Finite Group Databasebeta

← #13167075

Subgroups — #13167075

Order 1728.

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

normal subgroups only

This group has 368 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
1#11723678C211yesmaximal, characteristic
2C3×#63870C211yesmaximal, characteristic
3#11723102C211yesmaximal, characteristic
4#11722779C212yesmaximal
5#11722778C212yesmaximal
6#10593392—31nomaximal
7C2×C3×#634—21no—
8C2×C3⋊#622C2212yes—
9C3×C4×C3⋊Dic3C2211yescharacteristic
10C32×Dic3⋊C4—21no—
11#88815—21no—
12#88815C2212yes—
13C3×#3724C2211yescharacteristic
14C3×#3724C2211yescharacteristic
15C3×Dic3⋊Dic3—21no—
16C3×#889—61no—
17#63873—31no—
18#63870D311yescharacteristic
19#63866—61no—
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
27C3×#634—42no—
28C2×C32×Dic3—21no—
29C2×C32×Dic3—21no—
30C3⋊#622Q814yescomplemented
31#5064—91nomaximal
32C2×#634—21no—
33C2×C3×Dic6—61no—
34C2×C3×Dic6—61no—
35C2×C3×Dic6—121no—
36C2×#622—32no—
37C32×C4⋊C4—61no—
38C4×C3⋊Dic3C2×D311yescharacteristic
39C3×C4⋊Dic3—61no—
40C3×Dic3⋊C4—21no—
41C3×Dic3⋊C4—21no—
42C3×Dic3⋊C4—41no—
43C3×Dic3⋊C4—61no—
44C3×Dic3⋊C4—61no—
45C3×Dic3⋊C4—61no—
46C3×C4×Dic3—61no—
47C3×C4×Dic3—61no—
48#3724—32no—
49#3724—61no—
50#3724C2×D311yescharacteristic
51#3724C2×D311yescharacteristic
52Dic3⋊Dic3—21no—
53Dic3⋊Dic3—62no—
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#944—181no—
64#889—61no—
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
79#634—42no—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—21no—
82C2×C3×Dic3—21no—
83C2×C3×Dic3—21no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—122no—
93C3×Dic6—122no—
94C3×Dic6—122no—
95C3×Dic6—124no—
96#622—34no—
97#501—271noSylow
98C2×C33C4⋊Q811yescharacteristic
99C2×C33AGL(1,ℤ/8)11yescharacteristic
100C2×C33#13711yescharacteristic
101C2×C3×Q8—181no—
102C2×Dic6—61no—
103C2×Dic6—61no—
104C2×Dic6—121no—
105C3×C4⋊C4—61no—
106C3×C4⋊C4—61no—
107C3×C4⋊C4—91no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—121no—
110C3×C4⋊C4—181no—
111C3×C42—91no—
112C4⋊Dic3—61no—
113C4⋊Dic3—92no—
114C4⋊Dic3—181no—
115Dic3⋊C4—21no—
116Dic3⋊C4—61no—
117Dic3⋊C4—61no—
118Dic3⋊C4—61no—
119Dic3⋊C4—182no—
120Dic3⋊C4—182no—
121C4×Dic3—61no—
122C4×Dic3—61no—
123C2×C3⋊C8—92no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—41no—
127C22×C32—41no—
128C22×C32C2×#6711yescharacteristic
129C32×C4—22no—
130C32×C4—21no—
131C32×C4—41no—
132C32×C4—42no—
133C32×C4—81no—
134C32×C4—121no—
135C32×C4—121no—
136C3⋊Dic3—21no—
137C3⋊Dic3—42no—
138C3⋊Dic3D3×Q812yes—
139C3×Dic3—41no—
140C3×Dic3—41no—
141C3×Dic3—42no—
142C3×Dic3—41no—
143C3×Dic3—41no—
144C3×Dic3—62no—
145C3×Dic3—62no—
146C3×Dic3—62no—
147C3×Dic3—81no—
148C3×Dic3—121no—
149C3×Dic3—122no—
150C3×Dic3—124no—
151C3×Dic3—122no—
152C4⋊Q8—271no—
153#125—91no—
154#107—272no—
155#106—272no—
156#97—271no—
157C33#50111yescharacteristic, complemented, Sylow
158C3×Q8—362no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—61no—
166C2×C3×C4—61no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4—121no—
170C2×C3×C4—121no—
171C2×C3×C4—182no—
172C2×C3×C4GO+(4,𝔽2)11yescharacteristic
173C2×Dic3—21no—
174C2×Dic3—21no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—122no—
182Dic6—122no—
183Dic6—122no—
184Dic6—124no—
185C3⋊C8—94no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32#94411yescharacteristic
199C2×C32#94911yescharacteristic
200C2×C32#95111yescharacteristic
201C2×Q8—181no—
202C2×C8—272no—
203C4⋊C4—61no—
204C4⋊C4—91no—
205C4⋊C4—91no—
206C4⋊C4—181no—
207C4⋊C4—272no—
208C4⋊C4—541no—
209C42—91no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3C2×GO+(4,𝔽2)11yescharacteristic
215C3×C4—22no—
216C3×C4—21no—
217C3×C4—41no—
218C3×C4—42no—
219C3×C4—81no—
220C3×C4—92no—
221C3×C4—121no—
222C3×C4—121no—
223C3×C4—122no—
224C3×C4—121no—
225C3×C4—121no—
226C3×C4—181no—
227C3×C4—241no—
228C3×C4—362no—
229Dic3—41no—
230Dic3—41no—
231Dic3—62no—
232Dic3—62no—
233Dic3—62no—
234Dic3—121no—
235Dic3—122no—
236Dic3—124no—
237Dic3—122no—
238C32—21no—
239C32—21no—
240C32—41no—
241C32—41no—
242C32#506411yescharacteristic, complemented
243Q8—362no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—91no—
247C2×C4—91no—
248C2×C4—182no—
249C2×C4D32⋊D311yescharacteristic
250C8—274no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—41no—
261C2×C3—41no—
262C2×C3—41no—
263C2×C3#6422211yescharacteristic
264C2×C3#6422611yescharacteristic
265C2×C3#6423611yescharacteristic
266C22C2×D32⋊D311yescharacteristic
267C4—21no—
268C4—92no—
269C4—121no—
270C4—121no—
271C4—181no—
272C4—362no—
273C3—21no—
274C3—21no—
275C3—41no—
276C3—41no—
277C3#1059339211yescharacteristic, complemented
278C2#1172430211yescharacteristic
279C2#1172430911yescharacteristic
280C2#1172432211yescharacteristic
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
157. C33#501semidirect
242. C32#5064semidirect
277. C3#10593392semidirect

Maximal subgroups

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

subgroupindex
#117236782
C3×#638702
#117231022
#117227792
#117227782
#105933923
#50649