Finite Group Databasebeta

← #10814343

Subgroups — #10814343

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2354C211yesmaximal, characteristic, complemented
2C2×C4×#213C211yesmaximal, characteristic
3C4⋊C4×D12C211yesmaximal, characteristic, complemented
4#71604C211yesmaximal, characteristic, complemented
5#69657C212yesmaximal
6#69645C212yesmaximal, complemented
7#69584C211yesmaximal, characteristic, complemented
8#69574C211yesmaximal, characteristic, complemented
9#69477C211yesmaximal, characteristic, complemented
10#69474C212yesmaximal, complemented
11#69461C212yesmaximal, complemented
12#22758—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×C3×#117—21no—
15C22×#213C2211yescharacteristic
16C2×D3×C4⋊C4—21no—
17C2×C4×D12C2212yes—
18C2×C4×D12C2211yescharacteristic
19C3×#390C2212yes—
20C3×#388—22no—
21C3×#384C2211yescharacteristic
22C3×#380—21no—
23C3×C4×C4⋊C4C2212yes—
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#832C2212yes—
26C2×#832C2211yescharacteristic
27C4×#213C418yescomplemented
28#4987—21no—
29C2×#809C2212yes—
30C2×#809C2211yescharacteristic
31C2×#807C2211yescharacteristic
32#4935C2211yescharacteristic
33#4928C2212yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35C2×#803C2212yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37#4668—22no—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2212yes—
43C2×C4×D8—31no—
44#2354D311yescharacteristic, complemented
45D4×C4⋊C4—31no—
46#1936—32no—
47#1921—32no—
48#1879—32no—
49#1834—31no—
50#1802—31no—
51#1802—31no—
52#1770—31no—
53#1762—32no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×C4×D3—22no—
59C3×C4×D4—22no—
60C3×C4×D4C2×C414yescomplemented
61C3×#117—44no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42—22no—
72C2×C3×C42C2312yes—
73C2×C3×C42C2311yescharacteristic
74C2×#213—22no—
75C2×#213—22no—
76C2×#213C2×C414yes—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—22no—
84C22×C3⋊C8C2312yes—
85C22×C3⋊C8C2311yescharacteristic
86C22×C3⋊C8C2311yescharacteristic
87D3×C4⋊C4—44no—
88C4×D12—22no—
89C4×D12—24no—
90C4×D12—42no—
91C4×D12C2×C414yes—
92C3×#95—22no—
93C3×#95D412yes—
94#832—42no—
95#832C2×C414yescomplemented
96C3⋊#95—22no—
97#809—24no—
98#809C2×C414yes—
99#807C2×C414yescomplemented
100C4⋊(C3⋊C8)—22no—
101#803—42no—
102C4×C3⋊C8C2×C414yescomplemented
103C22×D8—31no—
104C2×C4×D4—31no—
105C2×C4×D4—32no—
106C2×C4×D4C2×D311yescharacteristic
107C2×#117—21no—
108C22×C4⋊C4—61no—
109C4×D8—38no—
110C2×#107—31no—
111C2×C4⋊C8—31no—
112C2×#102—31no—
113C2×#102—31no—
114C2×#102—32no—
115C2×#102—32no—
116C2×#97—32no—
117C2×C4×C8—31no—
118#390C2×D312yes—
119#389—32no—
120#388—22no—
121#385—62no—
122#384—31no—
123#384C2×D311yescharacteristic
124#380—21no—
125#380—61no—
126C4×C4⋊C4C2×D312yes—
127C4×C4⋊C4C2×D311yescharacteristic
128#340—32no—
129#340—32no—
130#340—32no—
131#340—32no—
132#336—32no—
133C24×C3—21no—
134C23×D3—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4C22×C412yes—
141C22×C3×C4—21no—
142C22×C3×C4—24no—
143C22×C3×C4—22no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—22no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12C22×C412yes—
157C2×D12#4112yes—
158C2×D12#4112yes—
159C2×C4×D3—24no—
160C2×C4×D3—42no—
161C2×C4×D3—42no—
162C2×C4×D3—44no—
163C2×C4×D3—44no—
164C2×C4×D3—82no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—81no—
172C3×C4⋊C4C22×C412yes—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—82no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—44no—
179C3×C42C22×C412yes—
180C3×C42#4112yes—
181#213—28no—
182(C2×D3)⋊C4—24no—
183(C2×D3)⋊C4—44no—
184C4⋊Dic3—21no—
185C4⋊Dic3—22no—
186C4⋊Dic3—22no—
187C4⋊Dic3—42no—
188C4⋊Dic3—81no—
189C4⋊Dic3C22×C412yes—
190Dic3⋊C4—44no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8C22×C412yes—
199C2×C3⋊C8C22×C412yes—
200C22×D4—31no—
201C22×D4C22×D311yescharacteristic
202C23×C4—21no—
203C23×C4—61no—
204C23×C4—62no—
205C2×D8—34no—
206C2×D8—62no—
207C2×D8—62no—
208C22×C8—31no—
209C22×C8—31no—
210C22×C8—32no—
211C4×D4—22no—
212C4×D4—34no—
213C4×D4—64no—
214C4×D4—62no—
215C4×D4—122no—
216C4×D4C4×D314yescomplemented
217#117—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—124no—
226C2×C4⋊C4#6711yescharacteristic
227C2×C4⋊C4#6711yescharacteristic
228C2×C4⋊C4C22×D312yes—
229C2×C4⋊C4C22×D311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—62no—
235C2×C42—22no—
236C2×C42C22×D312yes—
237C2×C42C22×D311yescharacteristic
238#107—34no—
239C4⋊C8—62no—
240#102—34no—
241#102—34no—
242#102—64no—
243#102—122no—
244#97—122no—
245C4×C8—34no—
246#95—22no—
247#95—62no—
248#95#6712yes—
249C23×C3—22no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3C22×D411yescharacteristic
257C22×D3—22no—
258C22×D3—22no—
259C22×D3—22no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C3×D4—22no—
265C3×D4—22no—
266C3×D4—41no—
267C3×D4—41no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—48no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4C4×D412yes—
284C2×C3×C4C4×D412yes—
285C2×C3×C4AGL(1,ℤ/8)12yes—
286C2×C3×C4AGL(1,ℤ/8)12yes—
287C2×C3×C4C23×C411yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4H(5,𝔽2)11yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×Dic3—22no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299D12—22no—
300D12—22no—
301D12—22no—
302D12#11714yescomplemented
303C4×D3—48no—
304C4×D3—88no—
305C3⋊C8—22no—
306C3⋊C8—22no—
307C3⋊C8—42no—
308C24—21no—
309C24—61no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—32no—
316C2×D4—32no—
317C2×D4—32no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4C2×C4×D312yes—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—61no—
328C22×C4—64no—
329C22×C4—62no—
330C22×C4—124no—
331C22×C4—124no—
332C22×C4—122no—
333C22×C4—122no—
334C22×C4—242no—
335C22×C4C2×#6712yes—
336C22×C4C2×#6712yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6711yescharacteristic
339C22×C4C23×D311yescharacteristic
340D8—68no—
341C2×C8—32no—
342C2×C8—32no—
343C2×C8—61no—
344C2×C8—61no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—62no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—32no—
356C4⋊C4—42no—
357C4⋊C4—61no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—81no—
361C4⋊C4—124no—
362C4⋊C4—122no—
363C4⋊C4—241no—
364C4⋊C4C2×C4×D312yes—
365C22⋊C4—24no—
366C22⋊C4—44no—
367C22⋊C4—64no—
368C22⋊C4—82no—
369C22⋊C4—124no—
370C42—22no—
371C42—22no—
372C42—44no—
373C42C2×C4×D312yes—
374C42#23512yes—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—81no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3C2×C4×D411yescharacteristic
383C22×C3#54611yescharacteristic
384C22×C3#54811yescharacteristic
385C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
386C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C3#57611yescharacteristic
388C22×C3#57811yescharacteristic
389C2×D3—24no—
390C2×D3—42no—
391C2×D3—42no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—42no—
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—82no—
401C3×C4#44214yes—
402C3×C4C2×#11711yescharacteristic
403C3×C4#52011yescharacteristic
404C3×C4#55211yescharacteristic
405C3×C4#55311yescharacteristic
406Dic3—42no—
407Dic3—82no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—62no—
416C23—62no—
417C23—62no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23—121no—
422C23C22×#6711yescharacteristic
423D4—22no—
424D4—22no—
425D4—34no—
426D4—41no—
427D4—41no—
428D4—62no—
429D4—62no—
430D4—62no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—48no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—62no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—128no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—248no—
454C2×C4C4×#6712yes—
455C2×C4C4×#6712yes—
456C2×C4#93212yes—
457C2×C4#94912yes—
458C2×C4C22×C4×D311yescharacteristic
459C2×C4C2×#23511yescharacteristic
460C2×C4D3×#4111yescharacteristic
461C2×C4D3×#4111yescharacteristic
462C2×C4#100911yescharacteristic
463C2×C4#101011yescharacteristic
464C8—62no—
465C8—62no—
466C8—122no—
467C2×C3—82no—
468C2×C3#235211yescharacteristic
469C2×C3C2×#44211yescharacteristic
470C2×C3#295111yescharacteristic
471C2×C3#332511yescharacteristic
472C2×C3#333011yescharacteristic
473C2×C3#336611yescharacteristic
474C2×C3#336711yescharacteristic
475D3—44no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—64no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—122no—
485C22—122no—
486C22—122no—
487C22—122no—
488C22—122no—
489C22—122no—
490C22C2×C4×#6711yescharacteristic
491C22C2×#93211yescharacteristic
492C22C2×#94911yescharacteristic
493C22#582911yescharacteristic
494C22#583111yescharacteristic
495C22#583511yescharacteristic
496C22#583811yescharacteristic
497C4—22no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4—122no—
503C4—242no—
504C4#501414yes—
505C4D3×#11711yescharacteristic
506C4#552311yescharacteristic
507C4#552911yescharacteristic
508C4#553011yescharacteristic
509C3#2275811yescharacteristic, complemented, Sylow
510C2—82no—
511C2—124no—
512C2#8227211yescharacteristic
513C2C2×#501411yescharacteristic
514C2#8255711yescharacteristic
515C2#8352711yescharacteristic
516C2#8358111yescharacteristic
517C2#8365411yescharacteristic
518C2#8366811yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2354C2
socleC23×C3C22×D4
prime radicalC22×#213C22
cosocleC22×C3×C4C24
hypercenter#2354D3
hypocenterC3#22758
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
1. C3×#2354C2semidirect
3. C4⋊C4×D12C2semidirect
4. #71604C2semidirect
6. #69645C2semidirect
7. #69584C2semidirect
8. #69574C2semidirect
9. #69477C2semidirect
10. #69474C2semidirect
11. #69461C2semidirect
27. C4×#213C4semidirect
44. #2354D3semidirect
60. C3×C4×D4C2×C4semidirect
95. #832C2×C4semidirect
99. #807C2×C4semidirect
102. C4×C3⋊C8C2×C4semidirect
216. C4×D4C4×D3semidirect
302. D12#117semidirect
509. C3#22758semidirect

Maximal subgroups

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

subgroupindex
C3×#23542
C2×C4×#2132
C4⋊C4×D122
#716042
#696572
#696452
#695842
#695742
#694772
#694742
#694612
#227583