Finite Group Databasebeta

← #13161786

Subgroups — #13161786

Order 1728.

456 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C3×#64128C211yesmaximal, characteristic
2#11723649C211yesmaximal, characteristic, complemented
3C3×#63656C212yesmaximal, complemented
4#11722068C212yesmaximal, complemented
5#11722066C211yesmaximal, characteristic, complemented
6#10591044—31nomaximal
7#10591042—32nomaximal
8C3×#3832—21no—
9#89040—21no—
10C33×Q16C2211yescharacteristic
11C3×#3755—21no—
12C3×#3746C2212yes—
13C3×#3745C2211yescharacteristic
14C3×C8×C3⋊D3C2211yescharacteristic
15#88812—21no—
16C3×C3⋊(C3⋊C16)C2212yes—
17#64128D311yescharacteristic
18C3×#919—32no—
19C3×#919—61no—
20#63802—31no—
21D3×Dic12—32no—
22#63656D312yescomplemented
23C3×#829—32no—
24C3×#829—32no—
25C3×#829—62no—
26#63549—32no—
27#63548—32no—
28#63547—32no—
29#63544—31no—
30#63543—32no—
31C33×Q8—21no—
32C3×C4⋊(C3⋊D3)—21no—
33C3×C4⋊(C3⋊D3)—41no—
34C3×C4×C3⋊D3—41no—
35C3×C4×C3⋊D3D411yescharacteristic
36C3×#634—22no—
37#6296—41no—
38#6294—41no—
39C33×C8C2311yescharacteristic
40C3×C3⋊(C3⋊C8)D411yescharacteristic
41#5195—91no—
42#4918—92no—
43#3832—21no—
44C3×#239—62no—
45C3×#239—62no—
46#3797—61no—
47D3×Dic6—62no—
48C32×Q16—21no—
49C32×Q16—22no—
50C32×Q16—41no—
51C32×Q16C2×D312yes—
52C32×Q16C2×D311yescharacteristic
53#3755—21no—
54#3746C2×D312yes—
55#3745C2×D311yescharacteristic
56C8×C3⋊D3C2×D311yescharacteristic
57C3×#215—62no—
58C3×#215—62no—
59C3×Dic12—32no—
60C3×Dic12—32no—
61C3×Dic12—62no—
62C3×D24—32no—
63C3×D24—61no—
64C3×D3×C8—32no—
65C3×D3×C8—61no—
66#3719—62no—
67#3718—61no—
68C3⋊(C3⋊C16)C2×D312yes—
69C3×C3⋊C16—32no—
70C3×C3⋊C16—32no—
71C3×C3⋊C16—62no—
72C2×C3×C3⋊D3—41no—
73C2×C3×C3⋊D3D811yescharacteristic
74C33×C4—41no—
75C33×C4C2×D411yescharacteristic
76C3×C3⋊Dic3—42no—
77C3×C3⋊Dic3D811yescharacteristic
78C3×#135—91no—
79#919—32no—
80#919—61no—
81D3×Q16—92no—
82C2×Dic12—91no—
83C3×Q32—92no—
84#829—32no—
85#829—32no—
86#829—62no—
87#828—92no—
88#827—92no—
89#812—91no—
90#800—92no—
91#798—92no—
92C32×Q8—21no—
93C32×Q8—22no—
94C32×Q8—22no—
95C32×Q8—42no—
96C32×Q8—42no—
97C4⋊(C3⋊D3)—21no—
98C4⋊(C3⋊D3)—41no—
99C4×C3⋊D3—41no—
100C4×C3⋊D3#6711yescharacteristic
101#634—22no—
102C3×D12—62no—
103C3×D12—62no—
104C3×D12—122no—
105C3×D12—122no—
106C3×C4×D3—32no—
107C3×C4×D3—61no—
108C3×C4×D3—122no—
109C3×C4×D3—122no—
110C3×Dic6—62no—
111C3×Dic6—62no—
112C3×Dic6—122no—
113#627—122no—
114#627—121no—
115#624—121no—
116D3×Dic3—122no—
117C32×C8—22no—
118C32×C8—21no—
119C32×C8—41no—
120C32×C8C22×D312yes—
121C32×C8C22×D311yescharacteristic
122C3⋊(C3⋊C8)#6711yescharacteristic
123C3×C3⋊C8—32no—
124C3×C3⋊C8—61no—
125#510—271noSylow
126C2×C33C2×D811yescharacteristic
127C3×C3⋊D3—21no—
128C3×C3⋊D3—81no—
129C3×#41—181no—
130#239—62no—
131#239—62no—
132D3×Q8—182no—
133C2×Dic6—181no—
134C3×Q16—21no—
135C3×Q16—22no—
136C3×Q16—41no—
137C3×Q16—92no—
138C3×Q16D3211yescharacteristic
139C3×Q16D3212yes—
140C3×SD16—181no—
141C3×D8—91no—
142C2×C3×C8—91no—
143#216—182no—
144#215—62no—
145#215—62no—
146Dic12—32no—
147Dic12—32no—
148Dic12—62no—
149Dic12—181no—
150D24—32no—
151D24—61no—
152D3×C8—32no—
153D3×C8—61no—
154C3×C16—92no—
155C3⋊C16—32no—
156C3⋊C16—32no—
157C3⋊C16—62no—
158C2×C3⋊D3—41no—
159C2×C3⋊D3#21311yescharacteristic
160C2×C3×D3—32no—
161C2×C3×D3—61no—
162C2×C3×D3—122no—
163C2×C3×D3—122no—
164C32×C4—22no—
165C32×C4—21no—
166C32×C4—41no—
167C32×C4—41no—
168C32×C4—42no—
169C32×C4—42no—
170C32×C4—82no—
171C32×C4—82no—
172C32×C4D3×D412yes—
173C32×C4C2×#6711yescharacteristic
174C3⋊Dic3—42no—
175C3⋊Dic3#21311yescharacteristic
176C3×Dic3—32no—
177C3×Dic3—61no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C3×Dic3—242no—
181#135—91no—
182C2×Q16—271no—
183Q32—92no—
184SD32—272no—
185M32—271no—
186C33#51011yescharacteristic, complemented, Sylow
187C3×Q8—22no—
188C3×Q8—22no—
189C3×Q8—21no—
190C3×Q8—42no—
191C3×Q8—42no—
192C3×Q8—182no—
193C3×D4—181no—
194C3×D4—361no—
195C2×C3×C4—91no—
196C2×C3×C4—361no—
197C2×Dic3—361no—
198D12—62no—
199D12—62no—
200D12—122no—
201D12—122no—
202C4×D3—32no—
203C4×D3—61no—
204C4×D3—122no—
205C4×D3—122no—
206C4×D3—362no—
207Dic6—62no—
208Dic6—62no—
209Dic6—122no—
210Dic6—362no—
211Dic6—361no—
212C3×C8—22no—
213C3×C8—21no—
214C3×C8—41no—
215C3×C8—91no—
216C3×C8C2×D3212yes—
217C3×C8C2×D3211yescharacteristic
218C3⋊C8—32no—
219C3⋊C8—61no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32D3×D812yes—
224C2×C32C2×#21311yescharacteristic
225C3⋊D3—21no—
226C3⋊D3—81no—
227C3×D3—62no—
228C3×D3—121no—
229C3×D3—242no—
230C3×D3—242no—
231#41—181no—
232C2×Q8—541no—
233Q16—92no—
234Q16—541no—
235Q16#111311yescharacteristic
236SD16—181no—
237D8—91no—
238C2×C8—91no—
239C16—92no—
240C22×C3—91no—
241C22×C3—361no—
242C2×D3—32no—
243C2×D3—61no—
244C2×D3—122no—
245C2×D3—122no—
246C3×C4—21no—
247C3×C4—22no—
248C3×C4—42no—
249C3×C4—42no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—82no—
253C3×C4—82no—
254C3×C4—91no—
255C3×C4—362no—
256C3×C4D3×#6712yes—
257C3×C4#381111yescharacteristic
258Dic3—32no—
259Dic3—61no—
260Dic3—122no—
261Dic3—122no—
262Dic3—242no—
263C32—22no—
264C32—21no—
265C32—41no—
266C32#491812yescomplemented
267C32#519511yescharacteristic, complemented
268Q8—21no—
269Q8—182no—
270Q8—1081no—
271D4—181no—
272D4—361no—
273C2×C4—91no—
274C2×C4—361no—
275C2×C4—1081no—
276C8—91no—
277C8C2×#111311yescharacteristic
278C2×C3—21no—
279C2×C3—22no—
280C2×C3—41no—
281C2×C3—181no—
282C2×C3—721no—
283C2×C3D3×#21312yes—
284C2×C3#6392611yescharacteristic
285D3—62no—
286D3—121no—
287D3—242no—
288D3—242no—
289C22—91no—
290C22—361no—
291C4—41no—
292C4—91no—
293C4—362no—
294C4#8904911yescharacteristic
295C3—21no—
296C3—22no—
297C3—41no—
298C3#1059104212yescomplemented
299C3#1059104411yescharacteristic, complemented
300C2—181no—
301C2—721no—
302C2#1172370211yescharacteristic
subgroupG/H
centerC2#11723702
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×Q16C22
socleC2×C33C2×D8
prime radicalC3×#64128C2
hypercenterQ16#1113
hypocenterC33#510
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
2. #11723649C2semidirect
3. C3×#63656C2semidirect
4. #11722068C2semidirect
5. #11722066C2semidirect
22. #63656D3semidirect
186. C33#510semidirect
266. C32#4918semidirect
267. C32#5195semidirect
298. C3#10591042semidirect
299. C3#10591044semidirect

Maximal subgroups

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

subgroupindex
C3×#641282
#117236492
C3×#636562
#117220682
#117220662
#105910443
#105910423