Finite Group Databasebeta

← #13167085

Subgroups — #13167085

Order 1728.

356 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 55 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
2C3×#63870C211yesmaximal, characteristic
3#11723102C211yesmaximal, characteristic
4#11722779C212yesmaximal
5#11722778C212yesmaximal
6#10593383—31nomaximal
7C2×#6296—21no—
8C2×C3⋊#622C2212yes—
9C3×C4⋊(C3⋊Dic3)—21no—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×Dic3⋊C4—21no—
12#88815C2212yes—
13C3×#3724C2211yescharacteristic
14C3×#3724C2211yescharacteristic
15C3×Dic3⋊Dic3—21no—
16C3×#889—61no—
17#63919—31no—
18#63918—61no—
19#63870D311yescharacteristic
20#63777—31no—
21#63743—32no—
22#63742—32no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—21no—
28C2×C32×Dic3—21no—
29#6296—42no—
30C3⋊#622Q814yescomplemented
31#5064—91nomaximal
32C2×#627—62no—
33C2×#627—61no—
34C2×#622—32no—
35C32×C4⋊C4—61no—
36C4⋊(C3⋊Dic3)—21no—
37C4×C3⋊Dic3C2×D311yescharacteristic
38C3×C4⋊Dic3—61no—
39C3×C4⋊Dic3—61no—
40C3×C4⋊Dic3—61no—
41C3×C4⋊Dic3—121no—
42C3×Dic3⋊C4—21no—
43C3×Dic3⋊C4—21no—
44C3×Dic3⋊C4—41no—
45C3×Dic3⋊C4—61no—
46C3×Dic3⋊C4—61no—
47C3×Dic3⋊C4—61no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50#3724—32no—
51#3724C2×D311yescharacteristic
52#3724C2×D311yescharacteristic
53Dic3⋊Dic3—21no—
54Dic3⋊Dic3—62no—
55C22×C33C2×D411yescharacteristic
56C33×C4—21no—
57C3×C3⋊Dic3—21no—
58C3×C3⋊Dic3—42no—
59C3×C3⋊Dic3C2×Q812yes—
60C32×Dic3—41no—
61C32×Dic3—41no—
62C3×#125—91no—
63#889—61no—
64#888—181no—
65#869—91no—
66#808—92no—
67#807—92no—
68#803—91no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4—61no—
74C2×C32×C4—61no—
75C2×C32×C4#6711yescharacteristic
76C2×C3⋊Dic3—22no—
77C2×C3⋊Dic3#6711yescharacteristic
78C2×C3⋊Dic3C22×D311yescharacteristic
79C2×C3×Dic3—21no—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—21no—
82C2×C3×Dic3—21no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—122no—
92#627—124no—
93#627—122no—
94#622—34no—
95#501—271noSylow
96C2×C33C4⋊Q811yescharacteristic
97C2×C33AGL(1,ℤ/8)11yescharacteristic
98C2×C33#13711yescharacteristic
99C2×Dic6—181no—
100C2×Dic6—182no—
101C3×C4⋊C4—61no—
102C3×C4⋊C4—61no—
103C3×C4⋊C4—91no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—121no—
106C3×C4⋊C4—181no—
107C3×C4⋊C4—181no—
108C3×C42—91no—
109C4⋊Dic3—61no—
110C4⋊Dic3—61no—
111C4⋊Dic3—61no—
112C4⋊Dic3—92no—
113C4⋊Dic3—121no—
114Dic3⋊C4—21no—
115Dic3⋊C4—61no—
116Dic3⋊C4—61no—
117Dic3⋊C4—61no—
118Dic3⋊C4—182no—
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—121no—
133C32×C4—121no—
134C3⋊Dic3—21no—
135C3⋊Dic3—42no—
136C3⋊Dic3D3×Q812yes—
137C3×Dic3—41no—
138C3×Dic3—41no—
139C3×Dic3—42no—
140C3×Dic3—41no—
141C3×Dic3—41no—
142C3×Dic3—62no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C3×Dic3—81no—
146C3×Dic3—121no—
147C3×Dic3—122no—
148C3×Dic3—122no—
149C3×Dic3—242no—
150C4⋊Q8—271no—
151#125—91no—
152#107—272no—
153#106—272no—
154#97—271no—
155C33#50111yescharacteristic, complemented, Sylow
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—121no—
167C2×C3×C4—121no—
168C2×C3×C4—182no—
169C2×C3×C4GO+(4,𝔽2)11yescharacteristic
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—122no—
179Dic6—364no—
180Dic6—362no—
181C3⋊C8—94no—
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#94411yescharacteristic
195C2×C32#94911yescharacteristic
196C2×C32#95111yescharacteristic
197C2×Q8—541no—
198C2×C8—272no—
199C4⋊C4—61no—
200C4⋊C4—91no—
201C4⋊C4—91no—
202C4⋊C4—181no—
203C4⋊C4—181no—
204C4⋊C4—272no—
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—121no—
218C3×C4—121no—
219C3×C4—122no—
220C3×C4—121no—
221C3×C4—121no—
222C3×C4—181no—
223C3×C4—241no—
224C3×C4—362no—
225Dic3—41no—
226Dic3—41no—
227Dic3—62no—
228Dic3—62no—
229Dic3—62no—
230Dic3—121no—
231Dic3—122no—
232Dic3—122no—
233Dic3—242no—
234C32—21no—
235C32—21no—
236C32—41no—
237C32—41no—
238C32#506411yescharacteristic, complemented
239Q8—1082no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—182no—
245C2×C4D32⋊D311yescharacteristic
246C8—274no—
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#6422211yescharacteristic
260C2×C3#6422411yescharacteristic
261C2×C3#6423911yescharacteristic
262C22C2×D32⋊D311yescharacteristic
263C4—21no—
264C4—92no—
265C4—121no—
266C4—121no—
267C4—181no—
268C4—362no—
269C3—21no—
270C3—21no—
271C3—41no—
272C3—41no—
273C3#1059338311yescharacteristic, complemented
274C2#1172430211yescharacteristic
275C2#1172430511yescharacteristic
276C2#1172432811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#501
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
30. C3⋊#622Q8semidirect
155. C33#501semidirect
238. C32#5064semidirect
273. C3#10593383semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#638702
#117231022
#117227792
#117227782
#105933833
#50649