Finite Group Databasebeta

← #10814342

Subgroups — #10814342

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic
2#82337C211yesmaximal, characteristic, complemented
3D4×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#72037C212yesmaximal
5#71604C211yesmaximal, characteristic, complemented
6#71604C211yesmaximal, characteristic, complemented
7#70610C212yesmaximal, complemented
8#70187C212yesmaximal, complemented
9#70152C212yesmaximal, complemented
10#69762C211yesmaximal, characteristic, complemented
11#69757C211yesmaximal, characteristic, complemented
12#22758—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×D4×Dic3C2212yes—
17C2×#926—21no—
18C22×C4⋊Dic3—21no—
19C3×C4×D8C418yescomplemented
20C2×C3×#107C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24#5221C2212yes—
25C2×#832C2212yes—
26C2×#832C2212yes—
27#5212—21no—
28#5212—21no—
29C2×#815C2212yes—
30#4984C2212yes—
31Dic3×C4⋊C4C2212yes—
32#4951—22no—
33#4950—22no—
34#4935C2211yescharacteristic
35#4935C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38#4551C2212yes—
39#4551C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×D8D311yescharacteristic
44#2354—31no—
45D4×C4⋊C4—31no—
46#1936—32no—
47#1921—32no—
48#1879—32no—
49#1834—31no—
50#1802—31no—
51#1802—31no—
52#1770—31no—
53#1762—32no—
54C22×C3×D4C2311yescharacteristic
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×Dic3—22no—
59C2×C3×D8—22no—
60C2×C3×D8—22no—
61C2×C3×D8C2×C414yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×C4×D4C2×C414yes—
67C3×C4×D4C2×C414yescomplemented
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76D4×Dic3—24no—
77D4×Dic3—42no—
78#926—44no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—44no—
85C2×C4⋊Dic3D411yescharacteristic
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2312yes—
89C2×C4×Dic3—22no—
90C2×C4×Dic3C2312yes—
91C22×C3⋊C8C2312yes—
92C3×#107C2×C414yescomplemented
93C3×#102C2×C414yes—
94C3×#102C2×C414yescomplemented
95C3×C4×C8C2×C414yescomplemented
96#832—24no—
97#832—42no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95D412yes—
101#815—42no—
102C4⋊(C3⋊C8)—22no—
103C22×D8C2×D311yescharacteristic
104C2×C4×D4—32no—
105C2×C4×D4C2×D311yescharacteristic
106C2×C4×D4C2×D311yescharacteristic
107C2×#117—61no—
108C22×C4⋊C4—61no—
109C4×D8Dic318yescomplemented
110C2×#107C2×D311yescharacteristic
111C2×C4⋊C8—31no—
112C2×#102—32no—
113C2×#102—32no—
114C2×#102C2×D311yescharacteristic
115C2×#102C2×D311yescharacteristic
116C2×#97—32no—
117C2×C4×C8C2×D311yescharacteristic
118#390—32no—
119#389—32no—
120#388—62no—
121#385—62no—
122#384—31no—
123#384—31no—
124#380—61no—
125#380—61no—
126C4×C4⋊C4—31no—
127C4×C4⋊C4—32no—
128#340—32no—
129#340—32no—
130#340—32no—
131#340—32no—
132#336—32no—
133C24×C3—21no—
134C24×C3—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4C22×C412yes—
144C2×C3×D4C22×C412yes—
145C2×C3×D4#4112yes—
146C2×C3×D4#4112yes—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—44no—
161C22×Dic3—44no—
162C22×Dic3—82no—
163C22×Dic3C2×D412yes—
164C22×Dic3C2×D412yes—
165C3×D8—28no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8C22×C412yes—
171C2×C3×C8C22×C412yes—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4C22×C412yes—
175C3×C4⋊C4C22×C412yes—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C42C22×C412yes—
179C3×C42#4112yes—
180#217—44no—
181#217—44no—
182#217—82no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—44no—
190C4⋊Dic3—42no—
191C4⋊Dic3—42no—
192C4⋊Dic3—81no—
193C4⋊Dic3—81no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196C4×Dic3—44no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C22×D4C22×D311yescharacteristic
201C22×D4C22×D311yescharacteristic
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—62no—
205C2×D8—22no—
206C2×D8—22no—
207C2×D8C2×Dic314yes—
208C22×C8—32no—
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—64no—
214C4×D4—122no—
215C4×D4C2×Dic314yes—
216C4×D4C2×Dic314yescomplemented
217#117—124no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—124no—
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C42—32no—
236C2×C42—62no—
237C2×C42C22×D311yescharacteristic
238#107C2×Dic314yescomplemented
239C4⋊C8—62no—
240#102—64no—
241#102—122no—
242#102C2×Dic314yes—
243#102C2×Dic314yescomplemented
244#97—122no—
245C4×C8C2×Dic314yescomplemented
246#95—32no—
247#95—62no—
248#95—62no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3C22×D411yescharacteristic
264C3×D4—22no—
265C3×D4—22no—
266C3×D4—22no—
267C3×D4—22no—
268C3×D4—22no—
269C3×D4—41no—
270C3×D4—41no—
271C3×D4#11714yescomplemented
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—48no—
281C2×C3×C4—48no—
282C2×C3×C4C4×D412yes—
283C2×C3×C4C4×D412yes—
284C2×C3×C4AGL(1,ℤ/8)12yes—
285C2×C3×C4AGL(1,ℤ/8)12yes—
286C2×C3×C4C23×C411yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4H(5,𝔽2)11yescharacteristic
291C2×C3×C42−1+411yescharacteristic
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—88no—
305C3×C8—22no—
306C3×C8—22no—
307C3⋊C8—42no—
308C24—21no—
309C24—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4#23512yes—
319C2×D4#23512yes—
320C2×D4C22×Dic312yes—
321C2×D4C22×Dic312yes—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—32no—
327C22×C4—32no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—124no—
335C22×C4—124no—
336C22×C4—242no—
337C22×C4D3×D411yescharacteristic
338C22×C4D3×D411yescharacteristic
339C22×C4C23×D311yescharacteristic
340D8—28no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C2×C8C22×Dic312yes—
349C2×C8C22×Dic312yes—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—62no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—124no—
359C4⋊C4—122no—
360C4⋊C4—122no—
361C4⋊C4—241no—
362C4⋊C4—241no—
363C4⋊C4C22×Dic312yes—
364C4⋊C4C22×Dic312yes—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—124no—
368C22⋊C4—124no—
369C22⋊C4—242no—
370C42—62no—
371C42—62no—
372C42—124no—
373C42#23712yes—
374C42C22×Dic312yes—
375C22×C3—24no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3C2×C4×D411yescharacteristic
390C22×C3#54611yescharacteristic
391C22×C3#54811yescharacteristic
392C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
393C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C3#57611yescharacteristic
395C22×C3#57811yescharacteristic
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4#44214yes—
400C3×C4C2×#11711yescharacteristic
401C3×C4#52011yescharacteristic
402C3×C4#55211yescharacteristic
403C3×C4#55311yescharacteristic
404Dic3—42no—
405Dic3—42no—
406Dic3—82no—
407Dic3—82no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C2×D3×D411yescharacteristic
423D4—22no—
424D4—22no—
425D4—22no—
426D4—22no—
427D4—22no—
428D4—41no—
429D4—41no—
430D4#92614yescomplemented
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—248no—
454C2×C4#91112yes—
455C2×C4#91112yes—
456C2×C4D4×Dic312yes—
457C2×C4D4×Dic312yes—
458C2×C4C2×#23511yescharacteristic
459C2×C4C2×#23511yescharacteristic
460C2×C4C2×#23711yescharacteristic
461C2×C4#100911yescharacteristic
462C2×C4#101011yescharacteristic
463C2×C4C23×Dic311yescharacteristic
464C8—22no—
465C8—22no—
466C8—122no—
467C2×C3—44no—
468C2×C3—82no—
469C2×C3#235211yescharacteristic
470C2×C3C2×#44211yescharacteristic
471C2×C3#295111yescharacteristic
472C2×C3#332511yescharacteristic
473C2×C3#333011yescharacteristic
474C2×C3#336611yescharacteristic
475C2×C3#336711yescharacteristic
476C22—24no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#555011yescharacteristic
491C22#555211yescharacteristic
492C22C2×#91111yescharacteristic
493C22C2×#91111yescharacteristic
494C22#576911yescharacteristic
495C22#577111yescharacteristic
496C22C2×D4×Dic311yescharacteristic
497C4—22no—
498C4—42no—
499C4—42no—
500C4—122no—
501C4—122no—
502C4—242no—
503C4—242no—
504C4#515214yes—
505C4#554111yescharacteristic
506C4#556511yescharacteristic
507C4C2×#92611yescharacteristic
508C4#582711yescharacteristic
509C3#2275811yescharacteristic, complemented, Sylow
510C2—44no—
511C2—82no—
512C2#7957011yescharacteristic
513C2#7957111yescharacteristic
514C2#7961211yescharacteristic
515C2#7961311yescharacteristic
516C2#8233511yescharacteristic
517C2C2×#515211yescharacteristic
518C2#8389811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
cosocleC22×C3×C4C24
hypercenterC2×C4×D8D3
hypocenterC3#22758
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. #82337C2semidirect
3. D4×C4⋊Dic3C2semidirect
5. #71604C2semidirect
6. #71604C2semidirect
7. #70610C2semidirect
8. #70187C2semidirect
9. #70152C2semidirect
10. #69762C2semidirect
11. #69757C2semidirect
19. C3×C4×D8C4semidirect
67. C3×C4×D4C2×C4semidirect
92. C3×#107C2×C4semidirect
94. C3×#102C2×C4semidirect
95. C3×C4×C8C2×C4semidirect
109. C4×D8Dic3semidirect
216. C4×D4C2×Dic3semidirect
238. #107C2×Dic3semidirect
243. #102C2×Dic3semidirect
245. C4×C8C2×Dic3semidirect
271. C3×D4#117semidirect
430. D4#926semidirect
509. C3#22758semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
#823372
D4×C4⋊Dic32
#720372
#716042
#706102
#701872
#701522
#697622
#697572
#227583