Finite Group Databasebeta

← #10804140

Subgroups — #10804140

Order 768.

1622 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5804C211yesmaximal, characteristic
2#73849C211yesmaximal, characteristic, complemented
3#73819C211yesmaximal, characteristic, complemented
4C3×#1781C211yesmaximal, characteristic, complemented
5#72130C211yesmaximal, characteristic, complemented
6#70036C214yesmaximal, complemented
7#70029C214yesmaximal, complemented
8#69495C211yesmaximal, characteristic, complemented
9#69495C211yesmaximal, characteristic, complemented
10#21749—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C23×#67C2211yescharacteristic
13C2×#953C412yescomplemented
14C22×#217C2211yescharacteristic
15#5804—24no—
16#5804—24no—
17C2×#939C414yescomplemented
18C2×#937C2212yes—
19C2×#935C2212yes—
20C2×#929C412yescomplemented
21C2×D3×C22⋊C4C2212yescomplemented
22C2×#877C2212yescomplemented
23C2×C3×#99C2214yes—
24C3×#379C2211yescharacteristic
25C3×#379C2211yescharacteristic
26#5247C2211yescharacteristic
27#5247C2211yescharacteristic
28C2×#834C2214yes—
29#4957—24no—
30#4956—24no—
31#4955C2212yescomplemented
32#4943C2212yescomplemented
33#4527C2214yescomplemented
34#4527C2214yescomplemented
35C2×#534—31no—
36#2309—31no—
37#2308—31no—
38#1899—34no—
39#1898—34no—
40#1819—31no—
41#1819—31no—
42#1781—31no—
43#1781D311yescharacteristic, complemented
44C25×C3C2×C411yescharacteristic, complemented
45C25×C3C2311yescharacteristic
46C24×D3C2311yescharacteristic
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—44no—
50C22×C3×D4C2×C412yescomplemented
51C22×C3×D4C2312yes—
52C23×C3×C4C2×C411yescharacteristic, complemented
53C22×#67—44no—
54C22×#67C2×C414yes—
55C22×#67C2×C412yes—
56C22×#67C2×C412yes—
57C22×#67D414yes—
58C22×#67D414yes—
59C23×Dic3C2×C412yescomplemented
60C23×Dic3C2311yescharacteristic
61C22×C4×D3—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4C2312yes—
65C2×C3×C22⋊C4C2312yes—
66#953—28no—
67C2×#217—22no—
68C2×#217—22no—
69C2×#217—22no—
70C2×#217—22no—
71C2×#217C2×C412yescomplemented
72C2×#217C2×C414yescomplemented
73C2×#217C2312yes—
74C2×#217C2312yes—
75C2×#217C2312yes—
76#939—28no—
77#939—28no—
78#937—24no—
79#937—24no—
80#935—24no—
81#935—24no—
82#929—28no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4C2×C414yes—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4C2×C414yescomplemented
88C2×C4×Dic3—22no—
89D3×C22⋊C4—44no—
90#877—44no—
91C3×#99—44no—
92#834—44no—
93C3⋊#95—24no—
94C23×D4—31no—
95C23×D4C2×D311yescharacteristic
96#534—64no—
97#534—64no—
98C2×#123—32no—
99C2×#123—32no—
100C2×#121—34no—
101C2×#120—32no—
102C2×#120—32no—
103C2×#117—32no—
104C22×C22⋊C4—31no—
105C22×C22⋊C4—32no—
106C2×#99—34no—
107C2×#99C2×D314yes—
108#390—64no—
109#386—64no—
110#381—32no—
111#379—31no—
112#379—31no—
113#379—32no—
114#379C2×D311yescharacteristic
115#379C2×D311yescharacteristic
116#342—34no—
117#342—34no—
118C24×C3—24no—
119C24×C3—22no—
120C24×C3—42no—
121C24×C3—82no—
122C24×C3—82no—
123C24×C3C22×C412yes—
124C24×C3C22×C411yescharacteristic
125C24×C3C22×C411yescharacteristic
126C24×C3C2×D412yes—
127C24×C3C2×D412yes—
128C24×C3C2411yescharacteristic
129C23×D3—24no—
130C23×D3—82no—
131C23×D3C22×C412yes—
132C23×D3C2×D412yes—
133C23×D3C2×D412yes—
134C2×C3×D4—24no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—48no—
142C2×C3×D4—48no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—42no—
147C22×C3×C4C22×C412yes—
148C2×#67—28no—
149C2×#67—44no—
150C2×#67—44no—
151C2×#67—44no—
152C2×#67—44no—
153C2×#67—44no—
154C2×#67—44no—
155C2×#67—84no—
156C2×#67C22⋊C418yes—
157C2×#67C22⋊C418yes—
158C22×Dic3—24no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—82no—
164C22×Dic3C22×C412yes—
165C22×Dic3C22×C414yes—
166C22×Dic3C2×D412yes—
167C22×Dic3C2×D412yes—
168C2×C4×D3—44no—
169C2×C4×D3—84no—
170C3×C22⋊C4—24no—
171C3×C22⋊C4—24no—
172C3×C22⋊C4—82no—
173C3×C22⋊C4—82no—
174#217—24no—
175#217—22no—
176#217—22no—
177#217—28no—
178#217—24no—
179#217—24no—
180#217—82no—
181#217—82no—
182(C2×D3)⋊C4—28no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—84no—
185Dic3⋊C4—28no—
186Dic3⋊C4—44no—
187C4×Dic3—44no—
188C25—31no—
189C25C4×D311yescharacteristic, complemented
190C25C22×D311yescharacteristic
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—32no—
194C22×D4—32no—
195C22×D4—34no—
196C22×D4—34no—
197C22×D4—34no—
198C22×D4—44no—
199C22×D4—124no—
200C22×D4C4×D312yescomplemented
201C22×D4C22×D312yes—
202C23×C4—31no—
203C23×C4—32no—
204C23×C4—62no—
205C23×C4C4×D311yescharacteristic, complemented
206#123—68no—
207#123—64no—
208#123—64no—
209#121—68no—
210#121—68no—
211#120—64no—
212#120—64no—
213#120—68no—
214#117—124no—
215C2×C4⋊C4—34no—
216C2×C4⋊C4—62no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—32no—
220C2×C22⋊C4—34no—
221C2×C22⋊C4—32no—
222C2×C22⋊C4—34no—
223C2×C22⋊C4—32no—
224C2×C22⋊C4—32no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—64no—
229C2×C22⋊C4—62no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—124no—
232C2×C22⋊C4C22×D312yes—
233C2×C22⋊C4C22×D312yes—
234C2×C42—62no—
235#99—44no—
236#99—124no—
237#95—64no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—44no—
253C23×C3—44no—
254C23×C3—82no—
255C23×C3—82no—
256C23×C3—88no—
257C23×C3—88no—
258C23×C3C2×C22⋊C412yes—
259C23×C3C2×C22⋊C412yes—
260C23×C3C23×C411yescharacteristic
261C23×C3C22×D411yescharacteristic
262C23×C3C22×D411yescharacteristic
263C23×C3H(5,𝔽2)14yes—
264C22×D3—22no—
265C22×D3—22no—
266C22×D3—24no—
267C22×D3—24no—
268C22×D3—44no—
269C22×D3—44no—
270C22×D3—44no—
271C22×D3—88no—
272C22×D3C2×C22⋊C414yes—
273C3×D4—48no—
274C3×D4—44no—
275C3×D4—44no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284#67—48no—
285#67—48no—
286#67—88no—
287C2×Dic3—24no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—24no—
291C2×Dic3—24no—
292C2×Dic3—44no—
293C2×Dic3—44no—
294C2×Dic3—44no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—82no—
298C2×Dic3—82no—
299C2×Dic3C2×C22⋊C414yes—
300C4×D3—88no—
301C24—24no—
302C24—22no—
303C24—32no—
304C24—32no—
305C24—32no—
306C24—42no—
307C24—64no—
308C24—82no—
309C24—82no—
310C24—242no—
311C24C2×C4×D312yes—
312C24C2×C4×D311yescharacteristic
313C24C2×C4×D311yescharacteristic
314C24D3×D412yes—
315C24D3×D412yes—
316C24C23×D311yescharacteristic
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—38no—
323C2×D4—38no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—68no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4—124no—
332C2×D4—124no—
333C2×D4—124no—
334C2×D4—124no—
335C2×D4—244no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—32no—
340C22×C4—32no—
341C22×C4—32no—
342C22×C4—34no—
343C22×C4—42no—
344C22×C4—64no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—124no—
350C22×C4—244no—
351C22×C4—242no—
352C22×C4C2×C4×D312yes—
353C4⋊C4—68no—
354C4⋊C4—124no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—68no—
358C22⋊C4—68no—
359C22⋊C4—64no—
360C22⋊C4—62no—
361C22⋊C4—62no—
362C22⋊C4—64no—
363C22⋊C4—64no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—124no—
367C22⋊C4—244no—
368C22⋊C4—242no—
369C22⋊C4—242no—
370C42—124no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—44no—
381C22×C3—44no—
382C22×C3—44no—
383C22×C3—42no—
384C22×C3—44no—
385C22×C3—84no—
386C22×C3—84no—
387C22×C3—82no—
388C22×C3—82no—
389C22×C3—88no—
390C22×C3C22×C22⋊C411yescharacteristic
391C22×C3#51812yes—
392C22×C3#53412yes—
393C22×C3#53412yes—
394C2×D3—24no—
395C2×D3—28no—
396C2×D3—44no—
397C2×D3—44no—
398C2×D3—84no—
399C2×D3—84no—
400C3×C4—42no—
401C3×C4—82no—
402C3×C4—82no—
403Dic3—24no—
404Dic3—44no—
405Dic3—82no—
406Dic3—82no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—34no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—62no—
425C23—62no—
426C23—64no—
427C23—64no—
428C23—82no—
429C23—82no—
430C23—88no—
431C23—88no—
432C23—124no—
433C23—124no—
434C23—124no—
435C23—248no—
436C23D3×C22⋊C412yes—
437C23D3×C22⋊C412yes—
438C23C22×C4×D311yescharacteristic
439C23C2×D3×D411yescharacteristic
440C23C2×D3×D411yescharacteristic
441C23#100414yes—
442D4—48no—
443D4—44no—
444D4—44no—
445D4—128no—
446D4—128no—
447D4—248no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—34no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—64no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—124no—
463C2×C4—124no—
464C2×C4—124no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—248no—
468C2×C4—242no—
469C2×C4—242no—
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3—82no—
476C2×C3#230811yescharacteristic
477C2×C3#288312yes—
478D3—44no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—68no—
494C22—64no—
495C22—84no—
496C22—84no—
497C22—82no—
498C22—82no—
499C22—88no—
500C22—124no—
501C22—124no—
502C22—244no—
503C22—244no—
504C22C2×D3×C22⋊C411yescharacteristic
505C22#548612yes—
506C22#558912yes—
507C22#558912yes—
508C4—42no—
509C4—64no—
510C4—82no—
511C4—82no—
512C4—124no—
513C4—242no—
514C4—242no—
515C3#2174911yescharacteristic, complemented, Sylow
516C2—22no—
517C2—22no—
518C2—22no—
519C2—42no—
520C2—42no—
521C2—82no—
522C2—124no—
523C2#7384911yescharacteristic
524C2#7868812yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1781C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5804C2
cosocleC24×C3C24
hypercenter#1781D3
hypocenterC3#21749
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #73849C2semidirect
3. #73819C2semidirect
4. C3×#1781C2semidirect
5. #72130C2semidirect
6. #70036C2semidirect
7. #70029C2semidirect
8. #69495C2semidirect
9. #69495C2semidirect
13. C2×#953C4semidirect
17. C2×#939C4semidirect
20. C2×#929C4semidirect
21. C2×D3×C22⋊C4C22semidirect
22. C2×#877C22semidirect
31. #4955C22semidirect
32. #4943C22semidirect
33. #4527C22semidirect
34. #4527C22semidirect
43. #1781D3semidirect
44. C25×C3C2×C4semidirect
50. C22×C3×D4C2×C4semidirect
52. C23×C3×C4C2×C4semidirect
59. C23×Dic3C2×C4semidirect
71. C2×#217C2×C4semidirect
72. C2×#217C2×C4semidirect
87. C2×Dic3⋊C4C2×C4semidirect
189. C25C4×D3semidirect
200. C22×D4C4×D3semidirect
205. C23×C4C4×D3semidirect
515. C3#21749semidirect

Maximal subgroups

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

subgroupindex
C2×#58042
#738492
#738192
C3×#17812
#721302
#700362
#700292
#694952
#217493