Finite Group Databasebeta

← #10876172

Subgroups — #10876172

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2283C211yesmaximal, characteristic, complemented
2C2×#4939C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic, complemented
4#69558C212yesmaximal, complemented
5#69557C212yesmaximal, complemented
6#69521C211yesmaximal, characteristic
7#69521C211yesmaximal, characteristic
8#69505C211yesmaximal, characteristic, complemented
9#69504C211yesmaximal, characteristic, complemented
10#69463C212yesmaximal, complemented
11#69462C212yesmaximal, complemented
12#27588—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4—21no—
16C22×(C2×D3)⋊C4—21no—
17C22×C4⋊Dic3C2211yescharacteristic
18C3×#383—21no—
19C3×#382C2211yescharacteristic, complemented
20C3×#382C2211yescharacteristic, complemented
21C3×#381—21no—
22C3×#380C2211yescharacteristic
23C3×#380C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic, complemented
25C3×C4×C4⋊C4C2211yescharacteristic, complemented
26C3×C4×C22⋊C4—21no—
27C3×#376—21no—
28C2×#830C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#809C2212yes—
31C2×#809C2212yes—
32C2×#809C2212yes—
33C2×#809C2212yes—
34C2×#808C2212yescomplemented
35C2×#807C2212yescomplemented
36#4939—21no—
37#4939—21no—
38#4939C2212yes—
39#4939C2212yes—
40#4529C2212yescomplemented
41#4529C2212yescomplemented
42#4524C2212yescomplemented
43#4524C2212yescomplemented
44C2×#390—31no—
45#2283D311yescharacteristic, complemented
46#2092—32no—
47#2073—32no—
48#1926—32no—
49#1925—32no—
50#1853—31no—
51#1852—31no—
52#1817—31no—
53#1786—31no—
54#1786—31no—
55C24×D3—21no—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—21no—
60C22×D12—21no—
61C22×D12—24no—
62C22×D12—22no—
63C22×D12—22no—
64C22×D12—42no—
65C22×D12C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42D411yescharacteristic
77C2×C3×C42D411yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—44no—
85C2×(C2×D3)⋊C4—44no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—42no—
88C2×C4⋊Dic3C2312yes—
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2312yes—
91C3×#95—21no—
92C3×#95—21no—
93C3×#95—21no—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—21no—
98C3×#95—21no—
99#830—22no—
100#830—22no—
101#809—24no—
102#809—24no—
103#809—42no—
104#809—42no—
105#808—42no—
106#807—42no—
107C23×D4—31no—
108C22×C4⋊C4—31no—
109C22×C22⋊C4—61no—
110C22×C22⋊C4—61no—
111C22×C42C2×D311yescharacteristic
112C2×#107—32no—
113C2×#106—32no—
114C2×#102—32no—
115C2×#102—32no—
116C2×#102—32no—
117C2×#102—32no—
118C2×C22⋊C8—31no—
119C2×C22⋊C8—31no—
120#390—32no—
121#390—32no—
122#390—61no—
123#390—61no—
124#383—21no—
125#382C2×D311yescharacteristic, complemented
126#382C2×D311yescharacteristic, complemented
127#381—21no—
128#380C2×D311yescharacteristic
129#380C2×D311yescharacteristic
130C4×C4⋊C4C2×D311yescharacteristic, complemented
131C4×C4⋊C4C2×D311yescharacteristic, complemented
132C4×C22⋊C4—21no—
133#376—21no—
134#340—32no—
135#340—32no—
136#336—32no—
137#336—32no—
138C24×C3C2×D411yescharacteristic
139C23×D3—24no—
140C23×D3—22no—
141C23×D3—42no—
142C23×D3—42no—
143C23×D3—42no—
144C23×D3—42no—
145C23×D3—42no—
146C23×D3—42no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—41no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—22no—
168C22×Dic3—42no—
169C22×Dic3—42no—
170C22×Dic3—42no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—24no—
176C2×D12—22no—
177C2×D12—48no—
178C2×D12—48no—
179C2×D12—42no—
180C2×D12—42no—
181C2×D12#4114yes—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—42no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—21no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42—42no—
195C3×C42—42no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—44no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—42no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C25—61no—
210C22×D4—32no—
211C22×D4—64no—
212C22×D4—62no—
213C22×D4—62no—
214C22×D4—61no—
215C22×D4—122no—
216C23×C4—61no—
217C23×C4#6711yescharacteristic
218C23×C4#6711yescharacteristic
219C23×C4C22×D311yescharacteristic
220C22×C8—32no—
221C22×C8—32no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—122no—
225C2×C4⋊C4C22×D311yescharacteristic
226C2×C4⋊C4C22×D311yescharacteristic
227C2×C4⋊C4C22×D311yescharacteristic
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C22⋊C4—124no—
236C2×C22⋊C4—124no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42#6711yescharacteristic
242C2×C42#6711yescharacteristic
243C2×C42C22×D311yescharacteristic
244C2×C42C22×D311yescharacteristic
245#107—122no—
246#106—122no—
247#102—64no—
248#102—64no—
249#102—122no—
250#102—122no—
251C22⋊C8—62no—
252C22⋊C8—62no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3#12312yes—
266C23×C3#13512yes—
267C23×C3AGL(1,ℤ/8)12yes—
268C23×C3C22×D411yescharacteristic
269C22×D3—24no—
270C22×D3—48no—
271C22×D3—48no—
272C22×D3—48no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—84no—
280C22×D3—84no—
281C22×D3—84no—
282C22×D3—84no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—44no—
299C2×C3×C4—44no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4#12312yes—
313C2×C3×C4AGL(1,ℤ/8)12yes—
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4112yes—
317C2×C3×C4H(5,𝔽2)12yes—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—84no—
322D12—24no—
323D12—24no—
324D12—48no—
325D12—42no—
326D12—42no—
327C3⋊C8—42no—
328C3⋊C8—42no—
329C24—64no—
330C24—62no—
331C24—122no—
332C24—122no—
333C24—122no—
334C24—122no—
335C24—122no—
336C24—122no—
337C24C2×#6711yescharacteristic
338C2×D4—34no—
339C2×D4—64no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—128no—
346C2×D4—128no—
347C2×D4—122no—
348C2×D4—122no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—62no—
362C22×C4—122no—
363C22×C4—122no—
364C22×C4—122no—
365C22×C4#23512yes—
366C22×C4#23512yes—
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C23×D311yescharacteristic
373C2×C8—62no—
374C2×C8—62no—
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—122no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—124no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—124no—
392C42—22no—
393C42—22no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—42no—
399C42—42no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3#48214yes—
407C22×C3C2×#12311yescharacteristic
408C22×C3#52511yescharacteristic
409C22×C3#53511yescharacteristic
410C22×C3C2×#13511yescharacteristic
411C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
412C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C3#57511yescharacteristic
414C2×D3—44no—
415C2×D3—44no—
416C2×D3—44no—
417C2×D3—816no—
418C2×D3—82no—
419C2×D3—82no—
420C2×D3—82no—
421C2×D3—82no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4—81no—
430C3×C4—81no—
431C3×C4#55112yes—
432C3×C4#55112yes—
433Dic3—82no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—64no—
439C23—128no—
440C23—128no—
441C23—128no—
442C23—122no—
443C23—122no—
444C23—122no—
445C23—122no—
446C23—122no—
447C23—122no—
448C23—244no—
449C23—244no—
450C23—244no—
451C23—244no—
452C23#92912yes—
453C23#94912yes—
454C23#95012yes—
455C23C22×#6711yescharacteristic
456D4—64no—
457D4—64no—
458D4—128no—
459D4—122no—
460D4—122no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—244no—
494C2×C4#92912yes—
495C2×C4#94912yes—
496C2×C4C2×#23511yescharacteristic
497C2×C4C2×#23511yescharacteristic
498C2×C4D3×#4112yes—
499C2×C4#100912yes—
500C8—122no—
501C8—122no—
502C2×C3—24no—
503C2×C3#248211yescharacteristic
504C2×C3C2×#48211yescharacteristic
505C2×C3#309411yescharacteristic
506C2×C3#310511yescharacteristic
507C2×C3#311411yescharacteristic
508C2×C3#321111yescharacteristic
509C2×C3#321711yescharacteristic
510D3—84no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—124no—
518C22—124no—
519C22—124no—
520C22—2416no—
521C22—242no—
522C22—242no—
523C22—242no—
524C22—242no—
525C22#500314yes—
526C22C2×#92911yescharacteristic
527C22C2×#94911yescharacteristic
528C22C2×#94911yescharacteristic
529C22C2×#95011yescharacteristic
530C22#582211yescharacteristic
531C22#582811yescharacteristic
532C22#583011yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—242no—
543C4#552712yes—
544C4#552712yes—
545C3#2758811yescharacteristic, complemented, Sylow
546C2—24no—
547C2—244no—
548C2C2×#500311yescharacteristic
549C2#8221711yescharacteristic
550C2#8228211yescharacteristic
551C2#8314111yescharacteristic
552C2#8314311yescharacteristic
553C2#8314411yescharacteristic
554C2#8314611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2283C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#2283D3
hypocenterC3#27588
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
1. C3×#2283C2semidirect
3. #71531C2semidirect
4. #69558C2semidirect
5. #69557C2semidirect
8. #69505C2semidirect
9. #69504C2semidirect
10. #69463C2semidirect
11. #69462C2semidirect
19. C3×#382C22semidirect
20. C3×#382C22semidirect
24. C3×C4×C4⋊C4C22semidirect
25. C3×C4×C4⋊C4C22semidirect
34. C2×#808C22semidirect
35. C2×#807C22semidirect
40. #4529C22semidirect
41. #4529C22semidirect
42. #4524C22semidirect
43. #4524C22semidirect
45. #2283D3semidirect
125. #382C2×D3semidirect
126. #382C2×D3semidirect
130. C4×C4⋊C4C2×D3semidirect
131. C4×C4⋊C4C2×D3semidirect
545. C3#27588semidirect

Maximal subgroups

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

subgroupindex
C3×#22832
C2×#49392
#715312
#695582
#695572
#695212
#695052
#695042
#694632
#694622
#275883