Finite Group Databasebeta

← #10817421

Subgroups — #10817421

Order 768.

1170 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2366C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3C2×C4×#213C212yesmaximal, complemented
4C4⋊C4×D12C211yesmaximal, characteristic, complemented
5#71764C211yesmaximal, characteristic, complemented
6#71606C212yesmaximal, complemented
7#69657C212yesmaximal, complemented
8#69644C212yesmaximal, complemented
9C4⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
10C4×#809C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#22995—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2212yes—
16C22×#213C2212yes—
17C2×D3×C4⋊C4—21no—
18C2×C4×D12C2212yes—
19C2×C4×D12C2211yescharacteristic
20C3×#390—21no—
21C3×#390C2212yes—
22C3×#386—22no—
23C3×C4×C4⋊C4C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#832C2212yes—
26C2×#832C2212yes—
27C2×#832C2212yes—
28#5072C418yescomplemented
29C4×#213—28no—
30#4987—21no—
31C2×#809C2211yescharacteristic
32C2×#809C2211yescharacteristic
33C2×#807C2212yes—
34#4935C2211yescharacteristic
35#4928C2212yes—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37C2×C4×C3⋊C8C2212yes—
38C2×C4×C3⋊C8C2211yescharacteristic
39#4668—22no—
40#4529C2211yescharacteristic
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4524C2212yes—
44C2×#459—31no—
45C2×C4×D8—32no—
46#2366D311yescharacteristic, complemented
47D4×C4⋊C4—31no—
48#1957—31no—
49#1947—31no—
50#1934—32no—
51#1879—32no—
52#1862—32no—
53C8×C4⋊C4—31no—
54C4×#102—31no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C23×C3×C4—22no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3—22no—
62C3×#127—22no—
63C3×#127C2×C414yescomplemented
64C3×C4×D4—24no—
65C3×C4×D4—42no—
66C3×C4×D4—44no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77C2×#213—24no—
78C2×#213—24no—
79C2×#213C2×C418yescomplemented
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—22no—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89C22×C3⋊C8C2311yescharacteristic
90D3×C4⋊C4—44no—
91C4×D12—22no—
92C4×D12—24no—
93C4×D12—42no—
94C4×D12C2×C414yescomplemented
95C3×#95—21no—
96C3×#95D412yes—
97#832—24no—
98#832—42no—
99#832C2×C418yescomplemented
100C3⋊#95—22no—
101#809—22no—
102#809D414yescomplemented
103#807—24no—
104C4⋊(C3⋊C8)C2×C414yescomplemented
105C4×C3⋊C8—22no—
106C4×C3⋊C8—24no—
107C22×D8—32no—
108C2×#127C2×D311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—31no—
111C2×C4×D4—32no—
112C2×C4×D4C2×D312yes—
113C22×C4⋊C4—61no—
114#459—38no—
115C4×D8—68no—
116C2×#107—32no—
117C2×C4⋊C8—31no—
118C2×#102—32no—
119C2×#102—31no—
120C2×#102—32no—
121C2×#102—31no—
122C2×#102—32no—
123C2×C4×C8—32no—
124C2×C4×C8—31no—
125#390—21no—
126#390C2×D312yes—
127#389—32no—
128#386—22no—
129#385—62no—
130#384—31no—
131#380—61no—
132C4×C4⋊C4C2×D311yescharacteristic
133C4×C4⋊C4C2×D311yescharacteristic
134#340—31no—
135#340—32no—
136#340—31no—
137#336—32no—
138C24×C3—22no—
139C23×D3—21no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
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—
149C2×C3×D4—81no—
150C2×C3×D4—81no—
151C2×C3×D4C22×C414yes—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—44no—
155C22×C3×C4—44no—
156C22×C3×C4—82no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—21no—
163C22×Dic3—22no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12C22×C412yes—
168C2×D12C2×D412yes—
169C2×D12#4112yes—
170C2×C4×D3—24no—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—44no—
174C2×C4×D3—44no—
175C2×C4×D3—82no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4D814yescomplemented
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—44no—
184C3×C22⋊C4—82no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—42no—
188C3×C42C22×C412yes—
189C3×C42#4112yes—
190#213—216no—
191(C2×D3)⋊C4—24no—
192(C2×D3)⋊C4—44no—
193C4⋊Dic3—21no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4⋊Dic3—81no—
198C4⋊Dic3C22×C412yes—
199Dic3⋊C4—44no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8C22×C414yes—
208C2×C3⋊C8C2×D412yes—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—31no—
212C22×D4C22×D312yes—
213C23×C4—22no—
214C23×C4—61no—
215C23×C4—62no—
216C2×D8—38no—
217C2×D8—64no—
218C2×D8—64no—
219C22×C8—32no—
220C22×C8—31no—
221C22×C8—31no—
222#127—22no—
223#127C4×D314yescomplemented
224C4×D4—24no—
225C4×D4—34no—
226C4×D4—42no—
227C4×D4—44no—
228C4×D4—64no—
229C4×D4—62no—
230C4×D4—122no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—124no—
237C2×C4⋊C4#6711yescharacteristic
238C2×C4⋊C4#6711yescharacteristic
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—62no—
246C2×C42—21no—
247C2×C42C22×D312yes—
248C2×C42C22×D311yescharacteristic
249#107—64no—
250C4⋊C8—34no—
251#102—38no—
252#102—34no—
253#102—62no—
254#102—64no—
255#102—122no—
256C4×C8—64no—
257C4×C8—62no—
258#95—21no—
259#95—62no—
260#95#6712yes—
261C23×C3—24no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3C22×D411yescharacteristic
269C22×D3—22no—
270C22×D3—22no—
271C22×D3—22no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C3×D4—24no—
277C3×D4—24no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—48no—
281C3×D4—84no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—88no—
296C2×C3×C4C4×D412yes—
297C2×C3×C4C4×D412yes—
298C2×C3×C4C2×D814yes—
299C2×C3×C4C2×D812yes—
300C2×C3×C4#13512yes—
301C2×C3×C4C23×C411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4H(5,𝔽2)11yescharacteristic
306C2×C3×C42−1+411yescharacteristic
307C2×Dic3—22no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313D12—22no—
314D12—22no—
315D12—22no—
316D12C4×D414yescomplemented
317C4×D3—48no—
318C4×D3—88no—
319C3⋊C8—24no—
320C3⋊C8—22no—
321C3⋊C8—41no—
322C24—22no—
323C24—61no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—32no—
332C2×D4—32no—
333C2×D4—32no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—61no—
339C2×D4—81no—
340C2×D4—81no—
341C2×D4C2×C4×D314yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—61no—
347C22×C4—64no—
348C22×C4—62no—
349C22×C4—82no—
350C22×C4—124no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—124no—
354C22×C4—242no—
355C22×C4C2×#6712yes—
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6712yes—
358C22×C4C2×#6711yescharacteristic
359C22×C4C23×D311yescharacteristic
360D8—616no—
361C2×C8—34no—
362C2×C8—32no—
363C2×C8—61no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—61no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—32no—
375C4⋊C4—61no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—124no—
379C4⋊C4—122no—
380C4⋊C4—241no—
381C4⋊C4#21314yescomplemented
382C4⋊C4D3×D412yes—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—64no—
386C22⋊C4—82no—
387C22⋊C4—124no—
388C42—22no—
389C42—22no—
390C42—42no—
391C42C2×C4×D312yes—
392C42#23512yes—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3C2×C4×D411yescharacteristic
401C22×C3D4211yescharacteristic
402C22×C3#54811yescharacteristic
403C22×C3C22×D811yescharacteristic
404C22×C3C2×#13511yescharacteristic
405C22×C3#57611yescharacteristic
406C22×C3#57711yescharacteristic
407C2×D3—24no—
408C2×D3—42no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C3×C4—22no—
415C3×C4—24no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—81no—
419C3×C4C4×D814yescomplemented
420C3×C4C2×C4×D411yescharacteristic
421C3×C4#52011yescharacteristic
422C3×C4#54711yescharacteristic
423C3×C4#55311yescharacteristic
424Dic3—42no—
425Dic3—82no—
426C23—24no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—62no—
434C23—62no—
435C23—62no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23C22×#6711yescharacteristic
441D4—24no—
442D4—24no—
443D4—34no—
444D4—48no—
445D4—42no—
446D4—42no—
447D4—62no—
448D4—62no—
449D4—62no—
450D4—84no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—62no—
465C2×C4—88no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—128no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—248no—
473C2×C4C4×#6712yes—
474C2×C4C4×#6712yes—
475C2×C4C2×#21314yes—
476C2×C4C2×#21312yes—
477C2×C4#95012yes—
478C2×C4C22×C4×D311yescharacteristic
479C2×C4C2×#23511yescharacteristic
480C2×C4C2×D3×D411yescharacteristic
481C2×C4#100411yescharacteristic
482C2×C4D3×#4111yescharacteristic
483C2×C4#101011yescharacteristic
484C8—64no—
485C8—62no—
486C8—121no—
487C2×C3—84no—
488C2×C3#236111yescharacteristic
489C2×C3C2×C4×D811yescharacteristic
490C2×C3#294611yescharacteristic
491C2×C3D4×D811yescharacteristic
492C2×C3#328411yescharacteristic
493C2×C3#335811yescharacteristic
494C2×C3#336111yescharacteristic
495D3—44no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—64no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—122no—
505C22—122no—
506C22—122no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22C2×C4×#6711yescharacteristic
511C22C22×#21311yescharacteristic
512C22D4×#6711yescharacteristic
513C22C2×#95011yescharacteristic
514C22#583111yescharacteristic
515C22#583511yescharacteristic
516C22#583611yescharacteristic
517C4—24no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—122no—
523C4—242no—
524C4C4×#21314yescomplemented
525C4C4×D3×D411yescharacteristic
526C4#554611yescharacteristic
527C4#555111yescharacteristic
528C4#555811yescharacteristic
529C3#2299511yescharacteristic, complemented, Sylow
530C2—84no—
531C2—124no—
532C2C2×C4×#21311yescharacteristic
533C2#8239911yescharacteristic
534C2#8253711yescharacteristic
535C2D4×#21311yescharacteristic
536C2#8357911yescharacteristic
537C2#8358011yescharacteristic
538C2#8371311yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2366C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
cosocleC22×C3×C4C24
hypercenter#2366D3
hypocenterC3#22995
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. C3×#2366C2semidirect
3. C2×C4×#213C2semidirect
4. C4⋊C4×D12C2semidirect
5. #71764C2semidirect
6. #71606C2semidirect
7. #69657C2semidirect
8. #69644C2semidirect
9. C4⋊C4×C3⋊C8C2semidirect
10. C4×#809C2semidirect
11. #69466C2semidirect
28. #5072C4semidirect
46. #2366D3semidirect
63. C3×#127C2×C4semidirect
79. C2×#213C2×C4semidirect
94. C4×D12C2×C4semidirect
99. #832C2×C4semidirect
102. #809D4semidirect
104. C4⋊(C3⋊C8)C2×C4semidirect
180. C3×C4⋊C4D8semidirect
223. #127C4×D3semidirect
316. D12C4×D4semidirect
381. C4⋊C4#213semidirect
419. C3×C4C4×D8semidirect
524. C4C4×#213semidirect
529. C3#22995semidirect

Maximal subgroups

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

subgroupindex
C3×#23662
C2×#50722
C2×C4×#2132
C4⋊C4×D122
#717642
#716062
#696572
#696442
C4⋊C4×C3⋊C82
C4×#8092
#694662
#229953