Finite Group Databasebeta

← #10846023

Subgroups — #10846023

Order 768.

1070 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82417C211yesmaximal, characteristic
2C2×#5019C211yesmaximal, characteristic, complemented
3C2×#4780C212yesmaximal, complemented
4C2×#4688C211yesmaximal, characteristic, complemented
5#73800C211yesmaximal, characteristic, complemented
6C3×C4×#102C211yesmaximal, characteristic, complemented
7#70205C212yesmaximal, complemented
8#69771C211yesmaximal, characteristic, complemented
9#69466C211yesmaximal, characteristic, complemented
10#68884C212yesmaximal, complemented
11#68825C212yesmaximal, complemented
12#25393—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#938C2212yes—
15C22×#214C2212yes—
16C2×#930—21no—
17C22×ΓL(1,𝔽25)C2212yes—
18C2×#897C2212yes—
19C2×#897C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5212—21no—
27C2×(C2×D3)⋊C8C2212yes—
28C2×#816C2212yes—
29C2×#816C2212yes—
30#5019—24no—
31#4987C2212yes—
32C2×#810C2212yes—
33C2×#807C2212yes—
34#4961—22no—
35#4939—21no—
36#4935—21no—
37#4935C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4780—28no—
40#4688—24no—
41#4677—22no—
42C3×#340C2211yescharacteristic
43C3×#340C2211yescharacteristic
44#4529C2212yes—
45C2×#468—32no—
46C2×#461—31no—
47C2×#436—31no—
48#2542—31no—
49#2537—31no—
50#2036—32no—
51#2000—32no—
52#1956—31no—
53#1947—31no—
54#1894—32no—
55C4×#102D311yescharacteristic, complemented
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×#67—22no—
59C22×D12D411yescharacteristic
60C22×C4×D3C2312yes—
61C22×Dic6C2312yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73#938—24no—
74#938—42no—
75C2×#214—24no—
76C2×#214—24no—
77C2×#214—42no—
78#930—44no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4D412yes—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3D411yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C22×C3⋊C8C2312yes—
92C2×ΓL(1,𝔽25)—24no—
93C2×ΓL(1,𝔽25)—24no—
94C2×ΓL(1,𝔽25)—42no—
95#897—24no—
96#897—24no—
97#897—42no—
98#897—42no—
99C4×D12—22no—
100C4×D12—22no—
101#869—22no—
102#869—22no—
103C3×#102—22no—
104C3×#102D414yescomplemented
105C3×C4×C8—22no—
106C3×#95—21no—
107C3⋊#95—22no—
108(C2×D3)⋊C8—24no—
109#816—24no—
110#816—42no—
111#810—24no—
112#807—24no—
113C4⋊(C3⋊C8)D414yescomplemented
114C22×SD16—32no—
115C22×M16—32no—
116C2×C4⋊Q8—31no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—32no—
119C2×#121—32no—
120C2×#121—61no—
121C2×C4×D4—31no—
122C2×C4×D4C2×D311yescharacteristic
123#468—68no—
124#461—64no—
125#436—64no—
126C2×#107—32no—
127C2×C4⋊C8—31no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#102C2×D311yescharacteristic
132C2×#102C2×D311yescharacteristic
133C2×C22⋊C8—32no—
134C2×C4×C8C2×D311yescharacteristic
135#399—62no—
136#394—62no—
137#390—61no—
138#384—31no—
139#384—61no—
140#380—32no—
141#380—61no—
142C4×C4⋊C4C2×D311yescharacteristic
143#340—32no—
144#340C2×D311yescharacteristic
145#340C2×D311yescharacteristic
146C24×C3—21no—
147C23×D3C2×D412yes—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4C2×D412yes—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C2×#67—44no—
164C2×#67—44no—
165C2×#67—82no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3C2×D412yes—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8C2×D412yes—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4#4112yes—
196C3×C22⋊C4—42no—
197C3×C42—42no—
198C3×C42C2×D412yes—
199C3×C42#4112yes—
200#217—44no—
201#214—48no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—44no—
205(C2×D3)⋊C4—44no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—44no—
210C4⋊Dic3—44no—
211C4⋊Dic3—42no—
212C4⋊Dic3—42no—
213C4⋊Dic3—42no—
214C4⋊Dic3—81no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8C2×D414yes—
218ΓL(1,𝔽25)—48no—
219C22×Q8—32no—
220C22×D4—31no—
221C22×D4—62no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—32no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×M16—64no—
229C2×M16—64no—
230C2×M16—122no—
231C22×C8—32no—
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D311yescharacteristic
234C4⋊Q8—62no—
235C4⋊Q8—62no—
236C22⋊Q8—64no—
237C22⋊Q8—64no—
238C22⋊Q8—122no—
239C22⋊Q8—122no—
240#121—64no—
241#121—122no—
242#121—124no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—21no—
265C2×C42C22×D311yescharacteristic
266#107—64no—
267C4⋊C8—34no—
268#103—64no—
269#103—64no—
270#103—122no—
271#102—22no—
272#102D1214yescomplemented
273C22⋊C8—64no—
274C4×C8—22no—
275#95—21no—
276#95—62no—
277C23×C3—22no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—41no—
295C3×D4—41no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—84no—
308C2×C3×C4#12112yes—
309C2×C3×C4#12112yes—
310C2×C3×C4AGL(1,ℤ/8)12yes—
311C2×C3×C4#13712yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318#67—88no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328D12—44no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—42no—
333Dic6—42no—
334Dic6—42no—
335Dic6—42no—
336Dic6—42no—
337C3×C8—22no—
338C3×C8—41no—
339C3⋊C8—24no—
340C24—21no—
341C24—32no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—62no—
354C2×D4—62no—
355C2×D4—62no—
356C2×D4—124no—
357C2×D4—124no—
358C2×D4—242no—
359C2×D4C2×D1212yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—32no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4D3×D411yescharacteristic
376C22×C4D3×D411yescharacteristic
377C22×C4C23×D311yescharacteristic
378SD16—128no—
379M16—128no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—34no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8C2×D1212yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—124no—
397C4⋊C4—124no—
398C4⋊C4—122no—
399C4⋊C4—122no—
400C4⋊C4—122no—
401C4⋊C4—241no—
402C4⋊C4C2×D1212yes—
403C4⋊C4#23512yes—
404C22⋊C4—42no—
405C22⋊C4—64no—
406C22⋊C4—124no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C22⋊C4—124no—
410C42—42no—
411C42#23512yes—
412C42D3×D412yes—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—81no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3C2×#12111yescharacteristic
421C22×C3D4211yescharacteristic
422C22×C3#54611yescharacteristic
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3C2×#13711yescharacteristic
425C22×C3#57711yescharacteristic
426C22×C3#57811yescharacteristic
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C3×C4—22no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—81no—
438C3×C4#46814yescomplemented
439C3×C4#53211yescharacteristic
440C3×C4#54011yescharacteristic
441C3×C4D4211yescharacteristic
442C3×C4#55011yescharacteristic
443Dic3—42no—
444Dic3—82no—
445Dic3—82no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—62no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23C2×D3×D411yescharacteristic
461Q8—122no—
462Q8—122no—
463Q8—122no—
464Q8—122no—
465D4—22no—
466D4—22no—
467D4—41no—
468D4—41no—
469D4—124no—
470D4—248no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—84no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4#88412yes—
497C2×C4#88412yes—
498C2×C4#91112yes—
499C2×C4#91512yes—
500C2×C4C22×D1211yescharacteristic
501C2×C4C2×#23511yescharacteristic
502C2×C4C2×#23511yescharacteristic
503C2×C4C2×D3×D411yescharacteristic
504C2×C4#100911yescharacteristic
505C2×C4#100911yescharacteristic
506C8—22no—
507C8—41no—
508C8—64no—
509C2×C3—82no—
510C2×C3#242811yescharacteristic
511C2×C3C2×#46811yescharacteristic
512C2×C3#306311yescharacteristic
513C2×C3#327211yescharacteristic
514C2×C3#327611yescharacteristic
515C2×C3#332911yescharacteristic
516C2×C3#333011yescharacteristic
517D3—82no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22C2×#88411yescharacteristic
533C22D4×D1211yescharacteristic
534C22#555011yescharacteristic
535C22C2×#91111yescharacteristic
536C22C2×#91511yescharacteristic
537C22#577011yescharacteristic
538C22#577111yescharacteristic
539C4—22no—
540C4—42no—
541C4—42no—
542C4—81no—
543C4—122no—
544C4—242no—
545C4—242no—
546C4#478014yescomplemented
547C4#547711yescharacteristic
548C4#553511yescharacteristic
549C4D4×D1211yescharacteristic
550C4#555311yescharacteristic
551C3#2539311yescharacteristic, complemented, Sylow
552C2—82no—
553C2—242no—
554C2#7348711yescharacteristic
555C2C2×#478011yescharacteristic
556C2#7884111yescharacteristic
557C2#7960311yescharacteristic
558C2#7961311yescharacteristic
559C2#7967211yescharacteristic
560C2#7968511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC4×#102D3
hypocenterC3#25393
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#5019C2semidirect
3. C2×#4780C2semidirect
4. C2×#4688C2semidirect
5. #73800C2semidirect
6. C3×C4×#102C2semidirect
7. #70205C2semidirect
8. #69771C2semidirect
9. #69466C2semidirect
10. #68884C2semidirect
11. #68825C2semidirect
55. C4×#102D3semidirect
104. C3×#102D4semidirect
113. C4⋊(C3⋊C8)D4semidirect
272. #102D12semidirect
438. C3×C4#468semidirect
546. C4#4780semidirect
551. C3#25393semidirect

Maximal subgroups

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

subgroupindex
#824172
C2×#50192
C2×#47802
C2×#46882
#738002
C3×C4×#1022
#702052
#697712
#694662
#688842
#688252
#253933