Finite Group Databasebeta

← #13173421

Subgroups — #13173421

Order 1728.

736 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×C3×#3830C211yesmaximal, characteristic
2C2×#88993C212yesmaximal, complemented
3#11723113C212yesmaximal, complemented
4#11723105C212yesmaximal, complemented
5#10593824—31nomaximal
6C2×C33×D4C411yescharacteristic
7C2×C3×#638C411yescharacteristic
8C2×C3×#638C2211yescharacteristic
9C22×C3×C3⋊Dic3C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C3×#3830—22no—
12C3×#3830—22no—
13C2×C3×C4×C3⋊D3C411yescharacteristic
14C2×C3×#634C2211yescharacteristic
15#88993—22no—
16#88993C2214yes—
17C2×C3⋊#622—22no—
18C2×#3830D311yescharacteristic
19C2×C3×#237—62no—
20C2×#3792—32no—
21#63788—32no—
22#63780—32no—
23C23×C33C2×C411yescharacteristic
24C23×C33C2×C411yescharacteristic
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C33×D4—22no—
27C2×C33×C4C2×C411yescharacteristic
28C3×#638—22no—
29C3×#638—22no—
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3D412yes—
35C2×C3×C3⋊Dic3D412yes—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4×C3⋊D3—21no—
38C3×C4×C3⋊D3—21no—
39C3×#634—22no—
40C3⋊#622—24no—
41#5237—91nomaximal
42C2×C32×D4—22no—
43C2×C32×D4—42no—
44C2×C32×D4Dic311yescharacteristic
45C2×#638Dic311yescharacteristic
46C2×#638C2×D311yescharacteristic
47C22×C3⋊Dic3Dic311yescharacteristic
48C22×C3⋊Dic3C2×D311yescharacteristic
49#3830—22no—
50#3830—22no—
51C2×C4×C3⋊D3Dic311yescharacteristic
52C2×#634C2×D311yescharacteristic
53C2×C3×#67—62no—
54C2×C3×#67—62no—
55C22×C3×Dic3—62no—
56C22×C3×Dic3—62no—
57C3×#237—68no—
58C3×#237—68no—
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—62no—
61#3792—34no—
62#3792—62no—
63C2×#622—62no—
64C22×C33—21no—
65C22×C33—41no—
66C22×C33—41no—
67C22×C33C22⋊C412yes—
68C22×C33C22×C411yescharacteristic
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3—41no—
71C33×C4—21no—
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3C22⋊C411yescharacteristic
74C3×C3⋊Dic3C22⋊C411yescharacteristic
75C3×C3⋊Dic3C2×D412yes—
76C2×C3×#41—91no—
77C2×#237—62no—
78C2×#208—92no—
79#836—92no—
80#833—92no—
81C23×C32—22no—
82C23×C32—22no—
83C23×C32—42no—
84C23×C32—42no—
85C23×C32C2×Dic311yescharacteristic
86C23×C32C2×Dic311yescharacteristic
87C22×C3⋊D3C2×Dic311yescharacteristic
88C22×C3×D3—62no—
89C32×D4—22no—
90C32×D4—28no—
91C32×D4—48no—
92C2×C32×C4—22no—
93C2×C32×C4—42no—
94C2×C32×C4C2×Dic311yescharacteristic
95#638—22no—
96#638—22no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3C2×Dic311yescharacteristic
100C2×C3⋊Dic3C2×Dic311yescharacteristic
101C2×C3⋊Dic3#6712yes—
102C2×C3⋊Dic3#6712yes—
103C2×C3⋊Dic3C22×D311yescharacteristic
104C4×C3⋊D3—21no—
105C4×C3⋊D3—21no—
106#634—22no—
107C3×#67—68no—
108C3×#67—68no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—64no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—64no—
116C3×C4×D3—64no—
117C3×C4×D3—64no—
118C3×Dic6—68no—
119#622—64no—
120#413—271noSylow
121C2×C33—21no—
122C2×C33—22no—
123C2×C33—41no—
124C2×C33C2×C22⋊C411yescharacteristic
125C3×C3⋊D3—41no—
126C3×#41—182no—
127C3×#41—182no—
128C2×C3×Q8—91no—
129C2×C3×D4—22no—
130C2×C3×D4—42no—
131C2×C3×D4—91no—
132C2×C3×D4—91no—
133C2×C3×D4#15711yescharacteristic
134C22×C3×C4—91no—
135C22×C3×C4—91no—
136C22×C3×C4—91no—
137C2×#67—62no—
138C2×#67—62no—
139C22×Dic3—62no—
140C22×Dic3—62no—
141#237—68no—
142#237—68no—
143C2×C4×D3—62no—
144C2×Dic6—62no—
145#208—94no—
146#208—182no—
147C2×C3⋊C8—182no—
148C22×C32—21no—
149C22×C32—24no—
150C22×C32—22no—
151C22×C32—24no—
152C22×C32—44no—
153C22×C32—42no—
154C22×C32—44no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—84no—
160C22×C32—84no—
161C22×C32#21712yes—
162C22×C32C22×Dic311yescharacteristic
163C2×C3⋊D3—21no—
164C2×C3⋊D3—41no—
165C2×C3×D3—64no—
166C2×C3×D3—124no—
167C32×C4—21no—
168C32×C4—24no—
169C32×C4—44no—
170C3⋊Dic3—21no—
171C3⋊Dic3#21711yescharacteristic
172C3⋊Dic3#21711yescharacteristic
173C3⋊Dic3C2×#6712yes—
174C3×Dic3—62no—
175C3×Dic3—62no—
176C3×Dic3—64no—
177C3×Dic3—64no—
178C2×#41—91no—
179C2×M16—272no—
180#101—272no—
181#100—272no—
182C33#41311yescharacteristic, complemented, Sylow
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—91no—
188C23×C3C2×#15711yescharacteristic
189C23×C3C2×#15711yescharacteristic
190C22×D3—62no—
191C3×Q8—182no—
192C3×D4—28no—
193C3×D4—22no—
194C3×D4—48no—
195C3×D4—182no—
196C3×D4—182no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—92no—
202C2×C3×C4—92no—
203C2×C3×C4—91no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4C2×#15711yescharacteristic
209#67—68no—
210#67—68no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—64no—
214C2×Dic3—64no—
215C2×Dic3—62no—
216C2×Dic3—64no—
217C2×Dic3—64no—
218C4×D3—64no—
219C4×D3—64no—
220Dic6—68no—
221C3⋊C8—184no—
222C2×C32—21no—
223C2×C32—24no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—44no—
227C2×C32—42no—
228C2×C32—44no—
229C2×C32—41no—
230C2×C32—44no—
231C2×C32—84no—
232C2×C32—84no—
233C2×C32C2×#21711yescharacteristic
234C3⋊D3—41no—
235C3×D3—124no—
236#41—182no—
237#41—182no—
238C2×Q8—91no—
239C2×D4—91no—
240C2×D4—91no—
241C2×D4C3⋊#15711yescharacteristic
242C22×C4—91no—
243C22×C4—91no—
244C22×C4—91no—
245M16—274no—
246M16—542no—
247C2×C8—542no—
248C22×C3—24no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—24no—
252C22×C3—44no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—44no—
256C22×C3—44no—
257C22×C3—42no—
258C22×C3—44no—
259C22×C3—84no—
260C22×C3—84no—
261C22×C3—181no—
262C22×C3—361no—
263C22×C3C22⋊#15712yes—
264C22×C3C22×#15711yescharacteristic
265C2×D3—64no—
266C2×D3—124no—
267C3×C4—24no—
268C3×C4—21no—
269C3×C4—44no—
270C3×C4—92no—
271C3×C4—91no—
272C3×C4—91no—
273C3×C4—181no—
274Dic3—64no—
275Dic3—62no—
276Dic3—62no—
277Dic3—64no—
278C32—22no—
279C32—42no—
280C32#523711yescharacteristic, complemented
281C23—91no—
282C23C2×C3⋊#15711yescharacteristic
283C23C2×C3⋊#15711yescharacteristic
284Q8—182no—
285D4—22no—
286D4—182no—
287D4—182no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—92no—
291C2×C4—92no—
292C2×C4—91no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4C2×C3⋊#15711yescharacteristic
298C8—544no—
299C2×C3—24no—
300C2×C3—21no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—44no—
304C2×C3—44no—
305C2×C3—44no—
306C2×C3—41no—
307C2×C3—42no—
308C2×C3—84no—
309C2×C3—84no—
310C2×C3—361no—
311C2×C3C2×C22⋊#15711yescharacteristic
312D3—124no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—181no—
317C22—361no—
318C22#8899412yes—
319C22C22×C3⋊#15711yescharacteristic
320C4—21no—
321C4—92no—
322C4—91no—
323C4—91no—
324C4—181no—
325C3—22no—
326C3—42no—
327C3#1059382411yescharacteristic, complemented
328C2—21no—
329C2—22no—
330C2—41no—
331C2—361no—
332C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC2×C33×D4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3830C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. #11723113C2semidirect
4. #11723105C2semidirect
182. C33#413semidirect
280. C32#5237semidirect
327. C3#10593824semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#889932
#117231132
#117231052
#105938243
#52379