Finite Group Databasebeta

← #10846024

Subgroups — #10846024

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#83259C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic, complemented
3C2×#4812C212yesmaximal, complemented
4C2×#4688C211yesmaximal, characteristic, complemented
5#74368C211yesmaximal, characteristic
6C3×#1956C211yesmaximal, characteristic, complemented
7#70275C212yesmaximal, complemented
8#70272C212yesmaximal, complemented
9#70271C212yesmaximal, complemented
10C4×#821C211yesmaximal, characteristic, complemented
11#69567C211yesmaximal, characteristic, complemented
12#25393—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×#946C2212yes—
16C22×#215C2212yes—
17C22×ΓL(1,𝔽25)C2212yes—
18C2×#895—21no—
19C2×#895C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C3×#384C2211yescharacteristic
26C2×#835C2212yes—
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×(C2×D3)⋊C8C2212yes—
30#5083—24no—
31#4993—22no—
32#4991—22no—
33#4988—21no—
34#4987—21no—
35#4987C2212yes—
36#4975—21no—
37C2×#807C2212yes—
38C4×C4⋊Dic3C2211yescharacteristic
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4812—28no—
41#4688—24no—
42#4551C2211yescharacteristic
43#4551C2211yescharacteristic
44#4529C2212yes—
45C2×#468—32no—
46C2×#461—31no—
47C2×#436—31no—
48#2542—31no—
49#2537—31no—
50#2036—32no—
51#2000—32no—
52#1956D311yescharacteristic, complemented
53#1947—31no—
54#1894—32no—
55C4×#102—31no—
56C22×C3×Q8C2312yes—
57C22×D12—22no—
58C22×D12D411yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3C2312yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72#946—24no—
73#946—24no—
74#946—42no—
75#946—42no—
76C2×#215—24no—
77C2×#215—24no—
78C2×#215—42no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4D412yes—
85C2×C4⋊Dic3D411yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—21no—
92C22×C3⋊C8C2312yes—
93C2×ΓL(1,𝔽25)—24no—
94C2×ΓL(1,𝔽25)—24no—
95C2×ΓL(1,𝔽25)—42no—
96#895—24no—
97#895—42no—
98#895—44no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—22no—
103C3×#103—24no—
104C3×#103—42no—
105C3×C4×C8—22no—
106#835—24no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#821—22no—
110#821D414yescomplemented
111(C2×D3)⋊C8—24no—
112#807—24no—
113C4⋊(C3⋊C8)D414yescomplemented
114C22×SD16—32no—
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×D4—31no—
123#468—68no—
124#461—64no—
125#436—64no—
126C2×#107—32no—
127C2×C4⋊C8—31no—
128C2×#103—32no—
129C2×#103C2×D312yes—
130C2×#103C2×D312yes—
131C2×#102—31no—
132C2×#102—31no—
133C2×C22⋊C8—32no—
134C2×C4×C8C2×D311yescharacteristic
135#399—62no—
136#394—62no—
137#390—61no—
138#384—61no—
139#384C2×D311yescharacteristic
140#380—32no—
141#380—61no—
142C4×C4⋊C4—31no—
143#340—31no—
144#340—31no—
145#340—32no—
146C23×D3—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—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3C2×D412yes—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12—44no—
172C2×D12—44no—
173C2×D12—82no—
174C2×D12C2×D412yes—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×C4×D3—42no—
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—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—81no—
196C3×C42C2×D412yes—
197C3×C42#4112yes—
198#215—48no—
199(C2×D3)⋊C4—24no—
200(C2×D3)⋊C4—44no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—44no—
204(C2×D3)⋊C4—42no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3C2×D412yes—
211C4⋊Dic3#4112yes—
212Dic3⋊C4—44no—
213Dic3⋊C4—44no—
214C4×Dic3—42no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8C2×D414yes—
218ΓL(1,𝔽25)—48no—
219C22×Q8C22×D312yes—
220C22×D4—31no—
221C22×D4—31no—
222C22×D4—62no—
223C23×C4—32no—
224C23×C4—61no—
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—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—62no—
244C4×D4—62no—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#107—64no—
267C4⋊C8—34no—
268#103—24no—
269#103—42no—
270#103—64no—
271#102—34no—
272#102—62no—
273C22⋊C8—64no—
274C4×C8—22no—
275#95—62no—
276#95—61no—
277C23×C3C22×D411yescharacteristic
278C22×D3—22no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×Q8—42no—
295C3×Q8—42no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#12112yes—
305C2×C3×C4#12112yes—
306C2×C3×C4AGL(1,ℤ/8)12yes—
307C2×C3×C4#13712yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4H(5,𝔽2)11yescharacteristic
313C2×C3×C4H(5,𝔽2)11yescharacteristic
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326D12—22no—
327D12—22no—
328D12—41no—
329D12—41no—
330D12—44no—
331D12—88no—
332C4×D3—42no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—84no—
337C3×C8—22no—
338C3×C8—41no—
339C3⋊C8—24no—
340C24—32no—
341C24—61no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×D4—32no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—124no—
358C2×D4—124no—
359C2×D4—242no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—32no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4—122no—
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×C8D3×D412yes—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—32no—
392C4⋊C4—32no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—62no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—81no—
402C4⋊C4—124no—
403C4⋊C4—124no—
404C22⋊C4—64no—
405C22⋊C4—124no—
406C22⋊C4—124no—
407C22⋊C4—124no—
408C22⋊C4—124no—
409C22⋊C4—122no—
410C42—122no—
411C42#23512yes—
412C42D3×D412yes—
413C22×C3C2×#12111yescharacteristic
414C22×C3D4211yescharacteristic
415C22×C3#54611yescharacteristic
416C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C3C2×#13711yescharacteristic
418C22×C3#57711yescharacteristic
419C22×C3#57811yescharacteristic
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—81no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C3×C4—22no—
435C3×C4—82no—
436C3×C4—82no—
437C3×C4#46814yescomplemented
438C3×C4#53211yescharacteristic
439C3×C4#54011yescharacteristic
440C3×C4D4211yescharacteristic
441C3×C4#55011yescharacteristic
442Dic3—42no—
443Dic3—42no—
444Dic3—42no—
445Dic3—81no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23C2×D3×D411yescharacteristic
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465D4—62no—
466D4—62no—
467D4—121no—
468D4—124no—
469D4—121no—
470D4—248no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
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—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—244no—
496C2×C4#88412yes—
497C2×C4#88412yes—
498C2×C4#91412yes—
499C2×C4#91812yes—
500C2×C4C2×#23511yescharacteristic
501C2×C4C2×D3×D411yescharacteristic
502C2×C4C2×D3×D411yescharacteristic
503C2×C4#100411yescharacteristic
504C2×C4D3×#4111yescharacteristic
505C2×C4#100911yescharacteristic
506C8—22no—
507C8—41no—
508C8—64no—
509C2×C3#242811yescharacteristic
510C2×C3C2×#46811yescharacteristic
511C2×C3#306311yescharacteristic
512C2×C3#327211yescharacteristic
513C2×C3#327611yescharacteristic
514C2×C3#332911yescharacteristic
515C2×C3#333011yescharacteristic
516D3—82no—
517D3—82no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—241no—
529C22—241no—
530C22—241no—
531C22—241no—
532C22C2×#88411yescharacteristic
533C22#567611yescharacteristic
534C22#571711yescharacteristic
535C22C2×#91411yescharacteristic
536C22#576211yescharacteristic
537C22C2×#91811yescharacteristic
538C22#576611yescharacteristic
539C4—22no—
540C4—82no—
541C4—82no—
542C4—122no—
543C4—122no—
544C4—122no—
545C4—241no—
546C4#481214yescomplemented
547C4#554811yescharacteristic
548C4#556211yescharacteristic
549C4#559911yescharacteristic
550C4#560911yescharacteristic
551C3#2539311yescharacteristic, complemented, Sylow
552C2—242no—
553C2—242no—
554C2#7644411yescharacteristic
555C2C2×#481211yescharacteristic
556C2#7880011yescharacteristic
557C2#8161711yescharacteristic
558C2#8161811yescharacteristic
559C2#8177011yescharacteristic
560C2#8180511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radical#74368C2
hypercenter#1956D3
hypocenterC3#25393
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. #83259C2semidirect
2. C2×#5083C2semidirect
3. C2×#4812C2semidirect
4. C2×#4688C2semidirect
6. C3×#1956C2semidirect
7. #70275C2semidirect
8. #70272C2semidirect
9. #70271C2semidirect
10. C4×#821C2semidirect
11. #69567C2semidirect
52. #1956D3semidirect
110. #821D4semidirect
113. C4⋊(C3⋊C8)D4semidirect
437. C3×C4#468semidirect
546. C4#4812semidirect
551. C3#25393semidirect

Maximal subgroups

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

subgroupindex
#832592
C2×#50832
C2×#48122
C2×#46882
#743682
C3×#19562
#702752
#702722
#702712
C4×#8212
#695672
#253933