Finite Group Databasebeta

← #13140954

Subgroups — #13140954

Order 1728.

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

normal subgroups only

This group has 510 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
1#11721864C211yesmaximal, characteristic
2C2×#3756⋊C3C212yesmaximal, complemented
3#11721099C211yesmaximal, characteristic, complemented
4C2×#88437C212yesmaximal
5#11720303C211yesmaximal, characteristic, complemented
6#10591627—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×C4⋊#272C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10#88542—22no—
11C4×#1084C2211yescharacteristic
12#3756⋊C3—24no—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88449—22no—
15#88437—24no—
16C2×#3756D312yescomplemented
17C3×#947—32no—
18C3×#947—61no—
19C2×C3×#216—32no—
20#63866—32no—
21C2×#3716—32no—
22C2×#3715—32no—
23#63554—32no—
24C22×#272—21no—
25Q8×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084D412yes—
29C4⋊#272—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#634⋊C3—22no—
32#634⋊C3—22no—
33(C3⋊(C3⋊C8))⋊C3D414yes—
34#5190—92no—
35#5060—91no—
36C2×C32×Q8—21no—
37C2×C32×Q8C2×D312yes—
38C2×#634C2×D312yes—
39C2×C3×D12—32no—
40C2×C3×D12—61no—
41C2×C3×Dic6—32no—
42#3756—24no—
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×#216—64no—
45C3×(C2×D3)⋊C4—64no—
46C3×(C2×D3)⋊C4—64no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—32no—
50Dic3⋊Dic3—64no—
51#3716—64no—
52#3715—64no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×#272—41no—
55C2×#272—42no—
56C4×H(3,𝔽3)—41no—
57C4×H(3,𝔽3)C2×D412yes—
58#1084—22no—
59(C3⋊Dic3)⋊C3—42no—
60C2×C3×Q16—92no—
61C3×#124—91no—
62#947—32no—
63#947—61no—
64#944—92no—
65C2×#216—32no—
66C2×#216—92no—
67C2×#215—92no—
68C2×#214—92no—
69C2×#204—92no—
70#869—91no—
71#815—92no—
72#803—91no—
73C22×C3×D3—62no—
74C22×C3×D3—62no—
75C32×Q8—22no—
76C32×Q8—22no—
77C32×Q8—42no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4C22×D312yes—
82C2×C3⋊Dic3—22no—
83#634—22no—
84#634—22no—
85C2×C3×Dic3—34no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C3×D12—62no—
89C3×D12—62no—
90C3×D12—122no—
91C3×Dic6—62no—
92C3×Dic6—62no—
93C3⋊(C3⋊C8)#6714yes—
94C3×C3⋊C8—34no—
95#497—271noSylow
96C2×H(3,𝔽3)#12711yescharacteristic
97C2×H(3,𝔽3)#13712yes—
98#272—81no—
99C2×C3×Q8—32no—
100C2×C3×Q8—61no—
101C2×C3×Q8—92no—
102C2×C3×Q8D3211yescharacteristic
103C2×C3×D4—91no—
104C2×D12—32no—
105C2×D12—61no—
106C2×Dic6—32no—
107C2×Dic6—92no—
108C3×Q16—184no—
109C2×C3×C8—92no—
110C3×C22⋊C4—182no—
111C3×C42—91no—
112#216—64no—
113#216—184no—
114#215—184no—
115#214—184no—
116(C2×D3)⋊C4—64no—
117(C2×D3)⋊C4—64no—
118C4⋊Dic3—182no—
119Dic3⋊C4—184no—
120C4×Dic3—32no—
121C4×Dic3—61no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—92no—
124#204—184no—
125C22×C32—21no—
126C22×C32D3×D412yes—
127C2×C3×D3—122no—
128C2×C3×D3—124no—
129C2×C3×D3—124no—
130C2×C3×D3—122no—
131C32×C4—22no—
132C32×C4—42no—
133C32×C4—42no—
134C32×C4D3×D412yes—
135C32×C4C2×#6712yes—
136C3⋊Dic3—42no—
137C3×Dic3—64no—
138C3×Dic3—122no—
139C3×Dic3—122no—
140C2×Q16—92no—
141C2×SD16—272no—
142C4⋊Q8—271no—
143#124—91no—
144#97—271no—
145H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
146C23×C3—181no—
147C22×D3—62no—
148C22×D3—62no—
149C3×Q8—22no—
150C3×Q8—62no—
151C3×Q8—62no—
152C3×Q8—122no—
153C3×Q8—182no—
154C3×Q8—182no—
155C3×D4—182no—
156C2×C3×C4—21no—
157C2×C3×C4—32no—
158C2×C3×C4—62no—
159C2×C3×C4—62no—
160C2×C3×C4—61no—
161C2×C3×C4—92no—
162C2×C3×C4—182no—
163C2×C3×C4C2×D3211yescharacteristic
164C2×Dic3—34no—
165C2×Dic3—62no—
166C2×Dic3—62no—
167C2×Dic3—182no—
168D12—62no—
169D12—62no—
170D12—122no—
171Dic6—62no—
172Dic6—62no—
173Dic6—182no—
174Dic6—182no—
175C3×C8—94no—
176C3⋊C8—34no—
177C3⋊C8—94no—
178C2×C32—22no—
179C2×C32—21no—
180C2×C32#91814yes—
181C2×C32#94012yes—
182C3×D3—242no—
183C3×D3—242no—
184C2×Q8—92no—
185C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
186C2×D4—91no—
187Q16—184no—
188SD16—544no—
189C2×C8—92no—
190C4⋊C4—542no—
191C22⋊C4—182no—
192C42—91no—
193C22×C3—32no—
194C22×C3—61no—
195C22×C3—362no—
196C22×C3—361no—
197C22×C3#381111yescharacteristic
198C2×D3—122no—
199C2×D3—124no—
200C2×D3—122no—
201C2×D3—124no—
202C3×C4—32no—
203C3×C4—32no—
204C3×C4—41no—
205C3×C4—62no—
206C3×C4—122no—
207C3×C4—122no—
208C3×C4—182no—
209C3×C4—362no—
210C3×C4D3×#6712yes—
211Dic3—64no—
212Dic3—122no—
213Dic3—122no—
214Dic3—362no—
215C32—21no—
216C32#519012yescomplemented
217C23—181no—
218Q8—22no—
219Q8—182no—
220Q8—182no—
221D4—182no—
222C2×C4—21no—
223C2×C4—92no—
224C2×C4—182no—
225C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
226C8—94no—
227C2×C3—34no—
228C2×C3—32no—
229C2×C3—62no—
230C2×C3—61no—
231C2×C3—721no—
232C2×C3#6394512yes—
233C2×C3#6397911yescharacteristic
234D3—242no—
235D3—242no—
236C22—362no—
237C22—361no—
238C22#8867711yescharacteristic
239C4—41no—
240C4—182no—
241C4—362no—
242C4#8867612yes—
243C3—32no—
244C3—61no—
245C3#1059162911yescharacteristic
246C2—721no—
247C2#1172122712yes—
248C2#1172131711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊#272C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#497
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
3. #11721099C2semidirect
5. #11720303C2semidirect
16. C2×#3756D3semidirect
145. H(3,𝔽3)#497semidirect
216. C32#5190semidirect

Maximal subgroups

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

subgroupindex
#117218642
C2×#3756⋊C32
#117210992
C2×#884372
#117203032
#105916273