Finite Group Databasebeta

← #13156663

Subgroups — #13156663

Order 1728.

530 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 108 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#11723450C212yesmaximal
2#11722080C211yesmaximal, characteristic
3#11722018C211yesmaximal, characteristic
4C3×#63576C211yesmaximal, characteristic
5C3×#63573C212yesmaximal
6#10590874—31nomaximal
7#10590417—32nomaximal
8C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C3×C4⋊(C3⋊Dic3)C2212yes—
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×C4⋊Dic3C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88807—22no—
14Dic3⋊(C3⋊Dic3)—22no—
15#64099—32no—
16#63872—32no—
17#63872—32no—
18#63872—62no—
19C8×C3⋊Dic3—31no—
20C3×#817—32no—
21C3×#808—32no—
22C3×#807—32no—
23#63576—31no—
24#63576D311yescharacteristic
25#63575—32no—
26#63573D312yes—
27#63553—32no—
28#63553—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3—22no—
34C3×C3⋊(C3⋊C8)Q812yes—
35C32×C3⋊C8Q812yes—
36#4878—91no—
37#4858—92no—
38C2×C32×C8—31no—
39C32×C4⋊C4—32no—
40C4⋊(C3⋊Dic3)—62no—
41C4⋊(C3⋊Dic3)C2×D312yes—
42#3750—62no—
43C4×C3⋊Dic3—32no—
44C4×C3⋊Dic3—31no—
45C4×C3⋊Dic3—61no—
46C4×C3⋊Dic3—62no—
47C4×C3⋊Dic3—121no—
48C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
49C3×C4⋊Dic3—22no—
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—62no—
53C3×C4⋊Dic3C2×D312yes—
54C3×C4⋊Dic3C2×D312yes—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—61no—
58C2×C3×C3⋊C8C2×D312yes—
59Dic3⋊Dic3—62no—
60Dic3⋊Dic3—62no—
61Dic3⋊Dic3—62no—
62Dic3⋊Dic3—62no—
63Dic3⋊Dic3—64no—
64Dic3⋊Dic3—64no—
65C22×C33C2×D411yescharacteristic
66C33×C4C2×Q811yescharacteristic
67C33×C4C2×Q811yescharacteristic
68C3⋊(C3⋊Dic3)—21no—
69C3⋊(C3⋊Dic3)—21no—
70C3×C3⋊Dic3—42no—
71C32×Dic3—42no—
72#890—92no—
73#890—92no—
74#890—92no—
75#890—182no—
76C3×#107—91no—
77C3×#106—92no—
78#818—91no—
79#817—32no—
80C8×Dic3—92no—
81C8×Dic3—91no—
82C8×Dic3—181no—
83#808—32no—
84#807—32no—
85#807—92no—
86C2×C32×C4—21no—
87C2×C32×C4—22no—
88C2×C32×C4—41no—
89C2×C32×C4—62no—
90C2×C32×C4C22×D312yes—
91C2×C32×C4C22×D311yescharacteristic
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—31no—
94C2×C3⋊Dic3—31no—
95C2×C3⋊Dic3—32no—
96C2×C3⋊Dic3—32no—
97C2×C3⋊Dic3—61no—
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—121no—
102C2×C3⋊Dic3—121no—
103C2×C3×Dic3—24no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—64no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C32×C8—32no—
110C3⋊(C3⋊C8)Dic612yes—
111C3×C3⋊C8—22no—
112C3×C3⋊C8—34no—
113C3×C3⋊C8—62no—
114C3×C3⋊C8Dic614yes—
115#499—271noSylow
116C2×C33C4⋊Q811yescharacteristic
117C2×C33#13511yescharacteristic
118C2×C33#13511yescharacteristic
119C2×C3×C8—32no—
120C2×C3×C8—61no—
121C2×C3×C8—91no—
122C3×C4⋊C4—32no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—62no—
125C3×C4⋊C4—92no—
126C4⋊Dic3—32no—
127C4⋊Dic3—32no—
128C4⋊Dic3—62no—
129C4⋊Dic3—182no—
130C4⋊Dic3—182no—
131C4⋊Dic3—182no—
132C4⋊Dic3—184no—
133C4⋊Dic3D3212yes—
134Dic3⋊C4—182no—
135Dic3⋊C4—182no—
136Dic3⋊C4—182no—
137Dic3⋊C4—184no—
138C4×Dic3—91no—
139C4×Dic3—92no—
140C4×Dic3—181no—
141C4×Dic3—182no—
142C4×Dic3—361no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—61no—
145C2×C3⋊C8D3211yescharacteristic
146C22×C32—21no—
147C22×C32—22no—
148C22×C32—41no—
149C22×C32D3×D412yes—
150C22×C32D3×D411yescharacteristic
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—21no—
154C32×C4—21no—
155C32×C4—41no—
156C32×C4—41no—
157C32×C4—122no—
158C32×C4C2×Dic611yescharacteristic
159C32×C4C2×Dic612yes—
160C32×C4D3×Q811yescharacteristic
161C32×C4D3×Q812yes—
162C3⋊Dic3—42no—
163C3⋊Dic3—61no—
164C3⋊Dic3—61no—
165C3⋊Dic3—62no—
166C3⋊Dic3—62no—
167C3⋊Dic3—62no—
168C3⋊Dic3—62no—
169C3⋊Dic3—122no—
170C3⋊Dic3—122no—
171C3⋊Dic3—122no—
172C3⋊Dic3—122no—
173C3×Dic3—44no—
174C3×Dic3—42no—
175C3×Dic3—42no—
176C3×Dic3—124no—
177C3×Dic3—122no—
178C3×Dic3—122no—
179#125—272no—
180#107—91no—
181#107—271no—
182#106—92no—
183C4×C8—271no—
184C33#49911yescharacteristic, complemented, Sylow
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—64no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—182no—
192C2×C3×C4C2×D3212yes—
193C2×C3×C4C2×D3211yescharacteristic
194C2×Dic3—22no—
195C2×Dic3—64no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×Dic3—91no—
199C2×Dic3—91no—
200C2×Dic3—92no—
201C2×Dic3—92no—
202C2×Dic3—181no—
203C2×Dic3—181no—
204C2×Dic3—182no—
205C2×Dic3—182no—
206C2×Dic3—361no—
207C2×Dic3—361no—
208C3×C8—34no—
209C3×C8—62no—
210C3×C8—92no—
211C3⋊C8—34no—
212C3⋊C8—62no—
213C3⋊C8#62712yes—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—22no—
217C2×C32—21no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32#88811yescharacteristic
224C2×C32#88812yes—
225C2×C32#91211yescharacteristic
226C2×C32#91612yes—
227C2×C32#91612yes—
228C2×C32#91911yescharacteristic
229C2×C8—31no—
230C2×C8—91no—
231C4⋊C4—32no—
232C4⋊C4—92no—
233C4⋊C4—542no—
234C4⋊C4—542no—
235C42—271no—
236C22×C3—22no—
237C22×C3—21no—
238C22×C3—41no—
239C22×C3#380211yescharacteristic
240C22×C3#381112yes—
241C3×C4—21no—
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—22no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—124no—
248C3×C4—122no—
249C3×C4—122no—
250C3×C4—362no—
251C3×C4D3×Dic612yes—
252C3×C4D3×Dic612yes—
253C3×C4#379711yescharacteristic
254C3×C4C2×#62711yescharacteristic
255Dic3—42no—
256Dic3—124no—
257Dic3—122no—
258Dic3—122no—
259Dic3—182no—
260Dic3—182no—
261Dic3—181no—
262Dic3—181no—
263Dic3—182no—
264Dic3—182no—
265Dic3—362no—
266Dic3—362no—
267Dic3—362no—
268Dic3—362no—
269C32—21no—
270C32—22no—
271C32—41no—
272C32#485812yescomplemented
273C32#487811yescharacteristic, complemented
274C2×C4—62no—
275C2×C4—182no—
276C2×C4—271no—
277C2×C4—271no—
278C2×C4D3311yescharacteristic
279C8—32no—
280C8—92no—
281C2×C3—21no—
282C2×C3—22no—
283C2×C3—21no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—21no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3#6380911yescharacteristic
291C2×C3#6380911yescharacteristic
292C2×C3#6384012yes—
293C2×C3#6391911yescharacteristic
294C2×C3#6393612yes—
295C2×C3#6395012yes—
296C22#8895411yescharacteristic
297C4—122no—
298C4—362no—
299C4—541no—
300C4—541no—
301C4D3×#62711yescharacteristic
302C4#8895711yescharacteristic
303C3—21no—
304C3—22no—
305C3—41no—
306C3#1059041712yescomplemented
307C3#1059087411yescharacteristic, complemented
308C2#1172286111yescharacteristic
309C2#1172286311yescharacteristic
310C2#1172303711yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#499
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
184. C33#499semidirect
272. C32#4858semidirect
273. C32#4878semidirect
306. C3#10590417semidirect
307. C3#10590874semidirect

Maximal subgroups

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

subgroupindex
#117234502
#117220802
#117220182
C3×#635762
C3×#635732
#105908743
#105904173