Finite Group Databasebeta

← #13147418

Subgroups — #13147418

Order 1728.

537 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 59 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×Q8×#272C211yesmaximal, characteristic
2#11722448C211yesmaximal, characteristic
3#11722445C211yesmaximal, characteristic
4#11720790C214yesmaximal, complemented
5C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
6Q8×#272—21no—
7Q8×#272—21no—
8Q8×#272C412yescomplemented
9Q8×#272C2212yes—
10C2×C4×#272—21no—
11C2×C4×#272C2211yescharacteristic
12C2×#6229—21no—
13C2×#6229C411yescharacteristic, complemented
14C22⋊#1088—21no—
15#88631—21no—
16C4⋊#1088C2212yes—
17C4×#1088—21no—
18#88627—21no—
19#88626C2212yes—
20C2×C3×D3×Q8—62no—
21C22×#272D411yescharacteristic
22C2×#1088—22no—
23Q8×H(3,𝔽3)—21no—
24Q8×H(3,𝔽3)C2×C412yes—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2×C411yescharacteristic
27C2×#1084—21no—
28C2×#1084D411yescharacteristic
29C4×#272—22no—
30C4×#272—22no—
31C4×#272C2×C411yescharacteristic
32C4×#272D411yescharacteristic
33C4×#272D411yescharacteristic
34C4×#272C2311yescharacteristic
35#6229—21no—
36#6229—22no—
37#6229—22no—
38#6229C2×C412yes—
39#6188—22no—
40C3×#419—91nomaximal
41C2×C32×Q8—22no—
42C3×D3×Q8—64no—
43C3×D3×Q8—64no—
44C3×D3×Q8—64no—
45C3×D3×Q8—64no—
46C2×C3×C4×D3—62no—
47C2×C3×C4×D3—64no—
48C2×C3×Dic6—62no—
49C2×C3×Dic6—64no—
50C22×H(3,𝔽3)C22⋊C411yescharacteristic
51C2×#272—22no—
52C2×#272C22⋊C411yescharacteristic
53C2×#272C2×D411yescharacteristic
54#1088—42no—
55C4×H(3,𝔽3)—22no—
56C4×H(3,𝔽3)C22⋊C411yescharacteristic
57C4×H(3,𝔽3)C22×C411yescharacteristic
58#1084—22no—
59#1084C22⋊C411yescharacteristic
60#1084C2×D411yescharacteristic
61C22×C3×Q8—91no—
62C2×D3×Q8—62no—
63C2×C3×M16—91no—
64C3×#117—91no—
65C3×#103—94no—
66C22×C3×D3—62no—
67C32×Q8—24no—
68C32×Q8—24no—
69C2×C32×C4—24no—
70C2×C32×C4—22no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—64no—
73C3×C4×D3—62no—
74C3×C4×D3—62no—
75C3×C4×D3—62no—
76C3×C4×D3—62no—
77C3×C4×D3—68no—
78C3×C4×D3—68no—
79C3×Dic6—64no—
80C3×Dic6—68no—
81C3×Dic6—68no—
82C3×Dic6—64no—
83#419—91noSylow
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
86#272—21no—
87#272#13712yes—
88C2×C3×Q8—62no—
89C2×C3×Q8—92no—
90C2×C3×Q8—92no—
91C2×C3×Q8—91no—
92C2×C3×Q8—181no—
93C2×C3×Q8—181no—
94C2×C3×Q8—181no—
95C2×C3×Q8#15711yescharacteristic
96C22×C3×C4—91no—
97C22×C3×C4—181no—
98D3×Q8—64no—
99D3×Q8—64no—
100D3×Q8—64no—
101D3×Q8—64no—
102C2×C4×D3—62no—
103C2×C4×D3—64no—
104C2×Dic6—62no—
105C2×Dic6—64no—
106C3×M16—181no—
107C3×M16—181no—
108C2×C3×C8—92no—
109C3×C4⋊C4—92no—
110C3×C22⋊C4—181no—
111C3×C42—181no—
112C22×C32—22no—
113C2×C3×D3—62no—
114C2×C3×D3—62no—
115C2×C3×D3—68no—
116C32×C4—28no—
117C32×C4—22no—
118C32×C4—22no—
119C3×Dic3—62no—
120C3×Dic3—62no—
121C3×Dic3—68no—
122C22×Q8—91no—
123C2×M16—91no—
124#117—91no—
125#103—94no—
126H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
127C23×C3—91no—
128C22×D3—62no—
129C3×Q8—21no—
130C3×Q8—64no—
131C3×Q8—64no—
132C3×Q8—92no—
133C3×Q8—182no—
134C3×Q8—182no—
135C3×Q8—181no—
136C3×Q8C2×#15712yes—
137C2×C3×C4—21no—
138C2×C3×C4—64no—
139C2×C3×C4—62no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—182no—
146C2×C3×C4—182no—
147C2×C3×C4—181no—
148C2×C3×C4—182no—
149C2×C3×C4C2×#15711yescharacteristic
150C2×Dic3—62no—
151C2×Dic3—64no—
152C4×D3—62no—
153C4×D3—62no—
154C4×D3—62no—
155C4×D3—62no—
156C4×D3—68no—
157C4×D3—68no—
158Dic6—64no—
159Dic6—68no—
160Dic6—68no—
161Dic6—64no—
162C3×C8—182no—
163C2×C32—24no—
164C2×C32—22no—
165C3×D3—64no—
166C3×D3—64no—
167C2×Q8—92no—
168C2×Q8—92no—
169C2×Q8—91no—
170C2×Q8—181no—
171C2×Q8—181no—
172C2×Q8—181no—
173C2×Q8#108811yescharacteristic, complemented
174C22×C4—91no—
175C22×C4—181no—
176M16—181no—
177M16—181no—
178C2×C8—92no—
179C4⋊C4—92no—
180C22⋊C4—181no—
181C42—181no—
182C22×C3—62no—
183C22×C3—91no—
184C22×C3—91no—
185C22×C3—182no—
186C22×C3C22⋊#15711yescharacteristic
187C2×D3—62no—
188C2×D3—62no—
189C2×D3—68no—
190C3×C4—22no—
191C3×C4—68no—
192C3×C4—62no—
193C3×C4—62no—
194C3×C4—91no—
195C3×C4—91no—
196C3×C4—182no—
197C3×C4—362no—
198C3×C4C22⋊#15711yescharacteristic
199C3×C4C22×#15711yescharacteristic
200Dic3—62no—
201Dic3—62no—
202Dic3—68no—
203C32—22no—
204C23—91no—
205Q8—21no—
206Q8—92no—
207Q8—182no—
208Q8—182no—
209Q8—181no—
210Q8C2×#108812yes—
211C2×C4—21no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—182no—
218C2×C4—182no—
219C2×C4—181no—
220C2×C4—182no—
221C2×C4C2×#108811yescharacteristic
222C8—182no—
223C2×C3—21no—
224C2×C3—64no—
225C2×C3—62no—
226C2×C3—92no—
227C2×C3—181no—
228C2×C3C2×C22⋊#15711yescharacteristic
229D3—64no—
230D3—64no—
231C22—91no—
232C22—91no—
233C22—182no—
234C22C22⋊#108811yescharacteristic
235C4—22no—
236C4—91no—
237C4—91no—
238C4—182no—
239C4—362no—
240C4C22⋊#108811yescharacteristic
241C4C22×#108811yescharacteristic
242C3—62no—
243C3#1059382911yescharacteristic
244C2—21no—
245C2—92no—
246C2—181no—
247C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC22×#272D4
cosocleC4×#272C23
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#419
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
4. #11720790C2semidirect
5. C2×Q8×H(3,𝔽3)C4semidirect
8. Q8×#272C4semidirect
13. C2×#6229C4semidirect
126. H(3,𝔽3)#419semidirect
173. C2×Q8#1088semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
#117224482
#117224452
#117207902
C3×#4199