Finite Group Databasebeta

← #13165494

Subgroups — #13165494

Order 1728.

524 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 64 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#11722931C212yesmaximal, complemented
3#11722772C212yesmaximal, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5Dic3×#622C211yesmaximal, characteristic, complemented
6C4×#3774—31nomaximal
7C2×#6295C2212yes—
8C2×C3×#624—22no—
9C2×C3⋊#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88941C414yescomplemented
12#88803—22no—
13Dic3×C3⋊Dic3C2211yescharacteristic
14C3×#3724C2212yes—
15C2×#3774—31no—
16#63904—62no—
17C4×#625—32no—
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
28C3×#624—44no—
29C3⋊#622—21no—
30C3×#622C2×C412yescomplemented
31#4872—91nomaximal
32C2×C3×C4×D3—62no—
33C2×C3×C4×D3—62no—
34C2×#626—62no—
35C2×#625—32no—
36C2×#624—22no—
37C2×#622—31no—
38C2×#622C2×D311yescharacteristic
39#3774—34no—
40#3752—62no—
41C4×C3⋊Dic3—31no—
42C3×Dic3⋊C4—62no—
43#3724C2×D312yes—
44(C2×D3)⋊Dic3—62no—
45(C2×D3)⋊Dic3—62no—
46(C2×D3)⋊Dic3—122no—
47Dic32—62no—
48C22×C33C2×D411yescharacteristic
49C2×C3×C3⋊D3—24no—
50C2×C3×C3⋊D3—42no—
51C2×C3×C3⋊D3—42no—
52C3⋊(C3⋊Dic3)—21no—
53C3×C3⋊Dic3C22×C411yescharacteristic
54C3×C3⋊Dic3#4111yescharacteristic
55C32×Dic3—42no—
56C32×Dic3D812yescomplemented
57C4×#67—182no—
58C2×D24—91no—
59#893—92no—
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
73C2×C3×Dic3—22no—
74C2×C3×Dic3—22no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—42no—
77C2×C3×Dic3—62no—
78C3×C4×D3—124no—
79C3×C4×D3—124no—
80#626—64no—
81#626—64no—
82#625—34no—
83#625—62no—
84#624—44no—
85#622—61no—
86#622C4×D312yescomplemented
87C4×D8—271noSylow
88C2×C33C4×D411yescharacteristic
89C2×C33C2×D811yescharacteristic
90C2×C33#13511yescharacteristic
91C3×C3⋊D3—44no—
92C22×C3×C4—182no—
93C2×#67—182no—
94C2×D12—92no—
95C2×C4×D3—62no—
96C2×C4×D3—62no—
97C2×C3×C8—91no—
98C3×C4⋊C4—92no—
99#217—182no—
100(C2×D3)⋊C4—182no—
101(C2×D3)⋊C4—182no—
102(C2×D3)⋊C4—362no—
103Dic3⋊C4—62no—
104C4×Dic3—91no—
105C4×Dic3—182no—
106C2×C3⋊C8—91no—
107D24—94no—
108C22×C32—22no—
109C22×C32—42no—
110C22×C32D3×D411yescharacteristic
111C2×C3⋊D3—24no—
112C2×C3⋊D3—42no—
113C2×C3⋊D3—42no—
114C2×C3×D3—64no—
115C2×C3×D3—64no—
116C2×C3×D3—124no—
117C2×C3×D3—122no—
118C2×C3×D3—122no—
119C2×C3×D3—122no—
120C2×C3×D3—122no—
121C2×C3×D3—242no—
122C2×C3×D3—242no—
123C32×C4—32no—
124C32×C4—122no—
125C3⋊Dic3—61no—
126C3⋊Dic3—122no—
127C3⋊Dic3—242no—
128C3⋊Dic3C2×C4×D311yescharacteristic
129C3⋊Dic3#23911yescharacteristic
130C3×Dic3—24no—
131C3×Dic3—42no—
132C3×Dic3—42no—
133C3×Dic3—62no—
134C3×Dic3—62no—
135C3×Dic3—82no—
136C2×D8—271no—
137C4×D4—272no—
138#107—91no—
139#102—272no—
140C4×C8—271no—
141C33C4×D811yescharacteristic, complemented, Sylow
142C23×C3—182no—
143C22×D3—62no—
144C22×D3—62no—
145C22×D3—122no—
146C2×C3×C4—62no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—91no—
150C2×C3×C4—122no—
151C2×C3×C4—364no—
152#67—184no—
153#67—184no—
154C2×Dic3—22no—
155C2×Dic3—62no—
156C2×Dic3—91no—
157C2×Dic3—182no—
158C2×Dic3—362no—
159C2×Dic3GO+(4,𝔽2)11yescharacteristic
160D12—94no—
161D12—182no—
162C4×D3—124no—
163C4×D3—124no—
164C3×C8—92no—
165C3⋊C8—181no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—42no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32#89311yescharacteristic
173C2×C32D3×D811yescharacteristic
174C2×C32#91911yescharacteristic
175C3⋊D3—44no—
176C3×D3—124no—
177C3×D3—124no—
178C3×D3—244no—
179C2×D4—272no—
180C22×C4—182no—
181D8—274no—
182C2×C8—91no—
183C2×C8—271no—
184C4⋊C4—92no—
185C22⋊C4—542no—
186C42—271no—
187C22×C3—22no—
188C22×C3—42no—
189C22×C3—184no—
190C22×C3—362no—
191C22×C3—362no—
192C22×C3C2×GO+(4,𝔽2)11yescharacteristic
193C2×D3—64no—
194C2×D3—64no—
195C2×D3—124no—
196C2×D3—122no—
197C2×D3—122no—
198C2×D3—122no—
199C2×D3—122no—
200C2×D3—242no—
201C2×D3—242no—
202C3×C4—64no—
203C3×C4—91no—
204C3×C4—91no—
205C3×C4—122no—
206C3×C4—122no—
207C3×C4—242no—
208Dic3—42no—
209Dic3—62no—
210Dic3—62no—
211Dic3—181no—
212Dic3—362no—
213Dic3—722no—
214Dic3#377412yescomplemented
215C32—22no—
216C32—42no—
217C32#487211yescharacteristic, complemented
218C23—182no—
219D4—274no—
220D4—542no—
221C2×C4—31no—
222C2×C4—62no—
223C2×C4—91no—
224C2×C4—271no—
225C2×C4—364no—
226C8—92no—
227C8—541no—
228C2×C3—22no—
229C2×C3—22no—
230C2×C3—22no—
231C2×C3—42no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—364no—
235C2×C3#6422311yescharacteristic
236C2×C3C4×GO+(4,𝔽2)11yescharacteristic
237C2×C3C2×#377411yescharacteristic
238D3—124no—
239D3—124no—
240D3—244no—
241C22—184no—
242C22—362no—
243C22—362no—
244C22D3×GO+(4,𝔽2)11yescharacteristic
245C4—32no—
246C4—91no—
247C4—91no—
248C4—122no—
249C4—541no—
250C3—22no—
251C3—42no—
252C3C4×#377411yescharacteristic, complemented
253C2—364no—
254C2#1172411111yescharacteristic
255C2D3×#377411yescharacteristic
256C2#1172413711yescharacteristic
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. #11722931C2semidirect
3. #11722772C2semidirect
4. C3×#63743C2semidirect
5. Dic3×#622C2semidirect
11. #88941C4semidirect
19. #63743D3semidirect
27. #6295C2×C4semidirect
30. C3×#622C2×C4semidirect
56. C32×Dic3D8semidirect
86. #622C4×D3semidirect
141. C33C4×D8semidirect
214. Dic3#3774semidirect
217. C32#4872semidirect
252. C3C4×#3774semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
#117229312
#117227722
C3×#637432
Dic3×#6222
C4×#37743
#48729