Finite Group Databasebeta

← #13140518

Subgroups — #13140518

Order 1728.

426 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11721846C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic, complemented
3#63659⋊C3C212yesmaximal
4#11720326C212yesmaximal, complemented
5#11720303C211yesmaximal, characteristic, complemented
6#10591488—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9C2×#6196—21no—
10#88552—21no—
11#88552—21no—
12C4×#1084C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×#936—32no—
17C3×#936—61no—
18#63918—32no—
19#63659D312yes—
20C3×#832—32no—
21#63566—32no—
22#63565—32no—
23#63554—32no—
24C23×H(3,𝔽3)—21no—
25D4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—21no—
28C2×#1084D411yescharacteristic
29C2×#1084D411yescharacteristic
30#6229—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32#6196—42no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5055—91no—
35#4765—92no—
36C2×C32×D4—21no—
37C2×C32×D4C2×D312yes—
38C2×C3×Dic6—32no—
39C2×C3×Dic6—61no—
40C2×#627—62no—
41C4⋊(C3⋊Dic3)C2×D312yes—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43C3×#217—62no—
44C3×#217—62no—
45C3×#217—62no—
46C3×#217—62no—
47C3×C4⋊Dic3—32no—
48C3×C4×Dic3—32no—
49C3×C4×Dic3—61no—
50C2×C3×C3⋊C8—32no—
51Dic3⋊Dic3—62no—
52C22×H(3,𝔽3)—41no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C4×H(3,𝔽3)#4112yes—
57#1084—21no—
58#1084—21no—
59#1084—41no—
60(C3⋊Dic3)⋊C3—42no—
61C3×#124—91no—
62#936—32no—
63#936—61no—
64#888—92no—
65#869—91no—
66C3×#102—92no—
67#835—92no—
68#832—32no—
69#832—92no—
70#816—92no—
71#815—92no—
72#810—92no—
73#803—91no—
74C23×C32—22no—
75C23×C32—22no—
76C32×D4—22no—
77C32×D4—22no—
78C32×D4—42no—
79C2×C32×C4—21no—
80C2×C32×C4C22×D312yes—
81C2×C3⋊Dic3—22no—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C3×Dic6—62no—
90C3×Dic6—62no—
91C3×Dic6—122no—
92#627—124no—
93C3⋊(C3⋊C8)—22no—
94C3×C3⋊C8—62no—
95#489—271noSylow
96C2×H(3,𝔽3)—81no—
97C2×H(3,𝔽3)#12411yescharacteristic
98C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
99C2×H(3,𝔽3)#13711yescharacteristic
100C2×C3×Q8—91no—
101C2×C3×D4—32no—
102C2×C3×D4—61no—
103C2×C3×D4D3211yescharacteristic
104C2×Dic6—32no—
105C2×Dic6—61no—
106C2×Dic6—182no—
107C2×Dic6—181no—
108C2×C3×C8—92no—
109C3×C4⋊C4—92no—
110C3×C22⋊C4—181no—
111C3×C22⋊C4—181no—
112C3×C42—91no—
113#217—62no—
114#217—62no—
115#217—62no—
116#217—62no—
117C4⋊Dic3—32no—
118C4⋊Dic3—92no—
119C4⋊Dic3—181no—
120Dic3⋊C4—182no—
121C4×Dic3—32no—
122C4×Dic3—61no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—92no—
125C22×C32—21no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—42no—
130C22×C32—42no—
131C22×C32—42no—
132C22×C32D3×D412yes—
133C32×C4—22no—
134C32×C4#23512yes—
135C32×C4#23712yes—
136C3⋊Dic3—42no—
137C3×Dic3—62no—
138C3×Dic3—62no—
139C3×Dic3—121no—
140C3×Dic3—121no—
141C3×Dic3—122no—
142C3×Dic3—122no—
143C3×Dic3—122no—
144C4⋊Q8—271no—
145#124—91no—
146#103—272no—
147#102—92no—
148#97—271no—
149H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
150C23×C3—21no—
151C23×C3—62no—
152C23×C3—62no—
153C3×Q8—182no—
154C3×D4—22no—
155C3×D4—62no—
156C3×D4—62no—
157C3×D4—122no—
158C2×C3×C4—32no—
159C2×C3×C4—61no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—182no—
163C2×C3×C4—181no—
164C2×C3×C4C2×D3211yescharacteristic
165C2×Dic3—32no—
166C2×Dic3—32no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—62no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×Dic3—182no—
173Dic6—62no—
174Dic6—62no—
175Dic6—122no—
176Dic6—364no—
177Dic6—362no—
178C3×C8—182no—
179C3⋊C8—62no—
180C3⋊C8—182no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—82no—
185C2×C32—82no—
186C2×C32#88512yes—
187C2×C32#91112yes—
188C2×C32#91512yes—
189C2×Q8—91no—
190C2×Q8—541no—
191C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
192C2×C8—92no—
193C4⋊C4—92no—
194C4⋊C4—541no—
195C22⋊C4—181no—
196C22⋊C4—181no—
197C42—91no—
198C22×C3—32no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3—61no—
203C22×C3—122no—
204C22×C3—122no—
205C22×C3—122no—
206C22×C3—122no—
207C22×C3—122no—
208C22×C3—122no—
209C22×C3#381111yescharacteristic
210C3×C4—32no—
211C3×C4—32no—
212C3×C4—62no—
213C3×C4—181no—
214C3×C4—181no—
215C3×C4—362no—
216C3×C4—361no—
217C3×C4#380412yes—
218Dic3—62no—
219Dic3—62no—
220Dic3—121no—
221Dic3—121no—
222Dic3—122no—
223Dic3—122no—
224Dic3—122no—
225Dic3—362no—
226C32—21no—
227C32#476512yescomplemented
228C23—21no—
229Q8—182no—
230Q8—1082no—
231D4—22no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C4—181no—
235C2×C4—182no—
236C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C8—182no—
238C2×C3—32no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—81no—
245C2×C3—242no—
246C2×C3—242no—
247C2×C3#6392711yescharacteristic
248C2×C3#6393111yescharacteristic
249C2×C3#6395311yescharacteristic
250C22—41no—
251C22—41no—
252C22—41no—
253C22#8867711yescharacteristic
254C4—181no—
255C4—181no—
256C4—361no—
257C4—362no—
258C4#8867112yes—
259C3—32no—
260C3—61no—
261C3#1059149611yescharacteristic
262C2—81no—
263C2#1172118911yescharacteristic
264C2#1172119111yescharacteristic
265C2#1172129411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#489
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721117C2semidirect
4. #11720326C2semidirect
5. #11720303C2semidirect
149. H(3,𝔽3)#489semidirect
227. C32#4765semidirect

Maximal subgroups

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

subgroupindex
#117218462
#117211172
#63659⋊C32
#117203262
#117203032
#105914883