Finite Group Databasebeta

← #13137441

Subgroups — #13137441

Order 1728.

510 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721721C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C2×#88439C212yesmaximal, complemented
4C2×#88437C212yesmaximal
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10590376—31nomaximal
7#10590372—31nomaximal
8C2×C4⋊#272C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12#88542—22no—
13C4⋊((C3⋊Dic3)⋊C3)—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88439—24no—
16#88437—24no—
17#64078D311yescharacteristic, complemented
18C3×#875—31no—
19C3×#875—31no—
20C3×#875—61no—
21C3×#869—31no—
22C2×#3719—32no—
23C2×#3718—32no—
24C2×#3716—32no—
25C2×#3715—32no—
26#63630D311yescharacteristic, complemented
27C3×#803—31no—
28C22×#272—21no—
29C2×C4×H(3,𝔽3)D412yes—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—21no—
32C4⋊#272—22no—
33#6229—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3D414yes—
38#5086—91no—
39#5060—91no—
40#4715—91no—
41C2×#634C2×D312yes—
42C2×C3×D12—31no—
43C2×C3×D12—31no—
44C2×C3×D12—61no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—31no—
47C2×C3×Dic6—31no—
48C2×C3×Dic6—61no—
49C32×C42—21no—
50C32×C42C2×D311yescharacteristic
51C32×C42C2×D311yescharacteristic
52C4⋊(C3⋊Dic3)—22no—
53C2×C3⋊(C3⋊C8)C2×D312yes—
54C3×(C2×D3)⋊C4—64no—
55C3×(C2×D3)⋊C4—62no—
56C3×(C2×D3)⋊C4—62no—
57C3×C4⋊Dic3—62no—
58C2×C3×C3⋊C8—32no—
59#3719—64no—
60#3718—64no—
61#3716—64no—
62#3715—64no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—41no—
65C2×#272—42no—
66C4×H(3,𝔽3)—22no—
67C4×H(3,𝔽3)C2×D412yes—
68#1084—41no—
69(C3⋊Dic3)⋊C3—42no—
70C3×C4⋊Q8—91no—
71C3×#124—91no—
72C2×#216—92no—
73C2×#216—92no—
74C2×#215—92no—
75C2×#214—92no—
76C2×Dic12—92no—
77C2×#204—92no—
78#875—31no—
79#875—31no—
80#875—61no—
81#869—31no—
82#869—91no—
83C3×#97—91no—
84#803—31no—
85#803—91no—
86C22×C3×D3—62no—
87C22×C3×D3—61no—
88C22×C3×D3—61no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4D1212yes—
92C2×C32×C4#6712yes—
93C2×C32×C4C22×D311yescharacteristic
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—22no—
96#634—22no—
97#634—22no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C3×D12—62no—
103C3×D12—62no—
104C3×D12—122no—
105C3×Dic6—62no—
106C3×Dic6—62no—
107C3×Dic6—62no—
108C3×Dic6—62no—
109C3×Dic6—122no—
110C3⋊(C3⋊C8)D1214yes—
111C3×C3⋊C8—34no—
112#497—271noSylow
113C2×H(3,𝔽3)#12711yescharacteristic
114C2×H(3,𝔽3)#13712yes—
115#272—81no—
116C2×C3×Q8—92no—
117C2×C3×Q8—91no—
118C2×C3×D4—91no—
119C2×D12—31no—
120C2×D12—31no—
121C2×D12—61no—
122C2×Dic6—31no—
123C2×Dic6—31no—
124C2×Dic6—32no—
125C2×Dic6—61no—
126C2×Dic6—92no—
127C2×C3×C8—92no—
128C3×C4⋊C4—182no—
129C3×C22⋊C4—182no—
130C3×C42—31no—
131C3×C42—31no—
132C3×C42—61no—
133C3×C42D3211yescharacteristic
134#216—184no—
135#216—184no—
136#215—184no—
137#214—184no—
138(C2×D3)⋊C4—62no—
139(C2×D3)⋊C4—62no—
140(C2×D3)⋊C4—64no—
141C4⋊Dic3—62no—
142C4⋊Dic3—182no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—92no—
145Dic12—184no—
146#204—184no—
147C22×C32—21no—
148C22×C32C2×D1211yescharacteristic
149C22×C32C2×#6711yescharacteristic
150C2×C3×D3—122no—
151C2×C3×D3—124no—
152C2×C3×D3—121no—
153C2×C3×D3—122no—
154C2×C3×D3—121no—
155C2×C3×D3—122no—
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—42no—
160C32×C4C2×D1212yes—
161C32×C4D3×D412yes—
162C3⋊Dic3—42no—
163C3×Dic3—122no—
164C3×Dic3—122no—
165C3×Dic3—121no—
166C3×Dic3—121no—
167C2×Q16—272no—
168C2×SD16—272no—
169C4⋊Q8—91no—
170#124—91no—
171#97—91no—
172H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
173C23×C3—181no—
174C22×D3—61no—
175C22×D3—61no—
176C22×D3—62no—
177C3×Q8—182no—
178C3×Q8—182no—
179C3×Q8—182no—
180C3×D4—182no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—62no—
186C2×C3×C4—61no—
187C2×C3×C4—182no—
188C2×C3×C4—181no—
189C2×C3×C4#62612yes—
190C2×C3×C4C2×D3211yescharacteristic
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—182no—
196D12—62no—
197D12—62no—
198D12—122no—
199Dic6—62no—
200Dic6—62no—
201Dic6—62no—
202Dic6—62no—
203Dic6—122no—
204Dic6—182no—
205Dic6—182no—
206C3×C8—94no—
207C3⋊C8—34no—
208C3⋊C8—94no—
209C2×C32—22no—
210C2×C32—21no—
211C2×C32#87411yescharacteristic
212C2×C32#90912yes—
213C2×C32#94011yescharacteristic
214C2×C32#94212yes—
215C3×D3—242no—
216C3×D3—241no—
217C3×D3—241no—
218C2×Q8—91no—
219C2×Q8—92no—
220C2×D4—91no—
221Q16—544no—
222SD16—544no—
223C2×C8—92no—
224C4⋊C4—182no—
225C22⋊C4—182no—
226C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—61no—
230C22×C3—362no—
231C22×C3—361no—
232C22×C3C2×#62611yescharacteristic
233C2×D3—121no—
234C2×D3—121no—
235C2×D3—122no—
236C2×D3—122no—
237C2×D3—122no—
238C2×D3—124no—
239C3×C4—22no—
240C3×C4—32no—
241C3×C4—32no—
242C3×C4—62no—
243C3×C4—62no—
244C3×C4—62no—
245C3×C4—122no—
246C3×C4—362no—
247C3×C4—361no—
248C3×C4D3×D1212yes—
249Dic3—121no—
250Dic3—121no—
251Dic3—122no—
252Dic3—122no—
253Dic3—362no—
254C32—21no—
255C32#471511yescharacteristic, complemented
256C32#508611yescharacteristic, complemented
257C23—181no—
258Q8—182no—
259Q8—182no—
260Q8—182no—
261D4—182no—
262C2×C4—181no—
263C2×C4—182no—
264C2×C4#620012yes—
265C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
266C8—94no—
267C2×C3—32no—
268C2×C3—32no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—62no—
272C2×C3—61no—
273C2×C3—721no—
274C2×C3#6383312yes—
275C2×C3#6391111yescharacteristic
276D3—241no—
277D3—241no—
278D3—242no—
279C22—362no—
280C22—361no—
281C22C2×#620011yescharacteristic
282C4—22no—
283C4—361no—
284C4—362no—
285C4#8865412yes—
286C3—31no—
287C3—31no—
288C3—61no—
289C3#1059037411yescharacteristic
290C2—721no—
291C2#1172106912yes—
292C2#1172115811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#497
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. C2×#88439C2semidirect
5. #63630⋊C3C2semidirect
17. #64078D3semidirect
26. #63630D3semidirect
172. H(3,𝔽3)#497semidirect
255. C32#4715semidirect
256. C32#5086semidirect

Maximal subgroups

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

subgroupindex
#117217212
#64078⋊C32
C2×#884392
C2×#884372
#63630⋊C32
#105903763
#105903723