Finite Group Databasebeta

← C2×#11720702 #13146665

Subgroups — C2×#11720702 #13146665

Order 1728.

491 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×#6196C211yesmaximal, characteristic
2C22×#6188C211yesmaximal, characteristic
3C2×#88449C211yesmaximal, characteristic
4#11720702C214yesmaximal, direct product factor
5C22×#1084C2211yescharacteristic
6C22×(C3⋊Dic3)⋊C3—21no—
7C22×(C3⋊Dic3)⋊C3—21no—
8C2×#6196—21no—
9C2×#6196—22no—
10C2×#6196C414yescomplemented
11C2×#6196C2212yes—
12C2×#6188—22no—
13C2×#6188C2212yes—
14#88449—21no—
15#88449C2212yes—
16C22×#627—61no—
17C2×Dic3⋊Dic3—61no—
18C23×H(3,𝔽3)D411yescharacteristic
19C2×#1084C2×C412yes—
20C2×#1084D411yescharacteristic
21C2×#1084D412yes—
22C2×#1084C2311yescharacteristic
23C2×(C3⋊Dic3)⋊C3—24no—
24C2×(C3⋊Dic3)⋊C3—22no—
25C2×(C3⋊Dic3)⋊C3—22no—
26C2×(C3⋊Dic3)⋊C3—42no—
27#6196—24no—
28#6196—22no—
29#6196C2×C414yescomplemented
30#6188—22no—
31C2×#816—91nomaximal
32C22×C3⋊Dic3—21no—
33C22×C3⋊Dic3—21no—
34C22×C3×Dic3—61no—
35C22×C3×Dic3—61no—
36C22×C3×Dic3—61no—
37C22×C3×Dic3—61no—
38C2×#627—64no—
39C2×#627—62no—
40C2×#627—62no—
41C2×#627—64no—
42Dic3⋊Dic3—62no—
43Dic3⋊Dic3—62no—
44C22×H(3,𝔽3)C22⋊C412yes—
45C22×H(3,𝔽3)SD1612yes—
46C22×H(3,𝔽3)Q1612yes—
47C22×H(3,𝔽3)C2×D411yescharacteristic
48#1084C22⋊C412yes—
49#1084C22×C411yescharacteristic
50#1084C2×D411yescharacteristic
51(C3⋊Dic3)⋊C3—24no—
52(C3⋊Dic3)⋊C3—42no—
53C22×Dic6—91no—
54C22×Dic6—181no—
55C22×C3×C8—91no—
56C2×C4⋊Dic3—91no—
57C2×Dic3⋊C4—181no—
58#816—94no—
59C23×C32—21no—
60C23×C32—21no—
61C2×C3⋊Dic3—22no—
62C2×C3⋊Dic3—24no—
63C2×C3⋊Dic3—22no—
64C2×C3⋊Dic3—42no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—62no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—61no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—64no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—122no—
77#627—64no—
78#627—68no—
79#627—64no—
80C2×#103—271noSylow
81C2×H(3,𝔽3)#10314yescomplemented
82C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
83C2×H(3,𝔽3)C2×SD1611yescharacteristic
84C2×H(3,𝔽3)C2×Q1611yescharacteristic
85C22×C3×C4—91no—
86C22×C3×C4—181no—
87C22×C3×C4—181no—
88C22×Dic3—61no—
89C22×Dic3—61no—
90C22×Dic3—61no—
91C22×Dic3—61no—
92C22×Dic3—181no—
93C22×Dic3—181no—
94C2×Dic6—92no—
95C2×Dic6—94no—
96C2×Dic6—182no—
97C2×Dic6—184no—
98C2×Dic6—181no—
99C2×Dic6—182no—
100C2×Dic6—184no—
101C2×Dic6—182no—
102C2×C3×C8—92no—
103C2×C3×C8—182no—
104C4⋊Dic3—92no—
105C4⋊Dic3—181no—
106Dic3⋊C4—182no—
107Dic3⋊C4—182no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—22no—
111C22×C32—22no—
112C22×C32—22no—
113C22×C32—22no—
114C22×C32—22no—
115C22×C32—22no—
116C3⋊Dic3—24no—
117C3⋊Dic3—42no—
118C3×Dic3—62no—
119C3×Dic3—62no—
120C3×Dic3—61no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123C3×Dic3—61no—
124C3×Dic3—64no—
125C3×Dic3—122no—
126C22×Q8—271no—
127C22×C8—91no—
128C2×C4⋊C4—271no—
129#103—274no—
130H(3,𝔽3)C2×#10311yescharacteristic, complemented, Sylow
131C23×C3—61no—
132C23×C3—61no—
133C23×C3GO+(4,𝔽2)11yescharacteristic
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—92no—
137C2×C3×C4—92no—
138C2×C3×C4—184no—
139C2×C3×C4—182no—
140C2×C3×C4—182no—
141C2×C3×C4—362no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—61no—
147C2×Dic3—61no—
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—64no—
152C2×Dic3—62no—
153C2×Dic3—122no—
154C2×Dic3—184no—
155C2×Dic3—182no—
156C2×Dic3—182no—
157C2×Dic3—362no—
158Dic6—94no—
159Dic6—184no—
160Dic6—182no—
161Dic6—184no—
162Dic6—184no—
163Dic6—188no—
164C3×C8—182no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—24no—
172C2×C32—24no—
173C2×Q8—272no—
174C2×Q8—274no—
175C2×Q8—542no—
176C2×Q8—541no—
177C22×C4—91no—
178C22×C4—181no—
179C22×C4—181no—
180C2×C8—92no—
181C2×C8—182no—
182C4⋊C4—272no—
183C4⋊C4—541no—
184C22×C3—62no—
185C22×C3—62no—
186C22×C3—62no—
187C22×C3—62no—
188C22×C3—62no—
189C22×C3—62no—
190C22×C3—61no—
191C22×C3—61no—
192C22×C3D32⋊C412yes—
193C22×C3#377512yes—
194C22×C3#377612yes—
195C22×C3C2×GO+(4,𝔽2)11yescharacteristic
196C3×C4—91no—
197C3×C4—91no—
198C3×C4—92no—
199C3×C4—184no—
200C3×C4—362no—
201Dic3—61no—
202Dic3—61no—
203Dic3—62no—
204Dic3—61no—
205Dic3—61no—
206Dic3—62no—
207Dic3—64no—
208Dic3—122no—
209Dic3—184no—
210Dic3—362no—
211C32—21no—
212C32—21no—
213C23#1088⋊C211yescharacteristic
214Q8—274no—
215Q8—542no—
216Q8—544no—
217C2×C4—92no—
218C2×C4—92no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—184no—
222C2×C4—182no—
223C2×C4—182no—
224C2×C4—362no—
225C8—182no—
226C2×C3—61no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3—61no—
230C2×C3—61no—
231C2×C3—61no—
232C2×C3—64no—
233C2×C3—64no—
234C2×C3#6374014yes—
235C2×C3C2×D32⋊C411yescharacteristic
236C2×C3C2×#377511yescharacteristic
237C2×C3C2×#377611yescharacteristic
238C22#8858712yes—
239C22#8858912yes—
240C22(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
241C22C2×#1088⋊C211yescharacteristic
242C4—91no—
243C4—91no—
244C4—92no—
245C4—184no—
246C4—362no—
247C3—61no—
248C3—61no—
249C3C2×#6374011yescharacteristic
250C2#1172070214yesdirect product factor
251C2C2×#8858711yescharacteristic
252C2C2×#8858911yescharacteristic
253C2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC23#1088⋊C2
commutator#1084C22×C4
FrattiniC22×C3C2×GO+(4,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)D4
socleC23×C3GO+(4,𝔽2)
cosocleC2×#1084C23
hypocenterH(3,𝔽3)C2×#103
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11720702C2direct (N × H)
10. C2×#6196C4semidirect
29. #6196C2×C4semidirect
81. C2×H(3,𝔽3)#103semidirect
130. H(3,𝔽3)C2×#103semidirect
250. C2#11720702direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61962
C22×#61882
C2×#884492
#117207022
C2×#8169