Finite Group Databasebeta

← #13165496

Subgroups — #13165496

Order 1728.

564 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 62 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×#88941C211yesmaximal, characteristic
2#11722938C212yesmaximal, complemented
3#11722772C212yesmaximal, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5Dic3×#622C211yesmaximal, characteristic, complemented
6#10593420—31nomaximal
7C2×#6295C2212yes—
8C2×Dic3×C3⋊D3—22no—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88941C414yescomplemented
12Dic3×C3⋊Dic3C2211yescharacteristic
13C3×#3724C2212yes—
14C3×#3722—22no—
15C2×#3774—31no—
16#63902—32no—
17#63881—62no—
18C4×#622—31no—
19#63743D311yescharacteristic, complemented
20#63739—32no—
21C22×C3×C3⋊D3—22no—
22C2×C3⋊(C3⋊Dic3)D411yescharacteristic
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C2×C32×Dic3—22no—
25C2×C32×Dic3D411yescharacteristic
26#6295—22no—
27#6295C2×C414yescomplemented
28Dic3×C3⋊D3—44no—
29C3⋊#622—21no—
30C3×#622C2×C412yescomplemented
31#4872—91nomaximal
32C2×C4×C3⋊D3—62no—
33C2×#626—62no—
34C2×#625—32no—
35C2×D3×Dic3—62no—
36C2×D3×Dic3—62no—
37C2×D3×Dic3—122no—
38C2×#622—31no—
39C2×#622C2×D311yescharacteristic
40#3774—34no—
41C4×C3⋊Dic3—31no—
42C3×(C2×D3)⋊C4—62no—
43C3×(C2×D3)⋊C4—62no—
44C3×Dic3⋊C4—62no—
45#3724C2×D312yes—
46#3722—22no—
47Dic32—62no—
48C22×C33C2×D411yescharacteristic
49C2×C3×C3⋊D3—24no—
50C2×C3×C3⋊D3—42no—
51C2×C3×C3⋊D3—42no—
52C3⋊(C3⋊Dic3)D812yescomplemented
53C3×C3⋊Dic3C22×C411yescharacteristic
54C3×C3⋊Dic3#4111yescharacteristic
55C32×Dic3—21no—
56C32×Dic3—42no—
57C2×D24—91no—
58#893—92no—
59#67⋊C4—182no—
60C3×#107—91no—
61C8×Dic3—91no—
62#809—92no—
63C22×C3⋊D3—22no—
64C22×C3×D3—62no—
65C22×C3×D3—62no—
66C22×C3×D3—122no—
67C2×C32×C4—31no—
68C2×C32×C4—62no—
69C2×C3⋊Dic3—31no—
70C2×C3⋊Dic3—62no—
71C2×C3⋊Dic3—122no—
72C2×C3⋊Dic3C22×D311yescharacteristic
73C4×C3⋊D3—124no—
74C2×C3×Dic3—22no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—62no—
79#626—64no—
80#626—64no—
81#625—34no—
82#625—62no—
83D3×Dic3—124no—
84D3×Dic3—124no—
85D3×Dic3—128no—
86#622—61no—
87#622C4×D312yescomplemented
88C4×D8—271noSylow
89C2×C33C4×D411yescharacteristic
90C2×C33C2×D811yescharacteristic
91C2×C33#13511yescharacteristic
92C3×C3⋊D3—44no—
93C2×#67—182no—
94C22×Dic3—182no—
95C2×D12—92no—
96C2×C4×D3—182no—
97C2×C4×D3—182no—
98C2×C4×D3—362no—
99C2×C3×C8—91no—
100C3×C4⋊C4—92no—
101C3×C22⋊C4—182no—
102(C2×D3)⋊C4—62no—
103(C2×D3)⋊C4—62no—
104Dic3⋊C4—62no—
105C4×Dic3—91no—
106C4×Dic3—182no—
107C2×C3⋊C8—91no—
108D24—94no—
109C22×C32—22no—
110C22×C32—42no—
111C22×C32D3×D411yescharacteristic
112C2×C3⋊D3—24no—
113C2×C3⋊D3—42no—
114C2×C3⋊D3—42no—
115C2×C3×D3—64no—
116C2×C3×D3—64no—
117C2×C3×D3—124no—
118C2×C3×D3—122no—
119C2×C3×D3—122no—
120C2×C3×D3—124no—
121C2×C3×D3—124no—
122C2×C3×D3—122no—
123C2×C3×D3—122no—
124C32×C4—61no—
125C32×C4—122no—
126C3⋊Dic3—32no—
127C3⋊Dic3—64no—
128C3⋊Dic3—124no—
129C3⋊Dic3C2×C4×D311yescharacteristic
130C3⋊Dic3#23911yescharacteristic
131C3×Dic3—42no—
132C3×Dic3—42no—
133C3×Dic3—44no—
134C3×Dic3—42no—
135C3×Dic3—62no—
136C3×Dic3—62no—
137C2×D8—271no—
138C4×D4—272no—
139#107—91no—
140#102—272no—
141C4×C8—271no—
142C33C4×D811yescharacteristic, complemented, Sylow
143C23×C3—182no—
144C22×D3—62no—
145C22×D3—62no—
146C22×D3—122no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—62no—
150C2×C3×C4—91no—
151C2×C3×C4—122no—
152#67—184no—
153#67—184no—
154C2×Dic3—22no—
155C2×Dic3—62no—
156C2×Dic3—91no—
157C2×Dic3—182no—
158C2×Dic3—362no—
159C2×Dic3—364no—
160C2×Dic3GO+(4,𝔽2)11yescharacteristic
161D12—94no—
162D12—182no—
163C4×D3—364no—
164C4×D3—364no—
165C4×D3—368no—
166C3×C8—92no—
167C3⋊C8—181no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32#89311yescharacteristic
175C2×C32D3×D811yescharacteristic
176C2×C32#91911yescharacteristic
177C3⋊D3—44no—
178C3×D3—124no—
179C3×D3—128no—
180C3×D3—124no—
181C2×D4—272no—
182C22×C4—542no—
183D8—274no—
184C2×C8—91no—
185C2×C8—271no—
186C4⋊C4—92no—
187C22⋊C4—182no—
188C42—271no—
189C22×C3—22no—
190C22×C3—42no—
191C22×C3—184no—
192C22×C3—362no—
193C22×C3—362no—
194C22×C3C2×GO+(4,𝔽2)11yescharacteristic
195C2×D3—64no—
196C2×D3—64no—
197C2×D3—124no—
198C2×D3—122no—
199C2×D3—122no—
200C2×D3—124no—
201C2×D3—124no—
202C2×D3—122no—
203C2×D3—122no—
204C3×C4—91no—
205C3×C4—91no—
206C3×C4—122no—
207C3×C4—122no—
208C3×C4—124no—
209C3×C4—122no—
210Dic3—21no—
211Dic3—42no—
212Dic3—62no—
213Dic3—62no—
214Dic3—92no—
215Dic3—184no—
216Dic3—364no—
217C32—22no—
218C32—42no—
219C32#487211yescharacteristic, complemented
220C23—182no—
221D4—274no—
222D4—542no—
223C2×C4—31no—
224C2×C4—62no—
225C2×C4—91no—
226C2×C4—271no—
227C2×C4—1084no—
228C8—92no—
229C8—541no—
230C2×C3—22no—
231C2×C3—22no—
232C2×C3—22no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—42no—
236C2×C3—364no—
237C2×C3#6422511yescharacteristic
238C2×C3C4×GO+(4,𝔽2)11yescharacteristic
239C2×C3#6423711yescharacteristic
240D3—124no—
241D3—128no—
242D3—124no—
243C22—184no—
244C22—362no—
245C22—362no—
246C22D3×GO+(4,𝔽2)11yescharacteristic
247C4—61no—
248C4—91no—
249C4—91no—
250C4—122no—
251C4—272no—
252C3—22no—
253C3—42no—
254C3#1059342011yescharacteristic, complemented
255C2—364no—
256C2#1172411111yescharacteristic
257C2#1172412711yescharacteristic
258C2#1172413411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×#88941C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C4×D8
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #11722938C2semidirect
3. #11722772C2semidirect
4. C3×#63743C2semidirect
5. Dic3×#622C2semidirect
11. #88941C4semidirect
19. #63743D3semidirect
27. #6295C2×C4semidirect
30. C3×#622C2×C4semidirect
52. C3⋊(C3⋊Dic3)D8semidirect
87. #622C4×D3semidirect
142. C33C4×D8semidirect
219. C32#4872semidirect
254. C3#10593420semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
#117229382
#117227722
C3×#637432
Dic3×#6222
#105934203
#48729