Finite Group Databasebeta

← #10876173

Subgroups — #10876173

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×#5221C211yesmaximal, characteristic
2#73863C211yesmaximal, characteristic, complemented
3C3×#1786C211yesmaximal, characteristic
4#72128C211yesmaximal, characteristic
5#70181C212yesmaximal, complemented
6#70174C212yesmaximal, complemented
7#70166C212yesmaximal, complemented
8#70155C212yesmaximal, complemented
9#69927C211yesmaximal, characteristic, complemented
10#69910C211yesmaximal, characteristic, complemented
11#69505C211yesmaximal, characteristic, complemented
12#27588—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#217—21no—
15C22×#217—21no—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4×Dic3C2211yescharacteristic
18C2×C3×#102C2212yes—
19C2×C3×#102C2212yes—
20C2×C3×C22⋊C8C2211yescharacteristic
21C3×#380C2211yescharacteristic
22#5221—21no—
23#5221—21no—
24#5221C2212yes—
25#5221C2212yes—
26C2×#832C2212yes—
27C2×#832C2212yes—
28#5212C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#818C2212yescomplemented
31#4977C2211yescharacteristic, complemented
32#4977C2211yescharacteristic, complemented
33Dic3×C4⋊C4C2211yescharacteristic, complemented
34Dic3×C4⋊C4C2211yescharacteristic, complemented
35C2×#808C2212yescomplemented
36#4943—21no—
37Dic3×C22⋊C4—21no—
38#4651—21no—
39#4648—21no—
40#4551C2212yescomplemented
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#1853—31no—
51#1852—31no—
52#1817—31no—
53#1786—31no—
54#1786D311yescharacteristic
55C25×C3—21no—
56C22×C3×D4—24no—
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—42no—
61C22×C3×D4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C23×Dic3D411yescharacteristic
65C23×Dic3D411yescharacteristic
66C22×C3×C8C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—44no—
76C2×#217—44no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—42no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
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×#102—24no—
92C3×#102—42no—
93C3×C22⋊C8—22no—
94C3×#95—21no—
95#832—24no—
96#832—42no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104#830—22no—
105#818—42no—
106#808—42no—
107C23×D4C2×D311yescharacteristic
108C22×C4⋊C4—31no—
109C22×C22⋊C4—61no—
110C22×C22⋊C4—61no—
111C22×C42—31no—
112C2×#107—32no—
113C2×#106—32no—
114C2×#102—32no—
115C2×#102—32no—
116C2×#102C2×D312yes—
117C2×#102C2×D312yes—
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#340—32no—
136#336—32no—
137#336—32no—
138C24×C3—24no—
139C24×C3—22no—
140C24×C3—42no—
141C24×C3—42no—
142C24×C3—42no—
143C24×C3—42no—
144C24×C3—42no—
145C24×C3—42no—
146C24×C3C2×D411yescharacteristic
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—48no—
154C2×C3×D4—48no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4#4114yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4#4112yes—
164C22×C3×C4#4112yes—
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—41no—
175C22×Dic3—42no—
176C22×Dic3—42no—
177C22×Dic3—42no—
178C22×Dic3C2×D411yescharacteristic
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C22⋊C4—42no—
188#217—44no—
189#217—44no—
190#217—44no—
191#217—42no—
192#217—44no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—42no—
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—21no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4C22×D312yes—
216C23×C4—31no—
217C23×C4—31no—
218C23×C4—61no—
219C23×C4C22×D311yescharacteristic
220C22×C8—32no—
221C22×C8C22×D312yes—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—122no—
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—122no—
247#102—24no—
248#102—42no—
249#102—64no—
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—24no—
266C23×C3—48no—
267C23×C3—48no—
268C23×C3—48no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—84no—
276C23×C3—84no—
277C23×C3—84no—
278C23×C3—84no—
279C23×C3#12312yes—
280C23×C3#13512yes—
281C23×C3AGL(1,ℤ/8)12yes—
282C23×C3C22×D411yescharacteristic
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—48no—
286C3×D4—42no—
287C3×D4—42no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4#12312yes—
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C4H(5,𝔽2)12yes—
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—21no—
314C2×Dic3—44no—
315C2×Dic3—44no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—84no—
326C2×Dic3AGL(1,ℤ/8)12yes—
327C3×C8—42no—
328C3⋊C8—42no—
329C24—24no—
330C24—22no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24D3×D411yescharacteristic
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4#23714yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—31no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—121no—
366C22×C4—122no—
367C22×C4—122no—
368C22×C4—122no—
369C22×C4#23512yes—
370C22×C4D3×D411yescharacteristic
371C22×C4#23712yes—
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—62no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—122no—
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—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—816no—
410C22×C3—82no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3#48214yes—
415C22×C3C2×#12311yescharacteristic
416C22×C3#52511yescharacteristic
417C22×C3#53511yescharacteristic
418C22×C3C2×#13511yescharacteristic
419C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
420C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
421C22×C3#57511yescharacteristic
422C3×C4—22no—
423C3×C4—81no—
424C3×C4—81no—
425C3×C4#55112yes—
426C3×C4#55112yes—
427Dic3—22no—
428Dic3—41no—
429Dic3—41no—
430Dic3—41no—
431Dic3—81no—
432Dic3—81no—
433Dic3—82no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—24no—
439C23—48no—
440C23—48no—
441C23—48no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23#88312yes—
453C23#91212yes—
454C23#91412yes—
455C23C2×D3×D411yescharacteristic
456D4—24no—
457D4—24no—
458D4—48no—
459D4—42no—
460D4—42no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—32no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
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—124no—
484C2×C4—124no—
485C2×C4—122no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—122no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—244no—
495C2×C4#88312yes—
496C2×C4C2×#23511yescharacteristic
497C2×C4C2×#23712yes—
498C2×C4C2×#23711yescharacteristic
499C2×C4#100412yes—
500C8—42no—
501C8—122no—
502C2×C3—24no—
503C2×C3—84no—
504C2×C3#248211yescharacteristic
505C2×C3C2×#48211yescharacteristic
506C2×C3#309411yescharacteristic
507C2×C3#310511yescharacteristic
508C2×C3#311411yescharacteristic
509C2×C3#321111yescharacteristic
510C2×C3#321711yescharacteristic
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—816no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22#478414yes—
526C22C2×#88311yescharacteristic
527C22#558611yescharacteristic
528C22#559211yescharacteristic
529C22C2×#91211yescharacteristic
530C22C2×#91411yescharacteristic
531C22D3×AGL(1,ℤ/8)11yescharacteristic
532C22#577311yescharacteristic
533C4—22no—
534C4—62no—
535C4—81no—
536C4—81no—
537C4—121no—
538C4—121no—
539C4—121no—
540C4—241no—
541C4—242no—
542C4—241no—
543C4#549512yes—
544C4#558712yes—
545C3#2758811yescharacteristic, complemented, Sylow
546C2—24no—
547C2—84no—
548C2#7383211yescharacteristic
549C2C2×#478411yescharacteristic
550C2#7893311yescharacteristic
551C2#7998911yescharacteristic
552C2#7999511yescharacteristic
553C2#8013511yescharacteristic
554C2#8014111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenter#1786D3
hypocenterC3#27588
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. #73863C2semidirect
5. #70181C2semidirect
6. #70174C2semidirect
7. #70166C2semidirect
8. #70155C2semidirect
9. #69927C2semidirect
10. #69910C2semidirect
11. #69505C2semidirect
30. C2×#818C22semidirect
31. #4977C22semidirect
32. #4977C22semidirect
33. Dic3×C4⋊C4C22semidirect
34. Dic3×C4⋊C4C22semidirect
35. C2×#808C22semidirect
40. #4551C22semidirect
41. #4550C22semidirect
42. #4550C22semidirect
43. #4529C22semidirect
545. C3#27588semidirect

Maximal subgroups

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

subgroupindex
C2×#52212
#738632
C3×#17862
#721282
#701812
#701742
#701662
#701552
#699272
#699102
#695052
#275883