Finite Group Databasebeta

← #13146377

Subgroups — #13146377

Order 1728.

264 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 28 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#11722393C211yesmaximal, characteristic
2#11720767C211yesmaximal, characteristic, complemented
3(C3⋊GU(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
4#10593221—31nomaximal
5#10591676—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7#88611C411yescharacteristic, complemented
8#88610C411yescharacteristic, complemented
9#88435—41nomaximal
10C3×#985—31no—
11C3×#985—61no—
12#64157D311yescharacteristic, complemented
13C3×#950—31no—
14#63815—31no—
15C3⋊(#131⋊C3)—31no—
16C3⋊GU(2,𝔽3)D311yescharacteristic, complemented
17C3×#837—31no—
18#63577—31no—
19#63569—31no—
20#63554—31no—
21C4×#272—41no—
22Q8⋊H(3,𝔽3)C2×C411yescharacteristic
23(C3⋊(C3⋊C8))⋊C3—41no—
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5405—91no—
26#5197—91no—
27#4823—91no—
28C32×#41C2×D311yescharacteristic
29C3×#235—31no—
30C3×#41⋊C3—21no—
31C3×#41⋊C3C2×D311yescharacteristic
32C3×GL(2,𝔽3)—31no—
33C3×GL(2,𝔽3)—61no—
34C3×#226—31no—
35C3×#226—61no—
36#3748—31no—
37C2×C3⋊(C3⋊C8)—31no—
38C3×#216—31no—
39C3×#215—31no—
40C3×#214—31no—
41C3×#213—31no—
42C3×C4×Dic3—31no—
43C3×#208—31no—
44C2×C3×C3⋊C8—31no—
45#3711—61no—
46#3711—121no—
47D3×C3⋊C8—61no—
48C2×#272—41no—
49C4×H(3,𝔽3)—41no—
50#1084—41no—
51#985—31no—
52#985—61no—
53C3×#135—91no—
54#950—31no—
55#948—31no—
56#948—91no—
57#948—91no—
58#907—91no—
59#131⋊C3—91no—
60GU(2,𝔽3)—91no—
61C3×#104—91no—
62#837—31no—
63#837—91no—
64#822—91no—
65#819—91no—
66#815—91no—
67#805—91no—
68#803—91no—
69C32×Q8C4×D311yescharacteristic
70C32×D4—31no—
71C2×C32×C4—31no—
72C3×#67—61no—
73C2×C3×Dic3—61no—
74C3×D12—31no—
75C3×C4×D3—61no—
76C3×C4×D3—121no—
77C3×C4×D3—241no—
78C3×Dic6—31no—
79C3×SL(2,𝔽3)—21no—
80C3×SL(2,𝔽3)C4×D311yescharacteristic
81C3⋊(C3⋊C8)—31no—
82C3⋊(C3⋊C8)—41no—
83C3⋊(C3⋊C8)S411yescharacteristic
84C3×C3⋊C8—31no—
85C3×C3⋊C8—31no—
86C3×C3⋊C8—61no—
87C3×C3⋊C8—121no—
88#444—271noSylow
89C2×H(3,𝔽3)—41no—
90#272—81no—
91C3×#41—31no—
92C3×#41—91no—
93C3×#41D3211yescharacteristic
94#235—31no—
95#41⋊C3—31no—
96#41⋊C3—61no—
97GL(2,𝔽3)—31no—
98GL(2,𝔽3)—61no—
99#226—31no—
100#226—61no—
101C3×Q16—91no—
102C3×SD16—91no—
103C3×SD16—91no—
104C3×D8—91no—
105C3×M16—91no—
106C2×C3×C8—91no—
107C3×C42—91no—
108#216—31no—
109#215—31no—
110#214—31no—
111#213—31no—
112C4×Dic3—31no—
113#208—31no—
114#208—91no—
115#208—91no—
116#208—181no—
117#208—181no—
118C2×C3⋊C8—31no—
119C2×C3⋊C8—91no—
120C2×C3⋊C8—91no—
121C2×C3⋊C8—181no—
122C2×C3⋊C8—181no—
123ΓL(1,𝔽25)—181no—
124ΓL(1,𝔽25)—361no—
125D3×C8—181no—
126C22×C32—31no—
127C2×C3×D3—61no—
128C2×C3×D3—121no—
129C2×C3×D3—241no—
130C32×C4—31no—
131C32×C4—41no—
132C32×C4—81no—
133C32×C4C2×S411yescharacteristic
134C3×Dic3—61no—
135C3×Dic3—121no—
136C3×Dic3—121no—
137C3×Dic3—241no—
138#135—91no—
139#131—91no—
140#131—271no—
141#108—271no—
142#104—91no—
143#104—271no—
144#97—271no—
145H(3,𝔽3)—41noSylow
146C3×Q8—31no—
147C3×Q8—91no—
148C3×Q8#62411yescharacteristic
149C3×D4—31no—
150C3×D4—31no—
151C3×D4—61no—
152C3×D4—91no—
153C3×D4—181no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—61no—
157C2×C3×C4—181no—
158C2×C3×C4—181no—
159#67—61no—
160C2×Dic3—61no—
161D12—31no—
162C4×D3—61no—
163C4×D3—121no—
164C4×D3—241no—
165Dic6—31no—
166SL(2,𝔽3)—31no—
167SL(2,𝔽3)—61no—
168C3×C8—91no—
169C3×C8—91no—
170C3×C8—181no—
171C3×C8—361no—
172C3⋊C8—31no—
173C3⋊C8—31no—
174C3⋊C8—31no—
175C3⋊C8—61no—
176C3⋊C8—91no—
177C3⋊C8—91no—
178C3⋊C8—91no—
179C3⋊C8—181no—
180C3⋊C8—361no—
181C2×C32—41no—
182C2×C32—61no—
183C2×C32—81no—
184C2×C32C4×S411yescharacteristic
185C3×D3—121no—
186C3×D3—241no—
187C3×D3—242no—
188#41—91no—
189#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
190Q16—91no—
191SD16—91no—
192SD16—91no—
193D8—91no—
194M16—91no—
195M16—271no—
196M16—541no—
197C2×C8—91no—
198C2×C8—271no—
199C2×C8—541no—
200C42—91no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—61no—
204C22×C3—181no—
205C2×D3—61no—
206C2×D3—121no—
207C2×D3—241no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—31no—
211C3×C4—61no—
212C3×C4—121no—
213C3×C4—181no—
214C3×C4—241no—
215C3×C4—361no—
216C3×C4D3×S411yescharacteristic
217Dic3—61no—
218Dic3—121no—
219Dic3—121no—
220Dic3—241no—
221C32—41no—
222C32—81no—
223C32#540511yescharacteristic, complemented
224Q8—91no—
225Q8#619911yescharacteristic
226D4—31no—
227D4—91no—
228D4—181no—
229C2×C4—31no—
230C2×C4—181no—
231C2×C4—181no—
232C8—91no—
233C8—91no—
234C8—91no—
235C8—181no—
236C8—271no—
237C2×C3—31no—
238C2×C3—61no—
239C2×C3—61no—
240C2×C3—121no—
241C2×C3—121no—
242C2×C3—241no—
243C2×C3—361no—
244C2×C3(C3⋊S4)⋊C411yescharacteristic
245D3—121no—
246D3—241no—
247D3—242no—
248C22—31no—
249C22—181no—
250C4—31no—
251C4—181no—
252C4—361no—
253C4((C3⋊S4)⋊C3)⋊C211yescharacteristic
254C3—31no—
255C3—121no—
256C3—241no—
257C3#1059322211yescharacteristic
258C2—61no—
259C2—361no—
260C2#6258⋊C411yescharacteristic
subgroupG/H
center, hypercenterC4((C3⋊S4)⋊C3)⋊C2
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
FrattiniC3×C4D3×S4
FittingC32×#41C2×D3
socleC2×C3(C3⋊S4)⋊C4
prime radical#11722393C2
cosocle#41⋊H(3,𝔽3)C22
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11720767C2semidirect
3. (C3⋊GU(2,𝔽3))⋊C3C2semidirect
7. #88611C4semidirect
8. #88610C4semidirect
12. #64157D3semidirect
16. C3⋊GU(2,𝔽3)D3semidirect
223. C32#5405semidirect

Maximal subgroups

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

subgroupindex
#117223932
#117207672
(C3⋊GU(2,𝔽3))⋊C32
#105932213
#105916763
#884354