Finite Group Databasebeta

← #13166321

Subgroups — #13166321

Order 1728.

396 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 62 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#11723404C211yesmaximal, characteristic
2#11723103C211yesmaximal, characteristic
3C3×#63778C211yesmaximal, characteristic
4#11722732C211yesmaximal, characteristic, complemented
5Dic3⋊#622C211yesmaximal, characteristic, complemented
6#11722724C211yesmaximal, characteristic, complemented
7Dic3×#622C211yesmaximal, characteristic, complemented
8#10593753—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
15C32×C4⋊Dic3C411yescharacteristic
16#88803C411yescharacteristic
17#88803C411yescharacteristic
18Dic3×C3⋊Dic3C2211yescharacteristic
19Dic3×C3⋊Dic3C2211yescharacteristic
20#64101—31no—
21#63841—62no—
22#63779—31no—
23#63778—31no—
24#63778D311yescharacteristic
25#63777—31no—
26C4⋊#622—31no—
27C4×#622—31no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×C4×C3⋊D3—21no—
34C3×C4×C3⋊D3—21no—
35C2×C32×Dic3C2×C411yescharacteristic
36C2×C32×Dic3C2×C411yescharacteristic
37C3⋊#622—21no—
38C3⋊#622—21no—
39C3×#622—21no—
40C3×#622—21no—
41#4720—91nomaximal
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×C3×C4×D3—62no—
44C2×#622—31no—
45C2×#622—31no—
46C2×#622C2×D311yescharacteristic
47C2×#622C2×D311yescharacteristic
48C32×C4⋊C4—31no—
49#3752—31no—
50#3752—31no—
51C4⋊(C3⋊Dic3)—31no—
52C4⋊(C3⋊Dic3)—62no—
53C4⋊(C3⋊Dic3)—122no—
54C4×C3⋊Dic3—31no—
55C4×C3⋊Dic3—31no—
56C3×C4⋊Dic3—22no—
57(C2×D3)⋊Dic3—62no—
58(C2×D3)⋊Dic3—62no—
59Dic32—62no—
60Dic32—62no—
61C22×C33C22×C411yescharacteristic
62C2×C3×C3⋊D3—21no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—21no—
65C33×C4—21no—
66C3⋊(C3⋊Dic3)—21no—
67C3⋊(C3⋊Dic3)—21no—
68C3×C3⋊Dic3#4111yescharacteristic
69C3×C3⋊Dic3#4111yescharacteristic
70C32×Dic3—21no—
71C32×Dic3—21no—
72#926—91no—
73#892—182no—
74C3×C22⋊C8—91no—
75#830—91no—
76#815—91no—
77Dic3⋊C8—91no—
78Dic3⋊C8—91no—
79C8×Dic3—91no—
80C22×C3⋊D3C4×D311yescharacteristic
81C22×C3×D3—62no—
82C2×C32×C4—22no—
83C2×C32×C4—31no—
84C2×C32×C4—31no—
85C2×C32×C4—42no—
86C2×C32×C4C4×D311yescharacteristic
87C2×C3⋊Dic3—31no—
88C2×C3⋊Dic3—31no—
89C2×C3⋊Dic3—62no—
90C2×C3⋊Dic3—62no—
91C2×C3⋊Dic3—122no—
92C2×C3⋊Dic3—122no—
93C2×C3⋊Dic3C22×D311yescharacteristic
94C4×C3⋊D3—21no—
95C4×C3⋊D3—21no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—62no—
99C3×C4×D3—64no—
100C3×C4×D3—64no—
101#622—21no—
102#622—21no—
103#622—61no—
104#622—61no—
105#433—271noSylow
106C2×C33#11711yescharacteristic
107C2×C33#13111yescharacteristic
108C2×C33#13111yescharacteristic
109C3×C3⋊D3—41no—
110C22×C3×C4—91no—
111C2×C4×D3—62no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C3×C4⋊C4—62no—
115#217—91no—
116#217—91no—
117(C2×D3)⋊C4—182no—
118(C2×D3)⋊C4—182no—
119C4⋊Dic3—91no—
120C4⋊Dic3—182no—
121C4⋊Dic3—362no—
122C4⋊Dic3#15711yescharacteristic
123C4×Dic3—91no—
124C4×Dic3—91no—
125C4×Dic3—182no—
126C4×Dic3—182no—
127C2×C3⋊C8—91no—
128C2×C3⋊C8—91no—
129C22×C32—22no—
130C22×C32—42no—
131C22×C32C2×C4×D311yescharacteristic
132C2×C3⋊D3—21no—
133C2×C3⋊D3—21no—
134C2×C3⋊D3—21no—
135C2×C3×D3—64no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C32×C4—21no—
139C32×C4—24no—
140C32×C4—44no—
141C32×C4—61no—
142C32×C4—61no—
143C3⋊Dic3—61no—
144C3⋊Dic3—61no—
145C3⋊Dic3—122no—
146C3⋊Dic3—122no—
147C3⋊Dic3—242no—
148C3⋊Dic3—242no—
149C3⋊Dic3#23711yescharacteristic
150C3⋊Dic3#23711yescharacteristic
151C3×Dic3—42no—
152C3×Dic3—42no—
153C3×Dic3—62no—
154C3×Dic3—62no—
155#117—271no—
156C4⋊C8—271no—
157C4⋊C8—271no—
158C22⋊C8—91no—
159C22⋊C8—271no—
160#97—271no—
161C4×C8—271no—
162C33#43311yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C22×D3—62no—
165C2×C3×C4—22no—
166C2×C3×C4—42no—
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—91no—
170C2×C3×C4—181no—
171C2×C3×C4—181no—
172C2×C3×C4C2×#15711yescharacteristic
173C2×Dic3—62no—
174C2×Dic3—91no—
175C2×Dic3—91no—
176C2×Dic3—182no—
177C2×Dic3—182no—
178C2×Dic3—362no—
179C2×Dic3—362no—
180C2×Dic3C2×#15711yescharacteristic
181C2×Dic3C2×#15711yescharacteristic
182C4×D3—64no—
183C4×D3—64no—
184C3×C8—181no—
185C3×C8—181no—
186C3⋊C8—181no—
187C3⋊C8—181no—
188C2×C32—22no—
189C2×C32—22no—
190C2×C32—22no—
191C2×C32—42no—
192C2×C32—42no—
193C2×C32—42no—
194C2×C32#87711yescharacteristic
195C2×C32#90111yescharacteristic
196C2×C32#90711yescharacteristic
197C3⋊D3—41no—
198C3×D3—124no—
199C22×C4—91no—
200C2×C8—91no—
201C2×C8—91no—
202C2×C8—271no—
203C2×C8—271no—
204C4⋊C4—31no—
205C4⋊C4—271no—
206C22⋊C4—271no—
207C22⋊C4—271no—
208C42—271no—
209C42—271no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—181no—
213C22×C3—181no—
214C22×C3—181no—
215C22×C3C22×#15711yescharacteristic
216C2×D3—64no—
217C2×D3—122no—
218C2×D3—122no—
219C3×C4—24no—
220C3×C4—21no—
221C3×C4—44no—
222C3×C4—91no—
223C3×C4—91no—
224C3×C4—122no—
225C3×C4—122no—
226Dic3—21no—
227Dic3—21no—
228Dic3—62no—
229Dic3—62no—
230Dic3—181no—
231Dic3—181no—
232Dic3—362no—
233Dic3—362no—
234Dic3—722no—
235Dic3—722no—
236C32—22no—
237C32—42no—
238C32#472011yescharacteristic, complemented
239C23—91no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—91no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4—271no—
246C2×C4—271no—
247C2×C4D3×#15711yescharacteristic
248C8—181no—
249C8—181no—
250C8—541no—
251C8—541no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—361no—
259C2×C3#6428611yescharacteristic
260C2×C3#6428711yescharacteristic
261C2×C3#6428911yescharacteristic
262D3—124no—
263C22—181no—
264C22—181no—
265C22—181no—
266C22C2×D3×#15711yescharacteristic
267C4—21no—
268C4—61no—
269C4—61no—
270C4—91no—
271C4—91no—
272C4—541no—
273C4—541no—
274C3—22no—
275C3—42no—
276C3#1059375311yescharacteristic, complemented
277C2—361no—
278C2#1172425011yescharacteristic
279C2#1172425311yescharacteristic
280C2#1172426111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722732C2semidirect
5. Dic3⋊#622C2semidirect
6. #11722724C2semidirect
7. Dic3×#622C2semidirect
162. C33#433semidirect
238. C32#4720semidirect
276. C3#10593753semidirect

Maximal subgroups

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

subgroupindex
#117234042
#117231032
C3×#637782
#117227322
Dic3⋊#6222
#117227242
Dic3×#6222
#105937533
#47209