Finite Group Databasebeta

← #10850475

Subgroups — #10850475

Order 768.

1618 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85786C211yesmaximal, characteristic
2C3×#2486C211yesmaximal, characteristic, complemented
3C2×#5053C214yesmaximal, complemented
4#82177C212yesmaximal, complemented
5C4×#834C211yesmaximal, characteristic, complemented
6#69539C214yesmaximal, complemented
7#69528C212yesmaximal, complemented
8#25772—31nomaximal, Sylow
9C2×#1004C2212yes—
10C2×C3×#123C2212yes—
11C2×C3×C4×D4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic, complemented
13C2×#930—21no—
14C2×#930C2212yes—
15C2×#929C2218yescomplemented
16#5555—22no—
17#5554—24no—
18D4×D12—22no—
19#5510—21no—
20#5493—21no—
21#5493C2212yes—
22#5477—21no—
23C3×#398—21no—
24C3×#397—21no—
25C3×#389—21no—
26C3×#381C2212yescomplemented
27C3×#380—21no—
28C3×#380C2211yescharacteristic, complemented
29C2×#834C2212yes—
30C4×#217C2212yescomplemented
31#5053—216no—
32#4993—24no—
33#4988—22no—
34#4975—22no—
35C2×#806C2218yescomplemented
36#4526C2212yescomplemented
37#3496—31no—
38C2×#458—34no—
39#2501—32no—
40#2486D311yescharacteristic, complemented
41#2022—34no—
42#1964—32no—
43C4×#99—31no—
44C22×C3×D4C2311yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C22×#67—22no—
49C22×#67C2314yes—
50#1004—28no—
51#1004—44no—
52C2×#237—24no—
53C2×D3×D4—24no—
54C2×#235—22no—
55C22×D12—21no—
56C22×D12D412yes—
57C3×#123—28no—
58C3×C4×D4—22no—
59C3×C4×D4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4D411yescharacteristic
69C2×C3×C22⋊C4C2312yes—
70C2×C3×C22⋊C4C2311yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×#217C2314yes—
73#930—22no—
74#930—22no—
75#930—22no—
76#930—22no—
77#930—22no—
78#930—22no—
79#930—22no—
80#930—22no—
81#929—216no—
82#929—48no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—28no—
85C2×(C2×D3)⋊C4D414yes—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3D412yes—
88C2×Dic3⋊C4—28no—
89C2×C4×Dic3—22no—
90#897—24no—
91#895—24no—
92#886—24no—
93#886—24no—
94#882—24no—
95#882—24no—
96#875—22no—
97#874—21no—
98C4×D12—22no—
99C4×D12—22no—
100#869—21no—
101C3×#95—22no—
102C3×#95—21no—
103C3×#95—22no—
104#834D418yescomplemented
105C3⋊#95—22no—
106#806—216no—
107C2×H(5,𝔽2)—32no—
108#547—62no—
109#546—64no—
110D42—62no—
111#537—61no—
112#534—32no—
113#534—61no—
114#532—61no—
115C2×#123—38no—
116C2×#123C2×D312yes—
117C2×#121—32no—
118C2×#121—61no—
119C2×C4×D4C2×D311yescharacteristic
120C22×C4⋊C4C2×D311yescharacteristic, complemented
121#458—616no—
122C2×#99—32no—
123C2×#99—38no—
124#399—64no—
125#398—21no—
126#397—21no—
127#390—62no—
128#389—21no—
129#384—62no—
130#381C2×D312yescomplemented
131#380—21no—
132#380C2×D311yescharacteristic, complemented
133C4×C22⋊C4—32no—
134#342—32no—
135C24×C3C2×D411yescharacteristic
136C24×C3C2411yescharacteristic
137C23×D3—22no—
138C23×D3C2×D414yes—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4C2×D414yes—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C2×#67—24no—
159C2×#67—28no—
160C2×#67—24no—
161C2×#67—24no—
162C2×#67—28no—
163C2×#67—44no—
164C2×#67—44no—
165C22×Dic3—22no—
166C22×Dic3—24no—
167C22×Dic3C2×D414yes—
168#237—416no—
169D3×D4—416no—
170#235—44no—
171#235—44no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—44no—
178C2×D12—82no—
179C2×C4×D3—24no—
180C2×Dic6—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4—41no—
186C3×C4⋊C4—44no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—44no—
191C3×C42—21no—
192C3×C42—21no—
193#217—24no—
194#217C2×D418yescomplemented
195(C2×D3)⋊C4—24no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—24no—
198(C2×D3)⋊C4—24no—
199(C2×D3)⋊C4—416no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204Dic3⋊C4—416no—
205C4×Dic3—44no—
206H(5,𝔽2)—68no—
207H(5,𝔽2)—124no—
208C2×#41—64no—
209C2×#41—62no—
210C22×D4—34no—
211C22×D4—32no—
212C22×D4—61no—
213C22×D4—64no—
214C22×D4—62no—
215C22×D4C22×D311yescharacteristic
216C23×C4#6711yescharacteristic
217C23×C4C22×D312yes—
218C23×C4C22×D311yescharacteristic
219C4⋊Q8—61no—
220#127—61no—
221#124—62no—
222#123—28no—
223#123—64no—
224#123—64no—
225#123—616no—
226#123—128no—
227C22⋊Q8—64no—
228#121—64no—
229#121—62no—
230#121—62no—
231#121—62no—
232#121—62no—
233#121—62no—
234#121—62no—
235#121—62no—
236#121—62no—
237#120—64no—
238#120—64no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—62no—
242C4×D4—62no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—68no—
252C2×C4⋊C4#6711yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—34no—
255C2×C22⋊C4—34no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—68no—
258C2×C22⋊C4#6711yescharacteristic
259C2×C22⋊C4C22×D312yes—
260C2×C22⋊C4C22×D311yescharacteristic
261C2×C42—62no—
262C2×C42#6711yescharacteristic
263#99—38no—
264#99—616no—
265#95—22no—
266#95—21no—
267#95—22no—
268#95—62no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3#12712yes—
276C23×C3C22×D412yes—
277C23×C3C22×D412yes—
278C23×C3C22×D411yescharacteristic
279C23×C3H(5,𝔽2)12yes—
280C22×D3—24no—
281C22×D3—28no—
282C22×D3—44no—
283C22×D3—44no—
284C22×D3—44no—
285C22×D3—82no—
286C22×D3—82no—
287C3×D4—28no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—48no—
306C2×C3×C4—81no—
307C2×C3×C4#12012yes—
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12712yes—
311#67—48no—
312#67—44no—
313#67—44no—
314#67—48no—
315C2×Dic3—24no—
316C2×Dic3—28no—
317C2×Dic3—28no—
318C2×Dic3—44no—
319C2×Dic3—44no—
320C2×Dic3—44no—
321D12—44no—
322D12—44no—
323D12—82no—
324D12—82no—
325D12—82no—
326C4×D3—48no—
327Dic6—44no—
328C24—34no—
329C24—62no—
330C24C2×#6711yescharacteristic
331C24C23×D311yescharacteristic
332#41—124no—
333#41—1216no—
334#41—124no—
335C2×Q8—62no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—64no—
342C2×D4—68no—
343C2×D4—64no—
344C2×D4—68no—
345C2×D4—64no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—124no—
349C2×D4—1216no—
350C2×D4—124no—
351C2×D4—124no—
352C2×D4—242no—
353C2×D4D3×D414yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—34no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—64no—
365C22×C4—62no—
366C22×C4—64no—
367C22×C4D3×D412yes—
368C22×C4D3×D412yes—
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—44no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—1216no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—24no—
388C22⋊C4—38no—
389C22⋊C4—44no—
390C22⋊C4—64no—
391C22⋊C4—64no—
392C22⋊C4—64no—
393C22⋊C4—64no—
394C22⋊C4—64no—
395C22⋊C4—1216no—
396C42—21no—
397C42—21no—
398C42—124no—
399C22×C3—22no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—22no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—81no—
410C22×C3C2×#12011yescharacteristic
411C22×C3C2×#12711yescharacteristic
412C22×C3#53511yescharacteristic
413C22×C3D4214yes—
414C2×D3—44no—
415C2×D3—44no—
416C2×D3—44no—
417C2×D3—88no—
418C2×D3—164no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—82no—
425C3×C4—82no—
426C3×C4#45812yescomplemented
427Dic3—48no—
428Dic3—44no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—64no—
436C23—68no—
437C23—124no—
438C23—124no—
439C23—124no—
440C23—242no—
441C23—242no—
442C23#94012yes—
443C23C2×D3×D412yes—
444C23C2×D3×D412yes—
445C23#100912yes—
446C23C22×#6711yescharacteristic
447Q8—124no—
448D4—28no—
449D4—124no—
450D4—128no—
451D4—124no—
452D4—124no—
453D4—128no—
454D4—124no—
455D4—242no—
456D4—242no—
457D4—242no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—48no—
476C2×C4—64no—
477C2×C4—68no—
478C2×C4—68no—
479C2×C4—81no—
480C2×C4—128no—
481C2×C4—124no—
482C2×C4—124no—
483C2×C4—124no—
484C2×C4#93711yescharacteristic
485C2×C4#93712yes—
486C2×C4#93711yescharacteristic
487C2×C4#94012yes—
488C2×C3—24no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—42no—
492C2×C3#243311yescharacteristic
493C2×C3C2×#45811yescharacteristic
494C2×C3#302811yescharacteristic
495D3—84no—
496C22—21no—
497C22—22no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—81no—
507C22—124no—
508C22—124no—
509C22—124no—
510C22—248no—
511C22—484no—
512C22#560914yes—
513C22C2×#93711yescharacteristic
514C22C2×#94011yescharacteristic
515C22#583011yescharacteristic
516C4—21no—
517C4—22no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—82no—
522C4—82no—
523C4—128no—
524C4—124no—
525C4#505312yescomplemented
526C3#2577211yescharacteristic, complemented, Sylow
527C2—21no—
528C2—24no—
529C2—21no—
530C2—42no—
531C2—244no—
532C2#8265311yescharacteristic
533C2C2×#505311yescharacteristic
534C2#8341711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2486C2
socleC22×C3C2×#120
prime radical#85786C2
hypercenter#2486D3
hypocenterC3#25772
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C3×#2486C2semidirect
3. C2×#5053C2semidirect
4. #82177C2semidirect
5. C4×#834C2semidirect
6. #69539C2semidirect
7. #69528C2semidirect
12. C22×C3×C4⋊C4C22semidirect
15. C2×#929C22semidirect
26. C3×#381C22semidirect
28. C3×#380C22semidirect
30. C4×#217C22semidirect
35. C2×#806C22semidirect
36. #4526C22semidirect
40. #2486D3semidirect
104. #834D4semidirect
120. C22×C4⋊C4C2×D3semidirect
130. #381C2×D3semidirect
132. #380C2×D3semidirect
194. #217C2×D4semidirect
426. C3×C4#458semidirect
525. C4#5053semidirect
526. C3#25772semidirect

Maximal subgroups

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

subgroupindex
#857862
C3×#24862
C2×#50532
#821772
C4×#8342
#695392
#695282
#257723