Finite Group Databasebeta

← #10849337

Subgroups — #10849337

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2#82478C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic, complemented
4C2×#4714C211yesmaximal, characteristic, complemented
5C3×#1985C211yesmaximal, characteristic, complemented
6#70458C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic, complemented
8#68683C212yesmaximal, complemented
9#68670C212yesmaximal, complemented
10#68630C212yesmaximal, complemented
11#68629C212yesmaximal
12#25660—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15#5585—24no—
16#5585—24no—
17C2×#894—21no—
18C2×#894—21no—
19C2×#893—21no—
20C2×#875C2211yescharacteristic
21C2×#875C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29#5229—21no—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#818C2212yes—
35C2×#816C2212yes—
36C2×#816C2212yes—
37#4979—21no—
38#4979—21no—
39#4975—21no—
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2212yes—
42#4928C2212yes—
43#4846—24no—
44#4714—24no—
45C3×#336C2212yes—
46C3×#336C2212yes—
47#4551C2212yes—
48C2×#559—31no—
49C2×#489—31no—
50C2×#474—31no—
51#2567—31no—
52#2042—32no—
53#2041—32no—
54#1985D311yescharacteristic, complemented
55#1983—31no—
56#1863—32no—
57#1862—32no—
58#1762—31no—
59C22×C3×Q8D411yescharacteristic
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2311yescharacteristic
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×C4×D3—21no—
66C22×Dic6—21no—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78#947—28no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94#894—28no—
95#894—28no—
96#893—28no—
97#875—22no—
98#875—22no—
99#875—22no—
100#875D414yes—
101#874—22no—
102#874—22no—
103C4×D12—22no—
104C4×D12—22no—
105C4×D12—22no—
106C4×D12D414yes—
107#869—22no—
108#869—22no—
109C4×Dic6—22no—
110C4×Dic6—22no—
111C3×C4⋊C8—22no—
112C3×#97—22no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115C3⋊#95—21no—
116#821—24no—
117#821—24no—
118#821—24no—
119#821—24no—
120#818—24no—
121#816—24no—
122#816—24no—
123#559—64no—
124#559—64no—
125C2×C4⋊Q8—31no—
126C2×#127—31no—
127C2×#124—31no—
128C2×#124—31no—
129C2×#124—61no—
130C2×#123—61no—
131C2×#123—61no—
132C2×C4×Q8—31no—
133C2×C4×Q8C2×D311yescharacteristic
134C2×C4×D4—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137#489—64no—
138#474—64no—
139C2×#107—32no—
140C2×C4⋊C8C2×D311yescharacteristic
141C2×#103—32no—
142C2×#103—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#97C2×D311yescharacteristic
148#395—61no—
149#395—61no—
150#391—61no—
151#389—32no—
152#389—32no—
153#384—61no—
154C4×C4⋊C4—31no—
155#340—32no—
156#336C2×D312yes—
157#336C2×D312yes—
158C23×D3—21no—
159C23×D3—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—41no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—22no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—24no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—24no—
182C2×D12—42no—
183C2×D12—42no—
184C2×D12—82no—
185C2×D12C2×D412yes—
186C2×D12C2×D412yes—
187C2×C4×D3—24no—
188C2×C4×D3—24no—
189C2×C4×D3—82no—
190C2×C4×D3—82no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—21no—
194C2×Dic6—42no—
195C2×Dic6—82no—
196C2×Dic6C2×D412yes—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C42—21no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42C2×D412yes—
211C3×C42C2×D412yes—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218C4⋊Dic3—21no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—42no—
223C4⋊Dic3C2×D412yes—
224C4⋊Dic3#4114yes—
225Dic3⋊C4—24no—
226Dic3⋊C4—82no—
227Dic3⋊C4—82no—
228C4×Dic3—24no—
229C4×Dic3—42no—
230C22×Q8—31no—
231C22×Q8—61no—
232C22×Q8D1211yescharacteristic
233C22×D4—31no—
234C22×D4—31no—
235C22×D4—61no—
236C22×D4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C22×C8C22×D312yes—
240C22×C8C22×D312yes—
241C4⋊Q8—62no—
242C4⋊Q8—62no—
243#127—62no—
244#127—62no—
245#124—34no—
246#124—62no—
247#124—62no—
248#124—68no—
249#124—62no—
250#123—68no—
251#123—68no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×D4—34no—
257C4×D4—62no—
258C4×D4—62no—
259C4×D4—68no—
260C4×D4—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4D1211yescharacteristic
269C2×C4⋊C4D1211yescharacteristic
270C2×C4⋊C4D1211yescharacteristic
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C42—61no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280C2×C42C22×D311yescharacteristic
281C2×C42C22×D311yescharacteristic
282#107—64no—
283C4⋊C8—22no—
284#103—64no—
285#103—64no—
286#102—64no—
287#102—64no—
288#102—64no—
289#102—64no—
290#97—22no—
291#95—61no—
292#95—62no—
293#95—62no—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—42no—
305C3×Q8—44no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12012yes—
320C2×C3×C4#13712yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4112yes—
324C2×C3×C42−1+412yes—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—24no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335D12—22no—
336D12—22no—
337D12—24no—
338D12—48no—
339D12—41no—
340D12—41no—
341D12—42no—
342D12—84no—
343C4×D3—84no—
344C4×D3—84no—
345Dic6—24no—
346Dic6—41no—
347Dic6—41no—
348Dic6—84no—
349C3×C8—42no—
350C3×C8—42no—
351C24—61no—
352C24—61no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—32no—
356C2×Q8—41no—
357C2×Q8—61no—
358C2×Q8—62no—
359C2×Q8—62no—
360C2×Q8—122no—
361C2×Q8—242no—
362C2×D4—32no—
363C2×D4—32no—
364C2×D4—64no—
365C2×D4—64no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—122no—
373C2×D4—122no—
374C2×D4—242no—
375C22×C4—64no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—64no—
380C22×C4—62no—
381C22×C4—242no—
382C22×C4—242no—
383C22×C4C2×D1211yescharacteristic
384C22×C4C2×D1211yescharacteristic
385C22×C4C2×D1211yescharacteristic
386C22×C4C2×D1211yescharacteristic
387C22×C4C2×D1211yescharacteristic
388C22×C4C2×D1211yescharacteristic
389C22×C4C23×D311yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—32no—
400C4⋊C4—34no—
401C4⋊C4—64no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—122no—
407C4⋊C4—242no—
408C4⋊C4—242no—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—64no—
420C42—122no—
421C42D3×D412yes—
422C42D3×D412yes—
423C22×C3C2×#12011yescharacteristic
424C22×C3#54812yes—
425C22×C3C2×#13711yescharacteristic
426C22×C3#57511yescharacteristic
427C22×C3#57612yes—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—82no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—82no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4#53211yescharacteristic
445C3×C4#53611yescharacteristic
446C3×C4#54712yes—
447Dic3—44no—
448Dic3—42no—
449Dic3—42no—
450Dic3—82no—
451C23—62no—
452C23—62no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—122no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23C22×D1211yescharacteristic
462Q8—44no—
463Q8—64no—
464Q8—121no—
465Q8—121no—
466Q8—244no—
467D4—62no—
468D4—62no—
469D4—64no—
470D4—128no—
471D4—121no—
472D4—121no—
473D4—122no—
474D4—244no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—64no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—244no—
498C2×C4—244no—
499C2×C4#88212yes—
500C2×C4#88212yes—
501C2×C4#90912yes—
502C2×C4C2×D3×D411yescharacteristic
503C2×C4C2×D3×D411yescharacteristic
504C2×C4#100712yes—
505C2×C4D3×#4112yes—
506C8—42no—
507C8—42no—
508C2×C3#243211yescharacteristic
509C2×C3#300711yescharacteristic
510C2×C3#300811yescharacteristic
511C2×C3#337012yes—
512C2×C3#337512yes—
513D3—82no—
514D3—82no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22—242no—
522C22—242no—
523C22—242no—
524C22—242no—
525C22C2×#88211yescharacteristic
526C22#557612yes—
527C22C2×#90911yescharacteristic
528C22#574811yescharacteristic
529C22#575212yes—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—122no—
537C4—124no—
538C4—122no—
539C4—242no—
540C4#563112yes—
541C4#566811yescharacteristic
542C4#572411yescharacteristic
543C3#2566011yescharacteristic, complemented, Sylow
544C2—242no—
545C2—242no—
546C2#7649811yescharacteristic
547C2#7864011yescharacteristic
548C2#7867011yescharacteristic
549C2#7997112yes—
550C2#7998412yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1985D3
hypocenterC3#25660
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #82478C2semidirect
3. C2×#4846C2semidirect
4. C2×#4714C2semidirect
5. C3×#1985C2semidirect
6. #70458C2semidirect
7. #69812C2semidirect
8. #68683C2semidirect
9. #68670C2semidirect
10. #68630C2semidirect
54. #1985D3semidirect
543. C3#25660semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#824782
C2×#48462
C2×#47142
C3×#19852
#704582
#698122
#686832
#686702
#686302
#686292
#256603