Finite Group Databasebeta

← #10849633

Subgroups — #10849633

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86262C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3#83252C211yesmaximal, characteristic, complemented
4#83212C211yesmaximal, characteristic
5#71839C211yesmaximal, characteristic, complemented
6#70678C214yesmaximal
7#70677C214yesmaximal, complemented
8#69877C211yesmaximal, characteristic, complemented
9C4×#825C211yesmaximal, characteristic
10#25688—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5728—21no—
14#5728—21no—
15#5698—22no—
16#5642—22no—
17#5642—22no—
18#5584—21no—
19#5584—21no—
20#5570—21no—
21#5570—21no—
22#5526C2212yes—
23#5516C2212yes—
24C4×#235C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#501—22no—
27C3×#500—21no—
28C3×#498—21no—
29C3×#428C2214yes—
30C3×C4×M16C2211yescharacteristic
31#5229—22no—
32C2×#825C2212yes—
33C2×#824C2214yes—
34#5094—21no—
35#5091—21no—
36#5087—21no—
37#5084—21no—
38#4977—22no—
39#4973C2214yes—
40#4966C2212yes—
41#4964C2212yes—
42C42×Dic3C2211yescharacteristic
43#4924C2211yescharacteristic
44#4555C2212yes—
45#4532C2214yes—
46#3509—31no—
47#3165D311yescharacteristic, complemented
48#3079—31no—
49#2862—31no—
50#2038—34no—
51#1994—31no—
52#1966—31no—
53#1857—34no—
54C4×#104—31no—
55C22×C3×Q8—21no—
56C2×#235C2311yescharacteristic
57C22×Dic6C2312yes—
58C2×C3×M16C2312yes—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8D411yescharacteristic
64C3×C4⋊Q8D411yescharacteristic
65C3×#125D412yes—
66C3×C22⋊Q8—21no—
67C3×C22⋊Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×#117C2312yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74#946—22no—
75#946—22no—
76Q8×Dic3—22no—
77Q8×Dic3—22no—
78C4×#67—21no—
79C4×#67—21no—
80#926C2311yescharacteristic
81C2×C4⋊Dic3D412yes—
82C2×C4⋊Dic3D412yes—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3C2312yes—
86C2×#208C2312yes—
87#898—24no—
88#898—24no—
89#897—22no—
90#897—22no—
91#892—22no—
92#892—22no—
93#890—24no—
94#890—24no—
95#890—22no—
96#890—22no—
97#887—22no—
98#887—22no—
99#886—24no—
100#877—24no—
101#875—22no—
102C4×D12D411yescharacteristic
103C4×D12D411yescharacteristic
104#872—21no—
105C42×D3—21no—
106#869—22no—
107#869—21no—
108#869—21no—
109#869—21no—
110#869—21no—
111C4×Dic6—22no—
112C4×Dic6—22no—
113C4×Dic6D411yescharacteristic
114C4×Dic6D411yescharacteristic
115C3×#107—24no—
116C3×#106—24no—
117C3×#97—21no—
118C3×C4×C8—21no—
119C3⋊#95—28no—
120#825—24no—
121#824—28no—
122#810—22no—
123#809—22no—
124#808—22no—
125#807—22no—
126C4⋊(C3⋊C8)—21no—
127C4⋊(C3⋊C8)—21no—
128#557—61no—
129#557—61no—
130#556—61no—
131#556—61no—
132#552—62no—
133#552—62no—
134#541—62no—
135#536—32no—
136#533—61no—
137#533—61no—
138#533C2×D311yescharacteristic
139C2×C4⋊Q8—31no—
140C2×C4⋊Q8C2×D311yescharacteristic
141#519—32no—
142C4×#41—31no—
143#501—22no—
144#500—21no—
145#498—21no—
146#477—61no—
147#476—61no—
148#475—61no—
149#474—61no—
150#428—32no—
151#428C2×D314yes—
152#423—31no—
153C2×#104—32no—
154#417—32no—
155C2×#101—34no—
156C4×M16C2×D311yescharacteristic
157#391—34no—
158#391—62no—
159#382—62no—
160C43—31no—
161#339—32no—
162#339—34no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—44no—
166C2×C3×Q8—41no—
167C2×C3×Q8—41no—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×#67—21no—
173C22×Dic3—22no—
174C22×Dic3C2×D414yes—
175#235—22no—
176#235—22no—
177C2×D12C2×D411yescharacteristic
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×C4×D3—21no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—24no—
184C2×Dic6—24no—
185C2×Dic6C2×D411yescharacteristic
186C3×M16—22no—
187C3×M16#4114yes—
188C2×C3×C8—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—22no—
200C3×C42—22no—
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205#217—21no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3C2×D411yescharacteristic
217C4⋊Dic3C2×D411yescharacteristic
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4—21no—
221Dic3⋊C4—21no—
222C4×Dic3—24no—
223C4×Dic3—24no—
224C4×Dic3—24no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C4×Dic3—22no—
229C4×Dic3—22no—
230C4×Dic3—42no—
231#208—24no—
232C2×C3⋊C8—22no—
233C2×#41—31no—
234C22×Q8—21no—
235C22×Q8—32no—
236C2×M16—32no—
237C2×M16C22×D312yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—62no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4⋊Q8#6711yescharacteristic
248C4⋊Q8#6711yescharacteristic
249#126—64no—
250#126—64no—
251#125—62no—
252#125—62no—
253#125—64no—
254#125—64no—
255#125#6712yes—
256#124—62no—
257#123—64no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—31no—
267C4×Q8—31no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8—62no—
274C4×D4—31no—
275C4×D4—31no—
276C4×D4—61no—
277C4×D4—61no—
278#117—31no—
279#117—64no—
280#117—62no—
281#117—62no—
282#117—61no—
283#117C22×D312yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4C22×D312yes—
288C2×C42—32no—
289C2×C42—61no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42C22×D311yescharacteristic
293#107—24no—
294#107—62no—
295#106—24no—
296#106—62no—
297C4⋊C8—61no—
298C4⋊C8—61no—
299#104—64no—
300#103—62no—
301#102—62no—
302#101—68no—
303#97—21no—
304C4×C8—21no—
305#95—68no—
306C23×C3C22×D411yescharacteristic
307C22×D3—21no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—82no—
311C3×Q8—82no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—44no—
317C2×C3×C4—42no—
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4#12011yescharacteristic
321C2×C3×C4#12011yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4112yes—
325C2×C3×C42−1+412yes—
326#67—42no—
327C2×Dic3—24no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—24no—
332C2×Dic3—28no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335D12—21no—
336D12—21no—
337C4×D3—42no—
338C4×D3—42no—
339C4×D3—42no—
340Dic6—21no—
341Dic6—21no—
342Dic6—42no—
343Dic6—42no—
344Dic6—44no—
345C3×C8—24no—
346C3⋊C8—42no—
347#41—62no—
348#41—62no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—31no—
352C2×Q8—44no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—62no—
356C2×Q8—64no—
357C2×Q8—64no—
358C2×Q8—62no—
359C2×D4—31no—
360C2×D4—61no—
361C22×C4—22no—
362C22×C4—34no—
363C22×C4—62no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4C23×D311yescharacteristic
370M16—22no—
371M16—64no—
372M16#23914yes—
373C2×C8—22no—
374C2×C8—62no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—31no—
382C4⋊C4—31no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—64no—
386C4⋊C4—64no—
387C4⋊C4—64no—
388C4⋊C4—64no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4C2×#6712yes—
396C4⋊C4C2×#6712yes—
397C22⋊C4—22no—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C42—22no—
404C42—64no—
405C42—64no—
406C42—64no—
407C42—61no—
408C42—61no—
409C42—61no—
410C42—62no—
411C42—62no—
412C42—122no—
413C42D3×D411yescharacteristic
414C42D3×D411yescharacteristic
415C42C2×#6711yescharacteristic
416C42C2×#6711yescharacteristic
417C22×C3—21no—
418C22×C3—21no—
419C22×C3C2×#12011yescharacteristic
420C22×C3#54712yes—
421C2×D3—41no—
422C2×D3—41no—
423C2×D3—41no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—82no—
429C3×C4#45412yes—
430C3×C4#45612yes—
431C3×C4#53211yescharacteristic
432C3×C4#53611yescharacteristic
433C3×C4#54812yes—
434Dic3—44no—
435Dic3—41no—
436Dic3—44no—
437Dic3—41no—
438Dic3—41no—
439C23—61no—
440C23C2×D3×D411yescharacteristic
441Q8—42no—
442Q8—42no—
443Q8—61no—
444Q8—61no—
445Q8—82no—
446Q8—82no—
447Q8—122no—
448Q8—122no—
449Q8—124no—
450D4—61no—
451D4—61no—
452D4—122no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—64no—
460C2×C4—68no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—64no—
464C2×C4—61no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4#93711yescharacteristic
471C2×C4#93711yescharacteristic
472C2×C4#93711yescharacteristic
473C2×C4#93711yescharacteristic
474C2×C4C2×D3×D411yescharacteristic
475C2×C4C2×#23912yes—
476C2×C4#100712yes—
477C2×C4C22×#6711yescharacteristic
478C8—24no—
479C8—122no—
480C2×C3—22no—
481C2×C3#243211yescharacteristic
482C2×C3C2×#45411yescharacteristic
483C2×C3C2×#45611yescharacteristic
484D3—81no—
485C22—21no—
486C22—21no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22#572612yes—
491C22C2×#93711yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—124no—
498C4—121no—
499C4—124no—
500C4—121no—
501C4—121no—
502C4#520112yes—
503C4#520512yes—
504C4#572411yescharacteristic
505C4#572511yescharacteristic
506C4#581712yes—
507C3#2568811yescharacteristic, complemented, Sylow
508C2—22no—
509C2—241no—
510C2#8325511yescharacteristic
511C2C2×#520111yescharacteristic
512C2C2×#520511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3165C2
socleC22×C3C2×#120
prime radicalC2×#235C23
hypercenter#3165D3
hypocenterC3#25688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
2. C3×#3165C2semidirect
3. #83252C2semidirect
5. #71839C2semidirect
7. #70677C2semidirect
8. #69877C2semidirect
47. #3165D3semidirect
507. C3#25688semidirect

Maximal subgroups

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

subgroupindex
#862622
C3×#31652
#832522
#832122
#718392
#706782
#706772
#698772
C4×#8252
#256883