Finite Group Databasebeta

← #10846019

Subgroups — #10846019

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
1C2×C3×#461C211yesmaximal, characteristic, complemented
2C2×#5174C212yesmaximal, complemented
3#83259C211yesmaximal, characteristic, complemented
4#82417C211yesmaximal, characteristic
5C2×#4838C211yesmaximal, characteristic, complemented
6#72054C212yesmaximal, complemented
7#71785C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic, complemented
9#70453C211yesmaximal, characteristic, complemented
10#70271C212yesmaximal, complemented
11#70244C212yesmaximal, complemented
12#25393—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#946C2212yes—
17C2×#946C2212yes—
18C2×#938C2212yes—
19C2×#930—21no—
20C22×ΓL(1,𝔽25)C2212yes—
21C2×C4×D12C2211yescharacteristic
22C3×#461—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29#5212—21no—
30#5174—28no—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#818C2212yes—
33#4993—22no—
34#4988—21no—
35#4987C2212yes—
36#4975—21no—
37#4961—22no—
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40C2×C4×C3⋊C8C2211yescharacteristic
41#4838—24no—
42#4551C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#468—32no—
46C2×#461D311yescharacteristic, complemented
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×#102—31no—
56C22×C3×Q8C2312yes—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×#67—22no—
60C22×D12D411yescharacteristic
61C22×C4×D3C2312yes—
62C2×C3×SD16—24no—
63C2×C3×SD16—24no—
64C2×C3×SD16—42no—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76#946—24no—
77#946—24no—
78#946—42no—
79#946—42no—
80#938—24no—
81#938—42no—
82#930—44no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4D412yes—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3D411yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×C4×Dic3—21no—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C2×ΓL(1,𝔽25)—24no—
99C2×ΓL(1,𝔽25)—24no—
100C2×ΓL(1,𝔽25)—42no—
101C4×D12—22no—
102C4×D12—22no—
103C3×C4⋊C8D414yescomplemented
104C3×#103—24no—
105#835—24no—
106#835—42no—
107#832—22no—
108#832D414yescomplemented
109C3⋊#95—21no—
110C3⋊#95—22no—
111(C2×D3)⋊C8—24no—
112#818—24no—
113C4×C3⋊C8—22no—
114C22×SD16C2×D312yes—
115C22×M16—32no—
116C2×C4⋊Q8C2×D311yescharacteristic
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—24no—
125#436—64no—
126C2×#107—32no—
127C2×C4⋊C8C2×D311yescharacteristic
128C2×#103—32no—
129C2×#103—32no—
130C2×#103C2×D312yes—
131C2×#102—31no—
132C2×#102—31no—
133C2×C22⋊C8—32no—
134C2×C4×C8—31no—
135#399—62no—
136#394—62no—
137#390—61no—
138#384—31no—
139#384—61no—
140#380—32no—
141#380—61no—
142C4×C4⋊C4—31no—
143#340—32no—
144#340—31no—
145#340—31no—
146C24×C3—21no—
147C23×D3C2×D412yes—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4C2×D412yes—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—82no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3C2×D412yes—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C3×SD16—48no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8C2×D414yes—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C42C2×D412yes—
194C3×C42#4112yes—
195#217—44no—
196(C2×D3)⋊C4—24no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—44no—
200C4⋊Dic3—22no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—42no—
206C4⋊Dic3—81no—
207C4⋊Dic3C2×D412yes—
208C4⋊Dic3#4112yes—
209Dic3⋊C4—44no—
210Dic3⋊C4—44no—
211C4×Dic3—42no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8C2×D412yes—
218ΓL(1,𝔽25)—48no—
219C22×Q8C22×D312yes—
220C22×D4—31no—
221C22×D4—62no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—32no—
225C2×SD16—24no—
226C2×SD16—24no—
227C2×SD16—42no—
228C2×M16—64no—
229C2×M16—64no—
230C2×M16—122no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D312yes—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
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—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
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—61no—
265C2×C42C22×D311yescharacteristic
266#107—64no—
267C4⋊C8#6714yescomplemented
268#103—24no—
269#103—64no—
270#103—122no—
271#102—34no—
272#102—62no—
273C22⋊C8—64no—
274C4×C8—62no—
275#95—62no—
276#95—61no—
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×Q8—42no—
293C3×Q8—42no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—84no—
309C2×C3×C4#12112yes—
310C2×C3×C4#12112yes—
311C2×C3×C4AGL(1,ℤ/8)12yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319#67—88no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332D12—44no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—42no—
337C3×C8—24no—
338C3⋊C8—22no—
339C3⋊C8—41no—
340C24—21no—
341C24—32no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
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×D4D3×D412yes—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—32no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—61no—
367C22×C4—61no—
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—48no—
379M16—128no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—32no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8C2×#6714yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—32no—
392C4⋊C4—32no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—62no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—124no—
401C4⋊C4—124no—
402C4⋊C4—122no—
403C4⋊C4—241no—
404C22⋊C4—42no—
405C22⋊C4—64no—
406C22⋊C4—124no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C22⋊C4—124no—
410C42—122no—
411C42#23712yes—
412C42C2×#6712yes—
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—82no—
436C3×C4—81no—
437C3×C4#46814yescomplemented
438C3×C4#53211yescharacteristic
439C3×C4#54011yescharacteristic
440C3×C4D4211yescharacteristic
441C3×C4#55011yescharacteristic
442Dic3—42no—
443Dic3—42no—
444Dic3—42no—
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—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465D4—22no—
466D4—22no—
467D4—41no—
468D4—41no—
469D4—124no—
470D4—248no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—84no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4#91412yes—
497C2×C4#91512yes—
498C2×C4#93812yes—
499C2×C4#93812yes—
500C2×C4C2×D3×D411yescharacteristic
501C2×C4C2×#23711yescharacteristic
502C2×C4#100411yescharacteristic
503C2×C4D3×#4111yescharacteristic
504C2×C4#100911yescharacteristic
505C2×C4C22×#6711yescharacteristic
506C8—24no—
507C8—62no—
508C8—121no—
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—
532C22#560911yescharacteristic
533C22#561211yescharacteristic
534C22C2×#91411yescharacteristic
535C22C2×#91511yescharacteristic
536C22#577811yescharacteristic
537C22#577911yescharacteristic
538C22C2×#93811yescharacteristic
539C4—22no—
540C4—81no—
541C4—82no—
542C4—122no—
543C4—122no—
544C4—122no—
545C4—242no—
546C4#517414yescomplemented
547C4#559611yescharacteristic
548C4#561911yescharacteristic
549C4D4×#6711yescharacteristic
550C4#583411yescharacteristic
551C3#2539311yescharacteristic, complemented, Sylow
552C2—82no—
553C2—242no—
554C2#8044611yescharacteristic
555C2#8044711yescharacteristic
556C2#8045411yescharacteristic
557C2#8045511yescharacteristic
558C2#8264811yescharacteristic
559C2C2×#517411yescharacteristic
560C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC2×#461D3
hypocenterC3#25393
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×C3×#461C2semidirect
2. C2×#5174C2semidirect
3. #83259C2semidirect
5. C2×#4838C2semidirect
6. #72054C2semidirect
7. #71785C2semidirect
8. C4×#832C2semidirect
9. #70453C2semidirect
10. #70271C2semidirect
11. #70244C2semidirect
46. C2×#461D3semidirect
103. C3×C4⋊C8D4semidirect
108. #832D4semidirect
267. C4⋊C8#67semidirect
437. C3×C4#468semidirect
546. C4#5174semidirect
551. C3#25393semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4612
C2×#51742
#832592
#824172
C2×#48382
#720542
#717852
C4×#8322
#704532
#702712
#702442
#253933