Finite Group Databasebeta

← #13146554

Subgroups — #13146554

Order 1728.

324 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 41 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#11721138C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic, complemented
3#11720785C211yesmaximal, characteristic, complemented
4#11720705C212yesmaximal
5#11720704C212yesmaximal, complemented
6C2×#6198C2211yescharacteristic
7C2×((C3⋊D3)⋊C3)⋊C4—21no—
8C2×#6196—21no—
9C2×#6188C2212yes—
10C4×#1084C2211yescharacteristic
11#3752⋊C3—21no—
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13#88449C2211yescharacteristic
14#88449C2212yes—
15#63918—61no—
16#63880—61no—
17C22×(C3⋊D3)⋊C3—21no—
18C2×C4×H(3,𝔽3)D411yescharacteristic
19C2×#1084D411yescharacteristic
20C2×#1084C2311yescharacteristic
21C2×(C3⋊Dic3)⋊C3—22no—
22C2×(C3⋊Dic3)⋊C3—21no—
23#6198—21no—
24#6198Q812yescomplemented
25((C3⋊D3)⋊C3)⋊C4—42no—
26#6196—42no—
27#6188—22no—
28#4846—91nomaximal
29C2×#627—62no—
30C2×#626—61no—
31C2×#624—61no—
32C2×D3×Dic3—61no—
33#3752—21no—
34C4⋊(C3⋊Dic3)—21no—
35C3×(C2×D3)⋊C4—61no—
36C3×C4⋊Dic3—61no—
37C3×C4×Dic3—61no—
38C3×C4×Dic3—61no—
39Dic3⋊Dic3—61no—
40Dic3⋊Dic3—62no—
41C22×H(3,𝔽3)C2×D411yescharacteristic
42C2×(C3⋊D3)⋊C3—22no—
43C2×(C3⋊D3)⋊C3—41no—
44C2×(C3⋊D3)⋊C3—41no—
45C4×H(3,𝔽3)—21no—
46#1084D812yes—
47#1084C2×Q811yescharacteristic
48#1084#4111yescharacteristic
49(C3⋊Dic3)⋊C3—42no—
50(C3⋊Dic3)⋊C3—41no—
51#888—181no—
52#67⋊C4—181no—
53C4×D12—91no—
54#869—91no—
55C3×C4⋊C8—91no—
56#821—92no—
57#818—92no—
58C22×C3⋊D3—21no—
59C22×C3×D3—61no—
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C3⋊Dic3—22no—
63C2×C3⋊Dic3—21no—
64C2×C3×Dic3—61no—
65C2×C3×Dic3—61no—
66C2×C3×Dic3—61no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—61no—
70#627—124no—
71#626—62no—
72#626—62no—
73#624—122no—
74D3×Dic3—122no—
75#474—271noSylow
76C2×H(3,𝔽3)C22⋊Q811yescharacteristic
77C2×H(3,𝔽3)C2×D811yescharacteristic
78C2×H(3,𝔽3)#13711yescharacteristic
79(C3⋊D3)⋊C3—42no—
80C2×#67—181no—
81C22×Dic3—181no—
82C2×D12—91no—
83C2×C4×D3—181no—
84C2×C4×D3—181no—
85C2×Dic6—181no—
86C2×Dic6—182no—
87C2×C3×C8—92no—
88C3×C4⋊C4—181no—
89C3×C22⋊C4—181no—
90C3×C42—91no—
91(C2×D3)⋊C4—61no—
92(C2×D3)⋊C4—181no—
93C4⋊Dic3—61no—
94C4⋊Dic3—91no—
95C4⋊Dic3—92no—
96C4⋊Dic3—181no—
97Dic3⋊C4—181no—
98Dic3⋊C4—182no—
99C4×Dic3—61no—
100C4×Dic3—61no—
101C22×C32—21no—
102C22×C32—21no—
103C2×C3⋊D3—22no—
104C2×C3⋊D3—41no—
105C2×C3⋊D3—41no—
106C2×C3×D3—62no—
107C2×C3×D3—121no—
108C2×C3×D3—121no—
109C32×C4—22no—
110C32×C4—41no—
111C3⋊Dic3—41no—
112C3⋊Dic3—42no—
113C3×Dic3—62no—
114C3×Dic3—62no—
115C3×Dic3—61no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118C3×Dic3—61no—
119C3×Dic3—121no—
120C3×Dic3—122no—
121C4⋊Q8—271no—
122C4×D4—271no—
123#107—272no—
124C4⋊C8—91no—
125#102—272no—
126H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
127C23×C3—181no—
128C22×D3—61no—
129C22×D3—181no—
130C2×C3×C4—61no—
131C2×C3×C4—61no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—181no—
135C2×C3×C4—182no—
136C2×C3×C4GO+(4,𝔽2)11yescharacteristic
137#67—182no—
138#67—182no—
139C2×Dic3—61no—
140C2×Dic3—61no—
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—62no—
144C2×Dic3—61no—
145C2×Dic3—182no—
146C2×Dic3—181no—
147C2×Dic3—362no—
148D12—92no—
149D12—181no—
150C4×D3—362no—
151C4×D3—362no—
152Dic6—364no—
153Dic6—362no—
154C3×C8—182no—
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32—21no—
161C3⋊D3—42no—
162C3×D3—122no—
163C2×Q8—541no—
164C2×D4—271no—
165C22×C4—541no—
166C2×C8—92no—
167C4⋊C4—181no—
168C4⋊C4—272no—
169C4⋊C4—271no—
170C22⋊C4—181no—
171C42—91no—
172C22×C3—61no—
173C22×C3—61no—
174C22×C3—182no—
175C22×C3—361no—
176C22×C3—361no—
177C22×C3C2×GO+(4,𝔽2)11yescharacteristic
178C2×D3—62no—
179C2×D3—121no—
180C2×D3—121no—
181C2×D3—182no—
182C2×D3—361no—
183C2×D3—361no—
184C3×C4—21no—
185C3×C4—62no—
186C3×C4—91no—
187C3×C4—91no—
188C3×C4—92no—
189C3×C4—121no—
190C3×C4—362no—
191C3×C4—361no—
192Dic3—62no—
193Dic3—62no—
194Dic3—61no—
195Dic3—61no—
196Dic3—61no—
197Dic3—61no—
198Dic3—122no—
199Dic3—121no—
200Dic3—362no—
201Dic3—361no—
202C32—21no—
203C32—21no—
204C23—181no—
205Q8—1082no—
206D4—272no—
207D4—541no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—181no—
211C2×C4—182no—
212C2×C4—1082no—
213C2×C4#1088⋊C211yescharacteristic
214C8—182no—
215C2×C3—61no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3—61no—
221C2×C3—362no—
222C2×C3D32⋊Q811yescharacteristic
223C2×C3#6422511yescharacteristic
224C2×C3#6423911yescharacteristic
225D3—122no—
226D3—362no—
227C22—182no—
228C22—361no—
229C22—361no—
230C22C2×#1088⋊C211yescharacteristic
231C4—21no—
232C4—92no—
233C4—91no—
234C4—91no—
235C4—361no—
236C4—362no—
237C3—61no—
238C3—61no—
239C3#1059336711yescharacteristic
240C2—362no—
241C2#1172229811yescharacteristic
242C2(((C3⋊D3)⋊C3)⋊C2)⋊Q811yescharacteristic
243C2#1172230711yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC2×#6198C22
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#474
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11721117C2semidirect
3. #11720785C2semidirect
5. #11720704C2semidirect
24. #6198Q8semidirect
126. H(3,𝔽3)#474semidirect

Maximal subgroups

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

subgroupindex
#117211382
#117211172
#117207852
#117207052
#117207042
#48469