Finite Group Databasebeta

← #13159936

Subgroups — #13159936

Order 1728.

1040 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C2×C3×#3745C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic, complemented
3Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592542—31nomaximal
9#10590753—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6292—21no—
14C2×#6292—21no—
15C2×Dic3×C3⋊D3—21no—
16C2×C33×C8C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C3×#3745—22no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88803—21no—
22#88791—22no—
23C2×#3745—31no—
24C2×#3745D311yescharacteristic, complemented
25#64083—31no—
26C4×C4⋊(C3⋊D3)—31no—
27C2×C3×D24—34no—
28#63911—34no—
29Dic3×D12—34no—
30C2×#3714—34no—
31C32×C4⋊C8—31no—
32#63648—31no—
33#63648—31no—
34#63648—34no—
35#63648—64no—
36C3×Dic3⋊C8D314yescomplemented
37#63563—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)D412yescomplemented
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51#6292—42no—
52#6292—42no—
53Dic3×C3⋊D3—42no—
54C33×C8—21no—
55C32×C3⋊C8—21no—
56#5150—91no—
57#4843—94no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4⋊(C3⋊D3)—61no—
62C2×C4⋊(C3⋊D3)—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—61no—
66C2×C3×D12—34no—
67C2×C3×D12—34no—
68C2×#626—64no—
69C2×#626—64no—
70C2×D3×Dic3—64no—
71C2×C32×C8—24no—
72C2×C32×C8—31no—
73C2×C32×C8C2×D314yes—
74C2×C32×C8C2×D311yescharacteristic
75C32×C42—31no—
76#3752—61no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—34no—
79C4⋊(C3⋊Dic3)—64no—
80#3745—22no—
81#3745—62no—
82C3×C4×Dic3C2×D314yes—
83C2×C3×C3⋊C8C2×D314yes—
84C3×D24—68no—
85(C2×D3)⋊Dic3—64no—
86#3714—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3⋊(C3⋊D3)—41no—
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C32×Dic3—21no—
101C32×Dic3D812yescomplemented
102C2×C3×D8—91no—
103#940—91no—
104D4×Dic3—91no—
105C2×#213—91no—
106C2×D24—34no—
107C2×D24—94no—
108#874—94no—
109C4×D12—94no—
110C3×C4⋊C8—34no—
111#832—91no—
112#821—94no—
113#821—94no—
114#821—91no—
115#821—184no—
116Dic3⋊C8C3⋊D311yescharacteristic, complemented
117C22×C3⋊D3—21no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—61no—
120C22×C3⋊D3—64no—
121C22×C3⋊D3—124no—
122C22×C3×D3—64no—
123C22×C3×D3—64no—
124C2×C32×C4—24no—
125C2×C32×C4—31no—
126C2×C32×C4—31no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—124no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—124no—
141C4⋊(C3⋊D3)—122no—
142C4⋊(C3⋊D3)—122no—
143C4⋊(C3⋊D3)#6712yescomplemented
144C4×C3⋊D3—122no—
145C2×C3×Dic3D1214yes—
146C2×C3×Dic3D1214yes—
147C3×D12—38no—
148C3×D12—64no—
149C3×D12—64no—
150C3×D12—64no—
151#626—128no—
152#626—128no—
153D3×Dic3—128no—
154C32×C8—21no—
155C32×C8—24no—
156C32×C8—44no—
157C32×C8—61no—
158C3×C3⋊C8—24no—
159#459—271noSylow
160C2×C33#12111yescharacteristic
161C2×C33C2×D811yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C3⋊(C3⋊D3)—81no—
164C3×C3⋊D3—42no—
165C3×C3⋊D3—81no—
166C2×C3×D4—91no—
167C2×C3×D4—91no—
168C2×#67—181no—
169C2×#67—181no—
170C22×Dic3—181no—
171C2×D12—34no—
172C2×D12—34no—
173C2×D12—94no—
174C2×D12—91no—
175C2×D12—184no—
176C2×D12—184no—
177C2×D12—184no—
178C2×C4×D3—184no—
179C3×D8—182no—
180C2×C3×C8—24no—
181C2×C3×C8—34no—
182C2×C3×C8D3214yes—
183C2×C3×C8C2×C3⋊D311yescharacteristic
184C3×C42—34no—
185#217—181no—
186#213—182no—
187(C2×D3)⋊C4—184no—
188C4⋊Dic3—94no—
189C4⋊Dic3—91no—
190C4⋊Dic3—184no—
191C4×Dic3C2×C3⋊D311yescharacteristic
192C2×C3⋊C8C2×C3⋊D311yescharacteristic
193D24—68no—
194D24—188no—
195C22×C32—24no—
196C22×C32C2×D1214yes—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—22no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—128no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—121no—
207C2×C3⋊D3—121no—
208C2×C3⋊D3—121no—
209C2×C3⋊D3—124no—
210C2×C3⋊D3—124no—
211C2×C3⋊D3—244no—
212C2×C3⋊D3—244no—
213C2×C3×D3—68no—
214C2×C3×D3—124no—
215C2×C3×D3—124no—
216C2×C3×D3—124no—
217C2×C3×D3—124no—
218C2×C3×D3—124no—
219C32×C4—24no—
220C32×C4—24no—
221C32×C4—32no—
222C32×C4—61no—
223C32×C4D3×D414yes—
224C32×C4#23711yescharacteristic
225C32×C4#23914yes—
226C32×C4C2×#6711yescharacteristic
227C3⋊Dic3—121no—
228C3⋊Dic3—124no—
229C3⋊Dic3—244no—
230C3×Dic3—24no—
231C3×Dic3D2418yescomplemented
232C2×D8—91no—
233C2×D8—271no—
234#127—271no—
235C4×D4—271no—
236C4⋊C8—31no—
237#102—271no—
238#102—271no—
239C33#45911yescharacteristic, complemented, Sylow
240C23×C3—181no—
241C23×C3—181no—
242C22×D3—64no—
243C22×D3—64no—
244C22×D3—184no—
245C22×D3—181no—
246C22×D3—364no—
247C3×D4—92no—
248C3×D4—181no—
249C3×D4—181no—
250C3×D4—181no—
251C2×C3×C4—24no—
252C2×C3×C4—34no—
253C2×C3×C4—34no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256#67—362no—
257#67—362no—
258C2×Dic3—184no—
259C2×Dic3—181no—
260C2×Dic3—364no—
261C2×Dic3—362no—
262C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
263C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
264D12—38no—
265D12—64no—
266D12—64no—
267D12—64no—
268D12—188no—
269D12—184no—
270D12—184no—
271D12—181no—
272D12—181no—
273D12—364no—
274D12—368no—
275D12—368no—
276C4×D3—368no—
277C3×C8—24no—
278C3×C8—21no—
279C3×C8—44no—
280C3×C8—64no—
281C3⋊C8—21no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32#89514yes—
286C2×C32C2×D2414yes—
287C2×C32#90814yes—
288C2×C32D3×D811yescharacteristic
289C2×C32#91111yescharacteristic
290C2×C32#93911yescharacteristic
291C3⋊D3—42no—
292C3⋊D3—81no—
293C3⋊D3—248no—
294C3⋊D3—241no—
295C3⋊D3—244no—
296C3×D3—128no—
297C3×D3—244no—
298C2×D4—91no—
299C2×D4—91no—
300C2×D4—271no—
301C2×D4—541no—
302C2×D4—541no—
303C22×C4—541no—
304D8—182no—
305D8—542no—
306C2×C8—31no—
307C2×C8D3×C3⋊D311yescharacteristic
308C4⋊C4—271no—
309C22⋊C4—541no—
310C42—31no—
311C22×C3—24no—
312C22×C3—182no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3D3×D1214yes—
319C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
320C2×D3—68no—
321C2×D3—124no—
322C2×D3—124no—
323C2×D3—124no—
324C2×D3—124no—
325C2×D3—124no—
326C2×D3—368no—
327C2×D3—364no—
328C2×D3—364no—
329C2×D3—361no—
330C2×D3—361no—
331C2×D3—361no—
332C2×D3—364no—
333C2×D3—724no—
334C2×D3—724no—
335C3×C4—24no—
336C3×C4—24no—
337C3×C4—38no—
338C3×C4—64no—
339C3×C4#379614yes—
340C3×C4D3×#6714yes—
341C3×C4D4×C3⋊D311yescharacteristic
342C3×C4#383211yescharacteristic
343Dic3—21no—
344Dic3—364no—
345Dic3—361no—
346Dic3—724no—
347Dic3#374512yescomplemented
348C32—24no—
349C32#484314yescomplemented
350C32#515011yescharacteristic, complemented
351C23—181no—
352C23—181no—
353C23—541no—
354D4—92no—
355D4—181no—
356D4—181no—
357D4—181no—
358D4—541no—
359D4—541no—
360D4—1082no—
361D4—1082no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—541no—
365C2×C4—1082no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—21no—
368C8—61no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—362no—
373C2×C3—721no—
374C2×C3D3×D2414yes—
375C2×C3#6379514yes—
376C2×C3#6388614yes—
377C2×C3#6410411yescharacteristic
378C2×C3C2×#374511yescharacteristic
379C2×C3#6411711yescharacteristic
380D3—128no—
381D3—244no—
382D3—728no—
383D3—721no—
384D3—724no—
385C22—182no—
386C22—361no—
387C22—361no—
388C22—361no—
389C22—361no—
390C22—361no—
391C22—1081no—
392C22—1081no—
393C22—1081no—
394C22D3×C4⋊(C3⋊D3)11yescharacteristic
395C4—32no—
396C4—61no—
397C4—1081no—
398C4#8901711yescharacteristic
399C4C3⋊D3×#6711yescharacteristic
400C3—24no—
401C3#1059075314yescomplemented
402C3#1059254211yescharacteristic, complemented
403C2—362no—
404C2—721no—
405C2—2161no—
406C2D3×#374511yescharacteristic
407C2#1172331611yescharacteristic
408C2#1172348211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3745C2semidirect
2. #11723517C2semidirect
3. Dic3×C4⋊(C3⋊D3)C2semidirect
4. C2×#88791C2semidirect
5. #11722191C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722034C2semidirect
24. C2×#3745D3semidirect
36. C3×Dic3⋊C8D3semidirect
48. C3×C4⋊(C3⋊D3)D4semidirect
101. C32×Dic3D8semidirect
116. Dic3⋊C8C3⋊D3semidirect
143. C4⋊(C3⋊D3)#67semidirect
231. C3×Dic3D24semidirect
239. C33#459semidirect
347. Dic3#3745semidirect
349. C32#4843semidirect
350. C32#5150semidirect
401. C3#10590753semidirect
402. C3#10592542semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37452
#117235172
Dic3×C4⋊(C3⋊D3)2
C2×#887912
#117221912
C32×Dic3⋊C82
#117220342
#105925423
#105907533