Finite Group Databasebeta

← #10772477

Subgroups — #10772477

Order 768.

940 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86205C211yesmaximal, characteristic
2C2×#5079C211yesmaximal, characteristic, complemented
3#79312C211yesmaximal, characteristic, complemented
4#79060C211yesmaximal, characteristic, complemented
5C3×#1548C211yesmaximal, characteristic, complemented
6#70754C211yesmaximal, characteristic, complemented
7#70498C211yesmaximal, characteristic, complemented
8#68495C214yesmaximal, complemented
9#68494C214yesmaximal, complemented
10#18637—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5817—22no—
13#5734C2212yes—
14#5733—22no—
15#5732—21no—
16#5730C2212yes—
17#5728—21no—
18#5726—21no—
19#5725—21no—
20#5514—22no—
21#5510—21no—
22#5504—21no—
23#5477C2211yescharacteristic
24C4×#235C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#431C2211yescharacteristic
27C3×#423C2211yescharacteristic
28C2×#835—22no—
29#5133—21no—
30#5121—21no—
31#5079—22no—
32#5079C2214yes—
33C2×C4×C3⋊C8C2211yescharacteristic
34#4851C2212yes—
35#4850—21no—
36#4846C2212yes—
37#4842—21no—
38#4841C2214yes—
39#4738—22no—
40#4693—21no—
41#4681—21no—
42C3×#333C2214yes—
43#4484C2212yes—
44#4483C2214yes—
45#4483C2212yes—
46#3511—31no—
47#3206—31no—
48C2×#487—31no—
49#3079—31no—
50#1646—34no—
51#1643—34no—
52#1548D311yescharacteristic, complemented
53#1492—31no—
54#1476—31no—
55C22×C3×Q8D412yes—
56C2×#239—22no—
57C2×#235—21no—
58C2×#235D411yescharacteristic
59C22×Dic6—21no—
60C2×C3×M16—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8D412yes—
64C3×C4⋊Q8C2312yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C42C2311yescharacteristic
68#947—22no—
69#947—24no—
70#946—22no—
71#946—24no—
72Q8×Dic3—22no—
73#930—24no—
74#930—21no—
75#930D412yes—
76C4×#67—24no—
77C4×#67—21no—
78C4×#67—21no—
79#926D411yescharacteristic
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C22×C3⋊C8—21no—
83C2×#204—24no—
84#898—24no—
85#897—22no—
86#897—22no—
87#895—22no—
88#895—22no—
89#895—22no—
90#894—24no—
91#893—22no—
92#893—22no—
93#892—22no—
94#892—22no—
95D3×C4⋊C4—22no—
96#890—22no—
97#887—22no—
98#875—21no—
99#874D411yescharacteristic
100C4×D12—21no—
101C4×D12D411yescharacteristic
102C4×D12C2311yescharacteristic
103C4×D12C2312yes—
104#872—21no—
105C42×D3—21no—
106#869—21no—
107#869—21no—
108#869—21no—
109#869D411yescharacteristic
110#869C2312yes—
111C4×Dic6D411yescharacteristic
112C4×Dic6C2311yescharacteristic
113C3×C4⋊C8—21no—
114C3×C4⋊C8C2312yes—
115C3×C4⋊C8C2312yes—
116C3×C22⋊C8—21no—
117C3×C4×C8—21no—
118#835—28no—
119#821—22no—
120#818—22no—
121#817—22no—
122#816—22no—
123#816—24no—
124C4×C3⋊C8—21no—
125C4×C3⋊C8C2312yes—
126#556—61no—
127#555—32no—
128#554—32no—
129#553—62no—
130#548—62no—
131#547—61no—
132#542—62no—
133#537—61no—
134#533—61no—
135#532—31no—
136#532—61no—
137C2×C4⋊Q8—31no—
138C2×C4⋊Q8C2×D311yescharacteristic
139#520—61no—
140C4×#41—31no—
141#492—61no—
142#487—34no—
143#487—61no—
144#487—62no—
145#477—61no—
146#476—61no—
147#475—32no—
148#474—32no—
149#461—34no—
150#448—62no—
151#431C2×D311yescharacteristic
152#428—61no—
153#423C2×D311yescharacteristic
154#417—61no—
155C2×#103—62no—
156C2×C4×C8—31no—
157#333C2×D314yes—
158Q8⋊C8—32no—
159D4⋊C8—34no—
160D4⋊C8—32no—
161C2×C3×Q8—22no—
162C2×C3×Q8—44no—
163C2×C3×Q8C2×D414yes—
164C22×C3×C4—22no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C2×#67—21no—
169C2×#67—22no—
170C22×Dic3—41no—
171#239—84no—
172#235—22no—
173#235—22no—
174#235—42no—
175#235—42no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—24no—
179C2×D12C2×D411yescharacteristic
180C2×D12C2×D412yes—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—22no—
185C2×C4×D3—24no—
186C2×Dic6—21no—
187C2×Dic6—22no—
188C2×Dic6—41no—
189C2×Dic6—82no—
190C2×Dic6C2×D411yescharacteristic
191C3×M16—42no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2411yescharacteristic
202#217—21no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—21no—
207(C2×D3)⋊C4—22no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—22no—
210C4⋊Dic3—24no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—41no—
215C4⋊Dic3—41no—
216C4⋊Dic3C2×D412yes—
217C4⋊Dic3C2×D411yescharacteristic
218C4⋊Dic3C2×D411yescharacteristic
219Dic3⋊C4—24no—
220Dic3⋊C4—21no—
221Dic3⋊C4—21no—
222C4×Dic3—24no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C4×Dic3—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—42no—
228#204—84no—
229C2×#41—31no—
230C2×#41—62no—
231C2×#41—61no—
232C22×Q8—61no—
233C22×Q8D1212yes—
234C2×SD16—64no—
235C2×M16—21no—
236C22×C8—61no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—31no—
240C4⋊Q8—32no—
241C4⋊Q8—61no—
242C4⋊Q8—61no—
243C4⋊Q8—61no—
244C4⋊Q8D1212yes—
245C4⋊Q8C22×D312yes—
246#127—31no—
247#126—64no—
248#125—62no—
249#124—62no—
250#124—64no—
251#124—61no—
252#123—64no—
253C22⋊Q8—62no—
254C22⋊Q8—62no—
255C22⋊Q8—62no—
256C22⋊Q8—64no—
257#121—32no—
258#121—62no—
259#121—62no—
260#121—62no—
261#121—64no—
262#121—61no—
263C4×Q8—31no—
264C4×Q8—31no—
265C4×Q8—62no—
266C4×Q8—62no—
267C4×D4—31no—
268C4×D4—31no—
269C4×D4—32no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—64no—
273C4×D4—61no—
274C4×D4—61no—
275C4×D4—61no—
276#117—31no—
277#117—62no—
278#117—62no—
279#117—61no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C42—61no—
286C2×C42C22×D311yescharacteristic
287#107—62no—
288#106—62no—
289C4⋊C8—21no—
290C4⋊C8C22×D312yes—
291C4⋊C8C22×D312yes—
292#103—64no—
293#103—62no—
294#103—68no—
295#102—62no—
296C22⋊C8—21no—
297C4×C8—21no—
298C4×C8—32no—
299C4×C8—61no—
300C23×C3#12011yescharacteristic
301C22×D3—21no—
302C22×D3—22no—
303C3×Q8—44no—
304C3×Q8—44no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—82no—
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4#13512yes—
313C2×C3×C4#13712yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317#67—42no—
318#67—44no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—22no—
323C2×Dic3—42no—
324C2×Dic3—82no—
325D12—21no—
326D12—21no—
327D12—24no—
328D12—42no—
329D12—42no—
330D12—84no—
331C4×D3—42no—
332C4×D3—42no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—84no—
337Dic6—21no—
338Dic6—21no—
339Dic6—42no—
340Dic6—44no—
341Dic6—82no—
342C3×C8—42no—
343C3×C8—42no—
344C3⋊C8—42no—
345#41—62no—
346#41—62no—
347#41—122no—
348#41—122no—
349#41—244no—
350C2×Q8—22no—
351C2×Q8—31no—
352C2×Q8—44no—
353C2×Q8—61no—
354C2×Q8—62no—
355C2×Q8—121no—
356C2×Q8—242no—
357C2×Q8C2×D1214yes—
358C2×D4—31no—
359C2×D4—32no—
360C2×D4—64no—
361C2×D4—62no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C22×C4—22no—
366C22×C4—64no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—121no—
372C22×C4C2×D1211yescharacteristic
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D411yescharacteristic
375SD16—244no—
376M16—42no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—62no—
380C2×C8—122no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—32no—
386C4⋊C4—31no—
387C4⋊C4—31no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—121no—
396C4⋊C4—121no—
397C22⋊C4—64no—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—62no—
403C22⋊C4—61no—
404C22⋊C4—62no—
405C42—64no—
406C42—61no—
407C42—61no—
408C42—61no—
409C42C2×D1211yescharacteristic
410C42D3×D411yescharacteristic
411C42D3×D411yescharacteristic
412C42C23×D311yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3#45212yes—
416C22×C3C2×#12011yescharacteristic
417C2×D3—41no—
418C2×D3—42no—
419C2×D3—41no—
420C2×D3—42no—
421C2×D3—41no—
422C2×D3—42no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—44no—
426C3×C4#45212yes—
427C3×C4#45512yes—
428C3×C4C2×#13511yescharacteristic
429C3×C4C2×#13711yescharacteristic
430C3×C4#57712yes—
431Dic3—41no—
432Dic3—42no—
433Dic3—41no—
434Dic3—41no—
435Dic3—82no—
436C23—61no—
437C23—62no—
438C23#88211yescharacteristic
439Q8—44no—
440Q8—44no—
441Q8—61no—
442Q8—61no—
443Q8—122no—
444Q8—124no—
445Q8—242no—
446D4—61no—
447D4—61no—
448D4—64no—
449D4—122no—
450D4—124no—
451D4—122no—
452D4—122no—
453D4—244no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—61no—
458C2×C4—62no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—82no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—244no—
469C2×C4—242no—
470C2×C4#88211yescharacteristic
471C2×C4#88211yescharacteristic
472C2×C4#88211yescharacteristic
473C2×C4#91612yes—
474C2×C4#91812yes—
475C2×C4C22×D1211yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C8—42no—
479C8—42no—
480C8—122no—
481C2×C3—22no—
482C2×C3C2×#45211yescharacteristic
483C2×C3#301011yescharacteristic
484C2×C3C2×#45511yescharacteristic
485D3—81no—
486D3—82no—
487C22—21no—
488C22—21no—
489C22—121no—
490C22—122no—
491C22—121no—
492C22—122no—
493C22—121no—
494C22—122no—
495C22#480812yes—
496C22C2×#88211yescharacteristic
497C4—21no—
498C4—21no—
499C4—44no—
500C4—121no—
501C4—121no—
502C4—121no—
503C4—122no—
504C4—242no—
505C4#474612yes—
506C4#480812yes—
507C4C2×#91611yescharacteristic
508C4C2×#91811yescharacteristic
509C4#577812yes—
510C3#1863711yescharacteristic, complemented, Sylow
511C2—22no—
512C2—241no—
513C2—242no—
514C2C2×#474611yescharacteristic
515C2C2×#480811yescharacteristic
516C2#7896011yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1548C2
socleC22×C3C2×#120
prime radical#86205C2
hypercenter#1548D3
hypocenterC3#18637
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. C2×#5079C2semidirect
3. #79312C2semidirect
4. #79060C2semidirect
5. C3×#1548C2semidirect
6. #70754C2semidirect
7. #70498C2semidirect
8. #68495C2semidirect
9. #68494C2semidirect
52. #1548D3semidirect
510. C3#18637semidirect

Maximal subgroups

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

subgroupindex
#862052
C2×#50792
#793122
#790602
C3×#15482
#707542
#704982
#684952
#684942
#186373