Finite Group Databasebeta

← C2×#11720790 #13147417

Subgroups — C2×#11720790 #13147417

Order 1728.

555 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C2×C4⋊#1088C211yesmaximal, characteristic
3C2×#88626C211yesmaximal, characteristic
4#11720790C214yesmaximal, direct product factor
5C22×#1088—21no—
6C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
7Q8×#272—21no—
8Q8×#272—21no—
9Q8×#272C412yescomplemented
10Q8×#272C2212yes—
11C2×C4×#272—21no—
12C2×C4×#272C2211yescharacteristic
13C2×#6229—21no—
14C2×#6229C411yescharacteristic, complemented
15C2×#6188—21no—
16C4⋊#1088—21no—
17C4⋊#1088C2212yes—
18#88626—21no—
19#88626C2212yes—
20C2×C3×D3×Q8—62no—
21C22×#272D411yescharacteristic
22C2×#1088—22no—
23C2×#1088—42no—
24Q8×H(3,𝔽3)—21no—
25Q8×H(3,𝔽3)C2×C412yescomplemented
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28C2×#1084—21no—
29C2×#1084D411yescharacteristic
30C4×#272—22no—
31C4×#272—22no—
32C4×#272C2×C411yescharacteristic
33C4×#272D411yescharacteristic
34C4×#272D411yescharacteristic
35C4×#272C2311yescharacteristic
36#6229—21no—
37#6229—22no—
38#6229—22no—
39#6229C2×C412yescomplemented
40#6188—22no—
41C2×C3×#103—91nomaximal
42C2×C32×Q8—22no—
43C3×D3×Q8—64no—
44C3×D3×Q8—64no—
45C3×D3×Q8—64no—
46C3×D3×Q8—64no—
47C2×C3×C4×D3—62no—
48C2×C3×C4×D3—64no—
49C2×C3×Dic6—62no—
50C2×C3×Dic6—64no—
51C22×H(3,𝔽3)C22⋊C411yescharacteristic
52C2×#272C22⋊C411yescharacteristic
53C2×#272SD1612yes—
54C2×#272Q1612yes—
55C2×#272C2×D411yescharacteristic
56#1088—42no—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)C22⋊C411yescharacteristic
59C4×H(3,𝔽3)C22×C411yescharacteristic
60#1084—22no—
61#1084C22⋊C411yescharacteristic
62#1084C2×D411yescharacteristic
63C22×C3×Q8—91no—
64C2×D3×Q8—62no—
65C22×C3×C8—91no—
66C2×C3×C4⋊C4—91no—
67C3×#103—94no—
68C22×C3×D3—62no—
69C32×Q8—24no—
70C32×Q8—24no—
71C2×C32×C4—24no—
72C2×C32×C4—22no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—64no—
75C3×C4×D3—62no—
76C3×C4×D3—62no—
77C3×C4×D3—62no—
78C3×C4×D3—62no—
79C3×C4×D3—68no—
80C3×C4×D3—68no—
81C3×Dic6—64no—
82C3×Dic6—68no—
83C3×Dic6—68no—
84C3×Dic6—64no—
85C2×#103—91noSylow
86C2×H(3,𝔽3)#10312yescomplemented
87C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
88#272#10312yescomplemented
89#272C2×SD1611yescharacteristic
90#272C2×Q1611yescharacteristic
91C2×C3×Q8—62no—
92C2×C3×Q8—92no—
93C2×C3×Q8—92no—
94C2×C3×Q8—91no—
95C2×C3×Q8—181no—
96C2×C3×Q8—181no—
97C2×C3×Q8—181no—
98C2×C3×Q8#15711yescharacteristic
99C22×C3×C4—91no—
100C22×C3×C4—181no—
101C22×C3×C4—181no—
102D3×Q8—64no—
103D3×Q8—64no—
104D3×Q8—64no—
105D3×Q8—64no—
106C2×C4×D3—62no—
107C2×C4×D3—64no—
108C2×Dic6—62no—
109C2×Dic6—64no—
110C2×C3×C8—92no—
111C2×C3×C8—181no—
112C2×C3×C8—181no—
113C3×C4⋊C4—92no—
114C3×C4⋊C4—181no—
115C22×C32—22no—
116C2×C3×D3—62no—
117C2×C3×D3—64no—
118C2×C3×D3—64no—
119C2×C3×D3—62no—
120C32×C4—28no—
121C32×C4—22no—
122C32×C4—22no—
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—68no—
126C22×Q8—91no—
127C22×C8—91no—
128C2×C4⋊C4—91no—
129#103—94no—
130H(3,𝔽3)C2×#10311yescharacteristic, complemented, Sylow
131C23×C3—91no—
132C22×D3—62no—
133C3×Q8—21no—
134C3×Q8—64no—
135C3×Q8—64no—
136C3×Q8—92no—
137C3×Q8—182no—
138C3×Q8—182no—
139C3×Q8—181no—
140C3×Q8C2×#15712yes—
141C2×C3×C4—21no—
142C2×C3×C4—64no—
143C2×C3×C4—62no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—182no—
150C2×C3×C4—182no—
151C2×C3×C4—181no—
152C2×C3×C4—182no—
153C2×C3×C4—362no—
154C2×C3×C4C2×#15711yescharacteristic
155C2×Dic3—62no—
156C2×Dic3—64no—
157C4×D3—62no—
158C4×D3—62no—
159C4×D3—62no—
160C4×D3—62no—
161C4×D3—68no—
162C4×D3—68no—
163Dic6—64no—
164Dic6—68no—
165Dic6—68no—
166Dic6—64no—
167C3×C8—182no—
168C2×C32—22no—
169C2×C32—24no—
170C3×D3—64no—
171C3×D3—62no—
172C3×D3—62no—
173C2×Q8—92no—
174C2×Q8—92no—
175C2×Q8—91no—
176C2×Q8—181no—
177C2×Q8—181no—
178C2×Q8—181no—
179C2×Q8#108811yescharacteristic, complemented
180C22×C4—91no—
181C22×C4—181no—
182C22×C4—181no—
183C2×C8—92no—
184C2×C8—181no—
185C2×C8—181no—
186C4⋊C4—92no—
187C4⋊C4—181no—
188C22×C3—62no—
189C22×C3—91no—
190C22×C3—91no—
191C22×C3—92no—
192C22×C3—92no—
193C22×C3C22⋊#15711yescharacteristic
194C2×D3—62no—
195C2×D3—62no—
196C2×D3—64no—
197C2×D3—64no—
198C3×C4—22no—
199C3×C4—68no—
200C3×C4—62no—
201C3×C4—62no—
202C3×C4—91no—
203C3×C4—91no—
204C3×C4—182no—
205C3×C4—362no—
206C3×C4C22⋊#15711yescharacteristic
207C3×C4C22×#15711yescharacteristic
208Dic3—62no—
209Dic3—62no—
210Dic3—68no—
211C32—22no—
212C23—91no—
213Q8—21no—
214Q8—92no—
215Q8—182no—
216Q8—182no—
217Q8—181no—
218Q8C2×#108812yescomplemented
219C2×C4—21no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—182no—
226C2×C4—182no—
227C2×C4—181no—
228C2×C4—182no—
229C2×C4—362no—
230C2×C4C2×#108811yescharacteristic
231C8—182no—
232C2×C3—64no—
233C2×C3—62no—
234C2×C3—91no—
235C2×C3—91no—
236C2×C3—92no—
237C2×C3#6378412yes—
238C2×C3C2×C22⋊#15711yescharacteristic
239D3—62no—
240D3—64no—
241D3—62no—
242C22—92no—
243C22—92no—
244C22—91no—
245C22—91no—
246C22C22⋊#108811yescharacteristic
247C4—22no—
248C4—91no—
249C4—91no—
250C4—182no—
251C4—362no—
252C4C22⋊#108811yescharacteristic
253C4C22×#108811yescharacteristic
254C3—62no—
255C3C2×#6378411yescharacteristic
256C2—91no—
257C2—92no—
258C2—91no—
259C2#1172079012yesdirect product factor
260C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#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)C2×#103
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
4. #11720790C2direct (N × H)
6. C2×Q8×H(3,𝔽3)C4semidirect
9. Q8×#272C4semidirect
14. C2×#6229C4semidirect
25. Q8×H(3,𝔽3)C2×C4semidirect
39. #6229C2×C4semidirect
86. C2×H(3,𝔽3)#103semidirect
88. #272#103semidirect
130. H(3,𝔽3)C2×#103semidirect
179. C2×Q8#1088semidirect
218. Q8C2×#1088semidirect
259. C2#11720790direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×C4⋊#10882
C2×#886262
#117207902
C2×C3×#1039