Finite Group Databasebeta

← #13146616

Subgroups — #13146616

Order 1728.

409 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 51 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×#88590C212yesmaximal
2#11721152C211yesmaximal, characteristic
3#11721147C211yesmaximal, characteristic
4#11720787C211yesmaximal, characteristic
5#11720707C212yesmaximal
6C22×#1088C2212yes—
7C2×C4×#272C2211yescharacteristic
8C2×#6199C2211yescharacteristic
9C2×#6199C2212yes—
10C2×#6199C2211yescharacteristic
11#88590—24no—
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13#3750⋊C3—21no—
14#88449—21no—
15#88448—21no—
16D3×C4⋊Dic3—61no—
17D3×Dic3⋊C4—61no—
18C22×#272C2311yescharacteristic
19C2×#1088—22no—
20C2×#1088—22no—
21C2×#1088Q814yes—
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×#1084D411yescharacteristic
24C4×#272—22no—
25C2×(C3⋊Dic3)⋊C3—21no—
26C2×(C3⋊Dic3)⋊C3—21no—
27C2×(C3⋊Dic3)⋊C3—22no—
28#6199—21no—
29#6199—22no—
30#6199—22no—
31#6199—22no—
32#6199D412yes—
33#4661—91nomaximal
34C2×C3×C4×D3—61no—
35C2×C3×C4×D3—61no—
36C2×D3×Dic3—61no—
37C2×D3×Dic3—61no—
38C2×D3×Dic3—62no—
39C4⋊(C3⋊Dic3)—21no—
40#3750—21no—
41C3×C4⋊Dic3—61no—
42C3×Dic3⋊C4—61no—
43#3724—61no—
44Dic3⋊Dic3—61no—
45Dic3⋊Dic3—62no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C2×#272C2×D411yescharacteristic
48C2×#272C2×Q812yes—
49C2×#272#4112yes—
50C2×#272#4111yescharacteristic
51#1088—24no—
52C4×H(3,𝔽3)—21no—
53#1084—21no—
54(C3⋊Dic3)⋊C3—22no—
55(C3⋊Dic3)⋊C3—42no—
56(C3⋊Dic3)⋊C3—41no—
57C2×C4⋊Dic3—91no—
58C2×C4⋊Dic3—92no—
59C2×Dic3⋊C4—91no—
60D3×C4⋊C4—181no—
61D3×C4⋊C4—181no—
62C3×#95—91no—
63C3⋊#95—92no—
64C22×C3×D3—61no—
65C22×C3×D3—61no—
66C2×C32×C4—21no—
67C2×C32×C4—21no—
68C2×C3⋊Dic3—21no—
69C2×C3⋊Dic3—21no—
70C2×C3⋊Dic3—22no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—61no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—61no—
76C3×C4×D3—64no—
77C3×C4×D3—122no—
78D3×Dic3—62no—
79D3×Dic3—62no—
80D3×Dic3—122no—
81D3×Dic3—122no—
82D3×Dic3—122no—
83#398—271noSylow
84C2×H(3,𝔽3)#12111yescharacteristic
85C2×H(3,𝔽3)#12311yescharacteristic
86C2×H(3,𝔽3)C4⋊Q811yescharacteristic
87#272C22⋊Q812yes—
88#272#12512yes—
89C22×C3×C4—92no—
90C22×C3×C4—91no—
91C22×Dic3—91no—
92C22×Dic3—91no—
93C22×Dic3—92no—
94C2×C4×D3—61no—
95C2×C4×D3—61no—
96C2×C4×D3—181no—
97C2×C4×D3—181no—
98C2×C4×D3—182no—
99C3×C4⋊C4—181no—
100C3×C4⋊C4—181no—
101C4⋊Dic3—61no—
102C4⋊Dic3—181no—
103C4⋊Dic3—181no—
104C4⋊Dic3—181no—
105C4⋊Dic3—184no—
106Dic3⋊C4—61no—
107Dic3⋊C4—181no—
108Dic3⋊C4—181no—
109Dic3⋊C4—182no—
110Dic3⋊C4—181no—
111C22×C32—21no—
112C22×C32—21no—
113C2×C3×D3—61no—
114C2×C3×D3—61no—
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117C2×C3×D3—61no—
118C2×C3×D3—61no—
119C2×C3×D3—62no—
120C2×C3×D3—62no—
121C32×C4—22no—
122C32×C4—41no—
123C3⋊Dic3—22no—
124C3⋊Dic3—42no—
125C3⋊Dic3—41no—
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—121no—
129C3×Dic3—122no—
130C3×Dic3—121no—
131C2×C4⋊C4—271no—
132C2×C4⋊C4—271no—
133C2×C4⋊C4—272no—
134#95—91no—
135#95—272no—
136H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
137C23×C3—91no—
138C22×D3—61no—
139C22×D3—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—94no—
143C2×C3×C4—91no—
144C2×C3×C4—182no—
145C2×C3×C4—181no—
146C2×C3×C4—182no—
147C2×C3×C4—182no—
148C2×C3×C4—182no—
149C2×C3×C4—181no—
150C2×C3×C4GO+(4,𝔽2)11yescharacteristic
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—61no—
154C2×Dic3—61no—
155C2×Dic3—62no—
156C2×Dic3—92no—
157C2×Dic3—181no—
158C2×Dic3—181no—
159C2×Dic3—182no—
160C2×Dic3—181no—
161C2×Dic3—182no—
162C2×Dic3—182no—
163C2×Dic3—182no—
164C4×D3—64no—
165C4×D3—122no—
166C4×D3—182no—
167C4×D3—182no—
168C4×D3—362no—
169C4×D3—362no—
170C4×D3—362no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—21no—
177C3×D3—62no—
178C3×D3—62no—
179C3×D3—62no—
180C3×D3—62no—
181C22×C4—92no—
182C22×C4—91no—
183C22×C4—271no—
184C22×C4—271no—
185C22×C4—272no—
186C4⋊C4—181no—
187C4⋊C4—181no—
188C4⋊C4—541no—
189C4⋊C4—541no—
190C4⋊C4—544no—
191C22×C3—61no—
192C22×C3—61no—
193C22×C3—92no—
194C22×C3—91no—
195C22×C3—91no—
196C22×C3—92no—
197C22×C3C2×GO+(4,𝔽2)11yescharacteristic
198C2×D3—62no—
199C2×D3—62no—
200C2×D3—61no—
201C2×D3—61no—
202C2×D3—61no—
203C2×D3—61no—
204C2×D3—62no—
205C2×D3—62no—
206C3×C4—21no—
207C3×C4—62no—
208C3×C4—121no—
209C3×C4—182no—
210C3×C4—184no—
211C3×C4—181no—
212C3×C4—362no—
213C3×C4—361no—
214Dic3—62no—
215Dic3—62no—
216Dic3—121no—
217Dic3—122no—
218Dic3—121no—
219Dic3—182no—
220Dic3—362no—
221Dic3—361no—
222C32—21no—
223C32—21no—
224C23—91no—
225C2×C4—94no—
226C2×C4—91no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4—182no—
230C2×C4—181no—
231C2×C4—182no—
232C2×C4—181no—
233C2×C4—272no—
234C2×C4—541no—
235C2×C4—542no—
236C2×C4—542no—
237C2×C4—542no—
238C2×C4#1088⋊C211yescharacteristic
239C2×C3—61no—
240C2×C3—61no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3—92no—
246C2×C3—92no—
247C2×C3#6422211yescharacteristic
248C2×C3#6423011yescharacteristic
249C2×C3#6423311yescharacteristic
250D3—62no—
251D3—62no—
252D3—62no—
253D3—62no—
254C22—92no—
255C22—91no—
256C22—91no—
257C22—92no—
258C22C2×#1088⋊C211yescharacteristic
259C4—21no—
260C4—184no—
261C4—182no—
262C4—181no—
263C4—362no—
264C4—361no—
265C3—61no—
266C3—61no—
267C3#1059333911yescharacteristic
268C2—92no—
269C2—92no—
270C2#1172230011yescharacteristic
271C2#1172230411yescharacteristic
272C2#1172231411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, prime radical, cosocleC22×#272C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#398
LCSG ▷ C22×#272 ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
136. H(3,𝔽3)#398semidirect

Maximal subgroups

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

subgroupindex
C2×#885902
#117211522
#117211472
#117207872
#117207072
#46619