Finite Group Databasebeta

← #13144038

Subgroups — #13144038

Order 1728.

781 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 99 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#11722676C211yesmaximal, characteristic
2#11721872C211yesmaximal, characteristic, complemented
3#11721788C211yesmaximal, characteristic, complemented
4#11721715C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
6#11720561C211yesmaximal, characteristic, complemented
7#11720522C211yesmaximal, characteristic, complemented
8C3×#4826—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10Q8×#272—21no—
11#88744—21no—
12#88744—22no—
13#88740—21no—
14#88740C2211yescharacteristic
15C2×#6229—21no—
16C2×#6229C2211yescharacteristic
17C2×#6229C2211yescharacteristic
18M16×H(3,𝔽3)C2211yescharacteristic
19C42×H(3,𝔽3)C2211yescharacteristic
20#88550—21no—
21#88546—21no—
22C4⋊#1084—22no—
23#88538C2211yescharacteristic
24#88530—21no—
25#88528—21no—
26C3×#1010—34no—
27C3×#951—34no—
28C3×#909—34no—
29C3×#869—34no—
30C32×#104D314yescomplemented
31C3×#824—34no—
32C3×#805—34no—
33Q8×H(3,𝔽3)D411yescharacteristic
34D4×H(3,𝔽3)—21no—
35D4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#6235—22no—
40#6235—21no—
41C2×#1084—22no—
42C2×#1084—21no—
43C2×#1084D412yes—
44C4⋊#272D411yescharacteristic
45C4×#272—22no—
46C4×#272—21no—
47#6229—22no—
48#6229—21no—
49#6229—21no—
50#6229—21no—
51#6229—21no—
52#6229D411yescharacteristic
53C8×H(3,𝔽3)—21no—
54#6180—21no—
55C3×#456—91no—
56#4826—34no—
57C32×#41C2×D314yes—
58C3×D3×Q8—64no—
59C3×#237—68no—
60C3×#237—64no—
61C3×#235—34no—
62C3×#235—64no—
63C2×C3×Dic6—34no—
64C2×C3×Dic6—34no—
65C2×C3×Dic6—64no—
66C32×M16C2×D314yes—
67C32×C42C2×D314yes—
68C3×#216—64no—
69C3×#214—64no—
70C3×C4⋊Dic3—68no—
71C3×#208—34no—
72C3×Dic12—64no—
73C3×#204—64no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×#272—21no—
77C4×H(3,𝔽3)—22no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)C2×D411yescharacteristic
80C4×H(3,𝔽3)C2×D411yescharacteristic
81#1084—22no—
82#1084—21no—
83#1084—41no—
84C3×2−1+4—91no—
85#1010—34no—
86C3×#137—91no—
87C3×#137—91no—
88C3×C4⋊Q8—91no—
89#951—34no—
90#909—34no—
91#869—34no—
92C3×#104—34no—
93C3×#104—91no—
94C3×#104C3⋊D311yescharacteristic
95C3×#101—91no—
96#824—34no—
97#805—34no—
98C32×Q8D1214yes—
99C32×D4—24no—
100C32×D4D1214yes—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4C22×D314yes—
104C3×#67—68no—
105C3×#67—64no—
106C2×C3×Dic3—38no—
107C2×C3×Dic3—68no—
108C2×C3×Dic3—64no—
109C3×D12—34no—
110C3×C4×D3—68no—
111C3×C4×D3—64no—
112C3×Dic6—34no—
113C3×Dic6—68no—
114C3×Dic6—64no—
115C3×Dic6—64no—
116C3×Dic6—64no—
117C3×Dic6—64no—
118C32×C8—24no—
119C3×C3⋊C8—64no—
120#456—91noSylow
121C2×H(3,𝔽3)—21no—
122C2×H(3,𝔽3)—41no—
123C2×H(3,𝔽3)#12011yescharacteristic
124#272—41no—
125C3×#41—34no—
126C3×#41—91no—
127C3×#41—181no—
128C3×#41—182no—
129C3×#41—181no—
130C3×#41C2×C3⋊D311yescharacteristic
131C2×C3×Q8—91no—
132C2×C3×Q8—91no—
133C2×C3×Q8—181no—
134C2×C3×Q8—181no—
135D3×Q8—64no—
136#237—68no—
137#237—64no—
138#235—34no—
139#235—64no—
140C2×Dic6—34no—
141C2×Dic6—34no—
142C2×Dic6—64no—
143C3×Q16—181no—
144C3×Q16—181no—
145C3×SD16—181no—
146C3×SD16—181no—
147C3×M16—34no—
148C3×M16—91no—
149C3×M16C2×C3⋊D311yescharacteristic
150C3×C4⋊C4—182no—
151C3×C42—34no—
152C3×C42C2×C3⋊D311yescharacteristic
153#216—64no—
154#214—64no—
155C4⋊Dic3—68no—
156#208—34no—
157Dic12—64no—
158#204—64no—
159C22×C32—24no—
160C22×C32D3×D414yes—
161C2×C3×D3—64no—
162C32×C4—28no—
163C32×C4—24no—
164C32×C4C2×D1214yes—
165C32×C4D3×D414yes—
166C3×Dic3—64no—
167C3×Dic3—68no—
168C3×Dic3—124no—
1692−1+4—91no—
170#137—91no—
171#137—91no—
172C4⋊Q8—91no—
173#104—91no—
174#104#27211yescharacteristic, complemented
175#101—91no—
176H(3,𝔽3)#45611yescharacteristic, complemented, Sylow
177C3×Q8—34no—
178C3×Q8—91no—
179C3×Q8—182no—
180C3×Q8—181no—
181C3×Q8—181no—
182C3×Q8—181no—
183C3×Q8—181no—
184C3×Q8C4⋊(C3⋊D3)11yescharacteristic
185C3×D4—21no—
186C3×D4—34no—
187C3×D4—64no—
188C3×D4—91no—
189C3×D4—182no—
190C3×D4—181no—
191C3×D4C4⋊(C3⋊D3)11yescharacteristic
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—34no—
195C2×C3×C4—64no—
196C2×C3×C4—64no—
197C2×C3×C4—92no—
198C2×C3×C4—182no—
199C2×C3×C4—182no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4C22×C3⋊D311yescharacteristic
203#67—64no—
204#67—68no—
205C2×Dic3—38no—
206C2×Dic3—68no—
207C2×Dic3—64no—
208D12—34no—
209C4×D3—64no—
210C4×D3—68no—
211Dic6—34no—
212Dic6—64no—
213Dic6—64no—
214Dic6—68no—
215Dic6—64no—
216Dic6—64no—
217C3×C8—21no—
218C3×C8—64no—
219C3×C8—181no—
220C3⋊C8—64no—
221C2×C32—24no—
222C2×C32—44no—
223C2×C32#88214yes—
224C3×D3—124no—
225#41—91no—
226#41—181no—
227#41—182no—
228#41—181no—
229#41C2×#27211yescharacteristic
230C2×Q8—91no—
231C2×Q8—91no—
232C2×Q8—181no—
233C2×Q8—181no—
234Q16—181no—
235Q16—181no—
236SD16—181no—
237SD16—181no—
238M16—91no—
239M16C2×#27211yescharacteristic
240C4⋊C4—182no—
241C42C2×#27211yescharacteristic
242C22×C3—21no—
243C22×C3—34no—
244C22×C3—64no—
245C22×C3—181no—
246C22×C3D4×C3⋊D311yescharacteristic
247C2×D3—64no—
248C3×C4—22no—
249C3×C4—21no—
250C3×C4—34no—
251C3×C4—34no—
252C3×C4—64no—
253C3×C4—68no—
254C3×C4—182no—
255C3×C4—181no—
256C3×C4—361no—
257C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
258C3×C4D4×C3⋊D311yescharacteristic
259Dic3—64no—
260Dic3—68no—
261Dic3—124no—
262C32#482614yescomplemented
263Q8—91no—
264Q8—182no—
265Q8—181no—
266Q8—181no—
267Q8—181no—
268Q8—181no—
269Q8C4⋊#27211yescharacteristic
270D4—21no—
271D4—91no—
272D4—182no—
273D4—181no—
274D4C4⋊#27211yescharacteristic
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—92no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4C22×#27211yescharacteristic
283C8—21no—
284C8—181no—
285C2×C3—21no—
286C2×C3—34no—
287C2×C3—41no—
288C2×C3—64no—
289C2×C3—124no—
290C2×C3—361no—
291C2×C3#6409111yescharacteristic
292D3—124no—
293C22—21no—
294C22—181no—
295C22D4×#27211yescharacteristic
296C4—22no—
297C4—21no—
298C4—182no—
299C4—181no—
300C4—361no—
301C4C2×C4⋊#27211yescharacteristic
302C4D4×#27211yescharacteristic
303C3—34no—
304C3#1059252511yescharacteristic
305C2—21no—
306C2—41no—
307C2—361no—
308C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#104C2
prime radical#11722676C2
hypercenterC3×#104C3⋊D3
hypocenterH(3,𝔽3)#456
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. #11721872C2semidirect
3. #11721788C2semidirect
4. #11721715C2semidirect
5. H(3,𝔽3)×#104C2semidirect
6. #11720561C2semidirect
7. #11720522C2semidirect
30. C32×#104D3semidirect
174. #104#272semidirect
176. H(3,𝔽3)#456semidirect
262. C32#4826semidirect

Maximal subgroups

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

subgroupindex
#117226762
#117218722
#117217882
#117217152
H(3,𝔽3)×#1042
#117205612
#117205222
C3×#48263