Finite Group Databasebeta

← #13159089

Subgroups — #13159089

Order 1728.

780 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C3×#64104C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic, complemented
3#11722177C211yesmaximal, characteristic, complemented
4C32×#808C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592565—31nomaximal
9#10590449—34nomaximal
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20Dic3×C3⋊Dic3—21no—
21#64104D311yescharacteristic
22#64101—31no—
23C3×#895—34no—
24#63841—34no—
25C32×#106—31no—
26#63648—31no—
27#63647—31no—
28#63636—31no—
29#63634—31no—
30#63634—34no—
31#63634—64no—
32C3×#808D314yescomplemented
33#63563—34no—
34#63562—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—42no—
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C2×C32×Dic3—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5035—91no—
51#4866—94no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×D12—34no—
56C2×C3×D12—64no—
57C2×C3×C4×D3—34no—
58C2×C32×C8—31no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—31no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63#3752—21no—
64#3752—61no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—64no—
68C4×C3⋊Dic3—61no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×(C2×D3)⋊C4—64no—
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8C2×D314yes—
75(C2×D3)⋊Dic3—64no—
76Dic32—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—21no—
89C32×Dic3—41no—
90C3×#121—91no—
91#926—91no—
92#895—34no—
93#892—94no—
94C3×#106—34no—
95#832—91no—
96#832—91no—
97#830—91no—
98#821—94no—
99(C2×D3)⋊C8—94no—
100#809—94no—
101#808C3⋊D311yescharacteristic, complemented
102#807—94no—
103#807—91no—
104#807—184no—
105C22×C3⋊D3—21no—
106C22×C3⋊D3#6711yescharacteristic
107C22×C3×D3—34no—
108C22×C3×D3—64no—
109C2×C32×C4—21no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—44no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119C2×C3⋊Dic3#6711yescharacteristic
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)—42no—
123C4×C3⋊D3—21no—
124C4×C3⋊D3—21no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—34no—
127C3×D12—64no—
128C3×D12—64no—
129C3×D12—128no—
130C3×C4×D3—64no—
131C3×C4×D3—64no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—68no—
134C3⋊(C3⋊C8)—61no—
135C3⋊(C3⋊C8)—64no—
136C3×C3⋊C8—24no—
137#482—271noSylow
138C2×C33#12311yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C3×C3⋊D3—41no—
142C3×C3⋊D3—81no—
143C2×C3×D4—91no—
144C2×C3×D4—181no—
145C22×C3×C4—91no—
146C2×D12—34no—
147C2×D12—64no—
148C2×C4×D3—34no—
149C2×C3×C8—34no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—34no—
152C3×C4⋊C4D3214yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154C3×C22⋊C4—181no—
155#217—181no—
156(C2×D3)⋊C4—64no—
157(C2×D3)⋊C4—184no—
158C4⋊Dic3—94no—
159C4⋊Dic3—91no—
160C4⋊Dic3—184no—
161C4⋊Dic3C2×C3⋊D311yescharacteristic
162C4×Dic3—181no—
163C4×Dic3—184no—
164C2×C3⋊C8—94no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—184no—
167C2×C3⋊C8C2×C3⋊D311yescharacteristic
168C22×C32—24no—
169C22×C32D3×D414yes—
170C22×C32C2×#6711yescharacteristic
171C2×C3⋊D3—21no—
172C2×C3⋊D3—21no—
173C2×C3⋊D3—21no—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3×D3—64no—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—124no—
181C2×C3×D3—124no—
182C2×C3×D3—124no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—41no—
186C32×C4—44no—
187C32×C4—84no—
188C32×C4—121no—
189C32×C4#23514yes—
190C32×C4#23711yescharacteristic
191C32×C4#23711yescharacteristic
192C32×C4#23914yes—
193C3⋊Dic3—21no—
194C3⋊Dic3—121no—
195C3⋊Dic3—124no—
196C3⋊Dic3—244no—
197C3×Dic3—44no—
198C3×Dic3—64no—
199#121—91no—
200#117—271no—
201#107—271no—
202#106—31no—
203#102—271no—
204#102—271no—
205C22⋊C8—271no—
206C33#48211yescharacteristic, complemented, Sylow
207C23×C3—91no—
208C23×C3—181no—
209C22×D3—34no—
210C22×D3—64no—
211C3×D4—181no—
212C3×D4—181no—
213C3×D4—362no—
214C2×C3×C4—24no—
215C2×C3×C4—21no—
216C2×C3×C4—24no—
217C2×C3×C4—44no—
218C2×C3×C4—64no—
219C2×C3×C4—91no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4C2×D3214yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—34no—
226C2×Dic3—184no—
227C2×Dic3—181no—
228C2×Dic3—364no—
229D12—64no—
230D12—64no—
231D12—128no—
232C4×D3—64no—
233C4×D3—64no—
234C3×C8—64no—
235C3⋊C8—21no—
236C3⋊C8—188no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32#89414yes—
243C2×C32#91414yes—
244C2×C32#91614yes—
245C2×C32#93211yescharacteristic
246C2×C32#93511yescharacteristic
247C2×C32#95011yescharacteristic
248C3⋊D3—41no—
249C3⋊D3—81no—
250C3×D3—124no—
251C3×D3—244no—
252C2×D4—91no—
253C2×D4—181no—
254C22×C4—91no—
255C2×C8—31no—
256C2×C8—271no—
257C4⋊C4—31no—
258C4⋊C4—271no—
259C4⋊C4D3×C3⋊D311yescharacteristic
260C22⋊C4—181no—
261C22⋊C4—541no—
262C42—541no—
263C22×C3—24no—
264C22×C3—181no—
265C22×C3—181no—
266C22×C3—181no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3D3×#6714yes—
271C22×C3D4×C3⋊D311yescharacteristic
272C2×D3—64no—
273C2×D3—64no—
274C2×D3—64no—
275C2×D3—124no—
276C2×D3—124no—
277C2×D3—124no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—41no—
281C3×C4—44no—
282C3×C4—84no—
283C3×C4—124no—
284C3×C4—181no—
285C3×C4#379514yes—
286C3×C4#379614yes—
287C3×C4#382811yescharacteristic
288C3×C4#383211yescharacteristic
289Dic3—41no—
290Dic3—64no—
291Dic3—361no—
292Dic3—364no—
293Dic3—724no—
294C32—24no—
295C32#486614yescomplemented
296C32#503511yescharacteristic, complemented
297C23—91no—
298C23—181no—
299D4—181no—
300D4—181no—
301D4—362no—
302C2×C4—21no—
303C2×C4—61no—
304C2×C4—91no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4—541no—
308C2×C4C2×D3×C3⋊D311yescharacteristic
309C8—61no—
310C8—541no—
311C2×C3—24no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3—361no—
315C2×C3—721no—
316C2×C3#6388414yes—
317C2×C3#6393414yes—
318C2×C3#6394714yes—
319C2×C3#6410311yescharacteristic
320C2×C3#6412311yescharacteristic
321C2×C3#6412511yescharacteristic
322D3—124no—
323D3—244no—
324C22—181no—
325C22—181no—
326C22—181no—
327C22—361no—
328C22—361no—
329C22—361no—
330C22C3⋊D3×#6711yescharacteristic
331C4—41no—
332C4—121no—
333C4—181no—
334C4—1081no—
335C4#8901711yescharacteristic
336C4#8902111yescharacteristic
337C3—24no—
338C3#1059044914yescomplemented
339C3#1059256511yescharacteristic, complemented
340C2—361no—
341C2—721no—
342C2#1172348811yescharacteristic
343C2#1172356211yescharacteristic
344C2#1172358311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723404C2semidirect
3. #11722177C2semidirect
4. C32×#808C2semidirect
5. #11722034C2semidirect
6. #11722033C2semidirect
7. #11722023C2semidirect
32. C3×#808D3semidirect
101. #808C3⋊D3semidirect
206. C33#482semidirect
295. C32#4866semidirect
296. C32#5035semidirect
338. C3#10590449semidirect
339. C3#10592565semidirect

Maximal subgroups

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

subgroupindex
C3×#641042
#117234042
#117221772
C32×#8082
#117220342
#117220332
#117220232
#105925653
#105904493