Finite Group Databasebeta

← #10876175

Subgroups — #10876175

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
1C2×#4988C211yesmaximal, characteristic
2#73863C211yesmaximal, characteristic, complemented
3C3×#1853C211yesmaximal, characteristic, complemented
4#71555C211yesmaximal, characteristic, complemented
5#70586C212yesmaximal, complemented
6#70569C212yesmaximal, complemented
7#69959C211yesmaximal, characteristic
8#69910C211yesmaximal, characteristic, complemented
9#69521C211yesmaximal, characteristic
10#68903C212yesmaximal, complemented
11#68791C212yesmaximal, complemented
12#27588—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×(C2×D3)⋊C4—21no—
16C22×(C2×D3)⋊C4—21no—
17C22×C4×Dic3C2211yescharacteristic
18C2×C3×#106C2212yescomplemented
19C2×C3×C22⋊C8C2211yescharacteristic
20C3×#380C2211yescharacteristic
21#5212C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#821C2212yes—
24C2×#821C2212yes—
25#4988—21no—
26#4988—21no—
27#4988C2212yes—
28#4988C2212yes—
29#4977C2211yescharacteristic, complemented
30#4977C2211yescharacteristic, complemented
31Dic3×C4⋊C4C2211yescharacteristic, complemented
32Dic3×C4⋊C4C2211yescharacteristic, complemented
33C2×#809C2212yes—
34C2×#809C2212yes—
35C2×#807C2212yescomplemented
36#4943—21no—
37Dic3×C22⋊C4—21no—
38#4651—21no—
39#4648—21no—
40C3×#340C2212yescomplemented
41#4550C2212yescomplemented
42#4550C2212yescomplemented
43#4529C2212yescomplemented
44C2×#390—31no—
45#2283—31no—
46#2092—32no—
47#2073—32no—
48#1926—32no—
49#1925—32no—
50#1853D311yescharacteristic, complemented
51#1852—31no—
52#1817—31no—
53#1786—31no—
54#1786—31no—
55C24×D3—21no—
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3D411yescharacteristic
59C23×Dic3D411yescharacteristic
60C22×D12—21no—
61C22×D12—24no—
62C22×D12—22no—
63C22×D12—22no—
64C22×D12—42no—
65C22×D12C2312yes—
66C22×C3×C8C2312yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×#217—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—44no—
79C2×(C2×D3)⋊C4—44no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C2×C4×Dic3C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91C3×#106—42no—
92C3×C22⋊C8—22no—
93C3×#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101#830—22no—
102#821—24no—
103#821—42no—
104#809—24no—
105#809—42no—
106#807—42no—
107C23×D4—31no—
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C22⋊C4—61no—
110C22×C22⋊C4—61no—
111C22×C42—31no—
112C2×#107—32no—
113C2×#106C2×D312yescomplemented
114C2×#102—32no—
115C2×#102—32no—
116C2×#102—32no—
117C2×#102—32no—
118C2×C22⋊C8—31no—
119C2×C22⋊C8C2×D311yescharacteristic
120#390—32no—
121#390—32no—
122#390—61no—
123#390—61no—
124#383—61no—
125#382—31no—
126#382—31no—
127#381—61no—
128#380—31no—
129#380C2×D311yescharacteristic
130C4×C4⋊C4—31no—
131C4×C4⋊C4—31no—
132C4×C22⋊C4—61no—
133#376—61no—
134#340—32no—
135#340C2×D312yescomplemented
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—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—41no—
167C22×Dic3C2×D411yescharacteristic
168C22×Dic3C2×D411yescharacteristic
169C22×Dic3C2×D411yescharacteristic
170C22×Dic3C2×D411yescharacteristic
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—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C22⋊C4—42no—
191#217—42no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—44no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C4×Dic3—21no—
203C4×Dic3—21no—
204C4×Dic3—42no—
205C4×Dic3—42no—
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—61no—
214C22×D4—62no—
215C22×D4—122no—
216C23×C4—21no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4C22×D311yescharacteristic
220C22×C8—32no—
221C22×C8C22×D312yes—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22×D312yes—
227C2×C4⋊C4C22×D311yescharacteristic
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—124no—
236C2×C22⋊C4—124no—
237C2×C42—31no—
238C2×C42—31no—
239C2×C42—31no—
240C2×C42—31no—
241C2×C42—62no—
242C2×C42—62no—
243C2×C42—61no—
244C2×C42—61no—
245#107—122no—
246#106—42no—
247#102—64no—
248#102—64no—
249#102—122no—
250#102—122no—
251C22⋊C8—22no—
252C22⋊C8—62no—
253#95—21no—
254#95—61no—
255#95—61no—
256#95—61no—
257#95—61no—
258#95—61no—
259#95—61no—
260#95—61no—
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—24no—
286C2×C3×C4—24no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—84no—
297C2×C3×C4#12312yes—
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4112yes—
301C2×C3×C4H(5,𝔽2)12yes—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—44no—
314C2×Dic3—44no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3AGL(1,ℤ/8)12yes—
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—
337C24D3×D411yescharacteristic
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—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—121no—
369C22×C4#23512yes—
370C22×C4D3×D411yescharacteristic
371C22×C4#23912yes—
372C22×C4C23×D311yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C22⋊C4—42no—
387C22⋊C4—124no—
388C22⋊C4—124no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—122no—
392C42—62no—
393C42—62no—
394C42—61no—
395C42—61no—
396C42—61no—
397C42—61no—
398C42—122no—
399C42—122no—
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—81no—
424C3×C4—81no—
425C3×C4—82no—
426C3×C4#55112yes—
427C3×C4#55112yes—
428Dic3—22no—
429Dic3—41no—
430Dic3—41no—
431Dic3—41no—
432Dic3—81no—
433Dic3—81no—
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#89412yes—
453C23#91412yes—
454C23#91612yes—
455C23C2×D3×D411yescharacteristic
456D4—64no—
457D4—64no—
458D4—128no—
459D4—122no—
460D4—122no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—32no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—84no—
487C2×C4—124no—
488C2×C4—124no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4#89412yes—
496C2×C4C2×#23511yescharacteristic
497C2×C4#100412yes—
498C2×C4C2×#23911yescharacteristic
499C2×C4D3×#4112yes—
500C8—42no—
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#486614yes—
526C22C2×#89411yescharacteristic
527C22#559211yescharacteristic
528C22#565611yescharacteristic
529C22C2×#91411yescharacteristic
530C22C2×#91611yescharacteristic
531C22D3×AGL(1,ℤ/8)11yescharacteristic
532C22#577711yescharacteristic
533C4—22no—
534C4—62no—
535C4—82no—
536C4—81no—
537C4—81no—
538C4—121no—
539C4—121no—
540C4—121no—
541C4—241no—
542C4—241no—
543C4#555612yes—
544C4#563712yes—
545C3#2758811yescharacteristic, complemented, Sylow
546C2—24no—
547C2—244no—
548C2#7461411yescharacteristic
549C2C2×#486611yescharacteristic
550C2#7921711yescharacteristic
551C2#8005711yescharacteristic
552C2#8014111yescharacteristic
553C2#8130911yescharacteristic
554C2#8141511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1853C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1853D3
hypocenterC3#27588
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. #73863C2semidirect
3. C3×#1853C2semidirect
4. #71555C2semidirect
5. #70586C2semidirect
6. #70569C2semidirect
8. #69910C2semidirect
10. #68903C2semidirect
11. #68791C2semidirect
18. C2×C3×#106C22semidirect
29. #4977C22semidirect
30. #4977C22semidirect
31. Dic3×C4⋊C4C22semidirect
32. Dic3×C4⋊C4C22semidirect
35. C2×#807C22semidirect
40. C3×#340C22semidirect
41. #4550C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
50. #1853D3semidirect
113. C2×#106C2×D3semidirect
135. #340C2×D3semidirect
545. C3#27588semidirect

Maximal subgroups

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

subgroupindex
C2×#49882
#738632
C3×#18532
#715552
#705862
#705692
#699592
#699102
#695212
#689032
#687912
#275883