Finite Group Databasebeta

← #13156676

Subgroups — #13156676

Order 1728.

514 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 106 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
1#11723447C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3#11722071C211yesmaximal, characteristic
4#11722054C212yesmaximal
5C3×#63576C212yesmaximal
6#10590563—31nomaximal
7#10590420—32nomaximal
8C2×C3×#627—21no—
9C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3C2212yes—
14#88806—21no—
15Dic3⋊(C3⋊Dic3)—21no—
16C3×#3724—21no—
17#64099—31no—
18C3×#888—32no—
19#63919D311yescharacteristic
20#63869—32no—
21#63869—61no—
22#63635—32no—
23C3×#807—32no—
24C3×#807—32no—
25#63576D312yes—
26#63573—32no—
27#63573—32no—
28#63573—62no—
29#63555—31no—
30#63554—32no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3—22no—
37C3×#627—42no—
38C3×C3⋊(C3⋊C8)Q814yescomplemented
39#5081—91no—
40#4876—92no—
41C2×C3×Dic6—62no—
42C2×#627—21no—
43C32×C4⋊C4—32no—
44C32×C4⋊C4—31no—
45C4⋊(C3⋊Dic3)—31no—
46C4⋊(C3⋊Dic3)—32no—
47C4⋊(C3⋊Dic3)—61no—
48C4⋊(C3⋊Dic3)—61no—
49C4⋊(C3⋊Dic3)—62no—
50C4⋊(C3⋊Dic3)—121no—
51#3750—61no—
52C4×C3⋊Dic3C2×D311yescharacteristic
53C2×C3⋊(C3⋊C8)C2×D312yes—
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3C2×D312yes—
57C3×C4⋊Dic3C2×D312yes—
58C3×C4⋊Dic3C2×D312yes—
59C3×Dic3⋊C4—62no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—62no—
65#3724—21no—
66#3724—62no—
67#3724—62no—
68Dic3⋊Dic3—62no—
69Dic3⋊Dic3—62no—
70C22×C33C2×D411yescharacteristic
71C33×C4C2×Q812yes—
72C3⋊(C3⋊Dic3)—41no—
73C3×C3⋊Dic3—21no—
74C3×C3⋊Dic3—21no—
75C32×Dic3—41no—
76C32×Dic3—42no—
77C3×C4⋊Q8—91no—
78#890—92no—
79#890—181no—
80#888—32no—
81#870—91no—
82C3×#107—92no—
83#817—92no—
84#815—92no—
85#808—92no—
86#808—92no—
87#808—182no—
88#807—32no—
89#807—32no—
90#803—91no—
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—61no—
95C2×C32×C4—62no—
96C2×C32×C4C22×D312yes—
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—122no—
102C2×C3⋊Dic3—122no—
103C2×C3⋊Dic3#6711yescharacteristic
104C2×C3⋊Dic3#6711yescharacteristic
105C2×C3×Dic3—22no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—42no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C3×Dic6—124no—
115#627—42no—
116C3⋊(C3⋊C8)Dic614yescomplemented
117C3×C3⋊C8—34no—
118C3×C3⋊C8—34no—
119C3×C3⋊C8—64no—
120#501—271noSylow
121C2×C33C4⋊Q811yescharacteristic
122C2×C33AGL(1,ℤ/8)11yescharacteristic
123C2×C33#13711yescharacteristic
124C2×C3×Q8—181no—
125C2×Dic6—62no—
126C2×C3×C8—92no—
127C3×C4⋊C4—32no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—62no—
132C3×C4⋊C4—181no—
133C3×C42—91no—
134C4⋊Dic3—91no—
135C4⋊Dic3—92no—
136C4⋊Dic3—181no—
137C4⋊Dic3—182no—
138C4⋊Dic3—182no—
139C4⋊Dic3—182no—
140C4⋊Dic3—361no—
141C4⋊Dic3D3211yescharacteristic
142C4⋊Dic3D3212yes—
143Dic3⋊C4—62no—
144Dic3⋊C4—181no—
145Dic3⋊C4—181no—
146Dic3⋊C4—182no—
147Dic3⋊C4—182no—
148C4×Dic3—32no—
149C4×Dic3—61no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—62no—
153C22×C32—22no—
154C22×C32—21no—
155C22×C32—41no—
156C22×C32D3×D412yes—
157C22×C32C2×#6711yescharacteristic
158C32×C4—22no—
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—42no—
162C32×C4—121no—
163C32×C4—122no—
164C32×C4C2×Dic612yes—
165C32×C4D3×Q812yes—
166C32×C4D3×Q812yes—
167C3⋊Dic3—21no—
168C3⋊Dic3—21no—
169C3⋊Dic3—121no—
170C3⋊Dic3—122no—
171C3⋊Dic3—122no—
172C3⋊Dic3—242no—
173C3⋊Dic3—242no—
174C3×Dic3—42no—
175C3×Dic3—42no—
176C3×Dic3—42no—
177C3×Dic3—42no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C3×Dic3—82no—
181C3×Dic3—121no—
182C3×Dic3—121no—
183C4⋊Q8—91no—
184#125—271no—
185#107—92no—
186#106—272no—
187#97—271no—
188C33#50111yescharacteristic, complemented, Sylow
189C3×Q8—362no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—41no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—62no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—122no—
200C2×C3×C4C2×D3211yescharacteristic
201C2×C3×C4C2×D3212yes—
202C2×Dic3—22no—
203C2×Dic3—21no—
204C2×Dic3—32no—
205C2×Dic3—32no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—182no—
209C2×Dic3—182no—
210C2×Dic3—181no—
211C2×Dic3—362no—
212C2×Dic3—362no—
213Dic6—124no—
214C3×C8—94no—
215C3⋊C8—34no—
216C3⋊C8—34no—
217C3⋊C8—64no—
218C2×C32—22no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—22no—
222C2×C32—21no—
223C2×C32—22no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32#88812yes—
228C2×C32#91112yes—
229C2×C32#91812yes—
230C2×C32#93211yescharacteristic
231C2×C32#94211yescharacteristic
232C2×C32#94411yescharacteristic
233C2×Q8—181no—
234C2×C8—92no—
235C4⋊C4—32no—
236C4⋊C4—31no—
237C4⋊C4—181no—
238C4⋊C4—271no—
239C4⋊C4—541no—
240C4⋊C4—541no—
241C42—91no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—41no—
245C22×C3D3×#6712yes—
246C22×C3#381111yescharacteristic
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—42no—
251C3×C4—122no—
252C3×C4—122no—
253C3×C4—122no—
254C3×C4—122no—
255C3×C4—181no—
256C3×C4—181no—
257C3×C4—242no—
258C3×C4D3×Dic612yes—
259C3×C4D3×Dic612yes—
260C3×C4#379712yes—
261Dic3—41no—
262Dic3—42no—
263Dic3—62no—
264Dic3—62no—
265Dic3—121no—
266Dic3—121no—
267Dic3—362no—
268Dic3—361no—
269Dic3—362no—
270Dic3—722no—
271Dic3—722no—
272C32—21no—
273C32—22no—
274C32—41no—
275C32#487612yescomplemented
276C32#508111yescharacteristic, complemented
277Q8—362no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—541no—
283C2×C4D3311yescharacteristic
284C8—94no—
285C2×C3—22no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3#6383912yes—
295C2×C3#6384011yescharacteristic
296C2×C3#6392711yescharacteristic
297C2×C3#6393212yes—
298C2×C3#6394511yescharacteristic
299C2×C3#6394812yes—
300C22#8895211yescharacteristic
301C4—122no—
302C4—121no—
303C4—181no—
304C4—181no—
305C4—1081no—
306C4#8895712yes—
307C3—22no—
308C3—21no—
309C3—41no—
310C3#1059042012yescomplemented
311C3#1059056311yescharacteristic, complemented
312C2#1172287211yescharacteristic
313C2#1172289011yescharacteristic
314C2#1172303411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#501
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
38. C3×C3⋊(C3⋊C8)Q8semidirect
116. C3⋊(C3⋊C8)Dic6semidirect
188. C33#501semidirect
275. C32#4876semidirect
276. C32#5081semidirect
310. C3#10590420semidirect
311. C3#10590563semidirect

Maximal subgroups

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

subgroupindex
#117234472
C3×#639192
#117220712
#117220542
C3×#635762
#105905633
#105904203