Finite Group Databasebeta

← #13181635

Subgroups — #13181635

Order 1728.

5516 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89080C211yesmaximal, characteristic, complemented
2C3×#64364C211yesmaximal, characteristic, complemented
3C32×#1007C211yesmaximal, characteristic, complemented
4#11724597C214yesmaximal, complemented
5C3⋊D3×#239C214yesmaximal, complemented
6D3×#3832C214yesmaximal, complemented
7#11724591C214yesmaximal, complemented
8#11724581C216yesmaximal, complemented
9#11724578C216yesmaximal, complemented
10#10596760—31nomaximal
11#10596689—34nomaximal
12C2×C33×Q8C2211yescharacteristic, complemented
13#89080C2214yescomplemented
14#89080C2214yescomplemented
15C2×C4⋊(C3⋊(C3⋊D3))C2213yescomplemented
16C2×C4×C3⋊(C3⋊D3)C2213yescomplemented
17C3×#3832C2214yescomplemented
18C3×Q8×C3⋊D3C2214yescomplemented
19C3×#3828C2216yescomplemented
20C32×#239C2214yescomplemented
21C32×D3×Q8C2214yescomplemented
22C32×#235C2216yescomplemented
23#89039C22112yescomplemented
24#89029C2214yescomplemented
25#89026C2216yescomplemented
26C3⋊D3×D12C22112yescomplemented
27D3×C4⋊(C3⋊D3)C22112yescomplemented
28C4×D3×C3⋊D3C22112yescomplemented
29#89022C22112yescomplemented
30#89020C22112yescomplemented
31#89019C2216yescomplemented
32#89018C22112yescomplemented
33#89017C2216yescomplemented
34#89015C2216yescomplemented
35C32×2−1+4—31no—
36#64366—34no—
37#64366—36no—
38#41×C3⋊D3—34no—
39#41×C3⋊D3—36no—
40#64364—34no—
41#64364D311yescharacteristic, complemented
42C2×#3832—31no—
43C2×#3832—34no—
44C2×#3832—34no—
45C2×#3832—64no—
46C3×#1007—34no—
47C3×#1007D314yescomplemented
48#64320—316no—
49D3×#239—316no—
50D3×#239—316no—
51#64316—316no—
52#64308—324no—
53#64306—324no—
54C22×C3⋊(C3⋊D3)C2313yes—
55C2×D3×C3⋊D3C23112yes—
56C33×Q8C2314yes—
57C2×C33×C4C2313yes—
58C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
59C4⋊(C3⋊(C3⋊D3))C23112yes—
60C4×C3⋊(C3⋊D3)C2316yes—
61C4×C3⋊(C3⋊D3)C2316yes—
62C3×#638C2314yes—
63C3×C4⋊(C3⋊D3)C2316yes—
64C3×C4×C3⋊D3C23112yes—
65C3×#634C2316yes—
66C32×#67C2314yes—
67C32×D12C2316yes—
68C32×C4×D3C23112yes—
69C32×Dic6C2316yes—
70#6293C2314yes—
71#6292C23112yes—
72#6291C23112yes—
73#6290C2314yes—
74#6289C23112yes—
75Dic3×C3⋊D3C2314yes—
76D3×C3⋊Dic3C2314yes—
77#5968—94no—
78#5968—91no—
79C32×#41—34no—
80C32×#41—36no—
81C2×C32×Q8—24no—
82C2×C32×Q8—34no—
83C2×C32×Q8C2×D314yescomplemented
84C2×C32×Q8C2×D311yescharacteristic, complemented
85#3832—34no—
86#3832—34no—
87#3832—316no—
88#3832—316no—
89#3832—36no—
90#3832—312no—
91#3832—616no—
92#3832—616no—
93#3832C2×D314yescomplemented
94Q8×C3⋊D3—36no—
95Q8×C3⋊D3C2×D314yescomplemented
96#3830—34no—
97#3830—36no—
98D4×C3⋊D3—312no—
99D4×C3⋊D3—36no—
100D4×C3⋊D3—312no—
101#3828—312no—
102#3828—312no—
103#3828C2×D316yescomplemented
104C2×C4⋊(C3⋊D3)—33no—
105C2×C4⋊(C3⋊D3)—312no—
106C2×C4⋊(C3⋊D3)—312no—
107C2×C4⋊(C3⋊D3)—612no—
108C2×C4×C3⋊D3—33no—
109C2×C4×C3⋊D3—312no—
110C2×C4×C3⋊D3—312no—
111C2×C4×C3⋊D3—612no—
112C3×#239—316no—
113C3×#239C2×D3116yescomplemented
114C3×D3×Q8—316no—
115C3×D3×Q8C2×D3116yescomplemented
116C3×#235—324no—
117C3×#235C2×D3124yescomplemented
118#3811—348no—
119#3805—316no—
120#3802—324no—
121D3×D12—348no—
122D3×D12—348no—
123C4×D32—348no—
124#3799—348no—
125#3799—348no—
126#3798—348no—
127#3797—324no—
128#3796—324no—
129#3796—324no—
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—43no—
132C2×C3⋊(C3⋊D3)C2416yes—
133C2×C3×C3⋊D3C2414yes—
134C2×C32×D3C2414yes—
135D3×C3⋊D3—412no—
136C33×C4C2416yes—
137C3⋊(C3⋊Dic3)C2411yescharacteristic
138C3⋊(C3⋊Dic3)C2411yescharacteristic
139C3×C3⋊Dic3C2414yes—
140C32×Dic3C2414yes—
141C3×2−1+4—34no—
142C3×2−1+4—91no—
143#1009—924no—
144#1009—916no—
145#1009—94no—
146#1009—96no—
147D3×#41—924no—
148D3×#41—916no—
149D3×#41—94no—
150D3×#41—96no—
151#1007—34no—
152#1007—916no—
153#1007—94no—
154#1007C3⋊D311yescharacteristic, complemented
155C2×#239—916no—
156C2×#239—94no—
157C2×#239—91no—
158C2×#239—94no—
159C2×#239—184no—
160C22×C3⋊D3—33no—
161C22×C3⋊D3—312no—
162C22×C3⋊D3—312no—
163C22×C3⋊D3—612no—
164C2×D32—348no—
165C32×Q8—216no—
166C32×Q8—36no—
167C32×Q8C22×D3116yes—
168C32×Q8C22×D314yes—
169C32×D4—34no—
170C32×D4—36no—
171C2×C32×C4—212no—
172C2×C32×C4—312no—
173C2×C32×C4C22×D3112yes—
174C2×C32×C4C22×D313yes—
175#638—34no—
176#638—312no—
177#638C22×D314yes—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—34no—
180C2×C3⋊Dic3—34no—
181C2×C3⋊Dic3—64no—
182C4⋊(C3⋊D3)—312no—
183C4⋊(C3⋊D3)—312no—
184C4⋊(C3⋊D3)—348no—
185C4⋊(C3⋊D3)—648no—
186C4⋊(C3⋊D3)C22×D316yes—
187C4×C3⋊D3—34no—
188C4×C3⋊D3—36no—
189C4×C3⋊D3—36no—
190C4×C3⋊D3—312no—
191C4×C3⋊D3—324no—
192C4×C3⋊D3—324no—
193C4×C3⋊D3—624no—
194C4×C3⋊D3—624no—
195C4×C3⋊D3C22×D3112yes—
196#634—34no—
197#634C22×D316yes—
198C3×#67—316no—
199C3×#67C22×D3116yes—
200C3×D12—324no—
201C3×D12C22×D3124yes—
202C3×C4×D3—348no—
203C3×C4×D3C22×D3148yes—
204C3×Dic6—324no—
205C3×Dic6C22×D3124yes—
206#627—316no—
207#626—348no—
208#626—348no—
209#625—316no—
210#624—348no—
211D3×Dic3—316no—
212D3×Dic3—316no—
213#585—271noSylow
214C2×C33—21no—
215C2×C33C2511yescharacteristic
216C3⋊(C3⋊D3)—26no—
217C3×C3⋊D3—24no—
218C32×D3—24no—
219C3×#41—316no—
220C3×#41—324no—
221C3×#41—94no—
222C3×#41—96no—
223C2×C3×Q8—24no—
224C2×C3×Q8—316no—
225C2×C3×Q8—94no—
226C2×C3×Q8D3214yescomplemented
227C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
228#239—316no—
229#239—948no—
230#239—924no—
231#239—916no—
232#239—916no—
233#239—94no—
234#239—94no—
235#239—96no—
236#239—912no—
237#239—1816no—
238#239—1816no—
239#239C2×C3⋊D314yescomplemented
240D3×Q8—316no—
241D3×Q8—924no—
242D3×Q8—96no—
243D3×Q8C2×C3⋊D314yescomplemented
244#237—924no—
245#237—916no—
246#237—94no—
247#237—96no—
248D3×D4—948no—
249D3×D4—924no—
250D3×D4—948no—
251D3×D4—912no—
252D3×D4—96no—
253D3×D4—912no—
254#235—324no—
255#235—948no—
256#235—948no—
257#235—912no—
258#235—912no—
259#235C2×C3⋊D316yescomplemented
260C2×D12—948no—
261C2×D12—912no—
262C2×D12—93no—
263C2×D12—912no—
264C2×D12—1812no—
265C2×C4×D3—948no—
266C2×C4×D3—912no—
267C2×C4×D3—93no—
268C2×C4×D3—912no—
269C2×C4×D3—1812no—
270C22×C32—24no—
271C22×C32—34no—
272C22×C32C23×D314yes—
273C22×C32C23×D311yescharacteristic
274C2×C3⋊D3—36no—
275C2×C3⋊D3—324no—
276C2×C3⋊D3—624no—
277C2×C3⋊D3—1212no—
278C2×C3⋊D3—123no—
279C2×C3⋊D3—1212no—
280C2×C3⋊D3—1224no—
281C2×C3⋊D3C23×D314yes—
282C2×C3×D3—316no—
283C2×C3×D3C23×D3116yes—
284D32—1248no—
285C32×C4—224no—
286C32×C4—34no—
287C32×C4C23×D3124yes—
288C32×C4C23×D316yes—
289C3⋊Dic3—31no—
290C3⋊Dic3—31no—
291C3⋊Dic3—34no—
292C3⋊Dic3—34no—
293C3⋊Dic3—64no—
294C3⋊Dic3—64no—
295C3⋊Dic3C23×D314yes—
296C3×Dic3—316no—
297C3×Dic3C23×D3116yes—
2982−1+4—31no—
2992−1+4—91no—
3002−1+4—274no—
301H(5,𝔽2)—274no—
302H(5,𝔽2)—276no—
303C2×#41—274no—
304C2×#41—274no—
305C2×#41—276no—
306C2×#41—271no—
307C33#58511yescharacteristic, complemented, Sylow
308C22×D3—948no—
309C22×D3—912no—
310C22×D3—93no—
311C22×D3—912no—
312C22×D3—1812no—
313C3×Q8—216no—
314C3×Q8—324no—
315C3×Q8—96no—
316C3×Q8C2×D32116yes—
317C3×Q8C22×C3⋊D314yes—
318C3×D4—316no—
319C3×D4—324no—
320C3×D4—94no—
321C3×D4—96no—
322C2×C3×C4—212no—
323C2×C3×C4—348no—
324C2×C3×C4—912no—
325C2×C3×C4C2×D32112yes—
326C2×C3×C4C22×C3⋊D313yes—
327#67—316no—
328#67—948no—
329#67—916no—
330#67—94no—
331#67—912no—
332#67C22×C3⋊D314yes—
333C2×Dic3—916no—
334C2×Dic3—94no—
335C2×Dic3—91no—
336C2×Dic3—94no—
337C2×Dic3—184no—
338D12—324no—
339D12—948no—
340D12—912no—
341D12—948no—
342D12—912no—
343D12—1848no—
344D12C22×C3⋊D316yes—
345C4×D3—348no—
346C4×D3—948no—
347C4×D3—916no—
348C4×D3—94no—
349C4×D3—924no—
350C4×D3—924no—
351C4×D3—96no—
352C4×D3—96no—
353C4×D3—912no—
354C4×D3—1824no—
355C4×D3—1824no—
356C4×D3C22×C3⋊D3112yes—
357Dic6—324no—
358Dic6—916no—
359Dic6—94no—
360Dic6C22×C3⋊D316yes—
361C2×C32—28no—
362C2×C32—24no—
363C2×C32—21no—
364C2×C32—24no—
365C2×C32—64no—
366C2×C32C24×D314yes—
367C2×C32C24×D311yescharacteristic
368C3⋊D3—24no—
369C3⋊D3—66no—
370C3⋊D3—624no—
371C3⋊D3—1224no—
372C3×D3—216no—
373C3×D3—616no—
374#41—34no—
375#41—36no—
376#41—94no—
377#41—96no—
378#41—2712no—
379#41—2712no—
380#41—2712no—
381#41—274no—
382#41—276no—
383#41—276no—
384#41—274no—
385#41—274no—
386C2×Q8—34no—
387C2×Q8—94no—
388C2×Q8—276no—
389C2×Q8D3×C3⋊D311yescharacteristic, complemented
390C2×D4—2712no—
391C2×D4—2712no—
392C2×D4—2712no—
393C2×D4—276no—
394C2×D4—273no—
395C22×C4—2712no—
396C22×C4—273no—
397C22×C3—24no—
398C22×C3—316no—
399C22×C3—94no—
400C22×C3C22×D3214yes—
401C22×C3C23×C3⋊D311yescharacteristic
402C2×D3—316no—
403C2×D3—924no—
404C2×D3—96no—
405C2×D3—1824no—
406C2×D3—3612no—
407C2×D3—3648no—
408C2×D3—3612no—
409C2×D3—363no—
410C2×D3—3624no—
411C2×D3C23×C3⋊D314yes—
412C3×C4—224no—
413C3×C4—316no—
414C3×C4—94no—
415C3×C4C22×D32124yes—
416C3×C4C23×C3⋊D316yes—
417Dic3—316no—
418Dic3—94no—
419Dic3—94no—
420Dic3—91no—
421Dic3—91no—
422Dic3—184no—
423Dic3—184no—
424Dic3C23×C3⋊D314yes—
425C32—24no—
426C32#596814yescomplemented
427C32#596811yescharacteristic, complemented
428C23—2712no—
429C23—273no—
430Q8—36no—
431Q8—96no—
432Q8—274no—
433Q8C2×D3×C3⋊D314yes—
434D4—34no—
435D4—36no—
436D4—94no—
437D4—96no—
438D4—2712no—
439D4—2712no—
440D4—274no—
441D4—2712no—
442C2×C4—312no—
443C2×C4—912no—
444C2×C4—2712no—
445C2×C4—274no—
446C2×C4—274no—
447C2×C4—271no—
448C2×C4—276no—
449C2×C4—276no—
450C2×C4C2×D3×C3⋊D313yes—
451C2×C3—28no—
452C2×C3—24no—
453C2×C3—21no—
454C2×C3—24no—
455C2×C3—616no—
456C2×C3—184no—
457C2×C3C23×D3214yes—
458C2×C3C24×C3⋊D311yescharacteristic
459D3—24no—
460D3—616no—
461D3—1824no—
462D3—186no—
463D3—3624no—
464C22—34no—
465C22—94no—
466C22—276no—
467C22—1083no—
468C22—10812no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—34no—
471C4—94no—
472C4—271no—
473C4—271no—
474C4C22×D3×C3⋊D316yes—
475C3—24no—
476C3#1059668914yescomplemented
477C3#1059676011yescharacteristic, complemented
478C2—21no—
479C2—64no—
480C2—184no—
481C2—546no—
482C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×C3⋊D3
commutator, socle, cosocleC2×C33C25
FittingC2×C33×Q8C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#585
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#89080C2semidirect
2. C3×#64364C2semidirect
3. C32×#1007C2semidirect
4. #11724597C2semidirect
5. C3⋊D3×#239C2semidirect
6. D3×#3832C2semidirect
7. #11724591C2semidirect
8. #11724581C2semidirect
9. #11724578C2semidirect
12. C2×C33×Q8C22semidirect
13. #89080C22semidirect
14. #89080C22semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C2×C4×C3⋊(C3⋊D3)C22semidirect
17. C3×#3832C22semidirect
18. C3×Q8×C3⋊D3C22semidirect
19. C3×#3828C22semidirect
20. C32×#239C22semidirect
21. C32×D3×Q8C22semidirect
22. C32×#235C22semidirect
23. #89039C22semidirect
24. #89029C22semidirect
25. #89026C22semidirect
26. C3⋊D3×D12C22semidirect
27. D3×C4⋊(C3⋊D3)C22semidirect
28. C4×D3×C3⋊D3C22semidirect
29. #89022C22semidirect
30. #89020C22semidirect
31. #89019C22semidirect
32. #89018C22semidirect
33. #89017C22semidirect
34. #89015C22semidirect
41. #64364D3semidirect
47. C3×#1007D3semidirect
83. C2×C32×Q8C2×D3semidirect
84. C2×C32×Q8C2×D3semidirect
93. #3832C2×D3semidirect
95. Q8×C3⋊D3C2×D3semidirect
103. #3828C2×D3semidirect
113. C3×#239C2×D3semidirect
115. C3×D3×Q8C2×D3semidirect
117. C3×#235C2×D3semidirect
154. #1007C3⋊D3semidirect
226. C2×C3×Q8D32semidirect
227. C2×C3×Q8C2×C3⋊D3semidirect
239. #239C2×C3⋊D3semidirect
243. D3×Q8C2×C3⋊D3semidirect
259. #235C2×C3⋊D3semidirect
307. C33#585semidirect
389. C2×Q8D3×C3⋊D3semidirect
426. C32#5968semidirect
427. C32#5968semidirect
476. C3#10596689semidirect
477. C3#10596760semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
C3×#643642
C32×#10072
#117245972
C3⋊D3×#2392
D3×#38322
#117245912
#117245812
#117245782
#105967603
#105966893