Finite Group Databasebeta

← #13147413

Subgroups — #13147413

Order 1728.

859 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 859 conjugacy classes of subgroups — too many to draw. Only its 75 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×D4×#272C211yesmaximal, characteristic
2C2×C22⋊#1088C214yesmaximal, complemented
3C2×C4⋊#1088C211yesmaximal, characteristic, complemented
4C2×C4×#1088C211yesmaximal, characteristic, complemented
5C23×#272C2212yes—
6C22×#1088C2212yes—
7C22×#1088C2212yes—
8C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
9C2×#6235C412yescomplemented
10D4×#272—22no—
11D4×#272—22no—
12C2×C4⋊#272C411yescharacteristic, complemented
13C2×C4×#272C2211yescharacteristic
14C22⋊#1088—28no—
15C4⋊#1088—22no—
16C4×#1088—22no—
17C2×C3×D3×D4—62no—
18C23×H(3,𝔽3)C2×C412yes—
19C22×#272—22no—
20C22×#272—22no—
21C22×#272—24no—
22C22×#272—24no—
23C22×#272C2×C412yes—
24C22×#272C2311yescharacteristic
25C2×#1088—22no—
26C2×#1088—22no—
27C2×#1088—22no—
28C2×#1088—24no—
29C2×#1088D414yescomplemented
30D4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)C2×C411yescharacteristic
32#6235—24no—
33C2×#1084C2×C411yescharacteristic
34C4⋊#272—22no—
35C4×#272D412yes—
36C4×#272D412yes—
37C3×#390—91nomaximal
38C23×C3×D3—64no—
39C2×C32×D4—22no—
40C2×C3×#67—64no—
41C3×D3×D4—68no—
42C3×D3×D4—68no—
43C2×C3×D12—62no—
44C2×C3×C4×D3—62no—
45C22×H(3,𝔽3)—22no—
46C22×H(3,𝔽3)—24no—
47C22×H(3,𝔽3)C22×C411yescharacteristic
48C2×#272—22no—
49C2×#272—24no—
50C2×#272—44no—
51C2×#272—42no—
52C2×#272—42no—
53C2×#272C2×D411yescharacteristic
54C2×#272C2×D412yes—
55C2×#272C2×D411yescharacteristic
56C2×#272#4112yes—
57#1088—24no—
58#1088—42no—
59C4×H(3,𝔽3)C22⋊C412yescomplemented
60#1084C22⋊C412yescomplemented
61C22×C3×D4—91no—
62C2×D3×D4—62no—
63C2×C3×C4⋊C4—91no—
64C2×C3×C22⋊C4—94no—
65C2×C3×C42—91no—
66C23×C32—24no—
67C22×C3×D3—62no—
68C22×C3×D3—64no—
69C22×C3×D3—68no—
70C22×C3×D3—68no—
71C22×C3×D3—128no—
72C22×C3×D3—128no—
73C32×D4—28no—
74C2×C32×C4—22no—
75C3×#67—616no—
76C2×C3×Dic3—62no—
77C3×D12—68no—
78C3×C4×D3—64no—
79C3×C4×D3—64no—
80#390—91noSylow
81C2×H(3,𝔽3)—42no—
82C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
83C2×H(3,𝔽3)C4×D412yes—
84#272—42no—
85#272#12112yes—
86#272#12411yescharacteristic
87#272#12711yescharacteristic
88C24×C3—92no—
89C23×D3—64no—
90C2×C3×D4—62no—
91C2×C3×D4—92no—
92C2×C3×D4—91no—
93C2×C3×D4—182no—
94C2×C3×D4—182no—
95C2×C3×D4#15711yescharacteristic
96C22×C3×C4—91no—
97C22×C3×C4—92no—
98C22×C3×C4—92no—
99C2×#67—64no—
100D3×D4—68no—
101D3×D4—68no—
102C2×D12—62no—
103C2×C4×D3—62no—
104C3×C4⋊C4—182no—
105C3×C22⋊C4—188no—
106C3×C42—182no—
107C22×C32—28no—
108C22×C32—22no—
109C22×C32—48no—
110C2×C3×D3—62no—
111C2×C3×D3—62no—
112C2×C3×D3—64no—
113C2×C3×D3—64no—
114C2×C3×D3—68no—
115C2×C3×D3—128no—
116C2×C3×D3—1216no—
117C2×C3×D3—128no—
118C2×C3×D3—128no—
119C32×C4—24no—
120C3×Dic3—64no—
121C22×D4—91no—
122C2×C4⋊C4—91no—
123C2×C22⋊C4—94no—
124C2×C42—91no—
125H(3,𝔽3)#39011yescharacteristic, complemented, Sylow
126C23×C3—64no—
127C23×C3—91no—
128C23×C3—92no—
129C23×C3—182no—
130C23×C3—182no—
131C23×C3—184no—
132C23×C3—184no—
133C23×C3C2×#15712yes—
134C22×D3—62no—
135C22×D3—64no—
136C22×D3—68no—
137C22×D3—68no—
138C22×D3—128no—
139C22×D3—128no—
140C3×D4—22no—
141C3×D4—68no—
142C3×D4—184no—
143C3×D4—182no—
144C2×C3×C4—62no—
145C2×C3×C4—91no—
146C2×C3×C4—92no—
147C2×C3×C4—92no—
148C2×C3×C4—94no—
149C2×C3×C4—182no—
150C2×C3×C4—182no—
151C2×C3×C4—184no—
152C2×C3×C4—182no—
153C2×C3×C4C2×#15711yescharacteristic
154#67—616no—
155C2×Dic3—62no—
156D12—68no—
157C4×D3—64no—
158C4×D3—64no—
159C2×C32—24no—
160C2×C32—22no—
161C2×C32—48no—
162C3×D3—64no—
163C3×D3—62no—
164C3×D3—62no—
165C3×D3—128no—
166C24—92no—
167C2×D4—92no—
168C2×D4—91no—
169C2×D4—182no—
170C2×D4—182no—
171C2×D4#108811yescharacteristic, complemented
172C22×C4—91no—
173C22×C4—92no—
174C22×C4—92no—
175C4⋊C4—182no—
176C22⋊C4—188no—
177C42—182no—
178C22×C3—24no—
179C22×C3—22no—
180C22×C3—68no—
181C22×C3—62no—
182C22×C3—92no—
183C22×C3—92no—
184C22×C3—91no—
185C22×C3—91no—
186C22×C3—128no—
187C22×C3—184no—
188C22×C3—182no—
189C22×C3—364no—
190C22×C3—362no—
191C22×C3—362no—
192C22×C3C22×#15711yescharacteristic
193C2×D3—62no—
194C2×D3—62no—
195C2×D3—64no—
196C2×D3—64no—
197C2×D3—68no—
198C2×D3—128no—
199C2×D3—1216no—
200C2×D3—128no—
201C2×D3—128no—
202C3×C4—64no—
203C3×C4—92no—
204C3×C4—184no—
205C3×C4—362no—
206C3×C4C22⋊#15712yes—
207Dic3—64no—
208C32—22no—
209C23—91no—
210C23—92no—
211C23—184no—
212C23—182no—
213C23—182no—
214C23—184no—
215C23C2×#108812yes—
216D4—22no—
217D4—184no—
218D4—182no—
219C2×C4—91no—
220C2×C4—92no—
221C2×C4—92no—
222C2×C4—94no—
223C2×C4—182no—
224C2×C4—182no—
225C2×C4—184no—
226C2×C4—182no—
227C2×C4C2×#108811yescharacteristic
228C2×C3—42no—
229C2×C3—64no—
230C2×C3—62no—
231C2×C3—92no—
232C2×C3—91no—
233C2×C3—91no—
234C2×C3—128no—
235C2×C3—362no—
236C2×C3D4×#15712yes—
237C2×C3C2×C22⋊#15711yescharacteristic
238D3—64no—
239D3—62no—
240D3—62no—
241D3—128no—
242C22—24no—
243C22—22no—
244C22—92no—
245C22—92no—
246C22—91no—
247C22—91no—
248C22—184no—
249C22—182no—
250C22—364no—
251C22—362no—
252C22—362no—
253C22C22×#108811yescharacteristic
254C4—92no—
255C4—184no—
256C4—362no—
257C4C22⋊#108812yescomplemented
258C3—62no—
259C3#1059382711yescharacteristic
260C2—42no—
261C2—92no—
262C2—91no—
263C2—91no—
264C2—362no—
265C2D4×#108812yes—
266C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×D4×#272C2
cosocleC22×#272C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#390
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#1088C2semidirect
3. C2×C4⋊#1088C2semidirect
4. C2×C4×#1088C2semidirect
8. C2×D4×H(3,𝔽3)C4semidirect
9. C2×#6235C4semidirect
12. C2×C4⋊#272C4semidirect
29. C2×#1088D4semidirect
59. C4×H(3,𝔽3)C22⋊C4semidirect
60. #1084C22⋊C4semidirect
125. H(3,𝔽3)#390semidirect
171. C2×D4#1088semidirect
257. C4C22⋊#1088semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×C22⋊#10882
C2×C4⋊#10882
C2×C4×#10882
C3×#3909