Finite Group Databasebeta

← #13156574

Subgroups — #13156574

Order 1728.

506 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723680C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3#11722037C212yesmaximal
4#11722020C211yesmaximal, characteristic
5C3×#63566C212yesmaximal
6#10591613—32nomaximal
7#10590904—31nomaximal
8C2×C32×Dic6C2212yes—
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2211yescharacteristic
13#88815—22no—
14C3×#3724—22no—
15C3×#944—32no—
16#63873D311yescharacteristic
17#63872—31no—
18#63869—32no—
19#63662—32no—
20#63642—31no—
21C3×#835—32no—
22C3×#816—32no—
23#63566D312yes—
24#63565—32no—
25#63565—32no—
26#63565—62no—
27#63554—32no—
28#63554—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3D412yes—
32C2×C32×Dic3—22no—
33C32×Dic6—22no—
34C32×Dic6—22no—
35C3⋊(C3⋊(C3⋊C8))—21no—
36C32×C3⋊C8—21no—
37#5082—91no—
38#4797—92no—
39C2×C32×Q8—32no—
40C2×C3×Dic6—22no—
41C2×C3×Dic6C2×D312yes—
42C2×C3×Dic6C2×D312yes—
43C2×C32×C8—31no—
44C4⋊(C3⋊Dic3)C2×D312yes—
45C4×C3⋊Dic3C2×D311yescharacteristic
46C2×C3⋊(C3⋊C8)—32no—
47C2×C3⋊(C3⋊C8)—31no—
48C2×C3⋊(C3⋊C8)—61no—
49C2×C3⋊(C3⋊C8)—62no—
50C2×C3⋊(C3⋊C8)—121no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—62no—
54C3×Dic3⋊C4—62no—
55C3×Dic3⋊C4—62no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—61no—
58C2×C3×C3⋊C8—21no—
59C2×C3×C3⋊C8C2×D312yes—
60#3724—22no—
61#3724—62no—
62Dic3⋊Dic3—62no—
63Dic3⋊Dic3—62no—
64C22×C33C2×D411yescharacteristic
65C33×C4#4111yescharacteristic
66C33×C4#4111yescharacteristic
67C3×C3⋊Dic3—22no—
68C3×C3⋊Dic3—42no—
69C32×Dic3—42no—
70C3×C4⋊Q8—91no—
71#944—32no—
72#890—92no—
73#870—91no—
74C3×#103—92no—
75#835—32no—
76#835—92no—
77#835—92no—
78#835—182no—
79#816—32no—
80#815—92no—
81#815—181no—
82#810—92no—
83#803—91no—
84C32×Q8—62no—
85C32×Q8—62no—
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#6712yes—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—32no—
97C2×C3×Dic3—32no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—122no—
103C3×Dic6—24no—
104C3×Dic6—22no—
105C3×Dic6—22no—
106C3×Dic6—22no—
107C3×Dic6—22no—
108C3×Dic6—42no—
109C32×C8—61no—
110C3⋊(C3⋊C8)—61no—
111C3⋊(C3⋊C8)—62no—
112C3⋊(C3⋊C8)—62no—
113C3⋊(C3⋊C8)—122no—
114C3⋊(C3⋊C8)—122no—
115C3×C3⋊C8—22no—
116C3×C3⋊C8—22no—
117#491—271noSylow
118C2×C33#12411yescharacteristic
119C2×C33#13712yes—
120C2×C3×Q8—32no—
121C2×C3×Q8—32no—
122C2×C3×Q8—62no—
123C2×Dic6D3212yes—
124C2×C3×C8—32no—
125C2×C3×C8—61no—
126C3×C4⋊C4—92no—
127C3×C4⋊C4—182no—
128C3×C42—91no—
129C4⋊Dic3—32no—
130C4⋊Dic3—32no—
131C4⋊Dic3—62no—
132C4⋊Dic3—182no—
133Dic3⋊C4—62no—
134Dic3⋊C4—62no—
135Dic3⋊C4—182no—
136Dic3⋊C4—182no—
137C4×Dic3—32no—
138C4×Dic3—61no—
139C2×C3⋊C8—91no—
140C2×C3⋊C8—92no—
141C2×C3⋊C8—181no—
142C2×C3⋊C8—182no—
143C2×C3⋊C8—361no—
144C2×C3⋊C8D3211yescharacteristic
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—41no—
148C22×C32D3×D412yes—
149C22×C32C2×#6711yescharacteristic
150C32×C4—21no—
151C32×C4—22no—
152C32×C4—21no—
153C32×C4—22no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4—122no—
157C32×C4#23512yes—
158C32×C4#23711yescharacteristic
159C32×C4#23712yes—
160C32×C4#23711yescharacteristic
161C3⋊Dic3—22no—
162C3⋊Dic3—42no—
163C3×Dic3—44no—
164C3×Dic3—42no—
165C3×Dic3—42no—
166C3×Dic3—62no—
167C3×Dic3—62no—
168C3×Dic3—122no—
169C3×Dic3—122no—
170C3×Dic3—122no—
171C3×Dic3—242no—
172C4⋊Q8—91no—
173#125—271no—
174#103—92no—
175#103—272no—
176#97—271no—
177C33#49111yescharacteristic, complemented, Sylow
178C3×Q8—64no—
179C3×Q8—62no—
180C3×Q8—62no—
181C3×Q8—62no—
182C3×Q8—62no—
183C3×Q8—122no—
184C2×C3×C4—21no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—92no—
190C2×C3×C4—122no—
191C2×C3×C4—182no—
192C2×C3×C4C2×D3211yescharacteristic
193C2×C3×C4C2×D3212yes—
194C2×Dic3—22no—
195C2×Dic3—32no—
196C2×Dic3—32no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—122no—
201Dic6—22no—
202Dic6—22no—
203C3×C8—62no—
204C3×C8—62no—
205C3⋊C8—21no—
206C3⋊C8—182no—
207C3⋊C8—181no—
208C3⋊C8—182no—
209C3⋊C8—362no—
210C3⋊C8—362no—
211C2×C32—22no—
212C2×C32—21no—
213C2×C32—22no—
214C2×C32—22no—
215C2×C32—22no—
216C2×C32—41no—
217C2×C32—42no—
218C2×C32#88512yes—
219C2×C32#91512yes—
220C2×C32#91812yes—
221C2×C32#93611yescharacteristic
222C2×C32#94212yes—
223C2×Q8—32no—
224C2×C8—31no—
225C2×C8—271no—
226C4⋊C4—92no—
227C4⋊C4—182no—
228C4⋊C4—542no—
229C42—91no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—41no—
233C22×C3#380211yescharacteristic
234C22×C3D3×#6712yes—
235C3×C4—21no—
236C3×C4—22no—
237C3×C4—21no—
238C3×C4—22no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—124no—
242C3×C4—122no—
243C3×C4—122no—
244C3×C4—182no—
245C3×C4—362no—
246C3×C4#379811yescharacteristic
247C3×C4#380412yes—
248C3×C4#380511yescharacteristic
249C3×C4#380512yes—
250Dic3—42no—
251Dic3—62no—
252Dic3—62no—
253Dic3—122no—
254Dic3—122no—
255Dic3—122no—
256Dic3—242no—
257C32—22no—
258C32—21no—
259C32—41no—
260C32#479712yescomplemented
261C32#508211yescharacteristic, complemented
262Q8—62no—
263Q8—62no—
264C2×C4—62no—
265C2×C4—92no—
266C2×C4—182no—
267C2×C4D3311yescharacteristic
268C8—61no—
269C8—541no—
270C2×C3—22no—
271C2×C3—22no—
272C2×C3—22no—
273C2×C3—22no—
274C2×C3—21no—
275C2×C3—42no—
276C2×C3—41no—
277C2×C3#6381112yes—
278C2×C3#6391511yescharacteristic
279C2×C3#6394612yes—
280C2×C3#6394812yes—
281C2×C3#6395912yes—
282C22#8895311yescharacteristic
283C4—122no—
284C4—182no—
285C4—362no—
286C4#8895911yescharacteristic
287C4#8896311yescharacteristic
288C3—22no—
289C3—21no—
290C3—41no—
291C3#1059090411yescharacteristic, complemented
292C3#1059161312yescomplemented
293C2#1172289212yes—
294C2#1172300611yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#491
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
177. C33#491semidirect
260. C32#4797semidirect
261. C32#5082semidirect
291. C3#10590904semidirect
292. C3#10591613semidirect

Maximal subgroups

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

subgroupindex
#117236802
C3×#638732
#117220372
#117220202
C3×#635662
#105916133
#105909043