Finite Group Databasebeta

← #13167041

Subgroups — #13167041

Order 1728.

356 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 51 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#11723679C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3#11723102C211yesmaximal, characteristic
4#11722775C212yesmaximal
5#11722775C212yesmaximal
6#10593378—31nomaximal
7C2×C3×#627C2212yes—
8C2×C3⋊#622C2212yes—
9C3×#3750—21no—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×C4⋊Dic3—21no—
12#88815—21no—
13#88815C2211yescharacteristic
14#88815C2211yescharacteristic
15C3×#3724—21no—
16C3×#888—61no—
17#63919D311yescharacteristic
18#63870—31no—
19#63867—61no—
20#63777—31no—
21#63740—32no—
22#63740—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—22no—
29C3×#627—24no—
30C3⋊#622—22no—
31#5082—91nomaximal
32C2×C3×Dic6—62no—
33C2×#627C2×D312yes—
34C2×#622—32no—
35C32×C4⋊C4—61no—
36#3750—21no—
37C4×C3⋊Dic3C2×D311yescharacteristic
38C3×C4⋊Dic3—21no—
39C3×C4⋊Dic3—21no—
40C3×C4⋊Dic3—41no—
41C3×Dic3⋊C4—62no—
42C3×Dic3⋊C4—61no—
43C3×Dic3⋊C4—61no—
44C3×Dic3⋊C4—121no—
45C3×C4×Dic3—61no—
46C3×C4×Dic3—61no—
47#3724—21no—
48#3724—31no—
49#3724—31no—
50#3724—61no—
51Dic3⋊Dic3—61no—
52Dic3⋊Dic3—61no—
53Dic3⋊Dic3—61no—
54Dic3⋊Dic3—61no—
55C22×C33C2×D411yescharacteristic
56C33×C4—21no—
57C3×C3⋊Dic3—21no—
58C3×C3⋊Dic3—41no—
59C3×C3⋊Dic3—41no—
60C3×C3⋊Dic3#4112yes—
61C32×Dic3—42no—
62C3×C4⋊Q8—91no—
63#889—181no—
64#888—61no—
65#870—91no—
66#835—92no—
67#835—92no—
68#803—91no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4—62no—
74C2×C32×C4#6711yescharacteristic
75C2×C3⋊Dic3—21no—
76C2×C3⋊Dic3—21no—
77C2×C3⋊Dic3#6711yescharacteristic
78C2×C3⋊Dic3C22×D311yescharacteristic
79C2×C3×Dic3—22no—
80C2×C3×Dic3—22no—
81C2×C3×Dic3—42no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—121no—
91C2×C3×Dic3—121no—
92C3×Dic6—124no—
93#627—24no—
94#622—62no—
95#491—271noSylow
96C2×C33#12411yescharacteristic
97C2×C33#13711yescharacteristic
98C2×C33#13711yescharacteristic
99C2×C3×Q8—92no—
100C2×Dic6—62no—
101C3×C4⋊C4—61no—
102C3×C4⋊C4—61no—
103C3×C4⋊C4—121no—
104C3×C4⋊C4—181no—
105C3×C4⋊C4—181no—
106C3×C42—91no—
107C4⋊Dic3—21no—
108C4⋊Dic3—91no—
109C4⋊Dic3—91no—
110C4⋊Dic3—181no—
111Dic3⋊C4—62no—
112Dic3⋊C4—61no—
113Dic3⋊C4—61no—
114Dic3⋊C4—121no—
115Dic3⋊C4—181no—
116Dic3⋊C4—181no—
117Dic3⋊C4—181no—
118Dic3⋊C4—181no—
119C4×Dic3—61no—
120C4×Dic3—61no—
121C2×C3⋊C8—92no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—41no—
125C22×C32—41no—
126C22×C32C2×#6711yescharacteristic
127C32×C4—22no—
128C32×C4—21no—
129C32×C4—41no—
130C32×C4—42no—
131C32×C4—81no—
132C32×C4—122no—
133C3⋊Dic3—21no—
134C3⋊Dic3—41no—
135C3⋊Dic3—41no—
136C3⋊Dic3#23712yes—
137C3×Dic3—42no—
138C3×Dic3—42no—
139C3×Dic3—62no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—82no—
143C3×Dic3—121no—
144C3×Dic3—121no—
145C3×Dic3—121no—
146C3×Dic3—122no—
147C3×Dic3—121no—
148C3×Dic3—121no—
149C3×Dic3—241no—
150C4⋊Q8—91no—
151#125—271no—
152#103—272no—
153#103—272no—
154#97—271no—
155C33#49111yescharacteristic, complemented, Sylow
156C3×Q8—184no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—62no—
162C2×C3×C4—62no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—122no—
166C2×C3×C4—181no—
167C2×C3×C4—181no—
168C2×C3×C4GO+(4,𝔽2)11yescharacteristic
169C2×Dic3—22no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—121no—
179C2×Dic3—121no—
180Dic6—124no—
181C3⋊C8—182no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32#93611yescharacteristic
195C2×C32#94211yescharacteristic
196C2×C32#94211yescharacteristic
197C2×Q8—92no—
198C2×C8—272no—
199C4⋊C4—61no—
200C4⋊C4—181no—
201C4⋊C4—181no—
202C4⋊C4—271no—
203C4⋊C4—271no—
204C4⋊C4—541no—
205C42—91no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3C2×GO+(4,𝔽2)11yescharacteristic
211C3×C4—22no—
212C3×C4—21no—
213C3×C4—41no—
214C3×C4—42no—
215C3×C4—81no—
216C3×C4—92no—
217C3×C4—122no—
218C3×C4—122no—
219C3×C4—181no—
220C3×C4—242no—
221C3×C4—361no—
222C3×C4—361no—
223Dic3—42no—
224Dic3—62no—
225Dic3—62no—
226Dic3—62no—
227Dic3—121no—
228Dic3—121no—
229Dic3—121no—
230Dic3—122no—
231Dic3—121no—
232Dic3—121no—
233Dic3—241no—
234C32—21no—
235C32—21no—
236C32—41no—
237C32—41no—
238C32#508211yescharacteristic, complemented
239Q8—184no—
240C2×C4—62no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4D32⋊D311yescharacteristic
246C8—542no—
247C2×C3—21no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—41no—
254C2×C3—41no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3#6422111yescharacteristic
260C2×C3#6422611yescharacteristic
261C2×C3#6423911yescharacteristic
262C22C2×D32⋊D311yescharacteristic
263C4—21no—
264C4—92no—
265C4—122no—
266C4—181no—
267C4—361no—
268C4—361no—
269C3—21no—
270C3—21no—
271C3—41no—
272C3—41no—
273C3#1059337811yescharacteristic, complemented
274C2#1172430111yescharacteristic
275C2#1172430711yescharacteristic
276C2#1172432611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#491
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
155. C33#491semidirect
238. C32#5082semidirect
273. C3#10593378semidirect

Maximal subgroups

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

subgroupindex
#117236792
C3×#639192
#117231022
#117227752
#105933783
#50829