Finite Group Databasebeta

← #10877264

Subgroups — #10877264

Order 768.

882 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C2×#4936C211yesmaximal, characteristic, complemented
3#71677C211yesmaximal, characteristic
4#71677C211yesmaximal, characteristic
5#71605C212yesmaximal, complemented
6#71530C211yesmaximal, characteristic, complemented
7#69654C212yesmaximal, complemented
8#69613C212yesmaximal, complemented
9#69505C211yesmaximal, characteristic, complemented
10#69504C211yesmaximal, characteristic, complemented
11#69468C212yesmaximal, complemented
12#27684—31nomaximal, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C3×#390C2211yescharacteristic
20C3×#389C2211yescharacteristic
21C3×#388—21no—
22C3×#386—21no—
23C3×#380C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C3×C4×C22⋊C4—21no—
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×C3⋊#95—21no—
29C2×C3⋊#95—21no—
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#808C2212yes—
33C2×#807C2212yes—
34#4936—21no—
35#4936—21no—
36#4936C2212yes—
37#4936C2212yes—
38C2×C4⋊(C3⋊C8)C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C4×#121D311yescharacteristic
45C2×#382—31no—
46#2101—32no—
47#2092—32no—
48#1949—32no—
49#1874—31no—
50#1874—31no—
51#1853—31no—
52#1852—31no—
53#1804—32no—
54#1797—31no—
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—21no—
62C23×Dic3—21no—
63C3×#121—22no—
64C3×#121Q814yes—
65C3×C4×D4—24no—
66C3×C4×D4—44no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42—24no—
77C2×C3×C42—21no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—42no—
85C2×C4⋊Dic3—42no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2312yes—
88C22×C3⋊C8C2312yes—
89C22×C3⋊C8C2312yes—
90C3×#95—21no—
91C3×#95—21no—
92C3×#95—21no—
93#832—42no—
94#832—42no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#830—22no—
100#830—22no—
101#808—42no—
102#807—42no—
103C4⋊(C3⋊C8)—24no—
104C2×#121C2×D311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4C2×D311yescharacteristic
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109C22×C42C2×D311yescharacteristic
110C2×#107—32no—
111C2×#106—32no—
112C2×C4⋊C8—32no—
113C2×#102—32no—
114C2×#102—32no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—31no—
117#390C2×D311yescharacteristic
118#389C2×D311yescharacteristic
119#388—21no—
120#386—21no—
121#382—32no—
122#382—32no—
123#382—61no—
124#382—61no—
125#380C2×D311yescharacteristic
126C4×C4⋊C4C2×D311yescharacteristic
127C4×C22⋊C4—21no—
128C2×#95—61no—
129C2×#95—61no—
130#340—32no—
131#340—32no—
132#340—32no—
133#340—32no—
134#336—32no—
135C24×C3—21no—
136C24×C3C2×D411yescharacteristic
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4C2×Q812yes—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×Q812yes—
164C22×C3×C4#4112yes—
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—42no—
169C22×Dic3—42no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C22×Dic3—42no—
173C22×Dic3—42no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4C2×Q812yes—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—42no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—42no—
188C3×C42#4114yes—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C22×D4—21no—
202C22×D4C22×D311yescharacteristic
203C23×C4—21no—
204C23×C4—61no—
205C23×C4—61no—
206C23×C4#6711yescharacteristic
207C23×C4#6711yescharacteristic
208C23×C4C22×D311yescharacteristic
209C22×C8—32no—
210C22×C8—32no—
211#121—22no—
212#121Dic614yes—
213C4×D4—24no—
214C4×D4—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—32no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—122no—
221C2×C4⋊C4—122no—
222C2×C4⋊C4C22×D311yescharacteristic
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42#6711yescharacteristic
233C2×C42#6711yescharacteristic
234C2×C42C22×D311yescharacteristic
235C2×C42C22×D311yescharacteristic
236#107—122no—
237#106—122no—
238C4⋊C8—64no—
239#102—122no—
240#102—122no—
241C22⋊C8—62no—
242C22⋊C8—62no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—62no—
247#95—62no—
248#95—62no—
249#95—62no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3C22⋊Q812yes—
262C23×C3#13512yes—
263C23×C3AGL(1,ℤ/8)12yes—
264C23×C3C22×D411yescharacteristic
265C3×D4—24no—
266C3×D4—48no—
267C3×D4—42no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—44no—
284C2×C3×C4—44no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—84no—
294C2×C3×C4C22⋊Q812yes—
295C2×C3×C4#12312yes—
296C2×C3×C4#12312yes—
297C2×C3×C4C22×Q811yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4112yes—
300C2×C3×C4H(5,𝔽2)12yes—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—84no—
308C2×Dic3—84no—
309C3⋊C8—42no—
310C3⋊C8—42no—
311C24—21no—
312C24C2×#6711yescharacteristic
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4C2×Dic612yes—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—122no—
337C22×C4—122no—
338C22×C4—122no—
339C22×C4—122no—
340C22×C4—122no—
341C22×C4—122no—
342C22×C4C2×Dic612yes—
343C22×C4#23512yes—
344C22×C4C2×#6711yescharacteristic
345C22×C4C2×#6711yescharacteristic
346C22×C4C2×#6711yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C22×C4C23×D311yescharacteristic
350C2×C8—62no—
351C2×C8—62no—
352C2×C8—62no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8—62no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—122no—
365C4⋊C4—122no—
366C4⋊C4C2×Dic612yes—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—42no—
376C42#23714yes—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3#47814yes—
391C22×C3C2×C22⋊Q811yescharacteristic
392C22×C3C2×#12311yescharacteristic
393C22×C3#53511yescharacteristic
394C22×C3C2×#13511yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3#57511yescharacteristic
397C22×C3#57511yescharacteristic
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—42no—
402C3×C4—41no—
403C3×C4—41no—
404C3×C4—81no—
405C3×C4—81no—
406C3×C4#55312yes—
407C3×C4#55412yes—
408Dic3—82no—
409Dic3—82no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23#92412yes—
422C23#93212yes—
423C23#95012yes—
424C23C22×#6711yescharacteristic
425D4—24no—
426D4—48no—
427D4—42no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—84no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—244no—
461C2×C4—244no—
462C2×C4#92412yes—
463C2×C4#93512yes—
464C2×C4#93512yes—
465C2×C4C22×Dic611yescharacteristic
466C2×C4C2×#23511yescharacteristic
467C2×C4C2×#23712yes—
468C2×C4#100912yes—
469C8—122no—
470C8—122no—
471C2×C3—24no—
472C2×C3—82no—
473C2×C3#248711yescharacteristic
474C2×C3C2×#47811yescharacteristic
475C2×C3#308011yescharacteristic
476C2×C3#309111yescharacteristic
477C2×C3#309411yescharacteristic
478C2×C3#321111yescharacteristic
479C2×C3#321311yescharacteristic
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22#500914yes—
494C22C2×#92411yescharacteristic
495C22C2×#93211yescharacteristic
496C22C2×#93511yescharacteristic
497C22C2×#95011yescharacteristic
498C22#582211yescharacteristic
499C22#582211yescharacteristic
500C22#583011yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—41no—
506C4—41no—
507C4—81no—
508C4—81no—
509C4—242no—
510C4—242no—
511C4#554312yes—
512C4#556512yes—
513C3#2768411yescharacteristic, complemented, Sylow
514C2—24no—
515C2—82no—
516C2C2×#500911yescharacteristic
517C2#8237411yescharacteristic
518C2#8250911yescharacteristic
519C2#8261611yescharacteristic
520C2#8291311yescharacteristic
521C2#8314411yescharacteristic
522C2#8330611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#121C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC4×#121D3
hypocenterC3#27684
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#4936C2semidirect
5. #71605C2semidirect
6. #71530C2semidirect
7. #69654C2semidirect
8. #69613C2semidirect
9. #69505C2semidirect
10. #69504C2semidirect
11. #69468C2semidirect
513. C3#27684semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C2×#49362
#716772
#716052
#715302
#696542
#696132
#695052
#695042
#694682
#276843