Finite Group Databasebeta

← #10828570

Subgroups — #10828570

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5625C211yesmaximal, characteristic
2C2×#5044C211yesmaximal, characteristic
3#82233C211yesmaximal, characteristic
4C2×#4721C211yesmaximal, characteristic
5C3×#1873C211yesmaximal, characteristic
6#71585C211yesmaximal, characteristic
7#70246C212yesmaximal, complemented
8#70238C212yesmaximal, complemented
9#69911C211yesmaximal, characteristic
10#68906C212yesmaximal, complemented
11#68828C212yesmaximal, complemented
12#24071—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C2×#944C2211yescharacteristic
17C2×#926C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C2×#924C2211yescharacteristic
20#5625—24no—
21#5625—24no—
22C2×#890—21no—
23C2×#890C2211yescharacteristic
24C2×#889—21no—
25C2×#887C2211yescharacteristic
26C2×#879—21no—
27C2×#879—21no—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×#835C2212yes—
31C2×#830C2211yescharacteristic
32C2×#817C2212yes—
33C2×#816C2212yes—
34#5044—24no—
35#4984C2212yes—
36#4984C2212yes—
37C2×#810C2212yes—
38C2×#807C2212yes—
39#4950—21no—
40#4950—21no—
41#4936—21no—
42#4935—21no—
43#4721—24no—
44C3×#340C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#541—31no—
49C2×#484—31no—
50C2×#483—31no—
51#2317—31no—
52#2059—32no—
53#2043—32no—
54#1873—31no—
55#1873D311yescharacteristic
56#1865—32no—
57#1863—32no—
58#1818—31no—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×Dic3—21no—
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2312yes—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8D414yes—
66C3×#117—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—21no—
73C2×#217—21no—
74C2×#217D411yescharacteristic
75C2×#217D411yescharacteristic
76Q8×Dic3—22no—
77Q8×Dic3—22no—
78#944—22no—
79#944—22no—
80#926—22no—
81#926—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3D411yescharacteristic
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×C4⋊Dic3C2311yescharacteristic
93#924—22no—
94#924D414yes—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105#890—28no—
106#890—22no—
107#890—22no—
108#889—28no—
109#887—22no—
110#887—22no—
111#879—28no—
112#879—28no—
113C3×#103—24no—
114C3×C22⋊C8—22no—
115#835—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119#817—24no—
120#816—24no—
121#810—24no—
122#807—24no—
123#541—64no—
124#541—64no—
125C2×C4⋊Q8—31no—
126C2×#126—61no—
127C2×#126—61no—
128C2×#125—31no—
129C2×#125—61no—
130C2×#125—61no—
131C2×C22⋊Q8—31no—
132C2×C22⋊Q8C2×D311yescharacteristic
133C2×C4×Q8—31no—
134C2×C4×Q8—31no—
135C2×#117—31no—
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138#484—64no—
139#483—64no—
140C2×#107—32no—
141C2×#106—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103C2×D312yes—
146C2×C22⋊C8—31no—
147C2×C22⋊C8C2×D311yescharacteristic
148#389—32no—
149#389—32no—
150#385—61no—
151#385—61no—
152#384—61no—
153#382—61no—
154#340—32no—
155#340C2×D312yes—
156#336—32no—
157#336—32no—
158C24×C3C2×D411yescharacteristic
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8C2×D412yes—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—42no—
176C22×Dic3—82no—
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179C22×Dic3C2×D411yescharacteristic
180C22×Dic3C2×D411yescharacteristic
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—21no—
184C2×Dic6C2×D412yes—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2×D412yes—
192C3×C4⋊C4#4114yes—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—24no—
195C3×C42—24no—
196#217—24no—
197#217—22no—
198#217—22no—
199C4⋊Dic3—24no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—21no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3—42no—
207C4⋊Dic3—82no—
208C4⋊Dic3—82no—
209C4⋊Dic3C2×D412yes—
210Dic3⋊C4—24no—
211Dic3⋊C4—24no—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215C4×Dic3—24no—
216C4×Dic3—24no—
217C4×Dic3—21no—
218C4×Dic3—21no—
219C4×Dic3—21no—
220C4×Dic3—21no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C22×Q8—31no—
225C22×Q8C22×D311yescharacteristic
226C23×C4—61no—
227C23×C4C22×D311yescharacteristic
228C22×C8—32no—
229C22×C8C22×D312yes—
230C4⋊Q8—62no—
231C4⋊Q8—62no—
232#126—68no—
233#126—68no—
234#125—62no—
235#125—68no—
236#125—68no—
237#125—62no—
238C22⋊Q8—22no—
239C22⋊Q8—34no—
240C22⋊Q8—62no—
241C22⋊Q8D1214yes—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8—62no—
246#117—24no—
247#117—62no—
248#117—62no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—31no—
271C2×C22⋊C4—31no—
272C2×C22⋊C4—61no—
273C2×C42—21no—
274C2×C42—31no—
275C2×C42—31no—
276C2×C42—61no—
277C2×C42—61no—
278#107—64no—
279#106—64no—
280#103—24no—
281#103—64no—
282#103—64no—
283#103—64no—
284C22⋊C8—22no—
285C22⋊C8—62no—
286#95—62no—
287#95—62no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3#12012yes—
293C23×C3C22×D411yescharacteristic
294C3×Q8—24no—
295C3×Q8—42no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4112yes—
312C2×C3×C42−1+412yes—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—22no—
317C2×Dic3—21no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—84no—
330Dic6—24no—
331Dic6—42no—
332C3×C8—42no—
333C3⋊C8—42no—
334C24D3×D411yescharacteristic
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—32no—
339C2×Q8—61no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8C2×D1212yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—122no—
356C22×C4—242no—
357C22×C4C2×D1212yes—
358C22×C4D3×D411yescharacteristic
359C22×C4D3×D412yes—
360C22×C4C23×D311yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8—62no—
367C4⋊C4—24no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—32no—
371C4⋊C4—64no—
372C4⋊C4—64no—
373C4⋊C4—64no—
374C4⋊C4—64no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—64no—
380C4⋊C4—61no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—122no—
384C4⋊C4—242no—
385C4⋊C4—242no—
386C4⋊C4C2×D1212yes—
387C4⋊C4#23714yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—62no—
391C22⋊C4—62no—
392C22⋊C4—64no—
393C42—24no—
394C42—64no—
395C42—64no—
396C42—61no—
397C42—61no—
398C42—61no—
399C42—61no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3C2×#12011yescharacteristic
406C22×C3#52511yescharacteristic
407C22×C3#53611yescharacteristic
408C22×C3#57712yes—
409C22×C3#57812yes—
410C3×C4—22no—
411C3×C4—44no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4#54712yes—
415C3×C4#54712yes—
416Dic3—42no—
417Dic3—41no—
418Dic3—41no—
419Dic3—41no—
420Dic3—41no—
421Dic3—42no—
422Dic3—82no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—21no—
427C23#88212yes—
428C23C2×D3×D411yescharacteristic
429Q8—24no—
430Q8—42no—
431Q8—64no—
432Q8—122no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—62no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—121no—
456C2×C4—122no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—244no—
462C2×C4#88212yes—
463C2×C4C22×D1211yescharacteristic
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×#23712yes—
466C2×C4#101012yes—
467C8—42no—
468C8—122no—
469C2×C3—42no—
470C2×C3#240611yescharacteristic
471C2×C3#301011yescharacteristic
472C2×C3#301111yescharacteristic
473C2×C3#312012yes—
474C2×C3#324212yes—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—42no—
479C22—42no—
480C22C2×#88211yescharacteristic
481C22#558611yescharacteristic
482C22#564811yescharacteristic
483C22#577012yes—
484C22#577912yes—
485C4—22no—
486C4—44no—
487C4—42no—
488C4—42no—
489C4—121no—
490C4—121no—
491C4—122no—
492C4—122no—
493C4—121no—
494C4—121no—
495C4—242no—
496C4#555512yes—
497C4#561112yes—
498C3#2407111yescharacteristic, complemented, Sylow
499C2—42no—
500C2#7468711yescharacteristic
501C2#7883611yescharacteristic
502C2#7896111yescharacteristic
503C2#8021612yes—
504C2#8150512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1873D3
hypocenterC3#24071
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
7. #70246C2semidirect
8. #70238C2semidirect
10. #68906C2semidirect
11. #68828C2semidirect
498. C3#24071semidirect

Maximal subgroups

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

subgroupindex
C2×#56252
C2×#50442
#822332
C2×#47212
C3×#18732
#715852
#702462
#702382
#699112
#689062
#688282
#240713