Finite Group Databasebeta

← #10877258

Subgroups — #10877258

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
1C2×C3×#382C211yesmaximal, characteristic, complemented
2C4×#930C211yesmaximal, characteristic
3#71560C211yesmaximal, characteristic
4#71560C211yesmaximal, characteristic
5#71556C211yesmaximal, characteristic, complemented
6#71555C211yesmaximal, characteristic, complemented
7#71530C211yesmaximal, characteristic, complemented
8#69558C212yesmaximal, complemented
9#69556C212yesmaximal, complemented
10#69542C212yesmaximal, complemented
11#69540C212yesmaximal, complemented
12#27684—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×C4×#67—21no—
18C2×C4×D12C2211yescharacteristic
19C3×#382—21no—
20C3×#382—21no—
21C3×#382C2212yes—
22C3×#382C2212yes—
23C2×C3×#95—21no—
24C2×C3×#95—21no—
25#5213—21no—
26#5212C2211yescharacteristic
27C2×#830C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#809C2212yes—
30C2×#809C2212yes—
31C2×#808C2212yes—
32C2×#807C2212yes—
33#4940—21no—
34#4939C2211yescharacteristic
35C4×(C2×D3)⋊C4—21no—
36C4×C4⋊Dic3C2211yescharacteristic
37#4928C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C4×#121—31no—
45C2×#382D311yescharacteristic, complemented
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—
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C42—24no—
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×#217—21no—
75#930—22no—
76#930Q814yes—
77C4×#67—44no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3—21no—
87C22×C3⋊C8C2312yes—
88C22×C3⋊C8C2312yes—
89C4×D12—24no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—22no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97#830—22no—
98#830—22no—
99#809—42no—
100#809—42no—
101#808—42no—
102#807—42no—
103C4⋊(C3⋊C8)—24no—
104C2×#121—31no—
105C2×C4×D4—31no—
106C2×C4×D4—61no—
107C22×C4⋊C4C2×D311yescharacteristic
108C22×C4⋊C4C2×D311yescharacteristic
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#390—31no—
118#389—31no—
119#388—61no—
120#386—61no—
121#382—21no—
122#382—21no—
123#382C2×D312yes—
124#382C2×D312yes—
125#380—31no—
126C4×C4⋊C4—31no—
127C4×C22⋊C4—61no—
128C2×#95—21no—
129C2×#95—21no—
130#340—32no—
131#340—32no—
132#340—32no—
133#340—32no—
134#336—32no—
135C24×C3C2×D411yescharacteristic
136C23×D3—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×Q812yes—
159C22×C3×C4#4112yes—
160C22×C3×C4C2411yescharacteristic
161C2×#67—24no—
162C2×#67—42no—
163C2×#67—42no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—21no—
170C2×D12C2×Q812yes—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—42no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42#4114yes—
184#217—42no—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3C2×Q812yes—
193Dic3⋊C4—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—31no—
202C22×D4—61no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—61no—
206C23×C4#6711yescharacteristic
207C23×C4#6711yescharacteristic
208C23×C4C22×D311yescharacteristic
209C22×C8—32no—
210C22×C8—32no—
211#121—34no—
212#121—62no—
213C4×D4—64no—
214C4×D4—124no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D312yes—
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—24no—
231C2×C42—61no—
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—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—61no—
248#95—61no—
249#95—61no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3C22⋊Q812yes—
255C23×C3#13512yes—
256C23×C3AGL(1,ℤ/8)12yes—
257C23×C3C22×D411yescharacteristic
258C22×D3—22no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—84no—
288C2×C3×C4—84no—
289C2×C3×C4C22⋊Q812yes—
290C2×C3×C4#12312yes—
291C2×C3×C4#12312yes—
292C2×C3×C4C22×Q811yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4H(5,𝔽2)12yes—
296#67—48no—
297C2×Dic3—22no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306D12—24no—
307D12—42no—
308C4×D3—84no—
309C3⋊C8—42no—
310C3⋊C8—42no—
311C24—61no—
312C24C2×#6711yescharacteristic
313C2×D4—32no—
314C2×D4—64no—
315C2×D4—62no—
316C2×D4—62no—
317C2×D4—61no—
318C2×D4—122no—
319C2×D4—122no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—122no—
340C22×C4—122no—
341C22×C4—122no—
342C22×C4D3×Q812yes—
343C22×C4#23912yes—
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—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—32no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—61no—
364C4⋊C4—62no—
365C4⋊C4—62no—
366C4⋊C4—122no—
367C22⋊C4—64no—
368C22⋊C4—122no—
369C22⋊C4—122no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—122no—
376C42#23514yes—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3#47814yes—
384C22×C3C2×C22⋊Q811yescharacteristic
385C22×C3C2×#12311yescharacteristic
386C22×C3#53511yescharacteristic
387C22×C3C2×#13511yescharacteristic
388C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
389C22×C3#57511yescharacteristic
390C22×C3#57511yescharacteristic
391C2×D3—42no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—81no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4—82no—
404C3×C4—82no—
405C3×C4#55312yes—
406C3×C4#55412yes—
407Dic3—42no—
408Dic3—81no—
409Dic3—81no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—62no—
415C23—121no—
416C23—121no—
417C23—121no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23#94612yes—
422C23#94912yes—
423C23#95012yes—
424C23C22×#6711yescharacteristic
425D4—64no—
426D4—128no—
427D4—122no—
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—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—62no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—244no—
462C2×C4#92912yes—
463C2×C4#92912yes—
464C2×C4#94612yes—
465C2×C4C2×#23512yes—
466C2×C4#100412yes—
467C2×C4C2×D3×Q811yescharacteristic
468C2×C4C2×#23911yescharacteristic
469C8—122no—
470C8—122no—
471C2×C3—24no—
472C2×C3#248711yescharacteristic
473C2×C3C2×#47811yescharacteristic
474C2×C3#308011yescharacteristic
475C2×C3#309111yescharacteristic
476C2×C3#309411yescharacteristic
477C2×C3#321111yescharacteristic
478C2×C3#321311yescharacteristic
479D3—82no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—122no—
487C22—122no—
488C22—122no—
489C22—241no—
490C22—241no—
491C22—241no—
492C22—241no—
493C22#506614yes—
494C22C2×#92911yescharacteristic
495C22#580511yescharacteristic
496C22C2×#94611yescharacteristic
497C22C2×#94911yescharacteristic
498C22C2×#95011yescharacteristic
499C22#582211yescharacteristic
500C22#582211yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—41no—
506C4—82no—
507C4—82no—
508C4—122no—
509C4—241no—
510C4—241no—
511C4#551112yes—
512C4#551412yes—
513C3#2768411yescharacteristic, complemented, Sylow
514C2—24no—
515C2—242no—
516C2#8219711yescharacteristic
517C2#8221511yescharacteristic
518C2#8221711yescharacteristic
519C2#8294611yescharacteristic
520C2#8294711yescharacteristic
521C2C2×#506611yescharacteristic
522C2#8301711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#382C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#382D3
hypocenterC3#27684
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
1. C2×C3×#382C2semidirect
5. #71556C2semidirect
6. #71555C2semidirect
7. #71530C2semidirect
8. #69558C2semidirect
9. #69556C2semidirect
10. #69542C2semidirect
11. #69540C2semidirect
45. C2×#382D3semidirect
513. C3#27684semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3822
C4×#9302
#715602
#715562
#715552
#715302
#695582
#695562
#695422
#695402
#276843