Finite Group Databasebeta

← #10829936

Subgroups — #10829936

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic, complemented
2C2×C4×#213C211yesmaximal, characteristic
3C2×#4841C211yesmaximal, characteristic, complemented
4#74369C211yesmaximal, characteristic, complemented
5C3×#1802C211yesmaximal, characteristic, complemented
6#70472C211yesmaximal, characteristic, complemented
7#70214C212yesmaximal
8#70178C212yesmaximal, complemented
9#69579C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68906C212yesmaximal, complemented
12#24185—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936C2212yes—
15C22×#213C2211yescharacteristic
16C2×#926—21no—
17C2×#924—21no—
18C22×#204C2212yes—
19#5542—24no—
20#5542—24no—
21C2×#897—21no—
22C2×#887C2212yes—
23C2×#879—22no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29C3×#384C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#816C2212yes—
32C4×#213—24no—
33#4994—22no—
34#4988C2212yes—
35#4987—21no—
36#4984C2212yes—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#809C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4841—24no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×C4×D8—31no—
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1956—31no—
54#1880—32no—
55#1863—32no—
56#1802D311yescharacteristic, complemented
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6D411yescharacteristic
62C22×Dic6C2312yes—
63C22×C3×C8C2312yes—
64C3×C4×D4—22no—
65C3×C4×D4D414yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—22no—
74#936—24no—
75#936—24no—
76C2×#213—22no—
77C2×#213—22no—
78C2×#213—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82#926—28no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88#924—28no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4D412yes—
91C2×C4×Dic3—22no—
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C2×#204—24no—
96C2×#204—24no—
97C2×#204—42no—
98#897—44no—
99#887—24no—
100#887—24no—
101#879—216no—
102C4×D12—22no—
103C4×D12—22no—
104#869—22no—
105#869D414yes—
106C4×Dic6—22no—
107C4×Dic6—22no—
108C3×C4⋊C8—22no—
109C3×#102—42no—
110#832—22no—
111C3⋊#95—22no—
112#816—24no—
113#810—24no—
114#810—42no—
115#809—22no—
116#807—22no—
117C4×C3⋊C8—22no—
118C22×SD16—32no—
119C22×D8—31no—
120#556—64no—
121#556—64no—
122C2×C4⋊Q8—31no—
123C2×#126—62no—
124C2×#124—32no—
125C2×C22⋊Q8—61no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—32no—
129C2×C4×D4—31no—
130C2×C4×D4C2×D311yescharacteristic
131C2×#117—61no—
132#461—64no—
133C4×D8—64no—
134C2×#107—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—32no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#102—31no—
140C2×#102—31no—
141C2×#102C2×D312yes—
142C2×C4×C8—31no—
143#396—62no—
144#390—32no—
145#389—32no—
146#384C2×D311yescharacteristic
147#380—61no—
148#340C2×D312yes—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C23×D3—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—21no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—22no—
168C22×Dic3—21no—
169C22×Dic3C2×D412yes—
170C22×Dic3C2×D412yes—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12#4112yes—
177C2×C4×D3—42no—
178C2×C4×D3—42no—
179C2×C4×D3—42no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6C2×D414yes—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—81no—
197C3×C4⋊C4C2×D412yes—
198C3×C22⋊C4—24no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#217—28no—
202#217—28no—
203#213—28no—
204(C2×D3)⋊C4—44no—
205(C2×D3)⋊C4—42no—
206(C2×D3)⋊C4—82no—
207C4⋊Dic3—22no—
208C4⋊Dic3—24no—
209C4⋊Dic3—24no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—44no—
213Dic3⋊C4—28no—
214Dic3⋊C4—24no—
215C4×Dic3—28no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8#4112yes—
224#204—48no—
225C22×Q8—31no—
226C22×Q8—32no—
227C22×D4—31no—
228C22×D4C22×D311yescharacteristic
229C23×C4—21no—
230C23×C4—61no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C2×D8—62no—
235C2×D8—62no—
236C2×D8—62no—
237C22×C8—31no—
238C22×C8—31no—
239C22×C8C22×D312yes—
240C4⋊Q8—34no—
241C4⋊Q8—62no—
242#126—616no—
243#124—64no—
244#124—64no—
245C22⋊Q8—68no—
246C22⋊Q8—124no—
247C4×Q8—64no—
248C4×Q8—64no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—22no—
252C4×D4—62no—
253C4×D4—62no—
254C4×D4D1214yes—
255#117—68no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4C22×D312yes—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—62no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—62no—
273C2×C42—32no—
274C2×C42—62no—
275C2×C42C22×D311yescharacteristic
276#107—62no—
277C4⋊C8—22no—
278#103—64no—
279#103—64no—
280#103—122no—
281#102—42no—
282#102—62no—
283#102—62no—
284C4×C8—62no—
285#95—62no—
286C23×C3—22no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3C22×D411yescharacteristic
294C22×D3—22no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—41no—
304C3×D4—41no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—48no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4#13512yes—
317C2×C3×C4AGL(1,ℤ/8)12yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336D12—22no—
337D12—22no—
338D12—41no—
339D12—41no—
340C4×D3—84no—
341Dic6—24no—
342Dic6—24no—
343Dic6—42no—
344Dic6—42no—
345Dic6—44no—
346C3×C8—42no—
347C3⋊C8—22no—
348C3⋊C8—41no—
349C24—21no—
350C24—61no—
351C2×Q8—34no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—62no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—32no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4C2×D1212yes—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—122no—
382C22×C4—122no—
383C22×C4—122no—
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D411yescharacteristic
386C22×C4C23×D311yescharacteristic
387SD16—128no—
388D8—68no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—32no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—68no—
403C4⋊C4—62no—
404C4⋊C4—64no—
405C4⋊C4—64no—
406C4⋊C4—64no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—81no—
410C4⋊C4—124no—
411C4⋊C4C2×D1212yes—
412C22⋊C4—24no—
413C22⋊C4—68no—
414C22⋊C4—68no—
415C22⋊C4—124no—
416C22⋊C4—122no—
417C22⋊C4—242no—
418C42—68no—
419C42—62no—
420C42—62no—
421C42C2×D1212yes—
422C42D3×D412yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#54811yescharacteristic
432C22×C3#55011yescharacteristic
433C22×C3C2×#13511yescharacteristic
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3#57711yescharacteristic
436C22×C3#57811yescharacteristic
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—82no—
447C3×C4#44914yes—
448C3×C4#52511yescharacteristic
449C3×C4#53711yescharacteristic
450C3×C4#54611yescharacteristic
451C3×C4#54711yescharacteristic
452Dic3—44no—
453Dic3—42no—
454Dic3—42no—
455Dic3—81no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—62no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23—121no—
470C23C2×D3×D411yescharacteristic
471Q8—64no—
472Q8—64no—
473Q8—122no—
474Q8—122no—
475Q8—124no—
476D4—22no—
477D4—22no—
478D4—41no—
479D4—41no—
480D4—62no—
481D4—62no—
482D4—121no—
483D4—121no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—244no—
506C2×C4#88212yes—
507C2×C4#88212yes—
508C2×C4#91112yes—
509C2×C4#91612yes—
510C2×C4C22×D1211yescharacteristic
511C2×C4C2×D3×D411yescharacteristic
512C2×C4#100411yescharacteristic
513C2×C4#100711yescharacteristic
514C2×C4D3×#4111yescharacteristic
515C2×C4D3×#4111yescharacteristic
516C8—42no—
517C8—62no—
518C8—121no—
519C2×C3—82no—
520C2×C3#240711yescharacteristic
521C2×C3C2×#44911yescharacteristic
522C2×C3#301311yescharacteristic
523C2×C3#336011yescharacteristic
524C2×C3#336711yescharacteristic
525C2×C3#339311yescharacteristic
526C2×C3#339711yescharacteristic
527D3—82no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—241no—
539C22—241no—
540C22—241no—
541C22—241no—
542C22C2×#88211yescharacteristic
543C22#559911yescharacteristic
544C22#563311yescharacteristic
545C22C2×#91111yescharacteristic
546C22C2×#91611yescharacteristic
547C22#577511yescharacteristic
548C22#577911yescharacteristic
549C4—22no—
550C4—42no—
551C4—82no—
552C4—124no—
553C4—122no—
554C4—122no—
555C4—241no—
556C4#478114yes—
557C4#551011yescharacteristic
558C4#552411yescharacteristic
559C4#561211yescharacteristic
560C4#563111yescharacteristic
561C3#2418511yescharacteristic, complemented, Sylow
562C2—82no—
563C2—242no—
564C2#7401511yescharacteristic
565C2C2×#478111yescharacteristic
566C2#7888711yescharacteristic
567C2#8032611yescharacteristic
568C2#8042111yescharacteristic
569C2#8099311yescharacteristic
570C2#8108711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1802D3
hypocenterC3#24185
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. C2×#5542C2semidirect
3. C2×#4841C2semidirect
4. #74369C2semidirect
5. C3×#1802C2semidirect
6. #70472C2semidirect
8. #70178C2semidirect
9. #69579C2semidirect
10. #68909C2semidirect
11. #68906C2semidirect
56. #1802D3semidirect
561. C3#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×C4×#2132
C2×#48412
#743692
C3×#18022
#704722
#702142
#701782
#695792
#689092
#689062
#241853