Finite Group Databasebeta

← #10772478

Subgroups — #10772478

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#85850C211yesmaximal, characteristic
2#82508C211yesmaximal, characteristic, complemented
3#79312C211yesmaximal, characteristic, complemented
4C2×#4681C211yesmaximal, characteristic, complemented
5C3×#1476C211yesmaximal, characteristic, complemented
6#70779C211yesmaximal, characteristic, complemented
7#69735C211yesmaximal, characteristic, complemented
8#68249C214yesmaximal, complemented
9#68183C214yesmaximal, complemented
10#18637—31nomaximal, Sylow
11C3×C4×#41C2211yescharacteristic
12#5833—21no—
13#5584—21no—
14#5579C2212yes—
15#5565—22no—
16#5563C2212yes—
17#5552—22no—
18#5551—21no—
19#5534—22no—
20#5523—21no—
21#5477—21no—
22#5477C2211yescharacteristic
23#5470—21no—
24C2×#869C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#431C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5232—21no—
29C2×#816—22no—
30#5021—21no—
31#5020C2212yes—
32#5008—21no—
33#5007C2212yes—
34#4995—21no—
35#4924C2211yescharacteristic
36#4841C2214yes—
37#4738—22no—
38#4693—21no—
39#4681—22no—
40#4681—21no—
41#4681C2214yes—
42C3×Q8⋊C8C2212yes—
43C3×D4⋊C8C2212yes—
44#4481C2214yes—
45#4459C2214yes—
46#3511—31no—
47#3206—31no—
48C2×#487—31no—
49#3079—31no—
50#1646—34no—
51#1643—34no—
52#1548—31no—
53#1492—31no—
54#1476D311yescharacteristic, complemented
55C2×C3×#41D411yescharacteristic
56C2×#235—21no—
57C2×#235—22no—
58C22×Dic6—21no—
59C22×Dic6D412yes—
60C22×C3×C8—21no—
61C3×C4×Q8D411yescharacteristic
62C3×C4×Q8C2311yescharacteristic
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×C4×D4D411yescharacteristic
66C3×C4×D4C2311yescharacteristic
67C3×#117—21no—
68C3×#117D411yescharacteristic
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71#946—22no—
72#939—24no—
73#938—22no—
74#930—22no—
75#930—21no—
76#930—22no—
77#930D412yes—
78#926—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83#924—22no—
84#924—22no—
85C2×#208—21no—
86C2×#204—24no—
87#898—24no—
88#896—24no—
89#892—22no—
90D3×C4⋊C4—22no—
91#885—24no—
92#883—24no—
93#67⋊C4—24no—
94#875—22no—
95#875—21no—
96#874D411yescharacteristic
97C4×D12—22no—
98C4×D12—22no—
99C4×D12—21no—
100C4×D12C2312yes—
101#870—22no—
102#869—21no—
103#869—21no—
104#869—21no—
105#869—21no—
106#869—21no—
107#869D412yes—
108#869D411yescharacteristic
109#869C2312yes—
110#869C2312yes—
111C4×Dic6—22no—
112C4×Dic6—22no—
113C3×C4⋊C8C2312yes—
114C3×C22⋊C8—21no—
115C3×C4×C8—21no—
116C3×C4×C8—21no—
117C3×C4×C8C2312yes—
118#835—22no—
119#832—22no—
120#816—28no—
121#816—24no—
122#808—22no—
123#807—22no—
124C4⋊(C3⋊C8)—21no—
125C4⋊(C3⋊C8)C2312yes—
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⋊Q8—31no—
139#520—61no—
140C4×#41C2×D311yescharacteristic
141#492—61no—
142#487—34no—
143#487—62no—
144#487—61no—
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#423—31no—
154#417—61no—
155C2×#103—62no—
156C2×C4×C8C2×D311yescharacteristic
157#333—34no—
158Q8⋊C8C2×D312yes—
159D4⋊C8—34no—
160D4⋊C8C2×D312yes—
161C3×#41—22no—
162C3×#41—22no—
163C2×C3×Q8C2×D411yescharacteristic
164C2×C3×D4—21no—
165C2×C3×D4C2×D411yescharacteristic
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C2×#67—22no—
173C2×#67—24no—
174C22×Dic3—22no—
175C22×Dic3—41no—
176#235—42no—
177#235—42no—
178#235—84no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12C2×D412yes—
182C2×C4×D3—22no—
183C2×C4×D3—24no—
184C2×Dic6—21no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—44no—
188C2×Dic6—41no—
189C2×Dic6—82no—
190C2×Dic6C2×D414yes—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—42no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4C2×D411yescharacteristic
197C3×C4⋊C4C2×D411yescharacteristic
198C3×C22⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—21no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—21no—
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207C3×C42C2411yescharacteristic
208#217—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—22no—
212(C2×D3)⋊C4—22no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—41no—
221C4⋊Dic3—41no—
222C4⋊Dic3C2×D412yes—
223Dic3⋊C4—24no—
224Dic3⋊C4—24no—
225C4×Dic3—24no—
226#208—42no—
227C2×C3⋊C8—22no—
228#204—84no—
229C2×#41—62no—
230C2×#41—61no—
231C2×#41D1211yescharacteristic
232C22×Q8—32no—
233C22×Q8—61no—
234C2×SD16—64no—
235C2×M16—61no—
236C22×C8—21no—
237C4⋊Q8—31no—
238C4⋊Q8—32no—
239C4⋊Q8—32no—
240C4⋊Q8—32no—
241C4⋊Q8—61no—
242C4⋊Q8—61no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246#127—31no—
247#126—64no—
248#125—62no—
249#124—62no—
250#124—64no—
251#124—61no—
252#123—64no—
253C22⋊Q8—64no—
254C22⋊Q8—62no—
255C22⋊Q8—62no—
256C22⋊Q8—62no—
257#121—32no—
258#121—62no—
259#121—62no—
260#121—64no—
261#121—62no—
262#121—61no—
263C4×Q8—62no—
264C4×Q8—62no—
265C4×Q8D1211yescharacteristic
266C4×Q8C22×D311yescharacteristic
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—32no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—64no—
273C4×D4—61no—
274C4×D4D1211yescharacteristic
275C4×D4C22×D311yescharacteristic
276#117—21no—
277#117—62no—
278#117—62no—
279#117D1211yescharacteristic
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C42—21no—
286C2×C42C22×D311yescharacteristic
287#107—62no—
288#106—62no—
289C4⋊C8—32no—
290C4⋊C8—61no—
291C4⋊C8C22×D312yes—
292#103—64no—
293#103—68no—
294#103—62no—
295#102—62no—
296C22⋊C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8C22×D312yes—
300C23×C3—21no—
301C23×C3#12011yescharacteristic
302C22×D3—22no—
303C3×Q8—21no—
304C3×Q8—21no—
305C3×D4—21no—
306C3×D4—21no—
307C3×D4—42no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12011yescharacteristic
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#13512yes—
320C2×C3×C4#13712yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324#67—44no—
325#67—84no—
326C2×Dic3—22no—
327C2×Dic3—24no—
328C2×Dic3—42no—
329C2×Dic3—82no—
330C2×Dic3—82no—
331D12—24no—
332D12—42no—
333D12—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—84no—
337Dic6—42no—
338Dic6—44no—
339Dic6—44no—
340Dic6—44no—
341Dic6—82no—
342C3×C8—42no—
343C3×C8—42no—
344C3⋊C8—42no—
345#41—22no—
346#41—22no—
347#41—122no—
348#41—122no—
349#41—244no—
350C2×Q8—34no—
351C2×Q8—61no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—124no—
355C2×Q8—121no—
356C2×Q8—242no—
357C2×Q8C2×D1211yescharacteristic
358C2×D4—21no—
359C2×D4—32no—
360C2×D4—62no—
361C2×D4—64no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4C2×D1211yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—121no—
372C22×C4C2×D1211yescharacteristic
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D411yescharacteristic
375SD16—244no—
376M16—122no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—42no—
380C2×C8—62no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—32no—
384C4⋊C4—64no—
385C4⋊C4—64no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—121no—
394C4⋊C4—121no—
395C4⋊C4C2×D1211yescharacteristic
396C4⋊C4C2×D1211yescharacteristic
397C22⋊C4—21no—
398C22⋊C4—21no—
399C22⋊C4—21no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—64no—
403C22⋊C4—62no—
404C22⋊C4—62no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—64no—
409C42C2×D1211yescharacteristic
410C42D3×D411yescharacteristic
411C42D3×D411yescharacteristic
412C42C23×D311yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3#45212yes—
419C22×C3C2×#12011yescharacteristic
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4#45212yes—
429C3×C4#45512yes—
430C3×C4C2×#13511yescharacteristic
431C3×C4C2×#13711yescharacteristic
432C3×C4#57712yes—
433Dic3—42no—
434Dic3—44no—
435Dic3—82no—
436C23—21no—
437C23—62no—
438C23#88211yescharacteristic
439Q8—21no—
440Q8—21no—
441Q8—124no—
442Q8—122no—
443Q8—124no—
444Q8—124no—
445Q8—242no—
446D4—21no—
447D4—21no—
448D4—42no—
449D4—64no—
450D4—124no—
451D4—122no—
452D4—122no—
453D4—244no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—62no—
463C2×C4—64no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—244no—
468C2×C4—242no—
469C2×C4—242no—
470C2×C4#88211yescharacteristic
471C2×C4#88211yescharacteristic
472C2×C4#88211yescharacteristic
473C2×C4#90412yes—
474C2×C4#90912yes—
475C2×C4C22×D1211yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C8—42no—
479C8—42no—
480C8—122no—
481C2×C3—22no—
482C2×C3—81no—
483C2×C3C2×#45211yescharacteristic
484C2×C3#301011yescharacteristic
485C2×C3C2×#45511yescharacteristic
486D3—82no—
487C22—21no—
488C22—21no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22#473312yes—
496C22C2×#88211yescharacteristic
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—122no—
503C4—124no—
504C4—242no—
505C4#473312yes—
506C4#482512yes—
507C4C2×#90411yescharacteristic
508C4C2×#90911yescharacteristic
509C4#577012yes—
510C3#1863711yescharacteristic, complemented, Sylow
511C2—22no—
512C2—81no—
513C2—242no—
514C2C2×#473311yescharacteristic
515C2#7883611yescharacteristic
516C2C2×#482511yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1476C2
socleC22×C3C2×#120
prime radical#85850C2
hypercenter#1476D3
hypocenterC3#18637
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #82508C2semidirect
3. #79312C2semidirect
4. C2×#4681C2semidirect
5. C3×#1476C2semidirect
6. #70779C2semidirect
7. #69735C2semidirect
8. #68249C2semidirect
9. #68183C2semidirect
54. #1476D3semidirect
510. C3#18637semidirect

Maximal subgroups

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

subgroupindex
#858502
#825082
#793122
C2×#46812
C3×#14762
#707792
#697352
#682492
#681832
#186373