Finite Group Databasebeta

← #10686607

Subgroups — #10686607

Order 768.

930 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×#5572C211yesmaximal, characteristic
2C3×#1479C211yesmaximal, characteristic, complemented
3#71621C211yesmaximal, characteristic, complemented
4#70782C211yesmaximal, characteristic
5#70782C211yesmaximal, characteristic
6#70767C211yesmaximal, characteristic, complemented
7#70766C211yesmaximal, characteristic, complemented
8#8323—31nomaximal, Sylow
9C22×D3×Q8C411yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×Q8×Dic3C411yescharacteristic
12#5572—22no—
13#5572—22no—
14#5572—22no—
15#5572—22no—
16C2×#892—21no—
17C2×#892C411yescharacteristic
18C2×#887—21no—
19C2×#887C411yescharacteristic
20C2×#872—21no—
21C2×#872C2211yescharacteristic
22C2×C4×Dic6—21no—
23C2×C4×Dic6C2211yescharacteristic
24C3×#431C2211yescharacteristic
25C3×#431C2211yescharacteristic
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C3×Q8⋊C8—22no—
29#4535—22no—
30#4484—22no—
31#4469—21no—
32#4469—21no—
33#4468—21no—
34#4468—21no—
35#4456—22no—
36C2×#519—31no—
37#1479—31no—
38#1479—31no—
39#1479—31no—
40#1479D311yescharacteristic, complemented
41#1458—31no—
42#1458—31no—
43C22×C3×Q8C2×C411yescharacteristic
44C2×D3×Q8—22no—
45C2×D3×Q8—22no—
46C2×D3×Q8—22no—
47C2×D3×Q8—22no—
48C2×D3×Q8—22no—
49C2×D3×Q8C814yescomplemented
50C22×C4×D3—21no—
51C22×C4×D3C2×C411yescharacteristic
52C22×Dic6—21no—
53C22×Dic6C2×C411yescharacteristic
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8D412yes—
57C3×C4×Q8D412yes—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4C2×C411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62Q8×Dic3—22no—
63Q8×Dic3—22no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4D411yescharacteristic
66C2×(C2×D3)⋊C4D411yescharacteristic
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3C2×C411yescharacteristic
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4D411yescharacteristic
72C2×Dic3⋊C4D411yescharacteristic
73C2×C4×Dic3—21no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—21no—
77C2×C4×Dic3C2×C411yescharacteristic
78#892—24no—
79#892—24no—
80#892—22no—
81#892—22no—
82#887—24no—
83#887—24no—
84#887—22no—
85#887—22no—
86#872—24no—
87#872—24no—
88#872—21no—
89#872—21no—
90#872—21no—
91#872—21no—
92C4×Dic6—24no—
93C4×Dic6—24no—
94C4×Dic6—21no—
95C4×Dic6—21no—
96C4×Dic6D412yes—
97C4×Dic6D412yes—
98C3×C4⋊C8—21no—
99C3×C4⋊C8—21no—
100C3×C22⋊C8—21no—
101C3×C22⋊C8—21no—
102C3×C4×C8—21no—
103C3×C4×C8—21no—
104#830—21no—
105#830—21no—
106C4⋊(C3⋊C8)—21no—
107C4⋊(C3⋊C8)—21no—
108C4×C3⋊C8—21no—
109C4×C3⋊C8—21no—
110C23×Q8—31no—
111#519—62no—
112#519—62no—
113#519—62no—
114#519—62no—
115C2×C4×Q8—31no—
116C2×C4×Q8—31no—
117C2×C4×Q8—31no—
118C2×C4×Q8—61no—
119C2×C4×Q8—61no—
120C2×C4×Q8C2×D311yescharacteristic
121C2×#117—31no—
122C2×#117—31no—
123C2×#117—61no—
124C2×#117—61no—
125#431—31no—
126#431—31no—
127#431C2×D311yescharacteristic
128#431C2×D311yescharacteristic
129Q8⋊C8—22no—
130Q8⋊C8—62no—
131Q8⋊C8—62no—
132Q8⋊C8—62no—
133#324—61no—
134#324—61no—
135#323—61no—
136#323—61no—
137C23×D3C22⋊C411yescharacteristic
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8C22⋊C412yes—
142C2×C3×Q8C2×C812yes—
143C2×C3×Q8M1612yes—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4C22×C411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3C22⋊C411yescharacteristic
152C22×Dic3C22⋊C411yescharacteristic
153C22×Dic3C22⋊C411yescharacteristic
154D3×Q8—28no—
155D3×Q8—28no—
156D3×Q8—28no—
157D3×Q8—28no—
158C2×C4×D3—24no—
159C2×C4×D3—24no—
160C2×C4×D3—24no—
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—21no—
164C2×C4×D3—21no—
165C2×C4×D3C2×C812yes—
166C2×C4×D3M1612yes—
167C2×Dic6—24no—
168C2×Dic6—24no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C2×Dic6—24no—
173C2×Dic6C22⋊C412yes—
174C2×Dic6C2×C812yes—
175C2×Dic6M1612yes—
176C2×C3×C8—41no—
177C2×C3×C8—41no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4C22⋊C412yes—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42C2×D411yescharacteristic
185C3×C42C2×D411yescharacteristic
186C3×C42C2×D411yescharacteristic
187C3×C42C2×D411yescharacteristic
188(C2×D3)⋊C4—24no—
189(C2×D3)⋊C4—21no—
190(C2×D3)⋊C4—21no—
191(C2×D3)⋊C4—21no—
192(C2×D3)⋊C4—21no—
193C4⋊Dic3—24no—
194C4⋊Dic3—21no—
195C4⋊Dic3C22⋊C412yes—
196Dic3⋊C4—24no—
197Dic3⋊C4—21no—
198Dic3⋊C4—21no—
199Dic3⋊C4—22no—
200Dic3⋊C4—22no—
201Dic3⋊C4—21no—
202Dic3⋊C4—21no—
203C4×Dic3—24no—
204C4×Dic3—24no—
205C4×Dic3—24no—
206C4×Dic3—21no—
207C4×Dic3—21no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C2×C3⋊C8—41no—
211C2×C3⋊C8—41no—
212C22×Q8—34no—
213C22×Q8—31no—
214C22×Q8—61no—
215C22×Q8—62no—
216C22×Q8—62no—
217C22×Q8—62no—
218C22×Q8—62no—
219C22×Q8—62no—
220C22×Q8C4×D311yescharacteristic
221C23×C4—31no—
222C23×C4—61no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—32no—
226C4×Q8—32no—
227C4×Q8—64no—
228C4×Q8—64no—
229C4×Q8—62no—
230C4×Q8—64no—
231C4×Q8—64no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×Q8—62no—
235C4×Q8—61no—
236C4×Q8—61no—
237C4×Q8D1212yes—
238C4×Q8#6712yes—
239#117—64no—
240#117—64no—
241#117—62no—
242#117—64no—
243#117—64no—
244#117—62no—
245#117—61no—
246#117—61no—
247#117—61no—
248#117—61no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4C4×D311yescharacteristic
257C2×C22⋊C4—31no—
258C2×C22⋊C4—31no—
259C2×C22⋊C4—61no—
260C2×C42—21no—
261C2×C42—31no—
262C2×C42—61no—
263C2×C42—61no—
264C2×C42—61no—
265C2×C42—61no—
266C2×C42C22×D311yescharacteristic
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—61no—
270C4⋊C8—61no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—61no—
274C22⋊C8—61no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—61no—
278C4×C8—61no—
279C23×C3C2×C22⋊C411yescharacteristic
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3C22⋊C812yes—
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8C22⋊C814yes—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4C2×C22⋊C411yescharacteristic
297C2×C3×C4C2×C22⋊C411yescharacteristic
298C2×C3×C4#12011yescharacteristic
299C2×C3×C4#12011yescharacteristic
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4C22×C811yescharacteristic
303C2×C3×C4C2×M1611yescharacteristic
304C2×C3×C4C2×M1611yescharacteristic
305C2×C3×C4C2×M1611yescharacteristic
306C2×Dic3—21no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—21no—
310C2×Dic3—22no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—22no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3C22⋊C812yes—
318C4×D3—216no—
319C4×D3—22no—
320C4×D3—22no—
321C4×D3—22no—
322C4×D3—22no—
323Dic6—216no—
324Dic6—22no—
325Dic6—22no—
326Dic6—22no—
327Dic6C22⋊C814yes—
328C3×C8—81no—
329C3×C8—81no—
330C3⋊C8—81no—
331C3⋊C8—81no—
332C24—31no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—32no—
337C2×Q8—32no—
338C2×Q8—32no—
339C2×Q8—64no—
340C2×Q8—64no—
341C2×Q8—64no—
342C2×Q8—68no—
343C2×Q8—68no—
344C2×Q8—68no—
345C2×Q8—68no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8D3×C812yes—
350C2×Q8ΓL(1,𝔽25)12yes—
351C2×Q8(C2×D3)⋊C412yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—31no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—64no—
360C22×C4—64no—
361C22×C4—64no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4C2×C4×D311yescharacteristic
369C22×C4D3×D411yescharacteristic
370C22×C4D3×D411yescharacteristic
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—121no—
374C2×C8—121no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—32no—
379C4⋊C4—64no—
380C4⋊C4—64no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—62no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—62no—
388C4⋊C4(C2×D3)⋊C412yes—
389C22⋊C4—64no—
390C22⋊C4—61no—
391C22⋊C4—61no—
392C22⋊C4—61no—
393C22⋊C4—61no—
394C42—22no—
395C42—22no—
396C42—64no—
397C42—64no—
398C42—64no—
399C42—61no—
400C42—62no—
401C42—61no—
402C42—62no—
403C42C2×D1211yescharacteristic
404C42D3×D411yescharacteristic
405C42D3×D411yescharacteristic
406C42C2×#6711yescharacteristic
407C22×C3—21no—
408C22×C3—21no—
409C22×C3#37911yescharacteristic
410C22×C3C2×C22⋊C811yescharacteristic
411C22×C3#40711yescharacteristic
412C2×D3—22no—
413C2×D3—42no—
414C2×D3—41no—
415C2×D3—41no—
416C2×D3—42no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—24no—
422C3×C4—42no—
423C3×C4C2×C22⋊C811yescharacteristic
424C3×C4C2×C22⋊C811yescharacteristic
425C3×C4#40711yescharacteristic
426C3×C4#40711yescharacteristic
427Dic3—22no—
428Dic3—24no—
429Dic3—41no—
430Dic3—41no—
431Dic3—42no—
432C23—32no—
433C23—61no—
434C23—61no—
435C23—121no—
436C23—121no—
437C23D3×C22⋊C411yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—22no—
441Q8—34no—
442Q8—616no—
443Q8—62no—
444Q8—62no—
445Q8—62no—
446Q8(C2×D3)⋊C814yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—32no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—616no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—61no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—122no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4D3×C22⋊C411yescharacteristic
472C2×C4#88211yescharacteristic
473C2×C4#88211yescharacteristic
474C2×C4C2×D3×C811yescharacteristic
475C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
476C2×C4D3×M1611yescharacteristic
477C2×C4D3×M1611yescharacteristic
478C2×C4C2×(C2×D3)⋊C411yescharacteristic
479C2×C4#93711yescharacteristic
480C2×C4#93711yescharacteristic
481C8—81no—
482C8—81no—
483C8—241no—
484C8—241no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3#177911yescharacteristic
488C2×C3#179611yescharacteristic
489C2×C3#179611yescharacteristic
490D3—42no—
491C22—21no—
492C22—21no—
493C22—62no—
494C22—122no—
495C22—122no—
496C22—121no—
497C22—121no—
498C22D3×C22⋊C811yescharacteristic
499C22#472611yescharacteristic
500C22#495511yescharacteristic
501C4—21no—
502C4—21no—
503C4—24no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—62no—
508C4—64no—
509C4—121no—
510C4—122no—
511C4—121no—
512C4D3×C22⋊C811yescharacteristic
513C4#472611yescharacteristic
514C4C2×(C2×D3)⋊C811yescharacteristic
515C4#512611yescharacteristic
516C3#832311yescharacteristic, complemented, Sylow
517C2—21no—
518C2—21no—
519C2—122no—
520C2#6995511yescharacteristic
521C2#7034711yescharacteristic
522C2#7041511yescharacteristic
subgroupG/H
centerC22#4955
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#379
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1479D3
hypocenterC3#8323
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. C3×#1479C2semidirect
3. #71621C2semidirect
6. #70767C2semidirect
7. #70766C2semidirect
40. #1479D3semidirect
49. C2×D3×Q8C8semidirect
516. C3#8323semidirect

Maximal subgroups

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

subgroupindex
C2×#55722
C3×#14792
#716212
#707822
#707672
#707662
#83233