Finite Group Databasebeta

← #13152093

Subgroups — #13152093

Order 1728.

2447 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 2447 conjugacy classes of subgroups — too many to draw. Only its 215 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#88676C213yesmaximal, complemented
3#11721876C211yesmaximal, characteristic, complemented
4#11721702C211yesmaximal, characteristic, complemented
5#11721321C213yesmaximal, complemented
6#11721316C213yesmaximal, complemented
7#11721310C213yesmaximal, complemented
8#10595527—31nomaximal
9D3×#953—31nomaximal
10C24×H(3,𝔽3)C2211yescharacteristic, complemented
11C23×(C3⋊D3)⋊C3—212no—
12C23×(C3⋊D3)⋊C3C2213yes—
13C23×(C3⋊D3)⋊C3C2213yescomplemented
14C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic, complemented
15C2×#6235C2213yescomplemented
16C2×#638⋊C3C2213yescomplemented
17#88676—212no—
18C2×#6200C2213yescomplemented
19C2×#6198C2213yescomplemented
20C2×((C3⋊D3)⋊C3)⋊C4C2213yescomplemented
21#88552C2213yescomplemented
22#3757⋊C3C2213yescomplemented
23#88450C2216yescomplemented
24C24×C3⋊D3D311yescharacteristic, complemented
25C24×C3×D3—31no—
26C2×#3811—33no—
27C2×D3×#67—33no—
28#64162D311yescharacteristic, complemented
29C3×#953—31no—
30C3×#953—31no—
31C3×#953—31no—
32C3×#953—61no—
33#63981—33no—
34#63981—33no—
35#63977—33no—
36#63976—33no—
37#63974—33no—
38D3×#217—33no—
39C23×H(3,𝔽3)—26no—
40C23×H(3,𝔽3)C2313yes—
41C22×#272C2311yescharacteristic, complemented
42C22×(C3⋊D3)⋊C3—232no—
43C22×(C3⋊D3)⋊C3—26no—
44C22×(C3⋊D3)⋊C3—212no—
45C22×(C3⋊D3)⋊C3—212no—
46C22×(C3⋊D3)⋊C3D4112yescomplemented
47C22×(C3⋊D3)⋊C3C2313yes—
48C22×(C3⋊D3)⋊C3C2311yescharacteristic
49C22×(C3⋊D3)⋊C3C2311yescharacteristic, complemented
50C2×((C3⋊D3)⋊C3)⋊C2—26no—
51#6235—26no—
52C2×#1084C2313yescomplemented
53#638⋊C3—26no—
54C2×(C3⋊Dic3)⋊C3C2313yescomplemented
55#6200—26no—
56#6198—26no—
57((C3⋊D3)⋊C3)⋊C4—26no—
58C2×#953—91no—
59D3×#120—91no—
60D3×#120—91no—
61C24×C32—21no—
62C24×C32C2×D311yescharacteristic, complemented
63C24×C32C2×D311yescharacteristic, complemented
64C23×C3⋊D3—212no—
65C23×C3⋊D3C2×D313yes—
66C23×C3⋊D3C2×D313yescomplemented
67C23×C3×D3—33no—
68C23×C3×D3—33no—
69C23×C3×D3—612no—
70C22×D32—31no—
71C22×D32—31no—
72C2×#638C2×D313yescomplemented
73C2×C3×#67—33no—
74C2×C3×#67—33no—
75C2×C3×#67—33no—
76C2×C3×#67—63no—
77#3811—612no—
78D3×#67—612no—
79C2×#626—33no—
80C2×#626—33no—
81C2×#626—33no—
82C2×#625—33no—
83C2×#624—33no—
84C2×D3×Dic3—33no—
85#3757C2×D313yescomplemented
86C3×#217—33no—
87C3×#217—33no—
88C3×#217—33no—
89C3×#217—63no—
90#3722—36no—
91(C2×D3)⋊Dic3—36no—
92C22×H(3,𝔽3)—28no—
93C22×H(3,𝔽3)—26no—
94C22×H(3,𝔽3)C2×D416yescomplemented
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×#272—23no—
97C2×(C3⋊D3)⋊C3—224no—
98C2×(C3⋊D3)⋊C3—26no—
99C2×(C3⋊D3)⋊C3—224no—
100C2×(C3⋊D3)⋊C3—23no—
101C2×(C3⋊D3)⋊C3C2×D416yes—
102C2×(C3⋊D3)⋊C3C2×D416yescomplemented
103((C3⋊D3)⋊C3)⋊C2—44no—
104#1084—23no—
105(C3⋊Dic3)⋊C3—23no—
106C25×C3—91no—
107C24×D3—31no—
108C24×D3—91no—
109C22×#67—93no—
110C2×D3×D4—93no—
111C2×D3×D4—93no—
112C3×#120—91no—
113C3×#120—91no—
114#953—31no—
115#953—31no—
116#953—31no—
117#953—61no—
118#953—91no—
119#953—93no—
120#953—93no—
121C2×#217—93no—
122#937—93no—
123#937—93no—
124#882—93no—
125#882—93no—
126D3×C22⋊C4—93no—
127D3×C22⋊C4—93no—
128C23×C32—26no—
129C23×C32—26no—
130C23×C32—23no—
131C23×C32—46no—
132C23×C32C22×D313yes—
133C23×C32C22×D313yes—
134C22×C3⋊D3—232no—
135C22×C3⋊D3—26no—
136C22×C3⋊D3—212no—
137C22×C3⋊D3—212no—
138C22×C3⋊D3#67112yescomplemented
139C22×C3⋊D3C22×D313yes—
140C22×C3⋊D3C22×D311yescharacteristic
141C22×C3⋊D3C22×D311yescharacteristic, complemented
142C22×C3×D3—31no—
143C22×C3×D3—312no—
144C22×C3×D3—33no—
145C22×C3×D3—31no—
146C22×C3×D3—31no—
147C22×C3×D3—31no—
148C22×C3×D3—61no—
149C22×C3×D3—612no—
150C22×C3×D3—612no—
151C22×C3×D3—632no—
152C22×C3×D3—66no—
153C2×D32—66no—
154C2×D32—66no—
155#638—26no—
156C2×C3⋊Dic3C22×D313yescomplemented
157C3×#67—66no—
158C3×#67—66no—
159C3×#67—66no—
160C3×#67—126no—
161C2×C3×Dic3—33no—
162C2×C3×Dic3—33no—
163C2×C3×Dic3—33no—
164C2×C3×Dic3—63no—
165#626—66no—
166#626—66no—
167#626—66no—
168#625—66no—
169#624—66no—
170D3×Dic3—66no—
171C2×#120—271noSylow
172C2×H(3,𝔽3)—26no—
173C2×H(3,𝔽3)C22×D413yes—
174#272—41no—
175(C3⋊D3)⋊C3—26no—
176(C3⋊D3)⋊C3—41no—
177(C3⋊D3)⋊C3#12014yescomplemented
178C24×C3—31no—
179C24×C3—31no—
180C24×C3—61no—
181C24×C3—93no—
182C24×C3—93no—
183C24×C3—1812no—
184C24×C3D3211yescharacteristic
185C23×D3—33no—
186C23×D3—33no—
187C23×D3—612no—
188C23×D3—93no—
189C23×D3—93no—
190C23×D3—91no—
191C23×D3—91no—
192C23×D3—91no—
193C23×D3—1812no—
194C2×C3×D4—93no—
195C2×C3×D4—93no—
196C2×#67—33no—
197C2×#67—33no—
198C2×#67—33no—
199C2×#67—63no—
200C2×#67—93no—
201C2×#67—93no—
202C2×#67—93no—
203C2×#67—93no—
204C2×#67—93no—
205C2×#67—1812no—
206C22×Dic3—93no—
207D3×D4—1812no—
208D3×D4—1812no—
209C2×D12—93no—
210C2×D12—93no—
211C2×C4×D3—93no—
212C2×C4×D3—93no—
213C3×C22⋊C4—93no—
214C3×C22⋊C4—93no—
215#217—33no—
216#217—33no—
217#217—33no—
218#217—63no—
219#217—93no—
220#217—96no—
221(C2×D3)⋊C4—96no—
222(C2×D3)⋊C4—96no—
223C22×C32—28no—
224C22×C32—28no—
225C22×C32—26no—
226C22×C32—26no—
227C22×C32—21no—
228C22×C32—26no—
229C22×C32—46no—
230C22×C32—48no—
231C22×C32D3×D416yescomplemented
232C22×C32C2×#6716yescomplemented
233C22×C32C23×D311yescharacteristic
234C22×C32C23×D311yescharacteristic
235C2×C3⋊D3—224no—
236C2×C3⋊D3—26no—
237C2×C3⋊D3—224no—
238C2×C3⋊D3—23no—
239C2×C3⋊D3C2×#6716yes—
240C2×C3⋊D3C2×#6716yescomplemented
241C2×C3×D3—36no—
242C2×C3×D3—36no—
243C2×C3×D3—624no—
244C2×C3×D3—624no—
245C2×C3×D3—66no—
246C2×C3×D3—63no—
247C2×C3×D3—63no—
248C2×C3×D3—63no—
249C2×C3×D3—123no—
250D32—124no—
251D32—124no—
252C3⋊Dic3—23no—
253C3×Dic3—63no—
254C3×Dic3—63no—
255C3×Dic3—63no—
256C3×Dic3—123no—
257C25—91no—
258C22×D4—273no—
259#120—91no—
260#120—91no—
261#120—273no—
262#120—273no—
263C2×C22⋊C4—273no—
264H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
265C23×C3—26no—
266C23×C3—33no—
267C23×C3—33no—
268C23×C3—66no—
269C23×C3—66no—
270C23×C3—63no—
271C23×C3—91no—
272C23×C3—93no—
273C23×C3—912no—
274C23×C3—91no—
275C23×C3—91no—
276C23×C3—126no—
277C23×C3—186no—
278C23×C3—1812no—
279C23×C3—1812no—
280C23×C3—1832no—
281C23×C3C2×D3213yes—
282C22×D3—31no—
283C22×D3—33no—
284C22×D3—312no—
285C22×D3—31no—
286C22×D3—31no—
287C22×D3—31no—
288C22×D3—61no—
289C22×D3—612no—
290C22×D3—612no—
291C22×D3—632no—
292C22×D3—66no—
293C22×D3—91no—
294C22×D3—912no—
295C22×D3—93no—
296C22×D3—91no—
297C22×D3—1812no—
298C22×D3—1812no—
299C22×D3—186no—
300C22×D3—1832no—
301C22×D3—186no—
302C22×D3—186no—
303C22×D3—186no—
304C3×D4—186no—
305C3×D4—186no—
306C2×C3×C4—93no—
307C2×C3×C4—93no—
308#67—66no—
309#67—66no—
310#67—66no—
311#67—126no—
312#67—186no—
313#67—186no—
314#67—186no—
315#67—186no—
316#67—186no—
317C2×Dic3—33no—
318C2×Dic3—33no—
319C2×Dic3—33no—
320C2×Dic3—63no—
321C2×Dic3—93no—
322C2×Dic3—186no—
323D12—186no—
324D12—186no—
325C4×D3—186no—
326C4×D3—186no—
327C2×C32—26no—
328C2×C32—26no—
329C2×C32—23no—
330C2×C32—46no—
331C2×C32C2×D3×D413yes—
332C2×C32C22×#6713yes—
333C3⋊D3—26no—
334C3⋊D3—41no—
335C3⋊D3#95314yescomplemented
336C3×D3—34no—
337C3×D3—66no—
338C3×D3—121no—
339C3×D3—121no—
340C3×D3—121no—
341C3×D3—241no—
342C24—93no—
343C24—93no—
344C24—1812no—
345C24—271no—
346C24((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
347C2×D4—93no—
348C2×D4—93no—
349C2×D4—273no—
350C2×D4—273no—
351C2×D4—5412no—
352C22×C4—273no—
353C22⋊C4—93no—
354C22⋊C4—93no—
355C22⋊C4—276no—
356C22×C3—28no—
357C22×C3—26no—
358C22×C3—31no—
359C22×C3—31no—
360C22×C3—36no—
361C22×C3—36no—
362C22×C3—66no—
363C22×C3—66no—
364C22×C3—68no—
365C22×C3—68no—
366C22×C3—61no—
367C22×C3—66no—
368C22×C3—96no—
369C22×C3—96no—
370C22×C3—126no—
371C22×C3—128no—
372C22×C3—186no—
373C22×C3—183no—
374C22×C3—1824no—
375C22×C3—1824no—
376C22×C3—183no—
377C22×C3D3×#6716yes—
378C22×C3C22×D3211yescharacteristic
379C2×D3—36no—
380C2×D3—36no—
381C2×D3—624no—
382C2×D3—624no—
383C2×D3—66no—
384C2×D3—63no—
385C2×D3—63no—
386C2×D3—63no—
387C2×D3—96no—
388C2×D3—96no—
389C2×D3—123no—
390C2×D3—1824no—
391C2×D3—186no—
392C2×D3—1824no—
393C2×D3—183no—
394C2×D3—364no—
395C2×D3—364no—
396C2×D3—364no—
397C3×C4—183no—
398C3×C4—183no—
399Dic3—63no—
400Dic3—63no—
401Dic3—63no—
402Dic3—123no—
403Dic3—183no—
404C32—21no—
405C32D3×#12011yescharacteristic, complemented
406C32C2×#95311yescharacteristic, complemented
407C23—26no—
408C23—91no—
409C23—93no—
410C23—912no—
411C23—91no—
412C23—91no—
413C23—1812no—
414C23—1812no—
415C23—1832no—
416C23—186no—
417C23—546no—
418C23C2×((C3⋊D3)⋊C3)⋊C213yes—
419D4—186no—
420D4—186no—
421D4—546no—
422D4—546no—
423C2×C4—93no—
424C2×C4—93no—
425C2×C4—546no—
426C2×C3—26no—
427C2×C3—33no—
428C2×C3—33no—
429C2×C3—66no—
430C2×C3—66no—
431C2×C3—63no—
432C2×C3—94no—
433C2×C3—126no—
434C2×C3—186no—
435C2×C3—361no—
436C2×C3—361no—
437C2×C3C2×D3×#6713yes—
438D3—34no—
439D3—66no—
440D3—94no—
441D3—121no—
442D3—121no—
443D3—121no—
444D3—186no—
445D3—241no—
446D3—361no—
447C22—28no—
448C22—26no—
449C22—96no—
450C22—96no—
451C22—1824no—
452C22—1824no—
453C22—186no—
454C22—183no—
455C22—183no—
456C22—1084no—
457C22#8867616yescomplemented
458C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
459C4—183no—
460C4—183no—
461C3—31no—
462C3—31no—
463C3—61no—
464C3D3×#95311yescharacteristic
465C2—26no—
466C2—94no—
467C2—186no—
468C2—361no—
469C2—361no—
470C2C2×#8867613yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#120
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×(C3⋊D3)⋊C3C2semidirect
2. C2×#88676C2semidirect
3. #11721876C2semidirect
4. #11721702C2semidirect
5. #11721321C2semidirect
6. #11721316C2semidirect
7. #11721310C2semidirect
10. C24×H(3,𝔽3)C22semidirect
13. C23×(C3⋊D3)⋊C3C22semidirect
14. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
15. C2×#6235C22semidirect
16. C2×#638⋊C3C22semidirect
18. C2×#6200C22semidirect
19. C2×#6198C22semidirect
20. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
21. #88552C22semidirect
22. #3757⋊C3C22semidirect
23. #88450C22semidirect
24. C24×C3⋊D3D3semidirect
28. #64162D3semidirect
41. C22×#272C23semidirect
46. C22×(C3⋊D3)⋊C3D4semidirect
49. C22×(C3⋊D3)⋊C3C23semidirect
52. C2×#1084C23semidirect
54. C2×(C3⋊Dic3)⋊C3C23semidirect
62. C24×C32C2×D3semidirect
63. C24×C32C2×D3semidirect
66. C23×C3⋊D3C2×D3semidirect
72. C2×#638C2×D3semidirect
85. #3757C2×D3semidirect
94. C22×H(3,𝔽3)C2×D4semidirect
102. C2×(C3⋊D3)⋊C3C2×D4semidirect
138. C22×C3⋊D3#67semidirect
141. C22×C3⋊D3C22×D3semidirect
156. C2×C3⋊Dic3C22×D3semidirect
177. (C3⋊D3)⋊C3#120semidirect
231. C22×C32D3×D4semidirect
232. C22×C32C2×#67semidirect
240. C2×C3⋊D3C2×#67semidirect
264. H(3,𝔽3)C2×#120semidirect
335. C3⋊D3#953semidirect
346. C24((C3⋊D3)⋊C3)⋊C2semidirect
405. C32D3×#120semidirect
406. C32C2×#953semidirect
457. C22#88676semidirect

Maximal subgroups

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

subgroupindex
C24×(C3⋊D3)⋊C32
C2×#886762
#117218762
#117217022
#117213212
#117213162
#117213102
#105955273
D3×#9533