4—1216no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—34no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—64no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4D3×C22⋊C412yes—
518C2×C4#67⋊C412yes—
519C2×C4#88314yes—
520C2×C4#88414yes—
521C2×C4C4×#6712yes—
522C2×C4#93714yes—
523C2×C4#93814yes—
524C2×C4C2×#23511yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4C2×D3×D411yescharacteristic
528C2×C4C2×#23711yescharacteristic
529C2×C4C22×#6711yescharacteristic
530C8—24no—
531C8—42no—
532C8—62no—
533C8—62no—
534C8—122no—
535C2×C3—24no—
536C2×C3—44no—
537C2×C3C2×#38611yescharacteristic
538C2×C3#288012yes—
539C2×C3#292911yescharacteristic
540C2×C3#293011yescharacteristic
541C2×C3#293312yes—
542D3—44no—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—416no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—68no—
556C22—64no—
557C22—122no—
558C22—122no—
559C22—124no—
560C22#495614yes—
561C22C2×D3×C22⋊C411yescharacteristic
562C22C2×#67⋊C411yescharacteristic
563C22C2×#90712yes—
564C22#574512yes—
565C22C2×C4×#6711yescharacteristic
566C4—42no—
567C4—64no—
568C4—124no—
569C4#495614yes—
570C4C2×#88311yescharacteristic
571C4C2×#88411yescharacteristic
572C4C2×#93711yescharacteristic
573C4C2×#93811yescharacteristic
574C3#2089911yescharacteristic, complemented, Sylow
575C2—24no—
576C2—44no—
577C2—124no—
578C2#7862611yescharacteristic
579C2#7862711yescharacteristic
580C2#7863112yes—
581C2C2×#495611yescharacteristic
582C2#8375712yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#407C2
prime radicalC2×#5791C2
cosocleC22×C3×C4C24
hypercenterC2×#407D3
hypocenterC3#20899
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C22×#907C2semidirect
3. C2×C3×#407C2semidirect
4. C22×#830C2semidirect
5. C2×#5129C2semidirect
6. #71980C2semidirect
7. #71980C2semidirect
8. #69958C2semidirect
9. #69957C2semidirect
28. C22×ΓL(1,𝔽25)C22semidirect
29. C22×D3×C8C22semidirect
36. C2×(C2×D3)⋊C8C22semidirect
37. #4550C22semidirect
38. #4550C22semidirect
39. #4550C22semidirect
43. C2×#407D3semidirect
76. C2×C3×M16D4semidirect
98. C22×C3⋊C8D4semidirect
258. C2×M16#67semidirect
307. C23×C3#131semidirect
491. C23#907semidirect
574. C3#20899semidirect

Maximal subgroups

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

subgroupindex
C2×#57912
C22×#9072
C2×C3×#4072
C22×#8302
C2×#51292
#719802
#699582
#699572
#208993