Finite Group Databasebeta

← #10764099

Subgroups — #10764099

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86190C211yesmaximal, characteristic
2#79293C211yesmaximal, characteristic, complemented
3#79052C211yesmaximal, characteristic, complemented
4C3×#1545C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic, complemented
6#70777C211yesmaximal, characteristic, complemented
7#70511C211yesmaximal, characteristic, complemented
8#68388C212yesmaximal, complemented
9#68386C212yesmaximal, complemented
10#68384C212yesmaximal, complemented
11#68381C212yesmaximal, complemented
12#17776—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14#5806—22no—
15#5806—22no—
16#5805C412yes—
17#5798C412yes—
18#5719C414yes—
19#5718—21no—
20#5718—21no—
21#5716C2212yes—
22#5716C2212yes—
23#5713—21no—
24#5713—21no—
25C4×#235C2211yescharacteristic
26#5470C2211yescharacteristic
27C3×#431C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5128—22no—
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4838—21no—
33#4837C2212yescomplemented
34#4835C2212yescomplemented
35#4833C2212yescomplemented
36D3×C4⋊C8C2212yescomplemented
37#4830—21no—
38#4728—22no—
39C8×D12—21no—
40#4679—21no—
41C3×#331C2212yes—
42C3×#331C2212yes—
43#4544C2212yes—
44#4544C2212yes—
45#4486C2212yescomplemented
46#4486C2211yescharacteristic, complemented
47#4486C2211yescharacteristic, complemented
48#4482C2211yescharacteristic, complemented
49#4482C2212yescomplemented
50#4482C2211yescharacteristic, complemented
51#3508—31no—
52#2991—31no—
53#2990—31no—
54#1561—31no—
55#1557—32no—
56#1553—32no—
57#1547—32no—
58#1545—31no—
59#1545D311yescharacteristic, complemented
60#1544—31no—
61#1540—32no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2×C412yes—
65C2×#237—22no—
66C2×#237—22no—
67C2×#235C2×C411yescharacteristic
68C2×C3×M16—21no—
69C3×#127—21no—
70C3×#127—21no—
71C3×#127C2×C412yes—
72C3×#127C2312yes—
73C2×C3×C42C2311yescharacteristic
74#953—24no—
75C2×#217—24no—
76#940—22no—
77#939—24no—
78#939—24no—
79#938—22no—
80#938—22no—
81#938C2×C414yes—
82#938C2×C414yes—
83#936—22no—
84#936—22no—
85#936—22no—
86#935—24no—
87#935—24no—
88D4×Dic3—22no—
89D4×Dic3—22no—
90D4×Dic3—22no—
91D4×Dic3—22no—
92D4×Dic3C2×C414yes—
93D4×Dic3C2×C414yes—
94C4×#67D412yes—
95C4×#67D412yes—
96#926C2×C412yes—
97#926C2×C411yescharacteristic
98#924—21no—
99#924—21no—
100C2×#208—21no—
101C2×ΓL(1,𝔽25)—22no—
102C2×D3×C8—22no—
103C4×D12D411yescharacteristic
104C4×D12D411yescharacteristic
105#872C2×C412yes—
106C42×D3C2312yes—
107#870C2×C412yes—
108#869C2311yescharacteristic
109#869C2311yescharacteristic
110C4×Dic6—21no—
111C4×Dic6—21no—
112C4×Dic6D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8C2312yes—
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×C22⋊C8—21no—
119C3×C4×C8—21no—
120#830—21no—
121(C2×D3)⋊C8—22no—
122(C2×D3)⋊C8—22no—
123#815—22no—
124Dic3⋊C8—22no—
125Dic3⋊C8—22no—
126C8×Dic3—22no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)C2312yes—
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4⋊(C3⋊C8)C2311yescharacteristic
131C4×C3⋊C8—21no—
132#559—61no—
133#559—61no—
134#550—61no—
135#550—61no—
136#546—62no—
137#546—62no—
138#540—34no—
139#535—32no—
140#533—31no—
141#532—32no—
142#532—32no—
143C2×#127C2×D311yescharacteristic
144#518—32no—
145C4×#41—31no—
146#446—61no—
147C8×Q8—61no—
148#436—61no—
149#436—62no—
150#435—62no—
151C8×D4—61no—
152#432—32no—
153#431—31no—
154#431—32no—
155#431C2×D311yescharacteristic
156#423—31no—
157#423—32no—
158#423C2×D311yescharacteristic
159C2×C4⋊C8—32no—
160#332—31no—
161#332—32no—
162#332—31no—
163#332—31no—
164#332—31no—
165#332—32no—
166#331—32no—
167#331—32no—
168#331C2×D312yes—
169#331C2×D312yes—
170C24×C3—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—44no—
177C2×C3×D4—81no—
178C2×C3×D4—81no—
179C2×C3×D4—81no—
180C2×C3×D4—81no—
181C2×C3×D4C22×C414yes—
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C2×#67—24no—
186C2×#67—24no—
187C2×#67C22⋊C412yes—
188C22×Dic3—24no—
189C22×Dic3—24no—
190#237—82no—
191#237—82no—
192#237—82no—
193#237—82no—
194#235—42no—
195C2×D12C22⋊C411yescharacteristic
196C2×C4×D3C22×C412yes—
197C2×C4×D3C2×D412yes—
198C2×C4×D3C2×D412yes—
199C2×Dic6—21no—
200C2×Dic6—21no—
201C2×Dic6C22⋊C411yescharacteristic
202C3×M16—81no—
203C2×C3×C8—22no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C3×C42C22×C411yescharacteristic
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209C3×C42C2411yescharacteristic
210#217—22no—
211#217—24no—
212#217—24no—
213#217—24no—
214#217—24no—
215#217C22⋊C412yes—
216(C2×D3)⋊C4C22⋊C412yes—
217(C2×D3)⋊C4C22⋊C412yes—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3C22⋊C411yescharacteristic
221C4⋊Dic3C22⋊C411yescharacteristic
222C4⋊Dic3C22×C412yes—
223C4⋊Dic3C22×C412yes—
224Dic3⋊C4—22no—
225Dic3⋊C4—22no—
226Dic3⋊C4C22⋊C412yes—
227Dic3⋊C4C22⋊C412yes—
228C4×Dic3—22no—
229C4×Dic3C22×C412yes—
230C4×Dic3C2×D412yes—
231C4×Dic3C2×D412yes—
232#208—81no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—22no—
236ΓL(1,𝔽25)—82no—
237D3×C8—82no—
238C2×#41—31no—
239C2×#41—62no—
240C2×#41—62no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C4×D312yes—
244C2×M16—21no—
245C2×M16—62no—
246C2×M16—61no—
247C22×C8—62no—
248C4⋊Q8—31no—
249C4⋊Q8—31no—
250#127—21no—
251#127—21no—
252#127—62no—
253#127C4×D312yes—
254#127C22×D312yes—
255#125—32no—
256#124—62no—
257#124—62no—
258#124—62no—
259#123—64no—
260#123—64no—
261C22⋊Q8—61no—
262C22⋊Q8—61no—
263#121—34no—
264#121—34no—
265#121—64no—
266#121—64no—
267#121—62no—
268#121—62no—
269#120—64no—
270C4×Q8—31no—
271C4×Q8—31no—
272C4×Q8—61no—
273C4×Q8—61no—
274C4×D4—32no—
275C4×D4—32no—
276C4×D4—34no—
277C4×D4—34no—
278C4×D4—31no—
279C4×D4—31no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—62no—
284#117—31no—
285#117—32no—
286#117—32no—
287C2×C22⋊C4—64no—
288C2×C42—32no—
289C2×C42C22×D311yescharacteristic
290C4⋊C8—21no—
291C4⋊C8—32no—
292C4⋊C8—31no—
293C4⋊C8—31no—
294C4⋊C8—62no—
295C4⋊C8—62no—
296C4⋊C8—61no—
297C4⋊C8C22×D311yescharacteristic
298C4⋊C8C22×D312yes—
299C4⋊C8C22×D311yescharacteristic
300C22⋊C8—21no—
301C22⋊C8—62no—
302C22⋊C8—62no—
303C22⋊C8—61no—
304#97—62no—
305C4×C8—21no—
306C4×C8—62no—
307C4×C8—61no—
308C23×C3—24no—
309C23×C3—44no—
310C23×C3—82no—
311C23×C3—82no—
312C23×C3C2×C22⋊C411yescharacteristic
313C22×D3C2×C22⋊C412yes—
314C3×D4—44no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—44no—
320C3×D4—82no—
321C3×D4—82no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4C2×C22⋊C411yescharacteristic
328C2×C3×C4C2×C22⋊C411yescharacteristic
329C2×C3×C4C2×C22⋊C411yescharacteristic
330C2×C3×C4C23×C411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333#67—42no—
334#67—84no—
335#67—84no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—84no—
339C2×Dic3—84no—
340C2×Dic3C2×C22⋊C412yes—
341C2×Dic3C2×C22⋊C412yes—
342C2×Dic3C2×C22⋊C412yes—
343D12—22no—
344C4×D3—22no—
345C4×D3—22no—
346C4×D3—22no—
347C4×D3—22no—
348C4×D3—42no—
349Dic6—22no—
350Dic6—41no—
351Dic6—41no—
352Dic6—41no—
353Dic6—41no—
354C3×C8—41no—
355C3×C8—42no—
356C3×C8—41no—
357C3⋊C8—42no—
358C3⋊C8—41no—
359C3⋊C8—41no—
360C24—22no—
361#41—122no—
362#41—242no—
363#41—242no—
364#41—242no—
365#41—242no—
366C2×Q8—31no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—32no—
375C2×D4—31no—
376C2×D4—44no—
377C2×D4—64no—
378C2×D4—64no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4C2×C4×D314yes—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4C2×C4×D311yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D411yescharacteristic
392M16—81no—
393M16—242no—
394M16—241no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—62no—
401C2×C8—242no—
402C4⋊C4—31no—
403C4⋊C4—31no—
404C4⋊C4—32no—
405C4⋊C4—32no—
406C4⋊C4—32no—
407C4⋊C4—32no—
408C4⋊C4—62no—
409C4⋊C4—62no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C22⋊C4—32no—
413C22⋊C4—32no—
414C22⋊C4—32no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C22⋊C4—62no—
420C42—32no—
421C42—32no—
422C42—32no—
423C42—62no—
424C42C2×C4×D311yescharacteristic
425C42D3×D411yescharacteristic
426C42D3×D411yescharacteristic
427C42C23×D311yescharacteristic
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—164no—
435C22×C3C22×C22⋊C411yescharacteristic
436C2×D3—22no—
437C2×D3—22no—
438C2×D3—22no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4#41312yes—
443C3×C4#57611yescharacteristic
444C3×C4#57611yescharacteristic
445C3×C4#57711yescharacteristic
446C3×C4#57711yescharacteristic
447Dic3—22no—
448Dic3—22no—
449Dic3—22no—
450Dic3—42no—
451Dic3—42no—
452C23—24no—
453C23—32no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23D3×C22⋊C411yescharacteristic
458Q8—62no—
459Q8—121no—
460Q8—121no—
461Q8—121no—
462Q8—121no—
463D4—44no—
464D4—42no—
465D4—42no—
466D4—44no—
467D4—42no—
468D4—42no—
469D4—62no—
470D4—82no—
471D4—82no—
472D4—122no—
473D4—244no—
474D4—244no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—122no—
490C2×C4—244no—
491C2×C4—244no—
492C2×C4D3×C22⋊C411yescharacteristic
493C2×C4D3×C22⋊C411yescharacteristic
494C2×C4D3×C22⋊C411yescharacteristic
495C2×C4C22×C4×D311yescharacteristic
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C8—42no—
499C8—41no—
500C8—41no—
501C8—121no—
502C8—121no—
503C8—122no—
504C2×C3—41no—
505C2×C3—84no—
506C2×C3C2×#41311yescharacteristic
507C2×C3#289711yescharacteristic
508C2×C3#289711yescharacteristic
509D3—42no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—164no—
520C22C2×D3×C22⋊C411yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—62no—
525C4—62no—
526C4—62no—
527C4—122no—
528C4—122no—
529C4#474512yes—
530C4#575711yescharacteristic
531C4#576211yescharacteristic
532C4#577411yescharacteristic
533C4#577511yescharacteristic
534C3#1777611yescharacteristic, complemented, Sylow
535C2—41no—
536C2—84no—
537C2—122no—
538C2C2×#474511yescharacteristic
539C2#7874211yescharacteristic
540C2#7892511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1545C2
socleC22×C3C22×C22⋊C4
prime radical#86190C2
cosocleC3×C42C24
hypercenter#1545D3
hypocenterC3#17776
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #79293C2semidirect
3. #79052C2semidirect
4. C3×#1545C2semidirect
5. #71759C2semidirect
6. #70777C2semidirect
7. #70511C2semidirect
8. #68388C2semidirect
9. #68386C2semidirect
10. #68384C2semidirect
11. #68381C2semidirect
33. #4837C22semidirect
34. #4835C22semidirect
35. #4833C22semidirect
36. D3×C4⋊C8C22semidirect
45. #4486C22semidirect
46. #4486C22semidirect
47. #4486C22semidirect
48. #4482C22semidirect
49. #4482C22semidirect
50. #4482C22semidirect
59. #1545D3semidirect
534. C3#17776semidirect

Maximal subgroups

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

subgroupindex
#861902
#792932
#790522
C3×#15452
#717592
#707772
#705112
#683882
#683862
#683842
#683812
#177763