Finite Group Databasebeta

← #10795124

Subgroups — #10795124

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5522C211yesmaximal, characteristic
2#74051C212yesmaximal, complemented
3C3×#1820C211yesmaximal, characteristic, complemented
4#71594C211yesmaximal, characteristic, complemented
5#69537C212yesmaximal
6#68959C214yesmaximal, complemented
7#68952C214yesmaximal, complemented
8#20723—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C22×C4×Dic3C2211yescharacteristic
12#5522—22no—
13#5522—22no—
14#5522—22no—
15#5522—22no—
16C2×#893C411yescharacteristic, complemented
17C2×#893C411yescharacteristic, complemented
18C2×#887C411yescharacteristic, complemented
19C2×#887C411yescharacteristic, complemented
20C2×#67⋊C4C2212yes—
21C2×#67⋊C4C2212yes—
22C2×#872C412yescomplemented
23C2×C42×D3C412yescomplemented
24C3×#381C2212yes—
25#5211C2212yes—
26C2×#806C2218yes—
27#4942C2214yes—
28Dic3×C22⋊C4C2214yes—
29#4651—22no—
30#4650—22no—
31#4649—22no—
32#4648—22no—
33C3×#342C2214yes—
34#4526C2214yes—
35C2×C4×#41—31no—
36#2283—32no—
37#1820—31no—
38#1820D311yescharacteristic, complemented
39#1799—34no—
40#1782—32no—
41C4×#99—34no—
42C23×C3×C4C2311yescharacteristic
43C22×#67C2312yes—
44C23×Dic3C2311yescharacteristic
45C23×Dic3C2311yescharacteristic
46C2×#235—22no—
47C2×#235—22no—
48C2×#235—28no—
49C22×D12C2×C411yescharacteristic
50C22×C4×D3C2×C412yes—
51C22×C4×D3C2×C412yes—
52C22×C4×D3C2×C412yes—
53C22×Dic6C2×C411yescharacteristic
54C3×#117—22no—
55C3×#117—22no—
56C2×C3×C4⋊C4C2×C411yescharacteristic
57C2×C3×C4⋊C4C2×C411yescharacteristic
58C2×C3×C22⋊C4C2314yes—
59C2×C3×C22⋊C4C2312yes—
60C2×C3×C42C2×C412yes—
61C2×#217C2314yes—
62C2×(C2×D3)⋊C4C2×C412yes—
63C2×(C2×D3)⋊C4C2×C412yes—
64C2×Dic3⋊C4—24no—
65C2×Dic3⋊C4C2×C412yes—
66C2×Dic3⋊C4C2×C412yes—
67C2×C4×Dic3—24no—
68C2×C4×Dic3C2×C412yes—
69C2×C4×Dic3C2×C412yes—
70C2×C4×Dic3C2×C411yescharacteristic
71C2×C4×Dic3C2×C411yescharacteristic
72C2×C4×Dic3C2×C411yescharacteristic
73C2×C4×Dic3C2×C411yescharacteristic
74C2×C4×Dic3D412yes—
75C2×C4×Dic3D412yes—
76C2×C4×Dic3D412yes—
77C2×C4×Dic3D412yes—
78#893—22no—
79#893—22no—
80#893—22no—
81#893—22no—
82#887—22no—
83#887—22no—
84#887—22no—
85#887—22no—
86#67⋊C4—24no—
87#67⋊C4—24no—
88#67⋊C4—24no—
89#67⋊C4—24no—
90#872—24no—
91#872—24no—
92C42×D3—22no—
93C42×D3—22no—
94C42×D3—22no—
95C42×D3—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102#806—216no—
103C22×#41—31no—
104C4×#41—62no—
105C4×#41—62no—
106C4×#41—62no—
107C4×#41—62no—
108C2×C4×Q8—31no—
109C2×C4×Q8—31no—
110C2×C4×D4—31no—
111C2×C4×D4—31no—
112C2×C4×D4—32no—
113C2×C4×D4—32no—
114C2×#117—32no—
115C2×#117C2×D311yescharacteristic
116C22×C42—31no—
117C22×C42—32no—
118C2×#99—38no—
119#383—62no—
120#382—62no—
121#381—32no—
122#381C2×D312yes—
123#380—34no—
124C4×C4⋊C4—62no—
125C4×C22⋊C4—34no—
126#376—62no—
127#342—34no—
128#342C2×D314yes—
129C24×C3C2411yescharacteristic
130C23×D3C22×C412yes—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4C22×C411yescharacteristic
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C22×C412yes—
136C22×C3×C4C22×C412yes—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D412yes—
139C2×#67—24no—
140C2×#67—24no—
141C2×#67#4118yes—
142C22×Dic3—24no—
143C22×Dic3—42no—
144C22×Dic3—42no—
145C22×Dic3C22×C411yescharacteristic
146C22×Dic3C22×C411yescharacteristic
147C22×Dic3C22×C412yes—
148C22×Dic3C22×C411yescharacteristic
149C22×Dic3C22×C411yescharacteristic
150C22×Dic3C2×D412yes—
151C22×Dic3C2×D412yes—
152C22×Dic3C2×D412yes—
153C22×Dic3C2×D412yes—
154#235—216no—
155#235—216no—
156C2×D12—24no—
157C2×D12—22no—
158C2×C4×D3—24no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—24no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—44no—
167C2×C4×D3—44no—
168C2×Dic6—22no—
169C2×Dic6—24no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—28no—
174C3×C42—22no—
175C3×C42—22no—
176#217—28no—
177(C2×D3)⋊C4—24no—
178(C2×D3)⋊C4—24no—
179Dic3⋊C4—24no—
180Dic3⋊C4—24no—
181Dic3⋊C4—48no—
182C4×Dic3—22no—
183C4×Dic3—22no—
184C4×Dic3—22no—
185C4×Dic3—22no—
186C4×Dic3—48no—
187C4×Dic3C22⋊C411yescharacteristic
188C4×Dic3C22⋊C411yescharacteristic
189C4×Dic3C22⋊C411yescharacteristic
190C4×Dic3C22⋊C411yescharacteristic
191C4×Dic3C22⋊C412yes—
192C4×Dic3C22⋊C412yes—
193C4×Dic3C22⋊C412yes—
194C4×Dic3C22⋊C412yes—
195C4×Dic3C22⋊C412yes—
196C4×Dic3C22⋊C412yes—
197C2×#41—68no—
198C2×#41—62no—
199C2×#41—62no—
200C22×Q8—31no—
201C22×D4—32no—
202C22×D4—31no—
203C23×C4—31no—
204C23×C4—31no—
205C23×C4—32no—
206C23×C4—32no—
207C23×C4—32no—
208C23×C4C22×D311yescharacteristic
209C4×Q8—62no—
210C4×Q8—62no—
211C4×Q8—62no—
212C4×Q8—62no—
213C4×D4—62no—
214C4×D4—62no—
215C4×D4—62no—
216C4×D4—62no—
217C4×D4—64no—
218C4×D4—64no—
219C4×D4—64no—
220C4×D4—64no—
221#117—22no—
222#117—22no—
223#117—64no—
224#117—64no—
225C2×C4⋊C4—32no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—64no—
228C2×C4⋊C4C4×D311yescharacteristic
229C2×C4⋊C4C4×D311yescharacteristic
230C2×C22⋊C4—32no—
231C2×C22⋊C4—32no—
232C2×C22⋊C4—34no—
233C2×C22⋊C4C22×D314yes—
234C2×C22⋊C4C22×D312yes—
235C2×C42—31no—
236C2×C42—31no—
237C2×C42—31no—
238C2×C42—31no—
239C2×C42—32no—
240C2×C42—32no—
241C2×C42—32no—
242C2×C42—32no—
243C2×C42—32no—
244C2×C42—32no—
245C2×C42—62no—
246C2×C42—62no—
247C2×C42—62no—
248C2×C42—62no—
249C2×C42—64no—
250C2×C42C4×D312yes—
251#99—616no—
252#95—22no—
253#95—22no—
254#95—64no—
255#95—64no—
256#95—62no—
257#95—62no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3C23×C411yescharacteristic
261C23×C3C22×D412yes—
262C23×C3C2×#4114yes—
263C22×D3—22no—
264C22×D3—22no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3#11714yes—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4C2×C22⋊C411yescharacteristic
280C2×C3×C4C2×C22⋊C411yescharacteristic
281C2×C3×C4C2×C22⋊C412yes—
282#67—216no—
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—24no—
286C2×Dic3—24no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—44no—
294C2×Dic3—44no—
295C2×Dic3—44no—
296C2×Dic3C2×C22⋊C411yescharacteristic
297C2×Dic3C2×C22⋊C411yescharacteristic
298C2×Dic3C2×C22⋊C412yes—
299C2×Dic3C2×C22⋊C412yes—
300C2×Dic3C2×C22⋊C411yescharacteristic
301C2×Dic3C2×C22⋊C411yescharacteristic
302C2×Dic3#11714yes—
303D12—28no—
304C4×D3—28no—
305C4×D3—28no—
306C4×D3—44no—
307C4×D3—44no—
308C4×D3—44no—
309C4×D3—44no—
310Dic6—28no—
311C24—32no—
312C24C23×D311yescharacteristic
313#41—616no—
314#41—616no—
315C2×Q8—64no—
316C2×Q8—62no—
317C2×D4—38no—
318C2×D4—64no—
319C2×D4—64no—
320C2×D4—64no—
321C2×D4—62no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—32no—
327C22×C4—32no—
328C22×C4—32no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—64no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—64no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—64no—
342C22×C4—124no—
343C22×C4—124no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4C2×C4×D311yescharacteristic
347C22×C4C2×C4×D311yescharacteristic
348C22×C4C2×C4×D312yes—
349C22×C4C2×C4×D312yes—
350C22×C4D3×D412yes—
351C22×C4D3×D412yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—64no—
355C4⋊C4—64no—
356C4⋊C4—128no—
357C22⋊C4—24no—
358C22⋊C4—28no—
359C22⋊C4—64no—
360C22⋊C4—64no—
361C22⋊C4—68no—
362C42—22no—
363C42—22no—
364C42—31no—
365C42—31no—
366C42—31no—
367C42—31no—
368C42—32no—
369C42—32no—
370C42—32no—
371C42—32no—
372C42—32no—
373C42—32no—
374C42—62no—
375C42—62no—
376C42—62no—
377C42—62no—
378C42—128no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—44no—
384C22×C3C22×C22⋊C411yescharacteristic
385C22×C3C2×#11712yes—
386C22×C3#52514yes—
387C2×D3—24no—
388C2×D3—44no—
389C2×D3—42no—
390C2×D3—42no—
391C2×D3—44no—
392C3×C4—22no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—42no—
396C3×C4—42no—
397C3×C4—84no—
398Dic3—21no—
399Dic3—21no—
400Dic3—22no—
401Dic3—22no—
402Dic3—24no—
403Dic3—84no—
404Dic3#41012yes—
405Dic3#41012yes—
406C23—34no—
407C23—41no—
408C23—41no—
409C23—62no—
410C23—62no—
411C23—122no—
412C23—122no—
413C23C22×C4×D311yescharacteristic
414C23C2×D3×D412yes—
415C23D3×#4114yes—
416Q8—68no—
417D4—68no—
418D4—616no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—32no—
430C2×C4—32no—
431C2×C4—34no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—68no—
438C2×C4—68no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—124no—
446C2×C4—124no—
447C2×C4—124no—
448C2×C4—124no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—124no—
454C2×C4—124no—
455C2×C4—124no—
456C2×C4D3×C22⋊C411yescharacteristic
457C2×C4D3×C22⋊C411yescharacteristic
458C2×C4D3×C22⋊C412yes—
459C2×C3—24no—
460C2×C3—22no—
461C2×C3#228011yescharacteristic
462C2×C3C2×#41011yescharacteristic
463C2×C3C2×#41011yescharacteristic
464D3—44no—
465C22—22no—
466C22—22no—
467C22—21no—
468C22—21no—
469C22—44no—
470C22—64no—
471C22—124no—
472C22—124no—
473C22—122no—
474C22—122no—
475C22C2×D3×C22⋊C411yescharacteristic
476C22D3×#11712yes—
477C22#565614yes—
478C4—21no—
479C4—21no—
480C4—22no—
481C4—32no—
482C4—32no—
483C4—42no—
484C4—42no—
485C4—64no—
486C4—61no—
487C4—61no—
488C4—62no—
489C4—62no—
490C4—84no—
491C4—244no—
492C3#2072311yescharacteristic, complemented, Sylow
493C2—24no—
494C2—22no—
495C2—124no—
496C2#7405011yescharacteristic
497C2D3×#41011yescharacteristic
498C2D3×#41011yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1820C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#235C22
cosocleC24×C3C24
hypercenter#1820D3
hypocenterC3#20723
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #74051C2semidirect
3. C3×#1820C2semidirect
4. #71594C2semidirect
6. #68959C2semidirect
7. #68952C2semidirect
16. C2×#893C4semidirect
17. C2×#893C4semidirect
18. C2×#887C4semidirect
19. C2×#887C4semidirect
22. C2×#872C4semidirect
23. C2×C42×D3C4semidirect
38. #1820D3semidirect
492. C3#20723semidirect

Maximal subgroups

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

subgroupindex
C2×#55222
#740512
C3×#18202
#715942
#695372
#689592
#689522
#207233