Finite Group Databasebeta

← #13146552

Subgroups — #13146552

Order 1728.

357 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 39 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#11721152C211yesmaximal, characteristic, complemented
2#11721142C211yesmaximal, characteristic
3#11720784C211yesmaximal, characteristic, complemented
4#11720705C212yesmaximal, complemented
5#11720704C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7C2×#6200—21no—
8C2×#6199—21no—
9C2×#6198C2211yescharacteristic
10C2×#6188C2212yes—
11#3752⋊C3—21no—
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13#88449C2211yescharacteristic
14#88449C2212yes—
15#63906—61no—
16D3×C4⋊Dic3—61no—
17C22×#272D411yescharacteristic
18C22×(C3⋊D3)⋊C3—21no—
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C2×#1084C2311yescharacteristic
21C4×#272—22no—
22C2×(C3⋊Dic3)⋊C3—22no—
23C2×(C3⋊Dic3)⋊C3—21no—
24#6200—42no—
25#6199—42no—
26#6198—22no—
27#6188—22no—
28#4736—91nomaximal
29C2×C3×C4×D3—61no—
30C2×C3×C4×D3—61no—
31C2×#626—61no—
32C2×#626—61no—
33C2×#625—61no—
34C2×D3×Dic3—62no—
35#3752—21no—
36C4⋊(C3⋊Dic3)—21no—
37C3×(C2×D3)⋊C4—61no—
38C3×C4⋊Dic3—61no—
39Dic3⋊Dic3—62no—
40Dic3⋊Dic3—61no—
41C22×H(3,𝔽3)C2×D411yescharacteristic
42C2×#272—21no—
43C2×#272—21no—
44C2×#272D812yes—
45C2×(C3⋊D3)⋊C3—41no—
46C2×(C3⋊D3)⋊C3—41no—
47C2×(C3⋊D3)⋊C3—41no—
48C4×H(3,𝔽3)—21no—
49#1084#4112yes—
50(C3⋊Dic3)⋊C3—42no—
51(C3⋊Dic3)⋊C3—41no—
52#930—91no—
53C2×C4⋊Dic3—91no—
54D3×C4⋊C4—181no—
55#884—181no—
56C3×C22⋊C8—91no—
57#821—92no—
58#818—92no—
59C22×C3⋊D3—21no—
60C22×C3×D3—61no—
61C22×C3×D3—61no—
62C22×C3×D3—61no—
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C3⋊Dic3—22no—
66C2×C3⋊Dic3—21no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—61no—
71C3×C4×D3—64no—
72C3×C4×D3—122no—
73#626—122no—
74#626—122no—
75#625—122no—
76D3×Dic3—124no—
77#480—271noSylow
78C2×H(3,𝔽3)#12311yescharacteristic
79C2×H(3,𝔽3)C2×D811yescharacteristic
80C2×H(3,𝔽3)#13711yescharacteristic
81#272—22no—
82(C3⋊D3)⋊C3—81no—
83C22×C3×C4—91no—
84C2×#67—181no—
85C2×#67—181no—
86C2×#67—181no—
87C22×Dic3—181no—
88C2×D12—91no—
89C2×D12—181no—
90C2×C4×D3—61no—
91C2×C4×D3—61no—
92C2×C4×D3—182no—
93C2×C3×C8—92no—
94C3×C4⋊C4—181no—
95C3×C22⋊C4—181no—
96(C2×D3)⋊C4—61no—
97(C2×D3)⋊C4—181no—
98C4⋊Dic3—61no—
99C4⋊Dic3—91no—
100C4⋊Dic3—92no—
101C4⋊Dic3—181no—
102Dic3⋊C4—181no—
103Dic3⋊C4—182no—
104C22×C32—21no—
105C22×C32—21no—
106C2×C3⋊D3—41no—
107C2×C3⋊D3—41no—
108C2×C3⋊D3—41no—
109C2×C3×D3—62no—
110C2×C3×D3—62no—
111C2×C3×D3—62no—
112C2×C3×D3—62no—
113C2×C3×D3—121no—
114C2×C3×D3—121no—
115C2×C3×D3—121no—
116C2×C3×D3—121no—
117C2×C3×D3—121no—
118C32×C4—22no—
119C32×C4—41no—
120C3⋊Dic3—42no—
121C3⋊Dic3—41no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—122no—
125C3×Dic3—121no—
126#121—271no—
127C2×C4⋊C4—271no—
128#107—272no—
129#102—272no—
130C22⋊C8—91no—
131H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C23×C3—181no—
134C22×D3—61no—
135C22×D3—61no—
136C22×D3—61no—
137C22×D3—181no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4—91no—
141C2×C3×C4—182no—
142C2×C3×C4—181no—
143C2×C3×C4—182no—
144C2×C3×C4GO+(4,𝔽2)11yescharacteristic
145#67—362no—
146#67—362no—
147#67—362no—
148C2×Dic3—61no—
149C2×Dic3—61no—
150C2×Dic3—62no—
151C2×Dic3—61no—
152C2×Dic3—182no—
153C2×Dic3—181no—
154C2×Dic3—362no—
155D12—182no—
156D12—362no—
157C4×D3—64no—
158C4×D3—122no—
159C4×D3—364no—
160C3×C8—182no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—21no—
164C2×C32—21no—
165C2×C32—21no—
166C2×C32—21no—
167C3⋊D3—81no—
168C3×D3—64no—
169C3×D3—122no—
170C3×D3—241no—
171C2×D4—271no—
172C2×D4—541no—
173C22×C4—91no—
174C22×C4—541no—
175C2×C8—92no—
176C4⋊C4—181no—
177C4⋊C4—272no—
178C4⋊C4—271no—
179C22⋊C4—181no—
180C22×C3—61no—
181C22×C3—61no—
182C22×C3—92no—
183C22×C3—181no—
184C22×C3—181no—
185C22×C3—361no—
186C22×C3—361no—
187C22×C3—361no—
188C22×C3C2×GO+(4,𝔽2)11yescharacteristic
189C2×D3—62no—
190C2×D3—62no—
191C2×D3—62no—
192C2×D3—62no—
193C2×D3—121no—
194C2×D3—121no—
195C2×D3—121no—
196C2×D3—121no—
197C2×D3—121no—
198C2×D3—361no—
199C2×D3—361no—
200C2×D3—361no—
201C3×C4—21no—
202C3×C4—62no—
203C3×C4—92no—
204C3×C4—121no—
205C3×C4—362no—
206C3×C4—361no—
207Dic3—62no—
208Dic3—62no—
209Dic3—122no—
210Dic3—121no—
211Dic3—362no—
212Dic3—361no—
213C32—21no—
214C32—21no—
215C23—91no—
216C23—181no—
217D4—542no—
218D4—1082no—
219C2×C4—91no—
220C2×C4—182no—
221C2×C4—181no—
222C2×C4—182no—
223C2×C4—1082no—
224C2×C4#1088⋊C211yescharacteristic
225C8—182no—
226C2×C3—61no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3—61no—
230C2×C3—61no—
231C2×C3—61no—
232C2×C3—182no—
233C2×C3—721no—
234C2×C3#6422511yescharacteristic
235C2×C3#6422611yescharacteristic
236C2×C3#6423311yescharacteristic
237D3—64no—
238D3—122no—
239D3—241no—
240D3—721no—
241C22—92no—
242C22—181no—
243C22—181no—
244C22—361no—
245C22—361no—
246C22—361no—
247C22C2×#1088⋊C211yescharacteristic
248C4—21no—
249C4—92no—
250C4—362no—
251C4—361no—
252C3—61no—
253C3—61no—
254C3#1059336811yescharacteristic
255C2—182no—
256C2—721no—
257C2#1172229411yescharacteristic
258C2#1172229811yescharacteristic
259C2#1172231411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radical#11721142C2
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#480
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
1. #11721152C2semidirect
3. #11720784C2semidirect
4. #11720705C2semidirect
5. #11720704C2semidirect
131. H(3,𝔽3)#480semidirect

Maximal subgroups

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

subgroupindex
#117211522
#117211422
#117207842
#117207052
#117207042
#47369