Finite Group Databasebeta

← #13159689

Subgroups — #13159689

Order 1728.

756 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 156 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
1Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic, complemented
3#11722187C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#11722055C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic
8#10592546—31nomaximal
9#10590772—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×Dic3×C3⋊D3—21no—
12C2×C33×C8C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20#88803—21no—
21C4×C4⋊(C3⋊D3)—31no—
22#64079—31no—
23Dic3×D12—34no—
24#63869—34no—
25C32×C4⋊C8—31no—
26#63648—31no—
27#63648D311yescharacteristic
28#63645—31no—
29#63644—31no—
30#63644—34no—
31#63644—64no—
32C3×#821—34no—
33C3×Dic3⋊C8D314yescomplemented
34#63575—34no—
35#63563—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)Q812yes—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45Dic3×C3⋊D3—42no—
46C33×C8—21no—
47C32×C3⋊C8—21no—
48#4847—94no—
49#4763—91no—
50C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
51C2×C4×C3⋊D3—61no—
52C2×C3×D12—34no—
53C2×D3×Dic3—64no—
54C2×C32×C8—24no—
55C2×C32×C8—31no—
56C2×C32×C8C2×D314yes—
57C2×C32×C8C2×D311yescharacteristic
58C32×C42—31no—
59#3752—61no—
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68#3750—61no—
69C3×C4⋊Dic3—34no—
70C3×C4×Dic3C2×D314yes—
71C2×C3×C3⋊C8C2×D314yes—
72#3724—64no—
73Dic3⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C33×C4C2×Q811yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—21no—
85C32×Dic3—21no—
86D4×Dic3—91no—
87#890—91no—
88C4×D12—94no—
89#870—94no—
90C3×C4⋊C8—34no—
91C3×#102—91no—
92#832—91no—
93#821—34no—
94#821—94no—
95#818—94no—
96#817—94no—
97#817—91no—
98#817—184no—
99Dic3⋊C8C3⋊D311yescharacteristic, complemented
100#807—91no—
101C22×C3⋊D3—21no—
102C22×C3×D3—64no—
103C2×C32×C4—24no—
104C2×C32×C4—31no—
105C2×C32×C4—31no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—124no—
114C2×C3⋊Dic3—124no—
115C4⋊(C3⋊D3)—21no—
116C4⋊(C3⋊D3)Dic612yes—
117C4×C3⋊D3—122no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3D1214yes—
120C2×C3×Dic3D1214yes—
121C3×D12—38no—
122C3×D12—64no—
123D3×Dic3—128no—
124C32×C8—28no—
125C32×C8—24no—
126C32×C8—21no—
127C32×C8—61no—
128C3×C3⋊C8—24no—
129#478—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C3×C3⋊D3—42no—
134C2×C3×D4—91no—
135C22×Dic3—181no—
136C2×D12—34no—
137C2×C4×D3—184no—
138C2×C3×C8—24no—
139C2×C3×C8—34no—
140C2×C3×C8D3214yes—
141C2×C3×C8C2×C3⋊D311yescharacteristic
142C3×C4⋊C4—91no—
143C3×C42—34no—
144#217—181no—
145(C2×D3)⋊C4—184no—
146C4⋊Dic3—34no—
147C4⋊Dic3—94no—
148C4⋊Dic3—94no—
149C4⋊Dic3—91no—
150C4⋊Dic3—91no—
151C4⋊Dic3—184no—
152C4⋊Dic3—184no—
153C4⋊Dic3—181no—
154Dic3⋊C4—184no—
155Dic3⋊C4—184no—
156Dic3⋊C4—181no—
157C4×Dic3C2×C3⋊D311yescharacteristic
158C2×C3⋊C8C2×C3⋊D311yescharacteristic
159C22×C32—24no—
160C22×C32C2×D1214yes—
161C22×C32D3×D411yescharacteristic
162C2×C3⋊D3—22no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3×D3—68no—
166C2×C3×D3—124no—
167C2×C3×D3—124no—
168C32×C4—24no—
169C32×C4—24no—
170C32×C4—61no—
171C32×C4—61no—
172C32×C4C2×Dic611yescharacteristic
173C32×C4#23714yes—
174C32×C4#23711yescharacteristic
175C32×C4D3×Q814yes—
176C3⋊Dic3—41no—
177C3⋊Dic3—121no—
178C3⋊Dic3—121no—
179C3⋊Dic3—124no—
180C3⋊Dic3—124no—
181C3⋊Dic3—244no—
182C3⋊Dic3—244no—
183C3×Dic3—24no—
184C3×Dic3—24no—
185C3×Dic3—124no—
186#125—271no—
187C4×D4—271no—
188#107—271no—
189#106—271no—
190C4⋊C8—31no—
191#102—91no—
192#102—271no—
193C33#47811yescharacteristic, complemented, Sylow
194C23×C3—181no—
195C22×D3—64no—
196C3×D4—92no—
197C3×D4—181no—
198C2×C3×C4—24no—
199C2×C3×C4—34no—
200C2×C3×C4—34no—
201C2×C3×C4—181no—
202C2×C3×C4C2×D3214yes—
203C2×C3×C4C22×C3⋊D311yescharacteristic
204C2×Dic3—64no—
205C2×Dic3—184no—
206C2×Dic3—184no—
207C2×Dic3—181no—
208C2×Dic3—181no—
209C2×Dic3—364no—
210C2×Dic3—364no—
211C2×Dic3—362no—
212C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
213C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
214D12—38no—
215D12—64no—
216C4×D3—368no—
217C3×C8—28no—
218C3×C8—24no—
219C3×C8—21no—
220C3×C8—64no—
221C3⋊C8—21no—
222C2×C32—24no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32#87811yescharacteristic
226C2×C32#89714yes—
227C2×C32#90414yes—
228C2×C32#90814yes—
229C2×C32#91111yescharacteristic
230C2×C32#91611yescharacteristic
231C3⋊D3—42no—
232C3×D3—128no—
233C2×D4—91no—
234C22×C4—541no—
235C2×C8—31no—
236C2×C8D3×C3⋊D311yescharacteristic
237C4⋊C4—91no—
238C4⋊C4—271no—
239C4⋊C4—271no—
240C4⋊C4—541no—
241C4⋊C4—541no—
242C22⋊C4—541no—
243C42—31no—
244C22×C3—24no—
245C22×C3—182no—
246C22×C3—361no—
247C22×C3—361no—
248C22×C3D3×D1214yes—
249C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
250C2×D3—68no—
251C2×D3—124no—
252C2×D3—124no—
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—64no—
256C3×C4—64no—
257C3×C4—361no—
258C3×C4D3×Dic614yes—
259C3×C4#380514yes—
260C3×C4#383011yescharacteristic
261C3×C4Q8×C3⋊D311yescharacteristic
262Dic3—21no—
263Dic3—21no—
264Dic3—124no—
265Dic3—364no—
266Dic3—364no—
267Dic3—361no—
268Dic3—361no—
269Dic3—724no—
270Dic3—724no—
271C32—24no—
272C32#476311yescharacteristic, complemented
273C32#484714yescomplemented
274C23—181no—
275D4—92no—
276D4—181no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—181no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4—1082no—
283C2×C4C2×D3×C3⋊D311yescharacteristic
284C8—21no—
285C8—61no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—362no—
290C2×C3#6379514yes—
291C2×C3#6380614yes—
292C2×C3#6389314yes—
293C2×C3#6410611yescharacteristic
294C2×C3#6411311yescharacteristic
295C2×C3#6411711yescharacteristic
296D3—128no—
297C22—182no—
298C22—361no—
299C22—361no—
300C22D3×C4⋊(C3⋊D3)11yescharacteristic
301C4—61no—
302C4—61no—
303C4—361no—
304C4—1081no—
305C4—1081no—
306C4C3⋊D3×Dic611yescharacteristic
307C4#8902911yescharacteristic
308C3—24no—
309C3#1059077214yescomplemented
310C3#1059254611yescharacteristic, complemented
311C2—362no—
312C2#1172331611yescharacteristic
313C2#1172333511yescharacteristic
314C2#1172348311yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#478
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
2. #11723454C2semidirect
3. #11722187C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. #11722055C2semidirect
33. C3×Dic3⋊C8D3semidirect
99. Dic3⋊C8C3⋊D3semidirect
193. C33#478semidirect
272. C32#4763semidirect
273. C32#4847semidirect
309. C3#10590772semidirect
310. C3#10592546semidirect

Maximal subgroups

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

subgroupindex
Dic3×C4⋊(C3⋊D3)2
#117234542
#117221872
C3×#636482
C32×Dic3⋊C82
#117220552
#117220342
#105925463
#105907723