Finite Group Databasebeta

← #10823388

Subgroups — #10823388

Order 768.

1156 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#4790C212yesmaximal, complemented
3#76479C211yesmaximal, characteristic
4C3×#2024C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#70273C212yesmaximal
7#69691C211yesmaximal, characteristic, complemented
8#68825C212yesmaximal
9#68781C212yesmaximal, complemented
10#68631C211yesmaximal, characteristic, complemented
11#68599C211yesmaximal, characteristic
12#23571—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—21no—
15C2×#946C2211yescharacteristic
16C2×Q8×Dic3—21no—
17C2×Q8×Dic3C2211yescharacteristic
18C22×#216C2212yes—
19#5728—28no—
20C2×#897C2212yes—
21C2×#892—21no—
22C2×#890—21no—
23C2×#887C2212yes—
24C2×C42×D3C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#103C2212yes—
27C3×#391C2211yescharacteristic
28C2×(C2×D3)⋊C8C2212yes—
29C2×#816C2212yes—
30C2×#816C2212yes—
31C2×C8×Dic3C2212yes—
32#4987C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34C2×#810C2212yes—
35C2×#807C2212yes—
36#4928C2211yescharacteristic
37#4790—28no—
38#4668—21no—
39#4666—21no—
40#4655—21no—
41#4649—21no—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×C4×Q16—32no—
48C4×C22⋊Q8—31no—
49#2057—31no—
50#2043—31no—
51#2036—32no—
52#2024D311yescharacteristic, complemented
53#2016—31no—
54#2001—32no—
55C4×#103—32no—
56C4×C22⋊C8—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×Dic6C2312yes—
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—24no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42C2311yescharacteristic
69#946—28no—
70#946—22no—
71#946—22no—
72Q8×Dic3—28no—
73Q8×Dic3—24no—
74C2×#216—24no—
75C2×#216—24no—
76C2×#216—24no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—21no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96#897—24no—
97#897—24no—
98#892—28no—
99#890—28no—
100#887—24no—
101#887D418yes—
102C42×D3—24no—
103#869—24no—
104C3×#103—24no—
105C3×#95—21no—
106C3×#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110(C2×D3)⋊C8—24no—
111#816—24no—
112#816—24no—
113C8×Dic3—24no—
114#810—24no—
115#807—24no—
116C22×Q16—32no—
117#533—68no—
118C2×C4⋊Q8—31no—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×#125—61no—
121C2×C22⋊Q8—31no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—61no—
124C2×C4×Q8—31no—
125C2×C4×Q8—32no—
126C2×C4×Q8—61no—
127C2×#117—61no—
128C22×C42—31no—
129C4×Q16—68no—
130C2×#107—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103C2×D312yes—
135C2×C22⋊C8—32no—
136C2×C4×C8—32no—
137#391C2×D311yescharacteristic
138#389—31no—
139#387—61no—
140#385—61no—
141#380—31no—
142C4×C4⋊C4—31no—
143C4×C4⋊C4—61no—
144C4×C22⋊C4—61no—
145#340—32no—
146#340C2×D312yes—
147#336—31no—
148#336C2×D311yescharacteristic
149C23×D3C2×D411yescharacteristic
150C2×C3×Q8—24no—
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—42no—
155C2×C3×Q8—42no—
156C2×C3×Q8#4112yes—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168C22×Dic3C2×D411yescharacteristic
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—22no—
177C2×C4×D3—41no—
178C2×C4×D3Q1614yescomplemented
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6C2×D414yes—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2×D414yes—
192C3×C42—22no—
193#216—216no—
194(C2×D3)⋊C4—28no—
195(C2×D3)⋊C4—24no—
196(C2×D3)⋊C4—42no—
197C4⋊Dic3—24no—
198C4⋊Dic3—24no—
199C4⋊Dic3—28no—
200C4⋊Dic3—21no—
201C4⋊Dic3—22no—
202C4⋊Dic3—42no—
203C4⋊Dic3#4112yes—
204Dic3⋊C4—24no—
205Dic3⋊C4—28no—
206Dic3⋊C4—42no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3—28no—
210C4×Dic3—42no—
211C4×Dic3—42no—
212C4×Dic3C2×D414yes—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8#4114yes—
216C22×Q8—21no—
217C22×Q8—32no—
218C22×Q8C22×D311yescharacteristic
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C2×Q16—64no—
223C2×Q16—64no—
224C2×Q16—64no—
225C22×C8—32no—
226C22×C8C22×D312yes—
227C4⋊Q8—24no—
228C4⋊Q8—64no—
229#125—68no—
230C22⋊Q8—64no—
231C22⋊Q8—62no—
232C22⋊Q8—68no—
233C22⋊Q8—62no—
234C22⋊Q8—64no—
235C4×Q8—38no—
236C4×Q8—64no—
237C4×Q8—64no—
238C4×Q8—68no—
239#117—68no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—61no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C4⋊C4C22×D312yes—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—61no—
255C2×C42—31no—
256C2×C42—31no—
257C2×C42—31no—
258C2×C42—64no—
259C2×C42—62no—
260C2×C42—61no—
261C2×C42—61no—
262C2×C42C22×D311yescharacteristic
263#107—64no—
264#103—24no—
265#103—64no—
266#103—64no—
267#103—64no—
268C22⋊C8—64no—
269C4×C8—64no—
270#95—21no—
271#95—21no—
272#95—61no—
273#95—61no—
274#95—61no—
275C23×C3C22×D411yescharacteristic
276C22×D3—21no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3C2×Q1612yes—
283C3×Q8—24no—
284C3×Q8—48no—
285C3×Q8—42no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4C2×Q1612yes—
300C2×C3×C4#13512yes—
301C2×C3×C4C22×D412yes—
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4112yes—
304C2×C3×C4C2×#4111yescharacteristic
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—24no—
309C2×Dic3—21no—
310C2×Dic3—22no—
311C2×Dic3—21no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3#12012yes—
320C2×Dic3#12012yes—
321C2×Dic3C2×Q1612yes—
322C2×Dic3#13712yes—
323C4×D3—28no—
324C4×D3—44no—
325C4×D3—42no—
326C4×D3—42no—
327Dic6—24no—
328Dic6—24no—
329Dic6—42no—
330Dic6—42no—
331C3×C8—42no—
332C3⋊C8—24no—
333C24—31no—
334C2×Q8—24no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—34no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8#23912yes—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—34no—
353C22×C4—31no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—121no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4C23×D311yescharacteristic
369Q16—616no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—34no—
374C2×C8—62no—
375C2×C8—62no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—32no—
380C4⋊C4—68no—
381C4⋊C4—64no—
382C4⋊C4—68no—
383C4⋊C4—64no—
384C4⋊C4—64no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—122no—
388C4⋊C4—122no—
389C4⋊C4D3×D414yes—
390C22⋊C4—68no—
391C22⋊C4—64no—
392C22⋊C4—122no—
393C42—22no—
394C42—34no—
395C42—62no—
396C42—62no—
397C42—68no—
398C42—122no—
399C42—122no—
400C22×C3C2×#12011yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#53211yescharacteristic
403C22×C3C22×Q1611yescharacteristic
404C22×C3C2×#13511yescharacteristic
405C22×C3C2×#13711yescharacteristic
406C22×C3#57511yescharacteristic
407C2×D3—22no—
408C2×D3—22no—
409C2×D3—22no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4—44no—
418C3×C4—81no—
419C3×C4#52512yes—
420C3×C4#53212yes—
421Dic3—22no—
422Dic3—42no—
423Dic3—44no—
424Dic3—41no—
425Dic3—81no—
426Dic3#45114yes—
427C23—32no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23—61no—
434C23C2×D3×D411yescharacteristic
435Q8—24no—
436Q8—48no—
437Q8—42no—
438Q8—64no—
439Q8—64no—
440Q8—122no—
441Q8—122no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—68no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—64no—
465C2×C4—62no—
466C2×C4—61no—
467C2×C4—124no—
468C2×C4—122no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4#91612yes—
478C2×C4D3×Q1612yes—
479C2×C4C2×D3×D412yes—
480C2×C4C2×#23911yescharacteristic
481C2×C4D3×#4111yescharacteristic
482C2×C4D3×#4112yes—
483C8—42no—
484C8—64no—
485C2×C3#238811yescharacteristic
486C2×C3C2×#45111yescharacteristic
487C2×C3#300211yescharacteristic
488C2×C3#310411yescharacteristic
489C2×C3#310711yescharacteristic
490C2×C3#317011yescharacteristic
491C2×C3#317211yescharacteristic
492D3—42no—
493C22—62no—
494C22—62no—
495C22—62no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22D3×#12011yescharacteristic
501C22#562711yescharacteristic
502C22#566811yescharacteristic
503C22C2×#91611yescharacteristic
504C22C2×D3×Q1611yescharacteristic
505C22#577311yescharacteristic
506C22D3×#13711yescharacteristic
507C4—22no—
508C4—22no—
509C4—34no—
510C4—44no—
511C4—42no—
512C4—62no—
513C4—81no—
514C4—122no—
515C4—121no—
516C4—124no—
517C4—241no—
518C4#562712yes—
519C4#566812yes—
520C3#2357111yescharacteristic, complemented, Sylow
521C2—122no—
522C2#7677711yescharacteristic
523C2#8014511yescharacteristic
524C2D3×#45111yescharacteristic
525C2#8099711yescharacteristic
526C2#8109211yescharacteristic
527C2#8159311yescharacteristic
528C2#8166511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2024C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2024D3
hypocenterC3#23571
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
2. C2×#4790C2semidirect
4. C3×#2024C2semidirect
5. #71782C2semidirect
7. #69691C2semidirect
9. #68781C2semidirect
10. #68631C2semidirect
52. #2024D3semidirect
178. C2×C4×D3Q16semidirect
520. C3#23571semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#47902
#764792
C3×#20242
#717822
#702732
#696912
#688252
#687812
#686312
#685992
#235713