Finite Group Databasebeta

← #13156683

Subgroups — #13156683

Order 1728.

514 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C4⋊#6296C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic
3#11722055C212yesmaximal
4C3×#63573C212yesmaximal
5C3×#63555C211yesmaximal, characteristic
6#10590892—32nomaximal
7#10590564—31nomaximal
8C2×#6296—21no—
9C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88806—21no—
16Dic3⋊(C3⋊Dic3)—21no—
17#64099—31no—
18#63919—31no—
19#63918—32no—
20#63869—32no—
21#63869—61no—
22#63645—32no—
23C3×#817—32no—
24C3×#815—32no—
25C3×#808—32no—
26#63575—32no—
27#63575—32no—
28#63575—62no—
29#63573D312yes—
30#63555D311yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—21no—
37#6296—42no—
38C32×C3⋊C8Q814yescomplemented
39#4856—92no—
40#4702—91no—
41C2×#627—62no—
42C2×#627—61no—
43C2×C32×C8—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—
51C4⋊(C3⋊Dic3)C2×D312yes—
52#3750—61no—
53C4×C3⋊Dic3C2×D311yescharacteristic
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C3×C4⋊Dic3C2×D312yes—
59C3×C4×Dic3—32no—
60C3×C4×Dic3—61no—
61C2×C3×C3⋊C8—22no—
62C2×C3×C3⋊C8C2×D312yes—
63C2×C3×C3⋊C8C2×D312yes—
64#3724—62no—
65#3724—62no—
66#3724—61no—
67Dic3⋊Dic3—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—
75C3×C3⋊Dic3—42no—
76C32×Dic3—41no—
77#890—92no—
78#890—181no—
79#888—92no—
80#870—91no—
81#869—91no—
82C3×#106—92no—
83C3×#97—91no—
84#818—92no—
85#818—92no—
86#818—182no—
87#817—32no—
88#815—32no—
89#808—32no—
90#807—92no—
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—61no—
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C22×D312yes—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—122no—
102C2×C3⋊Dic3—122no—
103C2×C3⋊Dic3D1211yescharacteristic
104C2×C3⋊Dic3D1211yescharacteristic
105C2×C3×Dic3—22no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—32no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114#627—124no—
115#627—122no—
116C32×C8—34no—
117C3×C3⋊C8—24no—
118C3×C3⋊C8Dic614yescomplemented
119C3×C3⋊C8Dic614yescomplemented
120#501—271noSylow
121C2×C33C4⋊Q811yescharacteristic
122C2×C33AGL(1,ℤ/8)11yescharacteristic
123C2×C33#13711yescharacteristic
124C2×Dic6—182no—
125C2×Dic6—181no—
126C2×C3×C8—32no—
127C2×C3×C8—32no—
128C2×C3×C8—62no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—92no—
132C3×C42—91no—
133C4⋊Dic3—32no—
134C4⋊Dic3—32no—
135C4⋊Dic3—62no—
136C4⋊Dic3—92no—
137C4⋊Dic3—91no—
138C4⋊Dic3—181no—
139C4⋊Dic3—182no—
140C4⋊Dic3—182no—
141C4⋊Dic3—181no—
142C4⋊Dic3—182no—
143C4⋊Dic3—361no—
144C4⋊Dic3D3211yescharacteristic
145Dic3⋊C4—181no—
146Dic3⋊C4—181no—
147Dic3⋊C4—182no—
148Dic3⋊C4—182no—
149Dic3⋊C4—182no—
150C4×Dic3—32no—
151C4×Dic3—61no—
152C2×C3⋊C8D3212yes—
153C22×C32—22no—
154C22×C32—21no—
155C22×C32—41no—
156C22×C32C2×D1211yescharacteristic
157C22×C32D3×D412yes—
158C32×C4—22no—
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—42no—
162C32×C4—121no—
163C32×C4C2×Dic612yes—
164C32×C4C2×Dic612yes—
165C32×C4D3×Q812yes—
166C3⋊Dic3—21no—
167C3⋊Dic3—21no—
168C3⋊Dic3—42no—
169C3⋊Dic3—121no—
170C3⋊Dic3—122no—
171C3⋊Dic3—122no—
172C3⋊Dic3—242no—
173C3⋊Dic3—242no—
174C3×Dic3—42no—
175C3×Dic3—42no—
176C3×Dic3—62no—
177C3×Dic3—62no—
178C3×Dic3—121no—
179C3×Dic3—121no—
180C3×Dic3—122no—
181C3×Dic3—122no—
182C3×Dic3—242no—
183C4⋊Q8—271no—
184#125—271no—
185#107—272no—
186#106—92no—
187#97—91no—
188C33#50111yescharacteristic, complemented, Sylow
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—41no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—182no—
197C2×C3×C4C2×D3211yescharacteristic
198C2×C3×C4C2×D3212yes—
199C2×Dic3—21no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—122no—
207C2×Dic3—182no—
208C2×Dic3—182no—
209C2×Dic3—181no—
210C2×Dic3—362no—
211C2×Dic3—362no—
212Dic6—364no—
213Dic6—362no—
214C3×C8—34no—
215C3×C8—34no—
216C3×C8—64no—
217C3⋊C8#62714yescomplemented
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—21no—
221C2×C32—22no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32#86911yescharacteristic
228C2×C32#88812yes—
229C2×C32#90811yescharacteristic
230C2×C32#90911yescharacteristic
231C2×C32#91412yes—
232C2×C32#91512yes—
233C2×Q8—541no—
234C2×C8—32no—
235C4⋊C4—31no—
236C4⋊C4—92no—
237C4⋊C4—271no—
238C4⋊C4—541no—
239C4⋊C4—541no—
240C4⋊C4—541no—
241C42—91no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—41no—
245C22×C3D3×D1212yes—
246C22×C3#381111yescharacteristic
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—42no—
251C3×C4—122no—
252C3×C4—122no—
253C3×C4—181no—
254C3×C4—181no—
255C3×C4—362no—
256C3×C4D3×Dic612yes—
257C3×C4D3×Dic612yes—
258C3×C4C2×#62712yes—
259Dic3—41no—
260Dic3—62no—
261Dic3—62no—
262Dic3—121no—
263Dic3—121no—
264Dic3—122no—
265Dic3—122no—
266Dic3—242no—
267Dic3—362no—
268Dic3—362no—
269Dic3—361no—
270Dic3—722no—
271Dic3—722no—
272C32—21no—
273C32—22no—
274C32—41no—
275C32#470211yescharacteristic, complemented
276C32#485612yescomplemented
277Q8—1082no—
278C2×C4—61no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—182no—
282C2×C4—541no—
283C2×C4D3311yescharacteristic
284C8—34no—
285C2×C3—21no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3#6379412yes—
295C2×C3#6380012yes—
296C2×C3#6384011yescharacteristic
297C2×C3#6391812yes—
298C2×C3#6393111yescharacteristic
299C2×C3#6394111yescharacteristic
300C22#8895511yescharacteristic
301C4—121no—
302C4—181no—
303C4—181no—
304C4—362no—
305C4—1081no—
306C4D3×#62712yes—
307C3—21no—
308C3—22no—
309C3—41no—
310C3#1059056411yescharacteristic, complemented
311C3#1059089212yescomplemented
312C2#1172288111yescharacteristic
313C2#1172289511yescharacteristic
314C2#1172303611yescharacteristic
subgroupG/H
centerC22#88955
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. C32×C3⋊C8Q8semidirect
118. C3×C3⋊C8Dic6semidirect
119. C3×C3⋊C8Dic6semidirect
188. C33#501semidirect
217. C3⋊C8#627semidirect
275. C32#4702semidirect
276. C32#4856semidirect
310. C3#10590564semidirect
311. C3#10590892semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234472
#117220552
C3×#635732
C3×#635552
#105908923
#105905643