Finite Group Databasebeta

← #11906128

Subgroups — #11906128

Order 1152.

3477 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64367C211yesmaximal, characteristic, complemented
2C2×C3×#1009C211yesmaximal, characteristic, complemented
3C3×Dic3×#41C211yesmaximal, characteristic, complemented
4#10595414C214yesmaximal, complemented
5#10595246C2112yesmaximal, complemented
6#10595042C213yesmaximal, complemented
7#10595029C213yesmaximal, complemented
8#10594856C213yesmaximal, complemented
9C2×#63860C213yesmaximal, complemented
10#87283—31nomaximal
11#85847—31nomaximal
12C2×C32×#41C2211yescharacteristic
13#64367—28no—
14C2×Q8×C3⋊D3C2212yescomplemented
15C2×#3830C2216yescomplemented
16C2×#3828C2213yescomplemented
17C22×#634C2213yescomplemented
18C3×#1009—28no—
19C2×C3×#239C2212yes—
20C2×C3×D3×D4C2216yes—
21C2×C3×#235C2213yes—
22C22×C3×D12C2213yes—
23C3×Q8×Dic3C2212yescomplemented
24C3×D4×Dic3C2216yescomplemented
25C3×#926C2213yescomplemented
26C2×C3×C4×Dic3C2213yescomplemented
27#63978C22112yescomplemented
28#63977C22112yescomplemented
29#63962C22112yescomplemented
30C2×(C2×D3)⋊Dic3C22112yescomplemented
31#63916C22112yescomplemented
32#63911C2213yescomplemented
33C4×#626C22112yescomplemented
34#63866C2213yescomplemented
35#63864C2211yescharacteristic, complemented
36#63860C2213yescomplemented
37#63860C2216yescomplemented
38#63860C22112yescomplemented
39#63857C22112yescomplemented
40C2×C3×H(5,𝔽2)—31no—
41C2×#1010—31no—
42C2×#1010—31no—
43C2×#1010—61no—
44C2×#1009D311yescharacteristic, complemented
45C3×C4×#41—31no—
46#5832—33no—
47#5831—34no—
48Dic3×#41D311yescharacteristic, complemented
49#5825—33no—
50#5806—312no—
51#5805—33no—
52C2×#936—33no—
53#5576—34no—
54#5554—312no—
55#5526—33no—
56#5525—33no—
57#5477—33no—
58C2×#875—33no—
59C23×C3×D3C2316yes—
60C32×#41D418yescomplemented
61C2×C32×Q8C2311yescharacteristic
62C2×C32×D4C2313yes—
63C22×C32×C4C2313yes—
64C2×#638C2316yes—
65C22×C3⋊Dic3C2316yes—
66Q8×C3⋊D3—28no—
67#3830—224no—
68#3828—212no—
69C2×C4⋊(C3⋊D3)C2311yescharacteristic
70C2×C4×C3⋊D3C2316yes—
71C2×#634—212no—
72C2×#634C2313yes—
73C2×#634C2316yes—
74C2×C3×#67C2316yes—
75C22×C3×Dic3C2316yes—
76C3×#239—28no—
77C3×D3×D4—224no—
78C3×#235—212no—
79C2×C3×D12—212no—
80C2×C3×D12C2313yes—
81C2×C3×D12C2316yes—
82C2×C3×C4×D3C2316yes—
83C2×C3×Dic6C2311yescharacteristic
84C2×#626C23112yes—
85C3×#217C2316yes—
86C3×C4⋊Dic3C2316yes—
87C3×C4×Dic3C2311yescharacteristic
88C3×C4×Dic3C2313yes—
89C3×C4×Dic3C2316yes—
90Dic3⋊Dic3C23112yes—
91#3722C2314yes—
92(C2×D3)⋊Dic3C23124yes—
93(C2×D3)⋊Dic3C23112yes—
94#3500—91noSylow
95C3×H(5,𝔽2)—68no—
96C2×C3×#41—21no—
97C2×C3×#41—32no—
98C2×C3×#41—33no—
99C2×C3×#41C2×D311yescharacteristic
100C2×C3×#41C2×D311yescharacteristic
101C22×C3×D4—36no—
102C22×C3×D4—33no—
103#1010—616no—
104#1010—68no—
105#1010—68no—
106#1009—28no—
107C2×#239C2×D312yes—
108C2×D3×Q8—32no—
109C2×D3×Q8—32no—
110C2×D3×Q8—62no—
111C2×#237—36no—
112C2×#237—36no—
113C2×#237—66no—
114C2×D3×D4C2×D316yes—
115C2×#235—33no—
116C2×#235—33no—
117C2×#235—63no—
118C2×#235C2×D313yes—
119C22×D12C2×D313yes—
120C22×Dic6—33no—
121C22×Dic6—33no—
122C22×Dic6—63no—
123C3×C4×Q8—32no—
124C3×C4×D4—36no—
125C3×#117—33no—
126C2×C3×C42—33no—
127#953—312no—
128C2×#217—312no—
129#947—31no—
130Q8×Dic3C2×D312yescomplemented
131#944—33no—
132#940—33no—
133#939—312no—
134#938—312no—
135#936—33no—
136#936—36no—
137#936—312no—
138#935—312no—
139D4×Dic3C2×D316yescomplemented
140#930—312no—
141C4×#67—312no—
142C2×(C2×D3)⋊C4—312no—
143#926C2×D313yescomplemented
144#924—312no—
145C2×C4×Dic3C2×D313yescomplemented
146#897—312no—
147#895—312no—
148#886—312no—
149#882—312no—
150#875—33no—
151#875—36no—
152#875—312no—
153#875—31no—
154#874—33no—
155C4×D12—312no—
156#869—33no—
157C23×C32C2413yes—
158C22×C3⋊D3C2412yes—
159C22×C3×D3—212no—
160C22×C3×D3—412no—
161C22×C3×D3C2416yes—
162C32×Q8C2×D414yes—
163C32×D4C2×D4112yes—
164C2×C32×C4C2×D4112yes—
165C2×C32×C4C2411yescharacteristic
166C2×C32×C4C2413yes—
167#638—212no—
168C2×C3⋊Dic3—212no—
169C2×C3⋊Dic3C2416yes—
170C4⋊(C3⋊D3)—22no—
171C4×C3⋊D3—212no—
172#634—26no—
173#634—212no—
174C3×#67—212no—
175C2×C3×Dic3—46no—
176C2×C3×Dic3C2412yes—
177C2×C3×Dic3C2416yes—
178C2×C3×Dic3C2412yes—
179C3×D12—26no—
180C3×D12—212no—
181C3×C4×D3—212no—
182C3×Dic6—22no—
183#626—412no—
184C2×2−1+4—91no—
185C2×H(5,𝔽2)—31no—
186#548—94no—
187#546—912no—
188#536—93no—
189#535—93no—
190#532—93no—
191C2×#124—93no—
192C4×#41—31no—
193C24×C3—36no—
194C23×D3C22×D316yes—
195C3×#41—28no—
196C3×#41—68no—
197C3×#41—612no—
198C3×#41D1218yescomplemented
199C3×#41#6718yescomplemented
200C2×C3×Q8—21no—
201C2×C3×Q8—31no—
202C2×C3×Q8C22×D311yescharacteristic
203C2×C3×Q8C22×D311yescharacteristic
204C2×C3×D4—23no—
205C2×C3×D4—36no—
206C2×C3×D4—33no—
207C2×C3×D4—36no—
208C2×C3×D4—624no—
209C2×C3×D4—612no—
210C2×C3×D4C22×D313yes—
211C2×C3×D4C22×D313yes—
212C22×C3×C4—23no—
213C22×C3×C4—36no—
214C22×C3×C4—36no—
215C22×C3×C4C22×D313yes—
216C22×C3×C4C22×D313yes—
217C2×#67—36no—
218C2×#67—36no—
219C2×#67—312no—
220C2×#67—66no—
221C2×#67C22×D316yes—
222C22×Dic3—36no—
223C22×Dic3—36no—
224C22×Dic3—66no—
225C22×Dic3C22×D316yes—
226#239—28no—
227D3×Q8—616no—
228D3×Q8—68no—
229D3×Q8—68no—
230#237—648no—
231#237—624no—
232#237—624no—
233D3×D4—224no—
234#235—212no—
235#235—624no—
236#235—612no—
237#235—612no—
238C2×D12—212no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—312no—
242C2×D12—61no—
243C2×D12C22×D313yes—
244C2×D12C22×D316yes—
245C2×C4×D3—36no—
246C2×C4×D3—36no—
247C2×C4×D3—66no—
248C2×C4×D3C22×D316yes—
249C2×Dic6—33no—
250C2×Dic6—36no—
251C2×Dic6—36no—
252C2×Dic6—33no—
253C2×Dic6—63no—
254C2×Dic6—66no—
255C2×Dic6—624no—
256C2×Dic6—612no—
257C2×Dic6—612no—
258C2×Dic6C22×D311yescharacteristic
259C3×C4⋊C4—36no—
260C3×C22⋊C4—36no—
261C3×C42—33no—
262C3×C42—36no—
263C3×C42—31no—
264#217—324no—
265#217—312no—
266#217C22×D316yes—
267(C2×D3)⋊C4—312no—
268(C2×D3)⋊C4—324no—
269(C2×D3)⋊C4—34no—
270(C2×D3)⋊C4—34no—
271C4⋊Dic3—312no—
272C4⋊Dic3C22×D316yes—
273Dic3⋊C4—312no—
274C4×Dic3C22×D311yescharacteristic
275C4×Dic3C22×D313yes—
276C4×Dic3C22×D316yes—
277C22×C32—26no—
278C22×C32C22×D416yes—
279C22×C32C2511yescharacteristic
280C2×C3⋊D3—22no—
281C2×C3⋊D3—24no—
282C2×C3×D3—26no—
283C2×C3×D3—212no—
284C2×C3×D3—812no—
285C32×C4C22×D412yes—
286C32×C4C22×D416yes—
287C3⋊Dic3—26no—
288C3×Dic3—22no—
289C3×Dic3—26no—
290C3×Dic3—22no—
2912−1+4—188no—
292H(5,𝔽2)—68no—
293C2×#41—32no—
294C2×#41—33no—
295C2×#41—96no—
296C2×#41—93no—
297C2×#41D3211yescharacteristic
298C22×Q8—92no—
299C22×Q8—93no—
300C22×D4—36no—
301C22×D4—33no—
302C4⋊Q8—93no—
303#127—93no—
304#124—93no—
305#124—96no—
306#124—912no—
307#124—91no—
308#123—912no—
309C22⋊Q8—912no—
310#121—912no—
311#121—912no—
312#120—912no—
313C4×Q8—32no—
314C4×D4—36no—
315C4×D4—912no—
316#117—33no—
317C2×C22⋊C4—912no—
318C2×C42—33no—
319C23×C3—23no—
320C23×C3—36no—
321C23×C3—612no—
322C23×C3—1212no—
323C23×C3C23×D313yes—
324C23×C3C23×D313yes—
325C22×D3—212no—
326C22×D3—32no—
327C22×D3—32no—
328C22×D3—412no—
329C22×D3—62no—
330C22×D3C23×D316yes—
331C3×Q8—24no—
332C3×Q8—62no—
333C3×Q8C2×D1214yes—
334C3×Q8C2×#6714yes—
335C3×D4—212no—
336C3×D4—612no—
337C3×D4—66no—
338C3×D4—612no—
339C3×D4C2×D12112yes—
340C3×D4C2×#67112yes—
341C2×C3×C4—21no—
342C2×C3×C4—23no—
343C2×C3×C4—212no—
344C2×C3×C4—32no—
345C2×C3×C4—36no—
346C2×C3×C4—32no—
347C2×C3×C4—612no—
348C2×C3×C4—126no—
349C2×C3×C4C2×D12112yes—
350C2×C3×C4C2×#67112yes—
351C2×C3×C4C23×D311yescharacteristic
352C2×C3×C4C23×D313yes—
353C2×C3×C4C23×D311yescharacteristic
354C2×C3×C4C23×D313yes—
355#67—212no—
356#67—624no—
357#67—612no—
358#67—612no—
359#67—1212no—
360C2×Dic3—36no—
361C2×Dic3—36no—
362C2×Dic3—46no—
363C2×Dic3—66no—
364C2×Dic3—624no—
365C2×Dic3—612no—
366C2×Dic3—612no—
367C2×Dic3C23×D312yes—
368C2×Dic3C23×D316yes—
369C2×Dic3C23×D312yes—
370D12—26no—
371D12—212no—
372D12—64no—
373D12—62no—
374D12—62no—
375D12—1212no—
376C4×D3—212no—
377C4×D3—624no—
378C4×D3—612no—
379C4×D3—612no—
380Dic6—22no—
381Dic6—612no—
382Dic6—624no—
383Dic6—66no—
384Dic6—612no—
385Dic6—66no—
386Dic6—612no—
387C2×C32—26no—
388C2×C32C23×D411yescharacteristic
389C2×C32#58512yes—
390C3⋊D3—42no—
391C3×D3—46no—
392C24—36no—
393#41—68no—
394#41—612no—
395#41—1824no—
396#41—1812no—
397#41#62618yescomplemented
398C2×Q8—31no—
399C2×Q8—96no—
400C2×Q8—93no—
401C2×Q8—188no—
402C2×Q8—1812no—
403C2×Q8C2×D3211yescharacteristic
404C2×D4—36no—
405C2×D4—36no—
406C2×D4—33no—
407C2×D4—624no—
408C2×D4—612no—
409C2×D4—912no—
410C2×D4—96no—
411C2×D4—91no—
412C2×D4C2×D3213yes—
413C22×C4—36no—
414C22×C4—36no—
415C22×C4—96no—
416C22×C4—96no—
417C22×C4C2×D3213yes—
418C4⋊C4—36no—
419C4⋊C4—912no—
420C22⋊C4—36no—
421C22⋊C4—94no—
422C22⋊C4—912no—
423C22⋊C4—924no—
424C42—31no—
425C42—33no—
426C42—36no—
427C22×C3—26no—
428C22×C3—26no—
429C22×C3—26no—
430C22×C3—21no—
431C22×C3—46no—
432C22×C3—66no—
433C22×C3—612no—
434C22×C3—2412no—
435C22×C3C22×D1216yes—
436C22×C3C22×#6716yes—
437C22×C3C24×D311yescharacteristic
438C22×C3C24×D311yescharacteristic
439C2×D3—212no—
440C2×D3—26no—
441C2×D3—64no—
442C2×D3—62no—
443C2×D3—64no—
444C2×D3—64no—
445C2×D3—62no—
446C2×D3—812no—
447C2×D3—124no—
448C3×C4—22no—
449C3×C4—26no—
450C3×C4—62no—
451C3×C4—66no—
452C3×C4—62no—
453C3×C4C22×D1212yes—
454C3×C4C22×D1216yes—
455C3×C4C22×#6712yes—
456C3×C4C22×#6716yes—
457Dic3—22no—
458Dic3—26no—
459Dic3—22no—
460Dic3—612no—
461Dic3—66no—
462Dic3—66no—
463C32#350011yescharacteristic, complemented, Sylow
464C23—36no—
465C23—612no—
466C23—92no—
467C23—1212no—
468C23C22×D3213yes—
469Q8—62no—
470Q8—1812no—
471Q8—186no—
472Q8C2×#62614yes—
473D4—612no—
474D4—612no—
475D4—66no—
476D4—1812no—
477D4—182no—
478D4—3612no—
479D4C2×#626112yes—
480C2×C4—32no—
481C2×C4—36no—
482C2×C4—32no—
483C2×C4—612no—
484C2×C4—96no—
485C2×C4—126no—
486C2×C4—1812no—
487C2×C4—1812no—
488C2×C4C2×#626112yes—
489C2×C4C22×D3211yescharacteristic
490C2×C4C22×D3213yes—
491C2×C3—26no—
492C2×C3—26no—
493C2×C3—22no—
494C2×C3—21no—
495C2×C3—46no—
496C2×C3—126no—
497C2×C3C23×D1211yescharacteristic
498C2×C3#596412yes—
499C2×C3#596612yes—
500C2×C3C23×#6711yescharacteristic
501D3—46no—
502D3—124no—
503D3—122no—
504D3—122no—
505C22—26no—
506C22—612no—
507C22—66no—
508C22—184no—
509C22—182no—
510C22—2412no—
511C22C22×#62616yes—
512C22C23×D3211yescharacteristic
513C4—62no—
514C4—66no—
515C4—62no—
516C4—186no—
517C4C22×#62616yes—
518C4C22×#62612yes—
519C3—21no—
520C3#8584711yescharacteristic, complemented
521C3#8728311yescharacteristic, complemented
522C2—26no—
523C2—126no—
524C2—362no—
525C2#1059669612yes—
526C2C23×#62611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC2×C32×#41C22
hypercenterC2×#41D32
hypocenterC32#3500
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#64367C2semidirect
2. C2×C3×#1009C2semidirect
3. C3×Dic3×#41C2semidirect
4. #10595414C2semidirect
5. #10595246C2semidirect
6. #10595042C2semidirect
7. #10595029C2semidirect
8. #10594856C2semidirect
9. C2×#63860C2semidirect
14. C2×Q8×C3⋊D3C22semidirect
15. C2×#3830C22semidirect
16. C2×#3828C22semidirect
17. C22×#634C22semidirect
23. C3×Q8×Dic3C22semidirect
24. C3×D4×Dic3C22semidirect
25. C3×#926C22semidirect
26. C2×C3×C4×Dic3C22semidirect
27. #63978C22semidirect
28. #63977C22semidirect
29. #63962C22semidirect
30. C2×(C2×D3)⋊Dic3C22semidirect
31. #63916C22semidirect
32. #63911C22semidirect
33. C4×#626C22semidirect
34. #63866C22semidirect
35. #63864C22semidirect
36. #63860C22semidirect
37. #63860C22semidirect
38. #63860C22semidirect
39. #63857C22semidirect
44. C2×#1009D3semidirect
48. Dic3×#41D3semidirect
60. C32×#41D4semidirect
130. Q8×Dic3C2×D3semidirect
139. D4×Dic3C2×D3semidirect
143. #926C2×D3semidirect
145. C2×C4×Dic3C2×D3semidirect
198. C3×#41D12semidirect
199. C3×#41#67semidirect
397. #41#626semidirect
463. C32#3500semidirect
520. C3#85847semidirect
521. C3#87283semidirect

Maximal subgroups

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

subgroupindex
C2×#643672
C2×C3×#10092
C3×Dic3×#412
#105954142
#105952462
#105950422
#105950292
#105948562
C2×#638602
#872833
#858473