Finite Group Databasebeta

← #10849636

Subgroups — #10849636

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#85830C211yesmaximal, characteristic
2#82508C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic, complemented
4#73308C211yesmaximal, characteristic, complemented
5C3×C4×#104C211yesmaximal, characteristic
6#70360C214yesmaximal
7#69877C211yesmaximal, characteristic, complemented
8#69396C211yesmaximal, characteristic, complemented
9#68764C214yesmaximal, complemented
10#25688—31nomaximal, Sylow
11C3×C4×#41C2211yescharacteristic
12#5825C2212yes—
13#5567—21no—
14#5567—21no—
15#5542—21no—
16#5542—21no—
17#5541—22no—
18#5541—22no—
19#5518—22no—
20#5516C2212yes—
21#5470—21no—
22#5470—21no—
23#5470C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#104C2212yes—
27C3×C4×M16C2211yescharacteristic
28C3×C43C2211yescharacteristic
29#5232C2212yes—
30C2×#824C2214yes—
31#5121C2214yes—
32#5021—21no—
33#5020—21no—
34#5008—21no—
35#5007—21no—
36#4995C2212yes—
37#4936—22no—
38#4929—22no—
39#4929C2214yes—
40#4924C2211yescharacteristic
41#4702—22no—
42#4682—21no—
43#4680—21no—
44C3×#339C2212yes—
45#4523C2214yes—
46#3509—31no—
47#3165—31no—
48#3079—31no—
49#2862—31no—
50#2038—34no—
51#1994—31no—
52#1966—31no—
53#1857—34no—
54C4×#104D311yescharacteristic
55C2×C3×#41C2311yescharacteristic
56C22×Dic6—21no—
57C22×Dic6C2312yes—
58C2×C3×M16C2312yes—
59C3×C4×Q8D411yescharacteristic
60C3×C4×Q8D411yescharacteristic
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×C4×D4D411yescharacteristic
64C3×C4×D4D411yescharacteristic
65C3×#117—21no—
66C3×#117C2311yescharacteristic
67C2×C3×C42—21no—
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72#944—22no—
73#936—22no—
74#935—24no—
75#926—22no—
76#926—22no—
77#926C2312yes—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3D412yes—
80C2×C4⋊Dic3D412yes—
81C2×C4⋊Dic3C2312yes—
82#924—22no—
83#924—22no—
84#924—21no—
85#924—21no—
86#924—22no—
87#924—22no—
88C2×#208C2312yes—
89#889—24no—
90#889—24no—
91#887—22no—
92#887—22no—
93#879—24no—
94#879—24no—
95#877—24no—
96#870—22no—
97#870—22no—
98#870D412yes—
99#869—21no—
100#869—21no—
101#869—21no—
102#869—21no—
103#869—21no—
104#869—21no—
105#869—21no—
106#869—21no—
107#869D411yescharacteristic
108#869D411yescharacteristic
109C4×Dic6—22no—
110C4×Dic6—22no—
111C4×Dic6—21no—
112C4×Dic6—21no—
113C4×Dic6—22no—
114C4×Dic6—22no—
115C3×#104—24no—
116C3×#97—21no—
117C3×C4×C8—21no—
118#835—22no—
119#832—22no—
120C3⋊#95—28no—
121#824—28no—
122#818—24no—
123#817—24no—
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—31no—
137#533—61no—
138#533—61no—
139C2×C4⋊Q8—31no—
140C2×C4⋊Q8—31no—
141#519—32no—
142C4×#41C2×D311yescharacteristic
143#501—62no—
144#500—61no—
145#498—61no—
146#477—61no—
147#476—61no—
148#475—61no—
149#474—61no—
150#428—34no—
151#428—32no—
152#423—31no—
153C2×#104C2×D312yes—
154#417—32no—
155C2×#101—34no—
156C4×M16C2×D311yescharacteristic
157#391—34no—
158#391—62no—
159#382—62no—
160C43C2×D311yescharacteristic
161#339—34no—
162#339C2×D312yes—
163C3×#41—22no—
164C3×#41—22no—
165C2×C3×Q8C2×D411yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4C2×D411yescharacteristic
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—22no—
176C22×Dic3C2×D414yes—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—24no—
182C2×Dic6—24no—
183C2×Dic6—44no—
184C2×Dic6—41no—
185C2×Dic6—41no—
186C3×M16—22no—
187C3×M16#4114yes—
188C2×C3×C8—22no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4C2×D411yescharacteristic
192C3×C4⋊C4C2×D411yescharacteristic
193C3×C22⋊C4—21no—
194C3×C22⋊C4—21no—
195C3×C22⋊C4—21no—
196C3×C42—24no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—42no—
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207#217—24no—
208#217—24no—
209#217—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—41no—
219C4⋊Dic3—41no—
220C4⋊Dic3C2×D412yes—
221C4⋊Dic3C2×D412yes—
222Dic3⋊C4—24no—
223Dic3⋊C4—24no—
224Dic3⋊C4—24no—
225Dic3⋊C4—24no—
226Dic3⋊C4—22no—
227Dic3⋊C4—22no—
228C4×Dic3—24no—
229C4×Dic3—24no—
230C4×Dic3—22no—
231#208—24no—
232C2×C3⋊C8—22no—
233C2×#41C22×D311yescharacteristic
234C22×Q8—32no—
235C22×Q8—61no—
236C2×M16—32no—
237C2×M16C22×D312yes—
238C4⋊Q8—31no—
239C4⋊Q8—31no—
240C4⋊Q8—62no—
241C4⋊Q8—61no—
242C4⋊Q8—61no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4⋊Q8—61no—
248C4⋊Q8—61no—
249#126—64no—
250#126—64no—
251#125—32no—
252#125—62no—
253#125—62no—
254#125—64no—
255#125—64no—
256#124—62no—
257#123—64no—
258C22⋊Q8—62no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262C22⋊Q8—61no—
263C22⋊Q8—61no—
264C4×Q8—62no—
265C4×Q8—62no—
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—61no—
271C4×Q8—61no—
272C4×Q8D1211yescharacteristic
273C4×Q8D1211yescharacteristic
274C4×D4—21no—
275C4×D4—21no—
276C4×D4D1211yescharacteristic
277C4×D4D1211yescharacteristic
278#117—21no—
279#117—32no—
280#117—64no—
281#117—62no—
282#117—62no—
283#117C22×D311yescharacteristic
284C2×C4⋊C4—32no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—61no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C22×D312yes—
292C2×C42C22×D311yescharacteristic
293#107—64no—
294#107—62no—
295#106—64no—
296#106—62no—
297C4⋊C8—61no—
298C4⋊C8—61no—
299#104—24no—
300#103—62no—
301#102—62no—
302#101—68no—
303#97—21no—
304C4×C8—21no—
305#95—68no—
306C23×C3—21no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—21no—
309C3×Q8—21no—
310C3×D4—21no—
311C3×D4—21no—
312C3×D4—42no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4#12011yescharacteristic
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4112yes—
331C2×C3×C42−1+412yes—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—28no—
336C2×Dic3—44no—
337C2×Dic3—42no—
338Dic6—42no—
339Dic6—42no—
340Dic6—44no—
341Dic6—42no—
342Dic6—42no—
343Dic6—82no—
344Dic6—82no—
345C3×C8—24no—
346C3⋊C8—42no—
347#41—22no—
348#41—22no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—62no—
352C2×Q8—64no—
353C2×Q8—62no—
354C2×Q8—64no—
355C2×Q8—124no—
356C2×Q8—121no—
357C2×Q8—121no—
358C2×Q8C2×D1211yescharacteristic
359C2×D4—21no—
360C2×D4C2×D1211yescharacteristic
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—34no—
366C22×C4—62no—
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4C23×D311yescharacteristic
370M16—22no—
371M16—64no—
372M16#23514yes—
373C2×C8—22no—
374C2×C8—62no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—32no—
378C4⋊C4—32no—
379C4⋊C4—64no—
380C4⋊C4—64no—
381C4⋊C4—64no—
382C4⋊C4—64no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C4⋊C4—62no—
387C4⋊C4—61no—
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—62no—
403C42—24no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—22no—
408C42—22no—
409C42—42no—
410C42—64no—
411C42—64no—
412C42—62no—
413C42C2×D1211yescharacteristic
414C42C2×D1211yescharacteristic
415C42D3×D411yescharacteristic
416C42D3×D411yescharacteristic
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3C2×#12011yescharacteristic
423C22×C3#54712yes—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—44no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4#45412yes—
431C3×C4#45612yes—
432C3×C4#53211yescharacteristic
433C3×C4#53611yescharacteristic
434C3×C4#54812yes—
435Dic3—44no—
436Dic3—42no—
437Dic3—42no—
438Dic3—82no—
439C23—21no—
440C23C2×D3×D411yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—122no—
444Q8—122no—
445Q8—124no—
446Q8—122no—
447Q8—122no—
448Q8—242no—
449Q8—242no—
450D4—21no—
451D4—21no—
452D4—42no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—64no—
467C2×C4—68no—
468C2×C4—124no—
469C2×C4—122no—
470C2×C4#88211yescharacteristic
471C2×C4#88211yescharacteristic
472C2×C4#88211yescharacteristic
473C2×C4#88211yescharacteristic
474C2×C4C22×D1211yescharacteristic
475C2×C4C2×#23512yes—
476C2×C4C2×D3×D411yescharacteristic
477C2×C4#101012yes—
478C8—24no—
479C8—122no—
480C2×C3—22no—
481C2×C3—81no—
482C2×C3#243211yescharacteristic
483C2×C3C2×#45411yescharacteristic
484C2×C3C2×#45611yescharacteristic
485C22—21no—
486C22—21no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22C2×#88211yescharacteristic
491C22#555112yes—
492C4—21no—
493C4—21no—
494C4—44no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—124no—
499C4—122no—
500C4—122no—
501C4—242no—
502C4#482612yes—
503C4#482712yes—
504C4#547711yescharacteristic
505C4#552611yescharacteristic
506C4#555212yes—
507C3#2568811yescharacteristic, complemented, Sylow
508C2—22no—
509C2—81no—
510C2#7348911yescharacteristic
511C2C2×#482711yescharacteristic
512C2C2×#482611yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#104C2
socleC22×C3C2×#120
prime radicalC2×C3×#41C23
hypercenterC4×#104D3
hypocenterC3#25688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
3. #78506C2semidirect
4. #73308C2semidirect
7. #69877C2semidirect
8. #69396C2semidirect
9. #68764C2semidirect
507. C3#25688semidirect

Maximal subgroups

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

subgroupindex
#858302
#825082
#785062
#733082
C3×C4×#1042
#703602
#698772
#693962
#687642
#256883