Finite Group Databasebeta

← #13165807

Subgroups — #13165807

Order 1728.

410 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×#88946C211yesmaximal, characteristic
2C2×#88945C212yesmaximal
3C2×C3×#3777C211yesmaximal, characteristic
4C2×Dic3×#157C211yesmaximal, characteristic
5#11722722C212yesmaximal
6#10593548—31nomaximal
7C22×C3⋊#157C2212yes—
8C22×C3⋊#157C2211yescharacteristic
9C22×C3×#157C2212yes—
10C22×C3×#157C2211yescharacteristic
11C2×Dic3×C3⋊D3C2211yescharacteristic
12#88946C414yescomplemented
13#88945—24no—
14C3×#3777—22no—
15Dic3×#157—22no—
16C2×C4×#157—31no—
17C2×#3777—31no—
18C2×#3777—32no—
19C2×#3777D311yescharacteristic
20#63781—32no—
21C22×C3×C3⋊D3C2311yescharacteristic
22C2×C3⋊#157—21no—
23C2×C3⋊#157—22no—
24C2×C3⋊#157—22no—
25C2×C3⋊#157—21no—
26C2×C3⋊#157C2×C414yes—
27C2×C3⋊#157D412yes—
28C2×C3×#157—22no—
29C2×C3×#157—22no—
30C2×C3×#157—22no—
31C2×C3×#157C2×C412yes—
32C2×C3×#157D412yes—
33C2×C3×#157Q812yes—
34C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
35C2×C32×Dic3Q811yescharacteristic
36Dic3×C3⋊D3—21no—
37Dic3×C3⋊D3—21no—
38#4972—91nomaximal
39C22×#157—31no—
40C22×#157—32no—
41C22×#157C2×D312yes—
42C22×#157C2×D311yescharacteristic
43C2×C4×C3⋊D3—31no—
44C2×D3×Dic3—121no—
45C4×#157—62no—
46#3777—22no—
47#3777—34no—
48#3777—64no—
49C22×C33C2×Q811yescharacteristic
50C2×C3×C3⋊D3C22×C411yescharacteristic
51C2×C3×C3⋊D3C2×D411yescharacteristic
52C2×C3×C3⋊D3C2×D411yescharacteristic
53C2×C3×C3⋊D3C2×Q811yescharacteristic
54C2×C3×C3⋊D3#4111yescharacteristic
55C2×C3×C3⋊D3#4111yescharacteristic
56C3⋊#157—24no—
57C3⋊#157—22no—
58C3×#157—42no—
59C3×#157C4⋊C414yescomplemented
60C3⋊(C3⋊Dic3)—21no—
61C32×Dic3—21no—
62C2×C3×C4⋊C4—91no—
63C2×C4⋊Dic3—91no—
64C2×Dic3⋊C4—92no—
65C2×C4×Dic3—91no—
66C3⋊#95—92no—
67C22×C3⋊D3C22×D311yescharacteristic
68C22×C3×D3—121no—
69C2×#157—22no—
70C2×#157—22no—
71C2×#157—22no—
72C2×#157—34no—
73C2×#157—32no—
74C2×#157—62no—
75C2×#157—61no—
76C2×#157—62no—
77C2×#157—61no—
78C2×#157Dic612yes—
79C2×#157C4×D312yes—
80C2×#157#6712yes—
81C2×C32×C4—31no—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—121no—
84C2×C3⋊Dic3—241no—
85C4×C3⋊D3—61no—
86C4×C3⋊D3—61no—
87C2×C3×Dic3—41no—
88D3×Dic3—122no—
89D3×Dic3—122no—
90#389—271noSylow
91C2×C33C4×Q811yescharacteristic
92C2×C33C22⋊Q811yescharacteristic
93C2×C33C4⋊Q811yescharacteristic
94C3×C3⋊D3C2×C4⋊C411yescharacteristic
95C3×C3⋊D3C4×D411yescharacteristic
96C3×C3⋊D3#12111yescharacteristic
97C3×C3⋊D3#12511yescharacteristic
98C22×C3×C4—91no—
99C22×C3×C4—92no—
100C22×Dic3—92no—
101C22×Dic3—91no—
102C22×Dic3—91no—
103C2×C4×D3—361no—
104C3×C4⋊C4—182no—
105C4⋊Dic3—94no—
106Dic3⋊C4—184no—
107C4×Dic3—182no—
108C22×C32—41no—
109C22×C32—81no—
110C22×C32D3×Q811yescharacteristic
111C2×C3⋊D3C2×Dic611yescharacteristic
112C2×C3⋊D3C2×C4×D311yescharacteristic
113C2×C3⋊D3D3×D411yescharacteristic
114C2×C3⋊D3#23711yescharacteristic
115C2×C3⋊D3#23911yescharacteristic
116C2×C3⋊D3C2×#6711yescharacteristic
117C2×C3×D3—121no—
118C2×C3×D3—121no—
119C2×C3×D3—121no—
120C2×C3×D3—121no—
121C2×C3×D3—121no—
122C2×C3×D3—121no—
123#157—42no—
124#157—64no—
125#157—62no—
126#157Dic3⋊C414yescomplemented
127C32×C4—61no—
128C3⋊Dic3—61no—
129C3⋊Dic3—122no—
130C3⋊Dic3—242no—
131C3×Dic3—42no—
132C2×C4⋊C4—91no—
133C2×C4⋊C4—271no—
134C2×C4⋊C4—272no—
135C2×C42—271no—
136#95—272no—
137C33#38911yescharacteristic, complemented, Sylow
138C23×C3—91no—
139C22×D3—121no—
140C2×C3×C4—92no—
141C2×C3×C4—92no—
142C2×C3×C4—92no—
143C2×C3×C4—121no—
144C2×C3×C4—182no—
145C2×C3×C4—182no—
146C2×C3×C4—182no—
147C2×Dic3—94no—
148C2×Dic3—92no—
149C2×Dic3—91no—
150C2×Dic3—181no—
151C2×Dic3—181no—
152C2×Dic3—181no—
153C2×Dic3—182no—
154C2×Dic3—181no—
155C2×Dic3—182no—
156C2×Dic3—361no—
157C2×Dic3—721no—
158C2×Dic3PSU(3,𝔽2)11yescharacteristic
159C4×D3—362no—
160C4×D3—362no—
161C2×C32—41no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—81no—
165C2×C32—81no—
166C2×C32—81no—
167C2×C32#88711yescharacteristic
168C2×C32#88811yescharacteristic
169C2×C32#94611yescharacteristic
170C3⋊D3#89011yescharacteristic
171C3⋊D3#89311yescharacteristic
172C3⋊D3C2×Dic3⋊C411yescharacteristic
173C3⋊D3#93811yescharacteristic
174C3×D3—121no—
175C3×D3—121no—
176C3×D3—121no—
177C3×D3—121no—
178C22×C4—91no—
179C22×C4—92no—
180C22×C4—271no—
181C22×C4—272no—
182C22×C4—271no—
183C4⋊C4—182no—
184C4⋊C4—274no—
185C4⋊C4—544no—
186C42—542no—
187C22×C3—41no—
188C22×C3—81no—
189C22×C3—91no—
190C22×C3—91no—
191C22×C3—91no—
192C22×C3—91no—
193C22×C3—91no—
194C22×C3—91no—
195C22×C3C2×PSU(3,𝔽2)11yescharacteristic
196C2×D3—121no—
197C2×D3—121no—
198C2×D3—121no—
199C2×D3—121no—
200C2×D3—121no—
201C2×D3—121no—
202C3×C4—94no—
203C3×C4—122no—
204C3×C4—362no—
205Dic3—21no—
206Dic3—184no—
207Dic3—182no—
208Dic3—181no—
209Dic3—362no—
210Dic3—722no—
211C32—41no—
212C32—81no—
213C32#497211yescharacteristic, complemented
214C23—91no—
215C2×C4—31no—
216C2×C4—92no—
217C2×C4—92no—
218C2×C4—92no—
219C2×C4—182no—
220C2×C4—182no—
221C2×C4—182no—
222C2×C4—274no—
223C2×C4—272no—
224C2×C4—271no—
225C2×C4—541no—
226C2×C4—541no—
227C2×C4—541no—
228C2×C4—542no—
229C2×C4—541no—
230C2×C4—542no—
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—81no—
235C2×C3—81no—
236C2×C3—81no—
237C2×C3—91no—
238C2×C3—91no—
239C2×C3—91no—
240C2×C3—91no—
241C2×C3C4×PSU(3,𝔽2)11yescharacteristic
242C2×C3C4⋊PSU(3,𝔽2)11yescharacteristic
243C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
244D3—121no—
245D3—121no—
246D3—121no—
247D3—121no—
248C22—91no—
249C22—91no—
250C22—91no—
251C22—91no—
252C22—91no—
253C22—91no—
254C22D3×PSU(3,𝔽2)11yescharacteristic
255C4—61no—
256C4—94no—
257C4—362no—
258C4—542no—
259C4—544no—
260C4—541no—
261C3—41no—
262C3—81no—
263C3#1059354811yescharacteristic, complemented
264C2—91no—
265C2—91no—
266C2—91no—
267C2—91no—
268C2#1172414811yescharacteristic
269C2#1172414911yescharacteristic
270C2#1172415411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×PSU(3,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×Q8
prime radical, cosocleC22×C3×C3⋊D3C23
hypocenterC33#389
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
12. #88946C4semidirect
59. C3×#157C4⋊C4semidirect
126. #157Dic3⋊C4semidirect
137. C33#389semidirect
213. C32#4972semidirect
263. C3#10593548semidirect

Maximal subgroups

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

subgroupindex
C2×#889462
C2×#889452
C2×C3×#37772
C2×Dic3×#1572
#117227222
#105935483
#49729