Finite Group Databasebeta

← #10813331

Subgroups — #10813331

Order 768.

982 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4860C212yesmaximal, complemented
2C2×#4834C211yesmaximal, characteristic
3#74324C211yesmaximal, characteristic
4#73638C211yesmaximal, characteristic
5C3×#1948C211yesmaximal, characteristic, complemented
6#70580C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69577C211yesmaximal, characteristic, complemented
9C4×#807C211yesmaximal, characteristic, complemented
10#68876C212yesmaximal, complemented
11#68869C212yesmaximal, complemented
12#22658—31nomaximal, Sylow
13C22×D3×C8C2212yes—
14C2×#892C2212yes—
15C2×#892C2212yes—
16C2×C4×D12C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×C4⋊C8C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C3×C4×C4⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C8C2212yes—
22C2×#817C2212yes—
23#4990—22no—
24#4988—21no—
25#4987C2212yes—
26#4987C2212yes—
27Dic3×C4⋊C4—21no—
28C2×#807C2211yescharacteristic
29C2×#807C2212yes—
30C2×#807C2212yes—
31C2×#807C2211yescharacteristic
32#4939—21no—
33#4935C2211yescharacteristic
34C4×C4⋊Dic3—21no—
35C4×C4⋊Dic3C2211yescharacteristic
36C2×C4×C3⋊C8C2211yescharacteristic
37#4860—28no—
38#4834C418yescomplemented
39#4670—22no—
40C3×#340C2212yes—
41#4551C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×C8×D4—31no—
45C2×#427—32no—
46#2354—31no—
47#2354—31no—
48#1948—31no—
49#1948D311yescharacteristic, complemented
50#1854—32no—
51#1854—32no—
52#1845—31no—
53#1808—32no—
54C4×#107—31no—
55C22×D12D411yescharacteristic
56C22×C4×D3C2312yes—
57C22×C3×C8C2312yes—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C42—21no—
64C2×C3×C42C2311yescharacteristic
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4D412yes—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×C4×Dic3—21no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×D3×C8—24no—
82C2×D3×C8—24no—
83C2×D3×C8C2×C418yes—
84#892—24no—
85#892—24no—
86#892—42no—
87#892—42no—
88C4×D12C2×C414yes—
89C4×D12Q814yes—
90C3×#106—24no—
91C3×C4⋊C8C2×C414yes—
92C3×#95—21no—
93C3⋊#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96(C2×D3)⋊C8C2×C418yes—
97#817—24no—
98#807—24no—
99#807—22no—
100#807—42no—
101#807D414yescomplemented
102C4×C3⋊C8C2×C414yes—
103C23×C8—32no—
104C2×C4×D4—31no—
105C2×#117—32no—
106C2×#117—32no—
107C8×D4—38no—
108#427—68no—
109C2×#107—31no—
110C2×#107—31no—
111C2×#107—32no—
112C2×#107—32no—
113C2×#106—32no—
114C2×#106C2×D312yes—
115C2×C4⋊C8C2×D311yescharacteristic
116C2×C22⋊C8—32no—
117C2×C4×C8—31no—
118#390—61no—
119#390—61no—
120#388—62no—
121#388—62no—
122#384—31no—
123#384C2×D311yescharacteristic
124#380—32no—
125#380—32no—
126C4×C4⋊C4—31no—
127C4×C4⋊C4—61no—
128C4×C4⋊C4—61no—
129C4×C4⋊C4C2×D311yescharacteristic
130#340—32no—
131#340—31no—
132#340—31no—
133#340C2×D312yes—
134C23×D3C2×D412yes—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—22no—
144C22×Dic3C2×D412yes—
145C2×D12—22no—
146C2×D12—22no—
147C2×D12C4⋊C414yes—
148C2×C4×D3—22no—
149C2×C4×D3—22no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3C22×C414yes—
153C2×C4×D3C2×Q814yes—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8C22×C414yes—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—42no—
163C3×C4⋊C4—81no—
164C3×C4⋊C4C2×D412yes—
165C3×C42—42no—
166C3×C42C22×C412yes—
167C3×C42C2×Q812yes—
168(C2×D3)⋊C4—44no—
169(C2×D3)⋊C4—44no—
170(C2×D3)⋊C4—82no—
171(C2×D3)⋊C4—82no—
172(C2×D3)⋊C4C4⋊C418yes—
173C4⋊Dic3—21no—
174C4⋊Dic3—22no—
175C4⋊Dic3—21no—
176C4⋊Dic3—22no—
177C4⋊Dic3—21no—
178C4⋊Dic3—42no—
179C4⋊Dic3—81no—
180C4⋊Dic3C4⋊C414yes—
181C4⋊Dic3C2×D412yes—
182C4×Dic3—44no—
183C4×Dic3—44no—
184C4×Dic3—42no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8C22×C412yes—
188C2×C3⋊C8C22×C412yes—
189C2×C3⋊C8C2×D412yes—
190C2×C3⋊C8#4112yes—
191D3×C8—216no—
192C22×D4—31no—
193C23×C4—32no—
194C22×C8—31no—
195C22×C8—31no—
196C22×C8—38no—
197C22×C8—64no—
198C22×C8—64no—
199C22×C8C22×D312yes—
200C4×D4—34no—
201C4×D4—34no—
202#117—64no—
203#117—64no—
204#117—122no—
205#117—122no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4C22×D312yes—
215C2×C4⋊C4C22×D311yescharacteristic
216C2×C4⋊C4C22×D311yescharacteristic
217C2×C22⋊C4—32no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4—62no—
221C2×C22⋊C4—62no—
222C2×C42—21no—
223C2×C42—62no—
224C2×C42—62no—
225C2×C42—61no—
226C2×C42C22×D311yescharacteristic
227#107—34no—
228#107—62no—
229#107—64no—
230#107—122no—
231#106—24no—
232#106—64no—
233C4⋊C8C4×D314yes—
234C22⋊C8—38no—
235C4×C8—34no—
236#95—21no—
237#95—62no—
238#95—62no—
239#95—61no—
240C23×C3C22×D411yescharacteristic
241C22×D3—22no—
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—22no—
245C22×D3—22no—
246C22×D3—22no—
247C22×D3C2×C4⋊C414yes—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4C2×C4⋊C412yes—
260C2×C3×C4C2×C4⋊C412yes—
261C2×C3×C4#13512yes—
262C2×C3×C4#13512yes—
263C2×C3×C4C23×C411yescharacteristic
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C22×Q811yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4H(5,𝔽2)11yescharacteristic
268C2×C3×C42−1+411yescharacteristic
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—41no—
280C2×Dic3C2×C4⋊C414yes—
281D12—28no—
282C4×D3—24no—
283C4×D3—24no—
284C4×D3—24no—
285C4×D3—24no—
286C3×C8—24no—
287C3⋊C8—22no—
288C3⋊C8C4×D414yes—
289C24—32no—
290C2×D4—34no—
291C2×D4—62no—
292C2×D4—62no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—32no—
297C22×C4—34no—
298C22×C4—34no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—62no—
306C22×C4D3×D411yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4C23×D311yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—32no—
312C2×C8—32no—
313C2×C8—32no—
314C2×C8—32no—
315C2×C8—616no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8C2×C4×D314yes—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—34no—
325C4⋊C4—32no—
326C4⋊C4—42no—
327C4⋊C4—61no—
328C4⋊C4—62no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—62no—
332C4⋊C4—81no—
333C4⋊C4—122no—
334C4⋊C4—241no—
335C4⋊C4D3×D412yes—
336C22⋊C4—38no—
337C22⋊C4—124no—
338C22⋊C4—124no—
339C22⋊C4—242no—
340C22⋊C4—242no—
341C42—42no—
342C42—124no—
343C42—124no—
344C42—122no—
345C42C2×C4×D312yes—
346C42D3×Q812yes—
347C22×C3C22×C4⋊C411yescharacteristic
348C22×C3D4211yescharacteristic
349C22×C3#54711yescharacteristic
350C22×C3C2×#13511yescharacteristic
351C22×C3C2×#13511yescharacteristic
352C22×C3#57611yescharacteristic
353C22×C3#57811yescharacteristic
354C2×D3—24no—
355C2×D3—24no—
356C2×D3—24no—
357C2×D3—42no—
358C2×D3—42no—
359C2×D3—42no—
360C2×D3—42no—
361C3×C4—22no—
362C3×C4—42no—
363C3×C4—81no—
364C3×C4—82no—
365C3×C4#42714yescomplemented
366C3×C4C2×C4×D411yescharacteristic
367C3×C4#52011yescharacteristic
368C3×C4D4×Q811yescharacteristic
369C3×C4#55411yescharacteristic
370Dic3—24no—
371Dic3—42no—
372Dic3—81no—
373Dic3—82no—
374C23—34no—
375C23—62no—
376C23—62no—
377C23—62no—
378C23—62no—
379C23—62no—
380C23—62no—
381C23C2×D3×D411yescharacteristic
382D4—68no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—34no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—64no—
398C2×C4—64no—
399C2×C4—64no—
400C2×C4—64no—
401C2×C4—62no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4D3×C4⋊C412yes—
411C2×C4D3×C4⋊C412yes—
412C2×C4#91612yes—
413C2×C4#91612yes—
414C2×C4C22×C4×D311yescharacteristic
415C2×C4C2×D3×D411yescharacteristic
416C2×C4C2×D3×Q811yescharacteristic
417C2×C4#100711yescharacteristic
418C2×C4D3×#4111yescharacteristic
419C2×C4#100911yescharacteristic
420C8—24no—
421C8—34no—
422C8—62no—
423C2×C3D4×C4⋊C411yescharacteristic
424C2×C3C2×#42711yescharacteristic
425C2×C3#291311yescharacteristic
426C2×C3#328311yescharacteristic
427C2×C3#328711yescharacteristic
428C2×C3#334111yescharacteristic
429C2×C3#334211yescharacteristic
430D3—44no—
431C22—64no—
432C22—64no—
433C22—64no—
434C22—122no—
435C22—122no—
436C22—122no—
437C22—122no—
438C22C2×D3×C4⋊C411yescharacteristic
439C22#560911yescharacteristic
440C22#563111yescharacteristic
441C22C2×#91611yescharacteristic
442C22C2×#91611yescharacteristic
443C22#577411yescharacteristic
444C22#577911yescharacteristic
445C4—22no—
446C4—42no—
447C4—64no—
448C4—82no—
449C4—81no—
450C4—122no—
451C4—242no—
452C4—241no—
453C4#486014yescomplemented
454C4C4×D3×D411yescharacteristic
455C4#557411yescharacteristic
456C4#569011yescharacteristic
457C4#572711yescharacteristic
458C3#2265811yescharacteristic, complemented, Sylow
459C2—124no—
460C2#7620211yescharacteristic
461C2C2×#486011yescharacteristic
462C2#7921011yescharacteristic
463C2#8039411yescharacteristic
464C2#8044911yescharacteristic
465C2#8102911yescharacteristic
466C2#8108211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC22×D12D4
cosocleC22×C3×C4C24
hypercenter#1948D3
hypocenterC3#22658
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. C2×#4860C2semidirect
5. C3×#1948C2semidirect
6. #70580C2semidirect
7. #70454C2semidirect
8. #69577C2semidirect
9. C4×#807C2semidirect
10. #68876C2semidirect
11. #68869C2semidirect
38. #4834C4semidirect
49. #1948D3semidirect
101. #807D4semidirect
365. C3×C4#427semidirect
453. C4#4860semidirect
458. C3#22658semidirect

Maximal subgroups

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

subgroupindex
C2×#48602
C2×#48342
#743242
#736382
C3×#19482
#705802
#704542
#695772
C4×#8072
#688762
#688692
#226583