Finite Group Databasebeta

← #10823154

Subgroups — #10823154

Order 768.

1362 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×Q16C211yesmaximal, characteristic
2C2×#4839C211yesmaximal, characteristic
3C2×#4802C212yesmaximal
4#76610C212yesmaximal
5C3×#2056C211yesmaximal, characteristic, complemented
6#72068C211yesmaximal, characteristic, complemented
7#70487C211yesmaximal, characteristic, complemented
8#70285C212yesmaximal, complemented
9#69681C212yesmaximal, complemented
10#68866C211yesmaximal, characteristic
11#68864C211yesmaximal, characteristic, complemented
12#23542—31nomaximal, Sylow
13C22×D3×Q8C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×#216C2212yes—
16C2×D3×Q16—28no—
17C2×D3×Q16—24no—
18C22×Dic12C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×#892C2212yes—
22C2×#872C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C3×#391C2212yes—
26#5229C2212yes—
27C2×#835C2212yes—
28C2×#816C2212yes—
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4973C2212yes—
32C2×#810C2212yes—
33#4929C2212yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35#4839—24no—
36#4802—28no—
37#4664—22no—
38#4664—22no—
39#4646—22no—
40#4646—22no—
41C3×#340C2211yescharacteristic
42#4551C2211yescharacteristic
43#4529C2212yes—
44C23×Q16—31no—
45C2×#424—31no—
46C2×#419—32no—
47#2310—32no—
48#2056—31no—
49#2056—32no—
50#2056—32no—
51#2056—31no—
52#2056—31no—
53#2056D311yescharacteristic, complemented
54#1735—31no—
55C22×C3×Q8C2312yes—
56C2×D3×Q8—24no—
57C2×D3×Q8—24no—
58C2×D3×Q8—24no—
59C2×D3×Q8—24no—
60C2×D3×Q8—24no—
61C2×D3×Q8D418yes—
62C22×C4×D3—22no—
63C22×C4×D3C2311yescharacteristic
64C22×Dic6—22no—
65C22×Dic6C2312yes—
66C2×C3×Q16—24no—
67C2×C3×Q16—22no—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72C2×#216—24no—
73C2×#216—24no—
74C2×#216—24no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4D411yescharacteristic
77C2×(C2×D3)⋊C4D411yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×Dic3⋊C4D411yescharacteristic
80C2×Dic3⋊C4D411yescharacteristic
81C2×C4×Dic3—22no—
82C2×C4×Dic3C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85D3×Q16—232no—
86C2×Dic12—24no—
87C2×Dic12—22no—
88C2×ΓL(1,𝔽25)—24no—
89C2×ΓL(1,𝔽25)—22no—
90C2×D3×C8—24no—
91C2×D3×C8D414yes—
92#892—24no—
93#892—24no—
94#872—22no—
95#872—22no—
96C3×C4⋊C8—22no—
97C3×#103—24no—
98C3×#95—22no—
99C3×#95—22no—
100#835—24no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#816—24no—
108Dic3⋊C8—22no—
109Dic3⋊C8—22no—
110#810—24no—
111C4⋊(C3⋊C8)—22no—
112C23×Q8—32no—
113C22×Q16—32no—
114C22×Q16—31no—
115C22×Q16—64no—
116C22×Q16—68no—
117C22×Q16C2×D311yescharacteristic
118C22×M16—31no—
119C23×C8—31no—
120C2×#117—31no—
121C2×#117—32no—
122#424—64no—
123C2×C4⋊C8—31no—
124C2×C4⋊C8—31no—
125C2×C4⋊C8—31no—
126C2×C4⋊C8C2×D311yescharacteristic
127#419—68no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#103C2×D312yes—
132#391—32no—
133#391—32no—
134#391—32no—
135#391—62no—
136#391—62no—
137#391C2×D312yes—
138#381—62no—
139#381—62no—
140#340—31no—
141#340—32no—
142#340C2×D311yescharacteristic
143C23×D3C2×D411yescharacteristic
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8C2×D414yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C22×Dic3C2×D411yescharacteristic
160D3×Q8—216no—
161D3×Q8—216no—
162D3×Q8—48no—
163D3×Q8—48no—
164C2×C4×D3—28no—
165C2×C4×D3—44no—
166C2×C4×D3—44no—
167C2×C4×D3C2×D414yes—
168C2×C4×D3C2×D412yes—
169C2×C4×D3#4112yes—
170C2×C4×D3#4114yes—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6—28no—
177C2×Dic6—44no—
178C2×Dic6—44no—
179C2×Dic6C2×D414yes—
180C3×Q16—28no—
181C2×C3×C8—22no—
182C2×C3×C8—21no—
183C2×C3×C8—22no—
184C2×C3×C8C2×D412yes—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C42—21no—
188C3×C42—21no—
189#216—216no—
190(C2×D3)⋊C4—22no—
191(C2×D3)⋊C4—22no—
192(C2×D3)⋊C4—44no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195Dic3⋊C4—22no—
196Dic3⋊C4—22no—
197C4×Dic3—21no—
198C4×Dic3—21no—
199C4×Dic3—44no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8C2×D412yes—
204Dic12—28no—
205ΓL(1,𝔽25)—44no—
206D3×C8—28no—
207C22×Q8—38no—
208C22×Q8—32no—
209C22×Q8—62no—
210C22×Q8—64no—
211C22×Q8—64no—
212C22×Q8—64no—
213C22×Q8—64no—
214C22×Q8—64no—
215C22×Q8C22×D312yes—
216C23×C4—31no—
217C23×C4—62no—
218C2×Q16—24no—
219C2×Q16—22no—
220C2×Q16—64no—
221C2×Q16—64no—
222C2×Q16—632no—
223C2×Q16—64no—
224C2×Q16—64no—
225C2×Q16—62no—
226C2×M16—64no—
227C2×M16—62no—
228C22×C8—31no—
229C22×C8—34no—
230C22×C8—31no—
231C22×C8—64no—
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D311yescharacteristic
234#117—62no—
235#117—62no—
236#117—64no—
237#117—64no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4C22×D312yes—
242C2×C22⋊C4—31no—
243C2×C22⋊C4—31no—
244C2×C22⋊C4—62no—
245C2×C42—31no—
246C2×C42—62no—
247C2×C42C22×D311yescharacteristic
248C4⋊C8—22no—
249C4⋊C8—62no—
250C4⋊C8—62no—
251C4⋊C8—62no—
252#103—24no—
253#103—64no—
254#103—64no—
255#103—64no—
256#95—22no—
257#95—22no—
258#95—62no—
259#95—62no—
260#95—62no—
261#95—62no—
262#95—62no—
263#95—62no—
264C23×C3C22×D411yescharacteristic
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—22no—
268C22×D3#12112yes—
269C22×D3#12114yes—
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8—42no—
273C3×Q8—42no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4C22×D412yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—24no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—22no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3#12114yes—
301C2×Dic3#12112yes—
302C4×D3—416no—
303C4×D3#12014yes—
304C4×D3#12318yes—
305C4×D3#12414yes—
306Dic6—24no—
307Dic6—24no—
308Dic6—42no—
309Dic6—42no—
310Dic6—416no—
311C3×C8—22no—
312C3×C8—41no—
313C3⋊C8—22no—
314C3⋊C8—41no—
315C24—31no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—34no—
322C2×Q8—68no—
323C2×Q8—616no—
324C2×Q8—616no—
325C2×Q8—62no—
326C2×Q8—62no—
327C2×Q8—62no—
328C2×Q8—62no—
329C2×Q8—62no—
330C2×Q8—124no—
331C2×Q8—124no—
332C2×Q8—128no—
333C2×Q8—128no—
334C2×Q8D3×D414yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—31no—
338C22×C4—32no—
339C22×C4—34no—
340C22×C4—34no—
341C22×C4—32no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—68no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—124no—
348C22×C4—124no—
349C22×C4D3×D411yescharacteristic
350C22×C4D3×D411yescharacteristic
351C22×C4C23×D311yescharacteristic
352Q16—28no—
353Q16—68no—
354Q16—616no—
355M16—124no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—32no—
360C2×C8—68no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—61no—
364C2×C8D3×D412yes—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—62no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C22⋊C4—62no—
372C22⋊C4—62no—
373C22⋊C4—124no—
374C42—21no—
375C42—21no—
376C42—61no—
377C42—61no—
378C42—124no—
379C22×C3C2×#12112yes—
380C22×C3C2×#12111yescharacteristic
381C22×C3#57811yescharacteristic
382C22×C3#57811yescharacteristic
383C22×C3#57812yes—
384C2×D3—24no—
385C2×D3—24no—
386C2×D3—22no—
387C2×D3—22no—
388C2×D3#39414yes—
389C3×C4—44no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—82no—
393C3×C4C2×#12011yescharacteristic
394C3×C4C2×#12312yes—
395C3×C4C2×#12411yescharacteristic
396Dic3—44no—
397Dic3—41no—
398Dic3—41no—
399Dic3—82no—
400Dic3#39414yes—
401C23—32no—
402C23—34no—
403C23—61no—
404C23—61no—
405C23—62no—
406C23C2×D3×D411yescharacteristic
407Q8—24no—
408Q8—24no—
409Q8—42no—
410Q8—42no—
411Q8—64no—
412Q8—64no—
413Q8—1216no—
414Q8—122no—
415Q8—122no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—34no—
423C2×C4—32no—
424C2×C4—38no—
425C2×C4—34no—
426C2×C4—34no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—62no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—64no—
438C2×C4—1216no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4C2×D3×D412yes—
445C2×C4C2×D3×D411yescharacteristic
446C2×C4D3×#4111yescharacteristic
447C2×C4D3×#4112yes—
448C8—22no—
449C8—41no—
450C8—62no—
451C8—121no—
452C2×C3C2×#39411yescharacteristic
453C2×C3#304412yes—
454C2×C3#304412yes—
455C2×C3#305911yescharacteristic
456C2×C3#305911yescharacteristic
457D3—24no—
458C22—34no—
459C22—62no—
460C22—64no—
461C22—62no—
462C22—64no—
463C22D3×#12111yescharacteristic
464C22D3×#12112yes—
465C22#576611yescharacteristic
466C22#577111yescharacteristic
467C22#577912yes—
468C4—34no—
469C4—41no—
470C4—44no—
471C4—41no—
472C4—82no—
473C4—121no—
474C4—124no—
475C4—121no—
476C4—242no—
477C4D3×#12011yescharacteristic
478C4D3×#12312yes—
479C4D3×#12411yescharacteristic
480C3#2354211yescharacteristic, complemented, Sylow
481C2—64no—
482C2D3×#39411yescharacteristic
483C2#8047812yes—
484C2#8060112yes—
485C2#8076011yescharacteristic
486C2#8090511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2056D3
hypocenterC3#23542
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
5. C3×#2056C2semidirect
6. #72068C2semidirect
7. #70487C2semidirect
8. #70285C2semidirect
9. #69681C2semidirect
11. #68864C2semidirect
53. #2056D3semidirect
480. C3#23542semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q162
C2×#48392
C2×#48022
#766102
C3×#20562
#720682
#704872
#702852
#696812
#688662
#688642
#235423