Finite Group Databasebeta

← #13167089

Subgroups — #13167089

Order 1728.

384 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 67 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×#63919C211yesmaximal, characteristic
3C4×C3⋊#622C211yesmaximal, characteristic
4#11722779C214yesmaximal
5#10593395—31nomaximal
6C2×#6296—21no—
7C2×C3×#627—21no—
8C2×C3⋊#622C2212yes—
9C3×C4⋊(C3⋊Dic3)—21no—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×C4⋊Dic3—21no—
12#88815C2212yes—
13C3×#3724C2212yes—
14C3×#888—61no—
15#63919—31no—
16#63919D311yescharacteristic
17#63918—61no—
18C4×#622—31no—
19#63743—34no—
20C2×C33×C4D411yescharacteristic
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C2×C32×Dic3—22no—
25#6296—42no—
26C3×#627—42no—
27C3⋊#622Q814yescomplemented
28#5078—91nomaximal
29C2×C3×Dic6—62no—
30C2×#627—21no—
31C2×#627—62no—
32C2×#627—61no—
33C2×#622—32no—
34C32×C4⋊C4—61no—
35C4⋊(C3⋊Dic3)—21no—
36C4×C3⋊Dic3C2×D311yescharacteristic
37C3×C4⋊Dic3—21no—
38C3×C4⋊Dic3—21no—
39C3×C4⋊Dic3—41no—
40C3×C4⋊Dic3—61no—
41C3×C4⋊Dic3—61no—
42C3×C4⋊Dic3—121no—
43C3×Dic3⋊C4—62no—
44C3×C4×Dic3—61no—
45C3×C4×Dic3—61no—
46#3724—32no—
47#3724C2×D312yes—
48Dic3⋊Dic3—62no—
49C22×C33C2×D411yescharacteristic
50C33×C4Q1612yes—
51C3×C3⋊Dic3—42no—
52C3×C3⋊Dic3D812yes—
53C3×C3⋊Dic3C2×Q812yes—
54C32×Dic3—42no—
55C3×C4⋊Q8—91no—
56#888—61no—
57#888—181no—
58#869—91no—
59#807—94no—
60C4×C3⋊C8—91no—
61C2×C32×C4—21no—
62C2×C32×C4—21no—
63C2×C32×C4—41no—
64C2×C32×C4—41no—
65C2×C32×C4—62no—
66C2×C32×C4#6711yescharacteristic
67C2×C3⋊Dic3—22no—
68C2×C3⋊Dic3#6711yescharacteristic
69C2×C3⋊Dic3C22×D311yescharacteristic
70C2×C3×Dic3—22no—
71C2×C3×Dic3—22no—
72C2×C3×Dic3—42no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—122no—
80C3×Dic6—124no—
81#627—42no—
82#627—124no—
83#627—122no—
84#622—34no—
85#500—271noSylow
86C2×C33C4⋊Q811yescharacteristic
87C2×C33C2×D811yescharacteristic
88C2×C33C2×Q1611yescharacteristic
89C2×C3×Q8—181no—
90C2×Dic6—62no—
91C2×Dic6—181no—
92C2×Dic6—182no—
93C3×C4⋊C4—61no—
94C3×C4⋊C4—61no—
95C3×C4⋊C4—92no—
96C3×C4⋊C4—121no—
97C3×C4⋊C4—181no—
98C3×C42—91no—
99C4⋊Dic3—21no—
100C4⋊Dic3—61no—
101C4⋊Dic3—61no—
102C4⋊Dic3—92no—
103C4⋊Dic3—121no—
104Dic3⋊C4—62no—
105Dic3⋊C4—182no—
106C4×Dic3—61no—
107C4×Dic3—61no—
108C2×C3⋊C8—92no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32C2×#6711yescharacteristic
114C32×C4—22no—
115C32×C4—22no—
116C32×C4—42no—
117C32×C4—42no—
118C32×C4—122no—
119C32×C4#21612yes—
120C3⋊Dic3—42no—
121C3⋊Dic3#21312yes—
122C3⋊Dic3D3×Q812yes—
123C3×Dic3—42no—
124C3×Dic3—42no—
125C3×Dic3—62no—
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—62no—
129C3×Dic3—82no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C3×Dic3—242no—
133C4⋊Q8—91no—
134C4⋊Q8—271no—
135#107—274no—
136C4×C8—271no—
137C33#50011yescharacteristic, complemented, Sylow
138C3×Q8—362no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—62no—
144C2×C3×C4—62no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—122no—
148C2×C3×C4—182no—
149C2×C3×C4GO+(4,𝔽2)11yescharacteristic
150C2×Dic3—22no—
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—61no—
154C2×Dic3—61no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—122no—
158Dic6—124no—
159Dic6—364no—
160Dic6—362no—
161C3⋊C8—94no—
162C2×C32—21no—
163C2×C32—21no—
164C2×C32—21no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—41no—
169C2×C32—41no—
170C2×C32—41no—
171C2×C32—41no—
172C2×C32—41no—
173C2×C32—41no—
174C2×C32C2×#21311yescharacteristic
175C2×C32C2×#21611yescharacteristic
176C2×C32#94411yescharacteristic
177C2×Q8—181no—
178C2×Q8—541no—
179C2×C8—272no—
180C4⋊C4—61no—
181C4⋊C4—92no—
182C4⋊C4—181no—
183C4⋊C4—272no—
184C42—91no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3C2×GO+(4,𝔽2)11yescharacteristic
190C3×C4—22no—
191C3×C4—22no—
192C3×C4—42no—
193C3×C4—42no—
194C3×C4—92no—
195C3×C4—92no—
196C3×C4—122no—
197C3×C4—122no—
198C3×C4—242no—
199C3×C4—362no—
200C3×C4#377612yes—
201Dic3—42no—
202Dic3—62no—
203Dic3—62no—
204Dic3—62no—
205Dic3—62no—
206Dic3—122no—
207Dic3—122no—
208Dic3—242no—
209C32—21no—
210C32—21no—
211C32—41no—
212C32—41no—
213C32#507811yescharacteristic, complemented
214Q8—362no—
215Q8—1082no—
216C2×C4—62no—
217C2×C4—91no—
218C2×C4—91no—
219C2×C4—182no—
220C2×C4D32⋊D311yescharacteristic
221C8—274no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—21no—
228C2×C3—41no—
229C2×C3—41no—
230C2×C3—41no—
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3#6422211yescharacteristic
235C2×C3#6422511yescharacteristic
236C2×C3C2×#377611yescharacteristic
237C22C2×D32⋊D311yescharacteristic
238C4—92no—
239C4—92no—
240C4—122no—
241C4—362no—
242C4#8893812yes—
243C3—21no—
244C3—21no—
245C3—41no—
246C3—41no—
247C3#1059339511yescharacteristic, complemented
248C2#1172430211yescharacteristic
249C2#1172430611yescharacteristic
250C2C2×#8893811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#500
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
27. C3⋊#622Q8semidirect
137. C33#500semidirect
213. C32#5078semidirect
247. C3#10593395semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#639192
C4×C3⋊#6222
#117227792
#105933953
#50789