Finite Group Databasebeta

← #10829955

Subgroups — #10829955

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3C2×C4×#204C211yesmaximal, characteristic
4#74369C211yesmaximal, characteristic
5C3×#1956C211yesmaximal, characteristic, complemented
6#70282C212yesmaximal
7#70246C212yesmaximal, complemented
8#70240C212yesmaximal
9#70238C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69588C211yesmaximal, characteristic, complemented
12#24187—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×Q8×Dic3C2212yes—
15C2×#944C2212yes—
16C22×#215C2212yes—
17C22×#204C2211yescharacteristic
18#5721—24no—
19#5721—24no—
20C2×#897—21no—
21C2×#890—21no—
22C2×#889—22no—
23C2×#887—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C2×#835C2212yes—
32C2×#821C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#5083—24no—
36#4994—22no—
37#4988C2212yes—
38#4987—21no—
39#4984C2212yes—
40C2×#810C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C4×#204—24no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#557—31no—
47C2×#461—31no—
48C2×C4×SD16—31no—
49#2538—31no—
50#2043—32no—
51#2013—32no—
52#1984—31no—
53#1956D311yescharacteristic, complemented
54#1880—32no—
55#1865—32no—
56#1803—31no—
57C22×C3×Q8C2312yes—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
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
72Q8×Dic3—24no—
73Q8×Dic3—24no—
74#944—24no—
75#944—24no—
76C2×#215—24no—
77C2×#215—24no—
78C2×#215—42no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4D412yes—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3C2312yes—
94C22×C3⋊C8C2312yes—
95C2×#204—22no—
96C2×#204—22no—
97C2×#204—22no—
98#897—44no—
99#890—28no—
100#889—216no—
101#887—28no—
102C4×D12—22no—
103C4×D12—22no—
104C4×Dic6—22no—
105C4×Dic6—22no—
106C4×Dic6—22no—
107C4×Dic6D414yes—
108C3×#103—24no—
109C3×#103—42no—
110C3×C4×C8—22no—
111#835—24no—
112C3⋊#95—22no—
113#821—22no—
114#817—22no—
115#816—22no—
116#810—42no—
117C4⋊(C3⋊C8)—22no—
118C22×SD16—32no—
119C22×SD16—31no—
120#557—64no—
121#557—64no—
122C2×C4⋊Q8—32no—
123C2×C4⋊Q8C2×D311yescharacteristic
124C2×#125—62no—
125C2×#125—61no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—32no—
129C2×C4×Q8—31no—
130C2×C4×Q8—61no—
131C2×C4×D4—31no—
132#461—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103—31no—
138C2×#103—32no—
139C2×#103C2×D312yes—
140C2×#103C2×D312yes—
141C2×#102—31no—
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384C2×D311yescharacteristic
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3C2×D412yes—
167C22×Dic3C2×D412yes—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12#4112yes—
174C2×C4×D3—42no—
175C2×C4×D3—42no—
176C2×C4×D3—42no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6SD1614yescomplemented
185C2×Dic6C2×D412yes—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8#4112yes—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—81no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#215—48no—
201(C2×D3)⋊C4—42no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—82no—
204C4⋊Dic3—28no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—44no—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—28no—
212Dic3⋊C4—28no—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215C4×Dic3—28no—
216C4×Dic3—24no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222#204—28no—
223C22×Q8—31no—
224C22×Q8—31no—
225C22×Q8C22×D312yes—
226C22×D4—31no—
227C23×C4—61no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—64no—
231C2×SD16—64no—
232C2×SD16—62no—
233C2×SD16—122no—
234C22×C8—32no—
235C22×C8C22×D311yescharacteristic
236C22×C8C22×D311yescharacteristic
237C4⋊Q8—22no—
238C4⋊Q8—64no—
239C4⋊Q8—64no—
240C4⋊Q8D1214yes—
241#125—616no—
242#125—68no—
243C22⋊Q8—124no—
244C4×Q8—34no—
245C4×Q8—68no—
246C4×Q8—64no—
247C4×Q8—64no—
248C4×Q8—62no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4C22×D312yes—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—61no—
269C2×C22⋊C4—62no—
270C2×C42—32no—
271C2×C42—62no—
272C2×C42—61no—
273C2×C42C22×D311yescharacteristic
274#106—62no—
275C4⋊C8—62no—
276#103—24no—
277#103—42no—
278#103—62no—
279#103—64no—
280#103—122no—
281#102—62no—
282C4×C8—22no—
283#95—62no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—42no—
295C3×Q8—42no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4C2×SD1612yes—
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×C42−1+411yescharacteristic
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3C2×SD1614yes—
329D12—22no—
330D12—22no—
331D12—41no—
332D12—41no—
333C4×D3—84no—
334Dic6—28no—
335Dic6—22no—
336Dic6—22no—
337Dic6—41no—
338Dic6—41no—
339C3×C8—22no—
340C3×C8—41no—
341C3⋊C8—42no—
342C24—61no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—34no—
349C2×Q8—32no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8C2×D1214yes—
358C2×D4—32no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—122no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4D3×D411yescharacteristic
375C22×C4D3×D411yescharacteristic
376C22×C4C23×D311yescharacteristic
377SD16—68no—
378SD16—128no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8#23712yes—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—32no—
393C4⋊C4—42no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—64no—
398C4⋊C4—62no—
399C4⋊C4—64no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—81no—
404C4⋊C4—124no—
405C22⋊C4—124no—
406C22⋊C4—122no—
407C22⋊C4—242no—
408C42—68no—
409C42—64no—
410C42—62no—
411C42—62no—
412C42C2×D1212yes—
413C42D3×D412yes—
414C22×C3C2×#12011yescharacteristic
415C22×C3#54811yescharacteristic
416C22×C3#55011yescharacteristic
417C22×C3C22×SD1611yescharacteristic
418C22×C3C2×#13711yescharacteristic
419C22×C3#57711yescharacteristic
420C22×C3#57811yescharacteristic
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—81no—
427C2×D3—81no—
428C3×C4—22no—
429C3×C4—44no—
430C3×C4—82no—
431C3×C4#44814yes—
432C3×C4#52511yescharacteristic
433C3×C4#53711yescharacteristic
434C3×C4#54611yescharacteristic
435C3×C4#54711yescharacteristic
436Dic3—24no—
437Dic3—42no—
438Dic3—42no—
439Dic3—42no—
440Dic3—81no—
441C23—62no—
442C23—121no—
443C23—121no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23—121no—
448C23C2×D3×D411yescharacteristic
449Q8—24no—
450Q8—24no—
451Q8—42no—
452Q8—42no—
453Q8—68no—
454Q8—62no—
455Q8—62no—
456Q8—121no—
457Q8—121no—
458D4—62no—
459D4—62no—
460D4—121no—
461D4—121no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—34no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—244no—
486C2×C4#88212yes—
487C2×C4#88212yes—
488C2×C4D3×SD1612yes—
489C2×C4#91812yes—
490C2×C4C22×D1211yescharacteristic
491C2×C4C2×D3×D411yescharacteristic
492C2×C4C2×#23711yescharacteristic
493C2×C4#100411yescharacteristic
494C2×C4D3×#4111yescharacteristic
495C2×C4#101011yescharacteristic
496C8—22no—
497C8—41no—
498C8—122no—
499C2×C3#240711yescharacteristic
500C2×C3C2×#44811yescharacteristic
501C2×C3#301311yescharacteristic
502C2×C3#335911yescharacteristic
503C2×C3#337211yescharacteristic
504C2×C3#339211yescharacteristic
505C2×C3#339811yescharacteristic
506D3—82no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—241no—
511C22—241no—
512C22—241no—
513C22—241no—
514C22C2×#88211yescharacteristic
515C22#567711yescharacteristic
516C22#571811yescharacteristic
517C22C2×D3×SD1611yescharacteristic
518C22#576211yescharacteristic
519C22C2×#91811yescharacteristic
520C22#576611yescharacteristic
521C4—22no—
522C4—44no—
523C4—64no—
524C4—82no—
525C4—122no—
526C4—122no—
527C4—122no—
528C4—241no—
529C4#480614yes—
530C4#555411yescharacteristic
531C4#555511yescharacteristic
532C4#560611yescharacteristic
533C4#560811yescharacteristic
534C3#2418711yescharacteristic, complemented, Sylow
535C2—242no—
536C2#7643511yescharacteristic
537C2C2×#480611yescharacteristic
538C2#7879511yescharacteristic
539C2#8162011yescharacteristic
540C2#8162311yescharacteristic
541C2#8175511yescharacteristic
542C2#8179711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1956C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1956D3
hypocenterC3#24187
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. C2×#5721C2semidirect
5. C3×#1956C2semidirect
7. #70246C2semidirect
9. #70238C2semidirect
10. #69788C2semidirect
11. #69588C2semidirect
53. #1956D3semidirect
184. C2×Dic6SD16semidirect
534. C3#24187semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×#50832
C2×C4×#2042
#743692
C3×#19562
#702822
#702462
#702402
#702382
#697882
#695882
#241873