Finite Group Databasebeta

← #13181632

Subgroups — #13181632

Order 1728.

5388 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#64364C211yesmaximal, characteristic, complemented
3C32×#1007C211yesmaximal, characteristic, complemented
4#11724596C214yesmaximal, complemented
5D3×Q8×C3⋊D3C214yesmaximal, complemented
6#11724592C214yesmaximal, complemented
7#11724590C214yesmaximal, complemented
8#11724578C216yesmaximal, complemented
9#11724563C216yesmaximal, complemented
10C3⋊D3×2−1+4—31nomaximal
11#10596687—34nomaximal
12C2×C33×Q8C2211yescharacteristic, complemented
13Q8×C3⋊(C3⋊D3)C2214yes—
14Q8×C3⋊(C3⋊D3)C2214yescomplemented
15C2×C4×C3⋊(C3⋊D3)C2213yes—
16C2×#6308C2213yescomplemented
17C3×#3832C2214yescomplemented
18C3×Q8×C3⋊D3C2214yescomplemented
19C3×#3828C2216yescomplemented
20C32×#239C2214yescomplemented
21C32×D3×Q8C2214yescomplemented
22C32×#235C2216yescomplemented
23#89039C2214yes—
24#89029C22112yescomplemented
25#89026C2216yes—
26C4×D3×C3⋊D3C22112yes—
27#89021C22112yescomplemented
28#89020C22112yescomplemented
29#89019C2216yes—
30#89017C2216yescomplemented
31C3⋊D3×Dic6C22112yescomplemented
32#89015C2216yescomplemented
33#89014C22112yescomplemented
34D3×#634C22112yescomplemented
35C32×2−1+4—31no—
36#64367—34no—
37#64367—36no—
38#41×C3⋊D3—34no—
39#41×C3⋊D3—36no—
40#64364—34no—
41#64364D311yescharacteristic, complemented
42C2×Q8×C3⋊D3—31no—
43C2×Q8×C3⋊D3—34no—
44C2×Q8×C3⋊D3—34no—
45C2×Q8×C3⋊D3—64no—
46C3×#1007—34no—
47C3×#1007D314yescomplemented
48#64319—316no—
49D32×Q8—316no—
50#64315—316no—
51#64315—316no—
52#64306—324no—
53#64298—324no—
54C22×C3⋊(C3⋊D3)C2311yescharacteristic
55C2×D3×C3⋊D3C2314yes—
56C33×Q8C2314yes—
57C2×C33×C4C2313yes—
58C2×C3⋊(C3⋊Dic3)C2313yes—
59C4×C3⋊(C3⋊D3)C2316yes—
60C4×C3⋊(C3⋊D3)C2316yes—
61#6308C23112yes—
62C3×#638C2314yes—
63C3×C4⋊(C3⋊D3)C2316yes—
64C3×C4×C3⋊D3C23112yes—
65C3×#634C2316yes—
66C32×#67C2314yes—
67C32×D12C2316yes—
68C32×C4×D3C23112yes—
69C32×Dic6C2316yes—
70#6293C23112yes—
71#6292C2314yes—
72#6291C2314yes—
73#6290C23112yes—
74#6289C2314yes—
75Dic3×C3⋊D3C23112yes—
76D3×C3⋊Dic3C23112yes—
77D3×2−1+4—94no—
78D3×2−1+4—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—36no—
86#3832C2×D314yescomplemented
87Q8×C3⋊D3—34no—
88Q8×C3⋊D3—34no—
89Q8×C3⋊D3—316no—
90Q8×C3⋊D3—316no—
91Q8×C3⋊D3—312no—
92Q8×C3⋊D3—36no—
93Q8×C3⋊D3—616no—
94Q8×C3⋊D3—616no—
95Q8×C3⋊D3C2×D314yescomplemented
96#3830—312no—
97#3830—312no—
98#3830—36no—
99D4×C3⋊D3—34no—
100D4×C3⋊D3—36no—
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×#634—33no—
109C2×#634—312no—
110C2×#634—312no—
111C2×#634—612no—
112C3×#239—316no—
113C3×#239C2×D3116yescomplemented
114C3×D3×Q8—316no—
115C3×D3×Q8C2×D3116yescomplemented
116C3×#235—324no—
117C3×#235C2×D3124yescomplemented
118#3811—316no—
119#3805—348no—
120#3802—324no—
121C4×D32—348no—
122#3798—348no—
123#3797—324no—
124#3796—324no—
125#3796—324no—
126#3795—348no—
127#3795—348no—
128D3×Dic6—348no—
129D3×Dic6—348no—
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)C2411yescharacteristic
133C2×C3⋊(C3⋊D3)C2411yescharacteristic
134C2×C3×C3⋊D3C2414yes—
135C2×C32×D3C2414yes—
136D3×C3⋊D3—28no—
137C33×C4C2416yes—
138C3⋊(C3⋊Dic3)C2416yes—
139C3×C3⋊Dic3C2414yes—
140C32×Dic3C2414yes—
141C3×2−1+4—34no—
142C3×2−1+4—91no—
143#1010—924no—
144#1010—916no—
145#1010—94no—
146#1010—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×D3×Q8—916no—
156C2×D3×Q8—94no—
157C2×D3×Q8—91no—
158C2×D3×Q8—94no—
159C2×D3×Q8—184no—
160C22×C3⋊D3—31no—
161C22×C3⋊D3—34no—
162C22×C3⋊D3—34no—
163C22×C3⋊D3—64no—
164C2×D32—316no—
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—312no—
176#638—34no—
177#638C22×D314yes—
178C2×C3⋊Dic3—33no—
179C2×C3⋊Dic3—312no—
180C2×C3⋊Dic3—312no—
181C2×C3⋊Dic3—612no—
182C4⋊(C3⋊D3)—34no—
183C4⋊(C3⋊D3)C22×D316yes—
184C4×C3⋊D3—312no—
185C4×C3⋊D3—36no—
186C4×C3⋊D3—36no—
187C4×C3⋊D3—34no—
188C4×C3⋊D3—324no—
189C4×C3⋊D3—324no—
190C4×C3⋊D3—624no—
191C4×C3⋊D3—624no—
192C4×C3⋊D3C22×D3112yes—
193#634—312no—
194#634—312no—
195#634—348no—
196#634—648no—
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—348no—
207#626—316no—
208#626—316no—
209#625—348no—
210#624—316no—
211D3×Dic3—348no—
212D3×Dic3—348no—
213C2×2−1+4—271noSylow
214C2×C33—21no—
215C2×C33C2511yescharacteristic
216C3⋊(C3⋊D3)—21no—
217C3⋊(C3⋊D3)2−1+412yescomplemented
218C3×C3⋊D3—24no—
219C32×D3—24no—
220C3×#41—316no—
221C3×#41—324no—
222C3×#41—94no—
223C3×#41—96no—
224C2×C3×Q8—24no—
225C2×C3×Q8—316no—
226C2×C3×Q8—94no—
227C2×C3×Q8D3214yescomplemented
228C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
229#239—316no—
230#239—924no—
231#239—96no—
232#239C2×C3⋊D314yescomplemented
233D3×Q8—316no—
234D3×Q8—924no—
235D3×Q8—948no—
236D3×Q8—916no—
237D3×Q8—916no—
238D3×Q8—94no—
239D3×Q8—94no—
240D3×Q8—912no—
241D3×Q8—96no—
242D3×Q8—1816no—
243D3×Q8—1816no—
244D3×Q8C2×C3⋊D314yescomplemented
245#237—924no—
246#237—948no—
247#237—948no—
248#237—912no—
249#237—912no—
250#237—96no—
251D3×D4—924no—
252D3×D4—916no—
253D3×D4—94no—
254D3×D4—96no—
255#235—324no—
256#235—948no—
257#235—948no—
258#235—912no—
259#235—912no—
260#235C2×C3⋊D316yescomplemented
261C2×C4×D3—948no—
262C2×C4×D3—912no—
263C2×C4×D3—93no—
264C2×C4×D3—912no—
265C2×C4×D3—1812no—
266C2×Dic6—948no—
267C2×Dic6—912no—
268C2×Dic6—93no—
269C2×Dic6—912no—
270C2×Dic6—1812no—
271C22×C32—24no—
272C22×C32—34no—
273C22×C32C23×D314yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—31no—
276C2×C3⋊D3—31no—
277C2×C3⋊D3—34no—
278C2×C3⋊D3—34no—
279C2×C3⋊D3—64no—
280C2×C3⋊D3—64no—
281C2×C3⋊D3—68no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—68no—
284C2×C3⋊D3—616no—
285C2×C3⋊D3C23×D314yes—
286C2×C3×D3—316no—
287C2×C3×D3C23×D3116yes—
288D32—632no—
289C32×C4—224no—
290C32×C4—34no—
291C32×C4C23×D3124yes—
292C32×C4C23×D316yes—
293C3⋊Dic3—36no—
294C3⋊Dic3—324no—
295C3⋊Dic3—624no—
296C3⋊Dic3C23×D314yes—
297C3×Dic3—316no—
298C3×Dic3C23×D3116yes—
2992−1+4—31no—
3002−1+4—91no—
3012−1+4—274no—
3022−1+4—274no—
3032−1+4—276no—
304C2×#41—274no—
305C2×#41—276no—
306C22×Q8—274no—
307C22×Q8—271no—
308C33C2×2−1+411yescharacteristic, complemented, Sylow
309C22×D3—916no—
310C22×D3—94no—
311C22×D3—91no—
312C22×D3—94no—
313C22×D3—184no—
314C3×Q8—216no—
315C3×Q8—324no—
316C3×Q8—96no—
317C3×Q8C2×D32116yes—
318C3×Q8C22×C3⋊D314yes—
319C3×D4—316no—
320C3×D4—324no—
321C3×D4—94no—
322C3×D4—96no—
323C2×C3×C4—212no—
324C2×C3×C4—348no—
325C2×C3×C4—912no—
326C2×C3×C4C2×D32112yes—
327C2×C3×C4C22×C3⋊D313yes—
328#67—316no—
329#67—916no—
330#67—948no—
331#67—912no—
332#67—94no—
333#67C22×C3⋊D314yes—
334C2×Dic3—948no—
335C2×Dic3—912no—
336C2×Dic3—93no—
337C2×Dic3—912no—
338C2×Dic3—1812no—
339D12—324no—
340D12—916no—
341D12—94no—
342D12C22×C3⋊D316yes—
343C4×D3—348no—
344C4×D3—916no—
345C4×D3—948no—
346C4×D3—912no—
347C4×D3—924no—
348C4×D3—924no—
349C4×D3—96no—
350C4×D3—96no—
351C4×D3—94no—
352C4×D3—1824no—
353C4×D3—1824no—
354C4×D3C22×C3⋊D3112yes—
355Dic6—324no—
356Dic6—948no—
357Dic6—948no—
358Dic6—912no—
359Dic6—912no—
360Dic6—1848no—
361Dic6C22×C3⋊D316yes—
362C2×C32—28no—
363C2×C32—24no—
364C2×C32—21no—
365C2×C32—24no—
366C2×C32—64no—
367C2×C32C24×D314yes—
368C2×C32C24×D311yescharacteristic
369C3⋊D3—24no—
370C3⋊D3—32no—
371C3⋊D3—38no—
372C3⋊D3—68no—
373C3⋊D3—68no—
374C3⋊D3—64no—
375C3⋊D3—61no—
376C3×D3—216no—
377C3×D3—616no—
378#41—34no—
379#41—36no—
380#41—94no—
381#41—96no—
382#41—2712no—
383#41—2712no—
384#41—2712no—
385#41—2712no—
386#41—276no—
387#41—276no—
388C2×Q8—34no—
389C2×Q8—94no—
390C2×Q8—2712no—
391C2×Q8—2712no—
392C2×Q8—276no—
393C2×Q8—274no—
394C2×Q8—274no—
395C2×Q8—273no—
396C2×Q8D3×C3⋊D311yescharacteristic, complemented
397C2×D4—274no—
398C2×D4—276no—
399C22×C4—2712no—
400C22×C4—273no—
401C22×C3—24no—
402C22×C3—316no—
403C22×C3—94no—
404C22×C3C22×D3214yes—
405C22×C3C23×C3⋊D311yescharacteristic
406C2×D3—316no—
407C2×D3—94no—
408C2×D3—91no—
409C2×D3—94no—
410C2×D3—91no—
411C2×D3—184no—
412C2×D3—184no—
413C2×D3—188no—
414C2×D3—1832no—
415C2×D3—188no—
416C2×D3—182no—
417C2×D3—1816no—
418C2×D3C23×C3⋊D314yes—
419C3×C4—224no—
420C3×C4—316no—
421C3×C4—94no—
422C3×C4C22×D32124yes—
423C3×C4C23×C3⋊D316yes—
424Dic3—316no—
425Dic3—924no—
426Dic3—96no—
427Dic3—1824no—
428Dic3C23×C3⋊D314yes—
429C32—24no—
430C32D3×2−1+414yescomplemented
431C32D3×2−1+411yescharacteristic, complemented
432C23—274no—
433C23—271no—
434Q8—36no—
435Q8—96no—
436Q8—2712no—
437Q8—2712no—
438Q8C2×D3×C3⋊D314yes—
439D4—34no—
440D4—36no—
441D4—94no—
442D4—96no—
443D4—274no—
444D4—274no—
445D4—2712no—
446C2×C4—312no—
447C2×C4—912no—
448C2×C4—274no—
449C2×C4—2712no—
450C2×C4—2712no—
451C2×C4—273no—
452C2×C4—276no—
453C2×C4—276no—
454C2×C4C2×D3×C3⋊D313yes—
455C2×C3—28no—
456C2×C3—24no—
457C2×C3—21no—
458C2×C3—24no—
459C2×C3—616no—
460C2×C3—184no—
461C2×C3C23×D3214yes—
462C2×C3C24×C3⋊D311yescharacteristic
463D3—24no—
464D3—616no—
465D3—98no—
466D3—92no—
467D3—188no—
468D3—188no—
469D3—184no—
470D3—181no—
471C22—34no—
472C22—94no—
473C22—271no—
474C22—271no—
475C22—542no—
476C22—548no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—34no—
479C4—94no—
480C4—276no—
481C4C22×D3×C3⋊D316yes—
482C3—24no—
483C3#1059668714yescomplemented
484C3C3⋊D3×2−1+411yescharacteristic, complemented
485C2—21no—
486C2—64no—
487C2—184no—
488C2—272no—
489C2—541no—
490C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×C3⋊D3
commutator, socle, cosocleC2×C33C25
FittingC2×C33×Q8C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C2×2−1+4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊(C3⋊D3)C2semidirect
2. C3×#64364C2semidirect
3. C32×#1007C2semidirect
4. #11724596C2semidirect
5. D3×Q8×C3⋊D3C2semidirect
6. #11724592C2semidirect
7. #11724590C2semidirect
8. #11724578C2semidirect
9. #11724563C2semidirect
12. C2×C33×Q8C22semidirect
14. Q8×C3⋊(C3⋊D3)C22semidirect
16. C2×#6308C22semidirect
17. C3×#3832C22semidirect
18. C3×Q8×C3⋊D3C22semidirect
19. C3×#3828C22semidirect
20. C32×#239C22semidirect
21. C32×D3×Q8C22semidirect
22. C32×#235C22semidirect
24. #89029C22semidirect
27. #89021C22semidirect
28. #89020C22semidirect
30. #89017C22semidirect
31. C3⋊D3×Dic6C22semidirect
32. #89015C22semidirect
33. #89014C22semidirect
34. D3×#634C22semidirect
41. #64364D3semidirect
47. C3×#1007D3semidirect
83. C2×C32×Q8C2×D3semidirect
84. C2×C32×Q8C2×D3semidirect
86. #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
217. C3⋊(C3⋊D3)2−1+4semidirect
227. C2×C3×Q8D32semidirect
228. C2×C3×Q8C2×C3⋊D3semidirect
232. #239C2×C3⋊D3semidirect
244. D3×Q8C2×C3⋊D3semidirect
260. #235C2×C3⋊D3semidirect
308. C33C2×2−1+4semidirect
396. C2×Q8D3×C3⋊D3semidirect
430. C32D3×2−1+4semidirect
431. C32D3×2−1+4semidirect
483. C3#10596687semidirect
484. C3C3⋊D3×2−1+4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊(C3⋊D3)2
C3×#643642
C32×#10072
#117245962
D3×Q8×C3⋊D32
#117245922
#117245902
#117245782
#117245632
C3⋊D3×2−1+43
#105966873