Finite Group Databasebeta

← #13181633

Subgroups — #13181633

Order 1728.

5444 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724682C211yesmaximal, characteristic, complemented
2C2×C3×#3832C211yesmaximal, characteristic, complemented
3C32×#1007C211yesmaximal, characteristic, complemented
4#11724597C214yesmaximal, complemented
5#11724596C214yesmaximal, complemented
6D3×#3832C214yesmaximal, complemented
7#11724592C214yesmaximal, complemented
8#11724580C216yesmaximal, complemented
9C3⋊D3×#235C216yesmaximal, complemented
10#10596760—31nomaximal
11#10596688—34nomaximal
12C2×C33×Q8C2211yescharacteristic, complemented
13#89080C2214yescomplemented
14Q8×C3⋊(C3⋊D3)C2214yescomplemented
15#89076C2216yescomplemented
16C3×#3832C2214yescomplemented
17C3×#3832C2214yescomplemented
18C2×C3×C4⋊(C3⋊D3)C2213yescomplemented
19C2×C3×C4×C3⋊D3C2213yescomplemented
20C32×#239C2214yescomplemented
21C32×D3×Q8C2214yescomplemented
22C32×#235C2216yescomplemented
23C3⋊D3×#67C22112yescomplemented
24#89028C2214yescomplemented
25#89026C22112yescomplemented
26C3⋊D3×D12C2216yescomplemented
27D3×C4⋊(C3⋊D3)C22112yescomplemented
28C4×D3×C3⋊D3C22112yescomplemented
29#89021C22112yescomplemented
30#89020C22112yescomplemented
31#89018C2216yescomplemented
32#89017C22112yescomplemented
33C3⋊D3×Dic6C2216yescomplemented
34#89014C2216yescomplemented
35C32×2−1+4—31no—
36#64366—34no—
37#64366—36no—
38#41×C3⋊D3—34no—
39#41×C3⋊D3—36no—
40#64364—31no—
41#64364—34no—
42#64364—34no—
43#64364—64no—
44C2×#3832—34no—
45C2×#3832D311yescharacteristic, complemented
46C3×#1007D314yescomplemented
47C2×C3×#239—34no—
48#64320—316no—
49#64319—316no—
50D3×#239—316no—
51#64315—316no—
52#64307—324no—
53D3×#235—324no—
54C22×C3×C3⋊D3C2313yes—
55C2×D3×C3⋊D3C23112yes—
56C33×Q8C2314yes—
57C2×C33×C4C2313yes—
58#6312C2314yes—
59C4⋊(C3⋊(C3⋊D3))C2316yes—
60C4×C3⋊(C3⋊D3)C23112yes—
61#6308C2316yes—
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C3×C4⋊(C3⋊D3)C23112yes—
64C3×C4×C3⋊D3C2316yes—
65C3×C4×C3⋊D3C2316yes—
66C32×#67C2314yes—
67C32×D12C2316yes—
68C32×C4×D3C23112yes—
69C32×Dic6C2316yes—
70#6293C2314yes—
71#6292C23112yes—
72#6291C2314yes—
73#6290C23112yes—
74#6289C2314yes—
75Dic3×C3⋊D3C23112yes—
76D3×C3⋊Dic3C2314yes—
77#5968—94no—
78#5964—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—316no—
87#3832—312no—
88#3832—36no—
89#3832—616no—
90#3832C2×D314yescomplemented
91#3832C2×D314yescomplemented
92Q8×C3⋊D3—34no—
93Q8×C3⋊D3—316no—
94Q8×C3⋊D3—36no—
95Q8×C3⋊D3—616no—
96#3830—34no—
97#3830—36no—
98D4×C3⋊D3—312no—
99D4×C3⋊D3—312no—
100D4×C3⋊D3—36no—
101#3828—36no—
102#3828—324no—
103#3828—312no—
104#3828—312no—
105#3828—624no—
106C2×C4⋊(C3⋊D3)—312no—
107C2×C4⋊(C3⋊D3)C2×D313yescomplemented
108C2×C4×C3⋊D3—312no—
109C2×C4×C3⋊D3C2×D313yescomplemented
110C3×#239—316no—
111C3×#239—316no—
112C3×#239C2×D3116yescomplemented
113C3×D3×Q8C2×D3116yescomplemented
114C3×#235C2×D3124yescomplemented
115C2×C3×D12—312no—
116C2×C3×C4×D3—312no—
117D3×#67—348no—
118#3804—316no—
119#3802—348no—
120D3×D12—324no—
121D3×D12—348no—
122C4×D32—348no—
123#3799—324no—
124#3798—348no—
125#3796—348no—
126#3795—324no—
127#3795—348no—
128D3×Dic6—324no—
129C22×C33C2411yescharacteristic
130C2×C3⋊(C3⋊D3)C2414yes—
131C2×C3×C3⋊D3—43no—
132C2×C3×C3⋊D3C2416yes—
133C2×C32×D3C2414yes—
134D3×C3⋊D3—412no—
135C33×C4C2416yes—
136C3⋊(C3⋊Dic3)C2414yes—
137C3×C3⋊Dic3C2411yescharacteristic
138C3×C3⋊Dic3C2411yescharacteristic
139C32×Dic3C2414yes—
140C3×2−1+4—34no—
141C2×C3×#41—91no—
142#1010—94no—
143#1009—924no—
144#1009—916no—
145#1009—94no—
146D3×#41—924no—
147D3×#41—916no—
148D3×#41—94no—
149D3×#41—94no—
150#1007—916no—
151#1007—94no—
152#1007—91no—
153#1007—184no—
154#1007C3⋊D311yescharacteristic, complemented
155C2×#239—34no—
156C2×#239—916no—
157#1004—96no—
158C2×#235—96no—
159C22×C3⋊D3—312no—
160C22×C3⋊D3C22×D313yes—
161C22×C3×D3—312no—
162C2×D32—348no—
163C32×Q8—216no—
164C32×Q8—36no—
165C32×Q8C22×D3116yes—
166C32×Q8C22×D314yes—
167C32×D4—34no—
168C32×D4—36no—
169C2×C32×C4—212no—
170C2×C32×C4—312no—
171C2×C32×C4C22×D3112yes—
172C2×C32×C4C22×D313yes—
173#638—34no—
174#638—312no—
175#638—34no—
176#638—316no—
177#638—616no—
178C2×C3⋊Dic3—34no—
179C2×C3⋊Dic3C22×D311yescharacteristic
180C4⋊(C3⋊D3)—312no—
181C4⋊(C3⋊D3)—36no—
182C4⋊(C3⋊D3)—324no—
183C4⋊(C3⋊D3)—624no—
184C4⋊(C3⋊D3)C22×D3112yes—
185C4×C3⋊D3—312no—
186C4×C3⋊D3—312no—
187C4×C3⋊D3—34no—
188C4×C3⋊D3—348no—
189C4×C3⋊D3—648no—
190C4×C3⋊D3C22×D316yes—
191C4×C3⋊D3C22×D316yes—
192#634—36no—
193#634—34no—
194#634—324no—
195#634—624no—
196C3×#67C22×D3116yes—
197C2×C3×Dic3—34no—
198C3×D12—348no—
199C3×D12C22×D3124yes—
200C3×C4×D3—324no—
201C3×C4×D3—324no—
202C3×C4×D3C22×D3148yes—
203C3×Dic6C22×D3124yes—
204#627—316no—
205#626—316no—
206#626—348no—
207#625—348no—
208#624—316no—
209D3×Dic3—316no—
210D3×Dic3—348no—
211#585—271noSylow
212C2×C33—21no—
213C2×C33C2511yescharacteristic
214C3⋊(C3⋊D3)—24no—
215C3×C3⋊D3—26no—
216C32×D3—24no—
217C3×#41—316no—
218C3×#41—324no—
219C3×#41—94no—
220C3×#41—94no—
221C2×C3×Q8—24no—
222C2×C3×Q8—316no—
223C2×C3×Q8D3214yescomplemented
224C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
225C2×C3×D4—93no—
226C22×C3×C4—93no—
227C2×#67—912no—
228#239—316no—
229#239—316no—
230#239—924no—
231#239—948no—
232#239—916no—
233#239—94no—
234#239—1816no—
235#239C2×C3⋊D314yescomplemented
236D3×Q8—924no—
237D3×Q8—916no—
238D3×Q8—94no—
239D3×Q8—1816no—
240D3×Q8C2×C3⋊D314yescomplemented
241#237—924no—
242#237—916no—
243#237—912no—
244#237—912no—
245D3×D4—924no—
246D3×D4—948no—
247D3×D4—948no—
248D3×D4—912no—
249D3×D4—912no—
250#235—948no—
251#235—948no—
252#235—924no—
253#235—96no—
254#235—94no—
255#235—96no—
256#235—912no—
257#235—96no—
258#235—1824no—
259#235C2×C3⋊D316yescomplemented
260C2×D12—312no—
261C2×D12—948no—
262C2×D12—96no—
263C2×C4×D3—312no—
264C2×C4×D3—948no—
265C2×C4×D3—912no—
266C2×Dic6—96no—
267C22×C32—24no—
268C22×C32—34no—
269C22×C32C23×D314yes—
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—34no—
272C2×C3⋊D3—316no—
273C2×C3⋊D3—43no—
274C2×C3⋊D3—616no—
275C2×C3⋊D3—1212no—
276C2×C3⋊D3C23×D316yes—
277C2×C3×D3—324no—
278C2×C3×D3—1212no—
279C2×C3×D3C23×D3116yes—
280D32—1248no—
281C32×C4—224no—
282C32×C4—34no—
283C32×C4C23×D3124yes—
284C32×C4C23×D316yes—
285C3⋊Dic3—34no—
286C3⋊Dic3—316no—
287C3⋊Dic3—616no—
288C3⋊Dic3C23×D311yescharacteristic
289C3⋊Dic3C23×D311yescharacteristic
290C3×Dic3—34no—
291C3×Dic3—34no—
292C3×Dic3C23×D3116yes—
2932−1+4—31no—
2942−1+4—274no—
2952−1+4—271no—
296H(5,𝔽2)—276no—
297H(5,𝔽2)—274no—
298C2×#41—91no—
299C2×#41—276no—
300C2×#41—274no—
301C2×#41—274no—
302C33#58511yescharacteristic, complemented, Sylow
303C23×C3—93no—
304C22×D3—312no—
305C22×D3—948no—
306C22×D3—912no—
307C3×Q8—216no—
308C3×Q8—324no—
309C3×Q8C2×D32116yes—
310C3×Q8C22×C3⋊D314yes—
311C3×D4—316no—
312C3×D4—324no—
313C3×D4—912no—
314C2×C3×C4—212no—
315C2×C3×C4—348no—
316C2×C3×C4—91no—
317C2×C3×C4—96no—
318C2×C3×C4—96no—
319C2×C3×C4C2×D32112yes—
320C2×C3×C4C22×C3⋊D313yes—
321#67—916no—
322#67—948no—
323#67—916no—
324#67—94no—
325#67—912no—
326#67—912no—
327#67—1816no—
328#67C22×C3⋊D314yes—
329C2×Dic3—34no—
330C2×Dic3—916no—
331C2×Dic3—912no—
332D12—348no—
333D12—948no—
334D12—94no—
335D12—924no—
336D12—96no—
337D12—1824no—
338D12C22×C3⋊D316yes—
339C4×D3—324no—
340C4×D3—324no—
341C4×D3—916no—
342C4×D3—948no—
343C4×D3—94no—
344C4×D3—948no—
345C4×D3—912no—
346C4×D3—94no—
347C4×D3—1848no—
348C4×D3C22×C3⋊D3112yes—
349Dic6—916no—
350Dic6—924no—
351Dic6—96no—
352Dic6—94no—
353Dic6—1824no—
354Dic6C22×C3⋊D316yes—
355C2×C32—21no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—44no—
359C2×C32—64no—
360C2×C32C24×D314yes—
361C2×C32C24×D311yescharacteristic
362C3⋊D3—26no—
363C3⋊D3—64no—
364C3⋊D3—616no—
365C3⋊D3—1216no—
366C3×D3—216no—
367C3×D3—624no—
368#41—34no—
369#41—36no—
370#41—94no—
371#41—94no—
372#41—2712no—
373#41—2712no—
374#41—274no—
375#41—276no—
376#41—276no—
377#41—2712no—
378#41—274no—
379#41—276no—
380C2×Q8—34no—
381C2×Q8—276no—
382C2×Q8—274no—
383C2×Q8D3×C3⋊D311yescharacteristic, complemented
384C2×D4—93no—
385C2×D4—2712no—
386C2×D4—276no—
387C2×D4—2712no—
388C2×D4—2712no—
389C22×C4—93no—
390C22×C4—2712no—
391C22×C3—24no—
392C22×C3—316no—
393C22×C3—96no—
394C22×C3—363no—
395C22×C3C22×D3214yes—
396C22×C3C23×C3⋊D311yescharacteristic
397C2×D3—324no—
398C2×D3—916no—
399C2×D3—94no—
400C2×D3—1212no—
401C2×D3—1816no—
402C2×D3—3612no—
403C2×D3—3648no—
404C2×D3C23×C3⋊D314yes—
405C3×C4—224no—
406C3×C4—316no—
407C3×C4—91no—
408C3×C4—91no—
409C3×C4C22×D32124yes—
410C3×C4C23×C3⋊D316yes—
411Dic3—34no—
412Dic3—34no—
413Dic3—916no—
414Dic3—94no—
415Dic3—1816no—
416Dic3C23×C3⋊D314yes—
417C32—24no—
418C32#596411yescharacteristic, complemented
419C32#596814yescomplemented
420C23—93no—
421C23—2712no—
422Q8—36no—
423Q8—274no—
424Q8—276no—
425Q8C2×D3×C3⋊D314yes—
426D4—34no—
427D4—36no—
428D4—912no—
429D4—2712no—
430D4—274no—
431D4—2712no—
432D4—274no—
433D4—276no—
434C2×C4—312no—
435C2×C4—91no—
436C2×C4—96no—
437C2×C4—96no—
438C2×C4—274no—
439C2×C4—2712no—
440C2×C4—274no—
441C2×C4—2712no—
442C2×C4C2×D3×C3⋊D313yes—
443C2×C3—24no—
444C2×C3—21no—
445C2×C3—24no—
446C2×C3—44no—
447C2×C3—616no—
448C2×C3—186no—
449C2×C3C23×D3214yes—
450C2×C3C24×C3⋊D311yescharacteristic
451D3—24no—
452D3—624no—
453D3—1816no—
454D3—184no—
455D3—3616no—
456C22—34no—
457C22—96no—
458C22—274no—
459C22—363no—
460C22—10812no—
461C22C22×D3×C3⋊D311yescharacteristic
462C4—34no—
463C4—91no—
464C4—91no—
465C4—274no—
466C4C22×D3×C3⋊D316yes—
467C3—24no—
468C3#1059668814yescomplemented
469C3#1059676011yescharacteristic, complemented
470C2—21no—
471C2—64no—
472C2—186no—
473C2—544no—
474C2C23×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. #11724682C2semidirect
2. C2×C3×#3832C2semidirect
3. C32×#1007C2semidirect
4. #11724597C2semidirect
5. #11724596C2semidirect
6. D3×#3832C2semidirect
7. #11724592C2semidirect
8. #11724580C2semidirect
9. C3⋊D3×#235C2semidirect
12. C2×C33×Q8C22semidirect
13. #89080C22semidirect
14. Q8×C3⋊(C3⋊D3)C22semidirect
15. #89076C22semidirect
16. C3×#3832C22semidirect
17. C3×#3832C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C3×C4×C3⋊D3C22semidirect
20. C32×#239C22semidirect
21. C32×D3×Q8C22semidirect
22. C32×#235C22semidirect
23. C3⋊D3×#67C22semidirect
24. #89028C22semidirect
25. #89026C22semidirect
26. C3⋊D3×D12C22semidirect
27. D3×C4⋊(C3⋊D3)C22semidirect
28. C4×D3×C3⋊D3C22semidirect
29. #89021C22semidirect
30. #89020C22semidirect
31. #89018C22semidirect
32. #89017C22semidirect
33. C3⋊D3×Dic6C22semidirect
34. #89014C22semidirect
45. C2×#3832D3semidirect
46. C3×#1007D3semidirect
83. C2×C32×Q8C2×D3semidirect
84. C2×C32×Q8C2×D3semidirect
90. #3832C2×D3semidirect
91. #3832C2×D3semidirect
107. C2×C4⋊(C3⋊D3)C2×D3semidirect
109. C2×C4×C3⋊D3C2×D3semidirect
112. C3×#239C2×D3semidirect
113. C3×D3×Q8C2×D3semidirect
114. C3×#235C2×D3semidirect
154. #1007C3⋊D3semidirect
223. C2×C3×Q8D32semidirect
224. C2×C3×Q8C2×C3⋊D3semidirect
235. #239C2×C3⋊D3semidirect
240. D3×Q8C2×C3⋊D3semidirect
259. #235C2×C3⋊D3semidirect
302. C33#585semidirect
383. C2×Q8D3×C3⋊D3semidirect
418. C32#5964semidirect
419. C32#5968semidirect
468. C3#10596688semidirect
469. C3#10596760semidirect

Maximal subgroups

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

subgroupindex
#117246822
C2×C3×#38322
C32×#10072
#117245972
#117245962
D3×#38322
#117245922
#117245802
C3⋊D3×#2352
#105967603
#105966883