Finite Group Databasebeta

← #10804168

Subgroups — #10804168

Order 768.

1446 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5828C211yesmaximal, characteristic
2D3×#407C212yesmaximal, complemented
3C3×#1785C211yesmaximal, characteristic, complemented
4#72137C211yesmaximal, characteristic, complemented
5#70115C214yesmaximal, complemented
6#70081C214yesmaximal, complemented
7#69974C212yesmaximal, complemented
8#21754—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C23×C4×D3C2211yescharacteristic
11#5828—22no—
12#5828—22no—
13#5828—22no—
14#5828—22no—
15C2×#946C2212yes—
16C2×#938C2212yes—
17C2×#937C412yes—
18C2×#935C412yes—
19C2×C4×#67C414yes—
20C2×#926C2211yescharacteristic
21C2×D3×M16C2214yes—
22C2×C3×#101C2212yescomplemented
23C2×C3×#100C2212yescomplemented
24C3×#407C2212yes—
25C2×#836C2212yescomplemented
26C2×#833C2212yescomplemented
27#5207C2212yes—
28#5126C2214yes—
29#4726—22no—
30#4726—22no—
31D3×C22⋊C8—22no—
32D3×C22⋊C8—22no—
33#4554C2218yescomplemented
34C2×#525—31no—
35C2×#407—32no—
36#1901—34no—
37#1900—34no—
38#1822—32no—
39#1785—31no—
40#1785D311yescharacteristic, complemented
41C24×D3C2×C411yescharacteristic
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—42no—
45C2×C3×#41—42no—
46C22×C3×Q8C2311yescharacteristic
47C22×C3×D4C2×C412yes—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4C2×C412yes—
50C23×C3×C4C2311yescharacteristic
51C22×#67C2×C414yes—
52C23×Dic3C2×C411yescharacteristic
53C22×C4×D3—42no—
54C22×C4×D3—42no—
55C22×C4×D3C2×C412yes—
56C22×C4×D3D412yes—
57C22×C4×D3D412yes—
58C22×C4×D3D412yes—
59C22×C4×D3D412yes—
60C22×C4×D3C2312yes—
61C2×C3×M16C2314yes—
62C2×#217C2×C412yes—
63C2×#217C2×C414yes—
64#946—22no—
65#946—22no—
66#946—22no—
67#946—22no—
68#938—22no—
69#938—22no—
70#938—22no—
71#938—22no—
72#937—22no—
73#937—24no—
74#937—22no—
75#935—22no—
76#935—24no—
77#935—22no—
78C4×#67—24no—
79C4×#67—24no—
80C4×#67—24no—
81C4×#67—24no—
82C2×(C2×D3)⋊C4C2×C414yes—
83#926—22no—
84#926—22no—
85C2×C4⋊Dic3C2312yes—
86C2×Dic3⋊C4C2×C414yes—
87C2×C4×Dic3C2×C412yes—
88C2×#208C2314yes—
89D3×M16—44no—
90D3×M16—44no—
91C2×ΓL(1,𝔽25)—24no—
92C2×D3×C8—24no—
93C3×#101—42no—
94C3×#100—42no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97#836—42no—
98#833—42no—
99#830—22no—
100#830—22no—
101(C2×D3)⋊C8—24no—
102(C2×D3)⋊C8—24no—
103C22×#41C2×D311yescharacteristic
104C24×C4—31no—
105C22×M16—34no—
106#525—62no—
107#525—62no—
108#525—62no—
109#525—62no—
110C2×#123—32no—
111C2×C22⋊Q8—32no—
112C2×#121—32no—
113C2×#120—32no—
114C2×C4×D4—34no—
115C2×#117—31no—
116C2×#101—32no—
117C2×#101C2×D312yescomplemented
118C2×#100—32no—
119C2×#100C2×D312yescomplemented
120#407—32no—
121#407—34no—
122#407—62no—
123#407—62no—
124#407C2×D312yes—
125C2×C22⋊C8—62no—
126C2×C22⋊C8—62no—
127#343—38no—
128C24×C3C22×C412yes—
129C24×C3C22×C411yescharacteristic
130C23×D3—44no—
131C23×D3—41no—
132C23×D3—41no—
133C23×D3C22⋊C411yescharacteristic
134C23×D3C22⋊C412yes—
135C23×D3C22⋊C411yescharacteristic
136C23×D3C22×C412yes—
137C3×#41—48no—
138C3×#41—48no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C22×C412yes—
155C22×C3×C4C22×C411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2411yescharacteristic
159C2×#67—24no—
160C2×#67—24no—
161C2×#67—48no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3C22⋊C411yescharacteristic
165C22×Dic3C22⋊C412yes—
166C22×Dic3C22⋊C411yescharacteristic
167C22×Dic3C22×C412yes—
168C22×Dic3C22×C414yes—
169C2×C4×D3—24no—
170C2×C4×D3—24no—
171C2×C4×D3—44no—
172C2×C4×D3—44no—
173C2×C4×D3—44no—
174C2×C4×D3—44no—
175C2×C4×D3—42no—
176C2×C4×D3—42no—
177C2×C4×D3C22⋊C412yes—
178C2×C4×D3C22⋊C412yes—
179C2×C4×D3C22⋊C412yes—
180C2×C4×D3C22⋊C412yes—
181C2×C4×D3C2×D412yes—
182C2×C4×D3C2×D412yes—
183C2×C4×D3C2×D412yes—
184C2×C4×D3C2×D412yes—
185C3×M16—24no—
186C3×M16—24no—
187C2×C3×C8—24no—
188#217—22no—
189#217—22no—
190#217—24no—
191#217—24no—
192(C2×D3)⋊C4—24no—
193(C2×D3)⋊C4—24no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196Dic3⋊C4—24no—
197Dic3⋊C4—24no—
198C4×Dic3—24no—
199#208—24no—
200#208—24no—
201C2×C3⋊C8—24no—
202ΓL(1,𝔽25)—48no—
203D3×C8—48no—
204C25—31no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—42no—
208C2×#41—42no—
209C22×Q8C22×D311yescharacteristic
210C22×D4—34no—
211C22×D4C4×D312yes—
212C22×D4C22×D311yescharacteristic
213C23×C4—31no—
214C23×C4—32no—
215C23×C4—32no—
216C23×C4—32no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4—32no—
220C23×C4—122no—
221C23×C4—122no—
222C23×C4C4×D312yes—
223C23×C4C22×D311yescharacteristic
224C2×M16—34no—
225C2×M16—64no—
226C2×M16—124no—
227C2×M16—124no—
228C2×M16C22×D314yes—
229C22×C8—64no—
230#123—62no—
231#123—64no—
232#123—62no—
233C22⋊Q8—62no—
234C22⋊Q8—62no—
235C22⋊Q8—62no—
236C22⋊Q8—62no—
237#121—62no—
238#121—62no—
239#121—62no—
240#121—62no—
241#120—62no—
242#120—64no—
243#120—62no—
244C4×D4—64no—
245C4×D4—64no—
246C4×D4—64no—
247C4×D4—64no—
248#117—62no—
249#117—62no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—32no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—34no—
255C2×C42—32no—
256#101—42no—
257#101—122no—
258#100—42no—
259#100—122no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—64no—
263C22⋊C8—64no—
264C22⋊C8—62no—
265C22⋊C8—62no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—42no—
272C23×C3—44no—
273C23×C3C2×C22⋊C411yescharacteristic
274C23×C3C2×C22⋊C411yescharacteristic
275C23×C3C23×C411yescharacteristic
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—24no—
279C22×D3—24no—
280C22×D3—24no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—41no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—48no—
288C22×D3—48no—
289C22×D3C2×C22⋊C412yes—
290C22×D3C2×C22⋊C412yes—
291C3×Q8—44no—
292C3×D4—44no—
293C3×D4—44no—
294C3×D4—44no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—48no—
303C2×C3×C4C2×C22⋊C412yes—
304C2×C3×C4C22×D412yes—
305C2×C3×C4H(5,𝔽2)12yes—
306C2×C3×C4H(5,𝔽2)12yes—
307#67—48no—
308#67—48no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—24no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—44no—
316C2×Dic3C2×C22⋊C412yes—
317C2×Dic3C2×C22⋊C412yes—
318C4×D3—24no—
319C4×D3—24no—
320C4×D3—24no—
321C4×D3—24no—
322C4×D3—42no—
323C4×D3—42no—
324C4×D3—42no—
325C4×D3—42no—
326C3×C8—44no—
327C3⋊C8—44no—
328C24—31no—
329C24—32no—
330C24—32no—
331C24—31no—
332C24—121no—
333C24—121no—
334C24—124no—
335C24C2×C4×D312yes—
336C24C2×C4×D311yescharacteristic
337#41—48no—
338#41—48no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—64no—
350C2×D4—64no—
351C2×D4—128no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—32no—
356C22×C4—31no—
357C22×C4—32no—
358C22×C4—31no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—32no—
366C22×C4—32no—
367C22×C4—34no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—124no—
373C22×C4—124no—
374C22×C4—124no—
375C22×C4—124no—
376C22×C4—122no—
377C22×C4—122no—
378C22×C4—121no—
379C22×C4—121no—
380C22×C4C2×C4×D312yes—
381C22×C4C2×C4×D311yescharacteristic
382C22×C4D3×D412yes—
383C22×C4D3×D412yes—
384C22×C4C23×D311yescharacteristic
385M16—24no—
386M16—24no—
387M16—64no—
388M16—64no—
389M16—128no—
390C2×C8—24no—
391C2×C8—64no—
392C2×C8—128no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C22⋊C4—64no—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—62no—
402C22⋊C4—62no—
403C42—64no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—44no—
413C22×C3—82no—
414C22×C3—82no—
415C22×C3C22×C22⋊C411yescharacteristic
416C22×C3#51811yescharacteristic
417C22×C3#51811yescharacteristic
418C2×D3—22no—
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—44no—
424C2×D3—44no—
425C2×D3—44no—
426C2×D3—48no—
427C2×D3—44no—
428C2×D3#41314yes—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4#53414yes—
432Dic3—22no—
433Dic3—22no—
434Dic3—44no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—32no—
441C23—32no—
442C23—42no—
443C23—44no—
444C23—64no—
445C23—64no—
446C23—64no—
447C23—62no—
448C23—62no—
449C23—121no—
450C23—122no—
451C23—121no—
452C23—122no—
453C23—122no—
454C23—122no—
455C23—128no—
456C23—128no—
457C23D3×C22⋊C411yescharacteristic
458C23D3×C22⋊C411yescharacteristic
459C23C22×C4×D311yescharacteristic
460Q8—44no—
461D4—44no—
462D4—44no—
463D4—44no—
464D4—128no—
465D4—128no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—48no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—121no—
488C2×C4—122no—
489C2×C4—121no—
490C2×C4—124no—
491C2×C4D3×C22⋊C412yes—
492C2×C4C2×D3×D412yes—
493C2×C4#100412yes—
494C2×C4#100412yes—
495C8—44no—
496C8—124no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—82no—
502C2×C3#230811yescharacteristic
503C2×C3C2×#41312yes—
504D3—24no—
505D3—42no—
506C22—22no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—34no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—44no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—62no—
521C22—82no—
522C22—82no—
523C22—124no—
524C22—124no—
525C22—124no—
526C22—128no—
527C22—124no—
528C22C2×D3×C22⋊C411yescharacteristic
529C22#548611yescharacteristic
530C22#548611yescharacteristic
531C4—22no—
532C4—42no—
533C4—62no—
534C4—62no—
535C4—124no—
536C4#558914yes—
537C3#2175411yescharacteristic, complemented, Sylow
538C2—21no—
539C2—21no—
540C2—41no—
541C2—41no—
542C2—64no—
543C2—82no—
544C2—122no—
545C2#7384911yescharacteristic
546C2D3×#41312yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1785C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5828C2
cosocleC22×C3×C4C24
hypercenter#1785D3
hypocenterC3#21754
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. D3×#407C2semidirect
3. C3×#1785C2semidirect
4. #72137C2semidirect
5. #70115C2semidirect
6. #70081C2semidirect
7. #69974C2semidirect
22. C2×C3×#101C22semidirect
23. C2×C3×#100C22semidirect
25. C2×#836C22semidirect
26. C2×#833C22semidirect
33. #4554C22semidirect
40. #1785D3semidirect
117. C2×#101C2×D3semidirect
119. C2×#100C2×D3semidirect
537. C3#21754semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
D3×#4072
C3×#17852
#721372
#701152
#700812
#699742
#217543