Finite Group Databasebeta

← #13158815

Subgroups — #13158815

Order 1728.

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

normal subgroups only

This group has 764 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
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic, complemented
4#11722056C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic
7#11722033C211yesmaximal, characteristic
8#10592765—31nomaximal
9#10590270—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C33×C42C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×Dic3⋊C4—21no—
21C32×#125—31no—
22C4×C4⋊(C3⋊D3)D311yescharacteristic
23C3×C4×D12—34no—
24C3×#870D314yescomplemented
25#63636—31no—
26#63636—31no—
27#63635—31no—
28#63634—31no—
29C4⋊(C3⋊(C3⋊C8))—31no—
30C4⋊(C3⋊(C3⋊C8))—34no—
31C4⋊(C3⋊(C3⋊C8))—64no—
32#63576—34no—
33#63574—34no—
34#63562—34no—
35#63562—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)Q812yes—
43C3×C4×C3⋊D3—42no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48#5066—94no—
49#5009—91no—
50C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
51C2×C4×C3⋊D3—21no—
52C2×C3×D12—34no—
53C2×C3×C4×D3—64no—
54C32×C4⋊C4—31no—
55C32×C4⋊C4—31no—
56C32×C4⋊C4—61no—
57C32×C4⋊C4—61no—
58C32×C42—24no—
59C32×C42C2×D314yes—
60C32×C42C2×D311yescharacteristic
61#3752—21no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×(C2×D3)⋊C4—64no—
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72C3×C4⋊Dic3C2×D314yes—
73C3×Dic3⋊C4—24no—
74C3×Dic3⋊C4—24no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C33×C4—21no—
80C33×C4—21no—
81C33×C4C2×Q811yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—41no—
84C32×Dic3—41no—
85C32×Dic3—41no—
86C3×#125—34no—
87C3×C4×D4—91no—
88C4×D12—34no—
89#870C3⋊D311yescharacteristic, complemented
90#832—91no—
91#832—91no—
92#809—94no—
93#809—94no—
94#808—94no—
95#808—91no—
96#807—94no—
97#807—91no—
98C4⋊(C3⋊C8)—94no—
99C4⋊(C3⋊C8)—91no—
100C4⋊(C3⋊C8)—184no—
101C22×C3⋊D3—21no—
102C22×C3×D3—64no—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4#6714yes—
109C2×C32×C4#6714yes—
110C2×C32×C4#6711yescharacteristic
111C2×C32×C4#6711yescharacteristic
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C4⋊(C3⋊D3)—21no—
116C4⋊(C3⋊D3)Dic612yes—
117C4×C3⋊D3—42no—
118C2×C3×Dic3—24no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—64no—
121C3×D12—38no—
122C3×D12—64no—
123C3×C4×D3—128no—
124C3⋊(C3⋊C8)—61no—
125C3⋊(C3⋊C8)—61no—
126C3⋊(C3⋊C8)—64no—
127C3⋊(C3⋊C8)—64no—
128C3⋊(C3⋊C8)—124no—
129C3⋊(C3⋊C8)—124no—
130#478—271noSylow
131C2×C33C22⋊Q811yescharacteristic
132C2×C33#13511yescharacteristic
133C2×C33AGL(1,ℤ/8)11yescharacteristic
134C3×C3⋊D3—42no—
135C2×C3×D4—91no—
136C22×C3×C4—181no—
137C2×D12—34no—
138C2×C4×D3—64no—
139C3×C4⋊C4—34no—
140C3×C4⋊C4—34no—
141C3×C4⋊C4—64no—
142C3×C4⋊C4—64no—
143C3×C4⋊C4—91no—
144C3×C22⋊C4—181no—
145C3×C42—24no—
146C3×C42D3214yes—
147C3×C42C2×C3⋊D311yescharacteristic
148(C2×D3)⋊C4—64no—
149C4⋊Dic3—34no—
150C4⋊Dic3C2×C3⋊D311yescharacteristic
151C4⋊Dic3C2×C3⋊D311yescharacteristic
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154C2×C3⋊C8—94no—
155C2×C3⋊C8—94no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—184no—
159C2×C3⋊C8—184no—
160C22×C32—24no—
161C22×C32C2×#6714yes—
162C22×C32C2×#6711yescharacteristic
163C2×C3⋊D3—22no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C2×C3×D3—68no—
167C2×C3×D3—124no—
168C2×C3×D3—124no—
169C32×C4—28no—
170C32×C4—21no—
171C32×C4—24no—
172C32×C4—21no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—24no—
176C32×C4—44no—
177C32×C4—121no—
178C32×C4—121no—
179C32×C4C2×Dic611yescharacteristic
180C32×C4#23511yescharacteristic
181C32×C4D3×Q814yes—
182C32×C4#23914yes—
183C3⋊Dic3—41no—
184C3×Dic3—44no—
185C3×Dic3—44no—
186C3×Dic3—124no—
187#125—31no—
188C4×D4—91no—
189#107—271no—
190#106—271no—
191C4⋊C8—271no—
192#102—271no—
193#102—271no—
194C33#47811yescharacteristic, complemented, Sylow
195C23×C3—181no—
196C22×D3—64no—
197C3×D4—92no—
198C3×D4—181no—
199C2×C3×C4—24no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—64no—
203C2×C3×C4—64no—
204C2×C3×C4—181no—
205C2×C3×C4—362no—
206C2×C3×C4#62514yes—
207C2×C3×C4#62514yes—
208C2×C3×C4#63811yescharacteristic
209C2×C3×C4#63811yescharacteristic
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—64no—
215D12—38no—
216D12—64no—
217C4×D3—128no—
218C3⋊C8—184no—
219C3⋊C8—184no—
220C3⋊C8—181no—
221C3⋊C8—181no—
222C3⋊C8—364no—
223C3⋊C8—364no—
224C2×C32—24no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32#92411yescharacteristic
228C2×C32#93211yescharacteristic
229C2×C32#94614yes—
230C2×C32#94914yes—
231C2×C32#95014yes—
232C2×C32#95011yescharacteristic
233C3⋊D3—42no—
234C3×D3—128no—
235C2×D4—91no—
236C22×C4—181no—
237C2×C8—271no—
238C2×C8—271no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—91no—
244C22⋊C4—181no—
245C42D3×C3⋊D311yescharacteristic
246C22×C3—24no—
247C22×C3—182no—
248C22×C3—361no—
249C22×C3—361no—
250C22×C3C2×#62514yes—
251C22×C3C2×#63811yescharacteristic
252C2×D3—68no—
253C2×D3—124no—
254C2×D3—124no—
255C3×C4—28no—
256C3×C4—24no—
257C3×C4—24no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—24no—
261C3×C4—24no—
262C3×C4—44no—
263C3×C4—124no—
264C3×C4—124no—
265C3×C4—361no—
266C3×C4D3×Dic614yes—
267C3×C4#379914yes—
268C3×C4Q8×C3⋊D311yescharacteristic
269C3×C4#383211yescharacteristic
270Dic3—41no—
271Dic3—41no—
272Dic3—124no—
273C32—24no—
274C32#500911yescharacteristic, complemented
275C32#506614yescomplemented
276C23—181no—
277D4—92no—
278D4—181no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—181no—
282C2×C4—362no—
283C2×C4#629011yescharacteristic
284C2×C4#629011yescharacteristic
285C2×C4C2×D3×C3⋊D311yescharacteristic
286C8—541no—
287C8—541no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—362no—
292C2×C3#6382214yes—
293C2×C3#6382314yes—
294C2×C3#6385614yes—
295C2×C3#6415511yescharacteristic
296C2×C3#6415811yescharacteristic
297C2×C3#6415911yescharacteristic
298D3—128no—
299C22—182no—
300C22—361no—
301C22—361no—
302C22C2×#629011yescharacteristic
303C4—21no—
304C4—21no—
305C4—121no—
306C4—121no—
307C4—361no—
308C4C3⋊D3×Dic611yescharacteristic
309C4#8901811yescharacteristic
310C3—24no—
311C3#1059027014yescomplemented
312C3#1059276511yescharacteristic, complemented
313C2—362no—
314C2#1172336311yescharacteristic
315C2#1172336511yescharacteristic
316C2#1172343511yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42D3×C3⋊D3
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C32×#870C2semidirect
3. C4⋊(C3⋊(C3⋊(C3⋊C8)))C2semidirect
4. #11722056C2semidirect
5. #11722053C2semidirect
24. C3×#870D3semidirect
89. #870C3⋊D3semidirect
194. C33#478semidirect
274. C32#5009semidirect
275. C32#5066semidirect
311. C3#10590270semidirect
312. C3#10592765semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C32×#8702
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
#117220562
#117220532
#117220332
#105927653
#105902703