Finite Group Databasebeta

← #13166307

Subgroups — #13166307

Order 1728.

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

normal subgroups only

This group has 412 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#11723469C211yesmaximal, characteristic
2#11723104C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4#11722726C211yesmaximal, characteristic
5#11722726C211yesmaximal, characteristic
6#11722724C211yesmaximal, characteristic, complemented
7Dic3×#622C211yesmaximal, characteristic, complemented
8#10593753—31nomaximal
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88859C411yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×Dic3⋊C4C411yescharacteristic
17Dic3⋊(C3⋊Dic3)C411yescharacteristic
18#88804C411yescharacteristic
19Dic3×C3⋊Dic3C2211yescharacteristic
20#64101—31no—
21#63836—62no—
22#63779—31no—
23#63778—31no—
24#63778—31no—
25#63777—31no—
26C4⋊#622D311yescharacteristic, complemented
27C4×#622—31no—
28C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3C2×C411yescharacteristic
34C2×C32×Dic3C2×C411yescharacteristic
35#6289—21no—
36#6289—21no—
37C3⋊#622—21no—
38C3⋊#622—21no—
39C3×#622—21no—
40C3×#622—21no—
41#4840—91nomaximal
42C2×C4×C3⋊D3—31no—
43C2×#624—62no—
44C2×#622—31no—
45C2×#622—31no—
46C2×#622C2×D311yescharacteristic
47C2×#622C2×D311yescharacteristic
48C32×C4⋊C4—31no—
49#3752—31no—
50#3752—31no—
51#3752—62no—
52#3752—122no—
53C4⋊(C3⋊Dic3)—31no—
54C4×C3⋊Dic3—31no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×Dic3⋊C4—22no—
57C3×C4×Dic3—62no—
58Dic3⋊Dic3—62no—
59#3722—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#892—182no—
73#872—91no—
74C3×C4⋊C8—91no—
75(C2×D3)⋊C8—91no—
76(C2×D3)⋊C8—91no—
77#815—91no—
78C8×Dic3—91no—
79C4⋊(C3⋊C8)—91no—
80C22×C3⋊D3—31no—
81C22×C3⋊D3—62no—
82C22×C3⋊D3—122no—
83C2×C32×C4—22no—
84C2×C32×C4—31no—
85C2×C32×C4—31no—
86C2×C32×C4—42no—
87C2×C32×C4C4×D311yescharacteristic
88C2×C3⋊Dic3—31no—
89C2×C3⋊Dic3—62no—
90C2×C3⋊Dic3—122no—
91C2×C3⋊Dic3C4×D311yescharacteristic
92C2×C3⋊Dic3C22×D311yescharacteristic
93C4×C3⋊D3—61no—
94C4×C3⋊D3—61no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99#624—64no—
100#624—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—
110C2×C4×D3—91no—
111C2×C4×D3—182no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C3×C4⋊C4—62no—
115C3×C42—91no—
116(C2×D3)⋊C4—91no—
117(C2×D3)⋊C4—91no—
118(C2×D3)⋊C4—182no—
119(C2×D3)⋊C4—182no—
120(C2×D3)⋊C4—362no—
121C4⋊Dic3—182no—
122Dic3⋊C4—91no—
123Dic3⋊C4#15711yescharacteristic
124C4×Dic3—62no—
125C4×Dic3—91no—
126C4×Dic3—182no—
127C2×C3⋊C8—91no—
128C2×C3⋊C8—91no—
129C22×C32—22no—
130C22×C32—42no—
131C22×C32C2×C4×D311yescharacteristic
132C2×C3⋊D3—61no—
133C2×C3⋊D3—64no—
134C2×C3⋊D3—61no—
135C2×C3⋊D3—61no—
136C2×C3⋊D3—124no—
137C2×C3⋊D3—122no—
138C2×C3⋊D3—122no—
139C2×C3⋊D3—242no—
140C2×C3⋊D3—242no—
141C32×C4—21no—
142C32×C4—42no—
143C32×C4—61no—
144C32×C4—61no—
145C32×C4—82no—
146C3⋊Dic3—21no—
147C3⋊Dic3—61no—
148C3⋊Dic3—122no—
149C3⋊Dic3—242no—
150C3⋊Dic3#23711yescharacteristic
151C3⋊Dic3#23911yescharacteristic
152C3×Dic3—24no—
153C3×Dic3—42no—
154C3×Dic3—62no—
155C3×Dic3—62no—
156C3×Dic3—122no—
157#117—271no—
158C4⋊C8—91no—
159C4⋊C8—271no—
160C22⋊C8—271no—
161C22⋊C8—271no—
162#97—271no—
163C4×C8—271no—
164C33#43311yescharacteristic, complemented, Sylow
165C22×D3—91no—
166C22×D3—182no—
167C22×D3—362no—
168C2×C3×C4—22no—
169C2×C3×C4—42no—
170C2×C3×C4—62no—
171C2×C3×C4—62no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4C2×#15711yescharacteristic
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—91no—
178C2×Dic3—182no—
179C2×Dic3—362no—
180C2×Dic3C2×#15711yescharacteristic
181C2×Dic3C2×#15711yescharacteristic
182C4×D3—181no—
183C4×D3—181no—
184C4×D3—184no—
185C4×D3—184no—
186C3×C8—181no—
187C3×C8—181no—
188C3⋊C8—181no—
189C3⋊C8—181no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—42no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32#89211yescharacteristic
197C2×C32#90111yescharacteristic
198C2×C32#90711yescharacteristic
199C3⋊D3—121no—
200C3⋊D3—124no—
201C3⋊D3—244no—
202C22×C4—271no—
203C2×C8—91no—
204C2×C8—91no—
205C2×C8—271no—
206C2×C8—271no—
207C4⋊C4—31no—
208C4⋊C4—271no—
209C22⋊C4—271no—
210C22⋊C4—271no—
211C42—91no—
212C42—271no—
213C22×C3—22no—
214C22×C3—42no—
215C22×C3C22×#15711yescharacteristic
216C2×D3—184no—
217C2×D3—181no—
218C2×D3—181no—
219C2×D3—181no—
220C2×D3—362no—
221C2×D3—362no—
222C2×D3—364no—
223C2×D3—722no—
224C2×D3—722no—
225C3×C4—21no—
226C3×C4—42no—
227C3×C4—64no—
228C3×C4—82no—
229C3×C4—91no—
230C3×C4—91no—
231C3×C4—122no—
232C3×C4—181no—
233Dic3—21no—
234Dic3—21no—
235Dic3—62no—
236Dic3—62no—
237Dic3—122no—
238Dic3—181no—
239Dic3—362no—
240Dic3—722no—
241C32—22no—
242C32—42no—
243C32#484011yescharacteristic, complemented
244C23—271no—
245C2×C4—31no—
246C2×C4—31no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—271no—
250C2×C4—541no—
251C2×C4—541no—
252C2×C4D3×#15711yescharacteristic
253C8—181no—
254C8—181no—
255C8—541no—
256C8—541no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—42no—
263C2×C3#6428611yescharacteristic
264C2×C3#6428711yescharacteristic
265C2×C3#6428911yescharacteristic
266D3—364no—
267D3—361no—
268D3—724no—
269C22—541no—
270C22—541no—
271C22—541no—
272C22C2×D3×#15711yescharacteristic
273C4—21no—
274C4—61no—
275C4—61no—
276C4—91no—
277C4—91no—
278C4—181no—
279C4—541no—
280C3—22no—
281C3—42no—
282C3#1059375311yescharacteristic, complemented
283C2—1081no—
284C2#1172424711yescharacteristic
285C2#1172425311yescharacteristic
286C2#1172426611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3⋊(C3⋊D3)C2×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
2. #11723104C2semidirect
3. C3×C4⋊#622C2semidirect
6. #11722724C2semidirect
7. Dic3×#622C2semidirect
26. C4⋊#622D3semidirect
164. C33#433semidirect
243. C32#4840semidirect
282. C3#10593753semidirect

Maximal subgroups

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

subgroupindex
#117234692
#117231042
C3×C4⋊#6222
#117227262
#117227242
Dic3×#6222
#105937533
#48409