Finite Group Databasebeta

← #13146703

Subgroups — #13146703

Order 1728.

519 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 47 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×C22⋊#1088C211yesmaximal, characteristic
2C2×#88590C211yesmaximal, characteristic, complemented
3#11721320C212yesmaximal
4#11720707C212yesmaximal, complemented
5#11720707C211yesmaximal, characteristic, complemented
6C23×#272C2211yescharacteristic
7C22×#1088C2211yescharacteristic
8C22×#1088C2211yescharacteristic
9C2×#6199C2212yes—
10C2×#6199C2212yes—
11C22⋊#1088—22no—
12#88590—22no—
13#3757⋊C3—22no—
14#88451—22no—
15D3×#217—62no—
16C23×H(3,𝔽3)D411yescharacteristic
17C22×#272—21no—
18C22×#272—22no—
19C22×#272—22no—
20C22×#272—21no—
21C22×#272D411yescharacteristic
22C22×#272C2311yescharacteristic
23C2×#1088—22no—
24C2×#1088—21no—
25C2×#1088—22no—
26C2×#1088D412yes—
27C2×(C3⋊Dic3)⋊C3—22no—
28C2×(C3⋊Dic3)⋊C3—22no—
29#6199—22no—
30#6199—22no—
31#6199—22no—
32#6199—22no—
33#4953—91nomaximal
34C23×C3×D3—62no—
35C2×D3×Dic3—62no—
36C2×D3×Dic3—62no—
37#3757—22no—
38C3×#217—62no—
39(C2×D3)⋊Dic3—62no—
40(C2×D3)⋊Dic3—62no—
41C22×H(3,𝔽3)—21no—
42C22×H(3,𝔽3)—22no—
43C22×H(3,𝔽3)C2×D411yescharacteristic
44C2×#272—21no—
45C2×#272—22no—
46C2×#272—41no—
47C2×#272—41no—
48C2×#272—42no—
49C2×#272C2×D411yescharacteristic
50C2×#272#4112yes—
51C2×#272#4112yes—
52C2×#272#4111yescharacteristic
53#1088—22no—
54#1088—41no—
55(C3⋊Dic3)⋊C3—42no—
56(C3⋊Dic3)⋊C3—42no—
57C2×C3×C22⋊C4—91no—
58C2×#217—92no—
59C2×C4⋊Dic3—91no—
60D3×C22⋊C4—182no—
61C3⋊#95—91no—
62C3⋊#95—92no—
63C23×C32—22no—
64C22×C3×D3—62no—
65C22×C3×D3—62no—
66C22×C3×D3—64no—
67C22×C3×D3—64no—
68C22×C3×D3—122no—
69C22×C3×D3—122no—
70C22×C3×D3—122no—
71C22×C3×D3—122no—
72C2×C3⋊Dic3—22no—
73C2×C3⋊Dic3—22no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—62no—
76D3×Dic3—122no—
77D3×Dic3—122no—
78D3×Dic3—122no—
79D3×Dic3—122no—
80#397—271noSylow
81C2×H(3,𝔽3)—41no—
82C2×H(3,𝔽3)#12111yescharacteristic
83C2×H(3,𝔽3)#12312yes—
84#272—41no—
85#272#12311yescharacteristic
86#272#12411yescharacteristic
87#272#12612yes—
88C24×C3—91no—
89C23×D3—62no—
90C22×C3×C4—91no—
91C22×C3×C4—91no—
92C22×Dic3—92no—
93C22×Dic3—92no—
94C2×C4×D3—182no—
95C2×C4×D3—182no—
96C3×C22⋊C4—182no—
97C3×C22⋊C4—182no—
98#217—62no—
99#217—182no—
100#217—182no—
101(C2×D3)⋊C4—182no—
102(C2×D3)⋊C4—182no—
103C4⋊Dic3—182no—
104C22×C32—24no—
105C22×C32—22no—
106C22×C32—42no—
107C22×C32—42no—
108C2×C3×D3—62no—
109C2×C3×D3—62no—
110C2×C3×D3—62no—
111C2×C3×D3—62no—
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—64no—
115C2×C3×D3—122no—
116C2×C3×D3—122no—
117C2×C3×D3—124no—
118C2×C3×D3—124no—
119C2×C3×D3—124no—
120C2×C3×D3—124no—
121C3⋊Dic3—42no—
122C3⋊Dic3—42no—
123C3×Dic3—122no—
124C3×Dic3—122no—
125C2×C4⋊C4—271no—
126C2×C22⋊C4—91no—
127C2×C22⋊C4—272no—
128#95—271no—
129#95—272no—
130H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
131C23×C3—62no—
132C23×C3—91no—
133C23×C3—91no—
134C23×C3—182no—
135C23×C3—182no—
136C23×C3—181no—
137C23×C3—181no—
138C23×C3GO+(4,𝔽2)11yescharacteristic
139C22×D3—62no—
140C22×D3—62no—
141C22×D3—64no—
142C22×D3—64no—
143C22×D3—122no—
144C22×D3—122no—
145C22×D3—122no—
146C22×D3—122no—
147C2×C3×C4—92no—
148C2×C3×C4—182no—
149C2×C3×C4—182no—
150C2×C3×C4—182no—
151C2×C3×C4—181no—
152C2×C3×C4—182no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—182no—
156C2×Dic3—182no—
157C2×Dic3—182no—
158C2×Dic3—182no—
159C2×Dic3—182no—
160C2×Dic3—182no—
161C4×D3—362no—
162C4×D3—362no—
163C4×D3—362no—
164C4×D3—362no—
165C2×C32—22no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—44no—
169C3×D3—62no—
170C3×D3—62no—
171C3×D3—62no—
172C3×D3—62no—
173C3×D3—124no—
174C24—91no—
175C22×C4—91no—
176C22×C4—91no—
177C22×C4—272no—
178C22×C4—272no—
179C4⋊C4—542no—
180C22⋊C4—182no—
181C22⋊C4—182no—
182C22⋊C4—542no—
183C22×C3—22no—
184C22×C3—21no—
185C22×C3—64no—
186C22×C3—62no—
187C22×C3—92no—
188C22×C3—92no—
189C22×C3—91no—
190C22×C3—91no—
191C22×C3—122no—
192C22×C3—122no—
193C22×C3—182no—
194C22×C3—181no—
195C22×C3—361no—
196C22×C3—361no—
197C22×C3—362no—
198C22×C3C2×GO+(4,𝔽2)11yescharacteristic
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—64no—
206C2×D3—122no—
207C2×D3—122no—
208C2×D3—124no—
209C2×D3—124no—
210C2×D3—124no—
211C2×D3—124no—
212C3×C4—182no—
213C3×C4—362no—
214C3×C4—362no—
215C3×C4—361no—
216Dic3—122no—
217Dic3—122no—
218Dic3—362no—
219Dic3—362no—
220C32—22no—
221C23—91no—
222C23—91no—
223C23—181no—
224C23—182no—
225C23—182no—
226C23—181no—
227C23#1088⋊C211yescharacteristic
228C2×C4—92no—
229C2×C4—182no—
230C2×C4—181no—
231C2×C4—182no—
232C2×C4—182no—
233C2×C4—182no—
234C2×C4—542no—
235C2×C4—542no—
236C2×C4—542no—
237C2×C4—542no—
238C2×C3—41no—
239C2×C3—62no—
240C2×C3—62no—
241C2×C3—62no—
242C2×C3—91no—
243C2×C3—91no—
244C2×C3—92no—
245C2×C3—124no—
246C2×C3—361no—
247C2×C3#6423312yes—
248C2×C3#6424211yescharacteristic
249D3—62no—
250D3—62no—
251D3—62no—
252D3—62no—
253D3—124no—
254C22—22no—
255C22—21no—
256C22—92no—
257C22—92no—
258C22—91no—
259C22—91no—
260C22—182no—
261C22—181no—
262C22—361no—
263C22—361no—
264C22—362no—
265C22C2×#1088⋊C211yescharacteristic
266C4—182no—
267C4—361no—
268C4—362no—
269C4—362no—
270C3—62no—
271C3#1059348611yescharacteristic
272C2—41no—
273C2—91no—
274C2—91no—
275C2—92no—
276C2—361no—
277C2#1172231412yes—
278C2#1172231611yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC22×#272C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC23×H(3,𝔽3)D4
prime radicalC23×#272C22
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#397
LCSG ▷ C22×#272 ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88590C2semidirect
4. #11720707C2semidirect
5. #11720707C2semidirect
130. H(3,𝔽3)#397semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#10882
C2×#885902
#117213202
#117207072
#49539