Finite Group Databasebeta

← #11723720

Subgroups — #11723720

Order 864.

486 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 87 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×#638C211yesmaximal, characteristic
2C22×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#3757C211yesmaximal, characteristic, complemented
4#88815C212yesmaximal, complemented
5#88814C212yesmaximal, complemented
6#63966—31nomaximal
7#63962—31nomaximal
8#63961—31nomaximal
9C23×C33C2211yescharacteristic
10C22×C3×C3⋊D3C2211yescharacteristic
11C3×#638—22no—
12C2×C3×C3⋊Dic3—22no—
13C2×C3×C3⋊Dic3C2212yes—
14C2×C3×C3⋊Dic3C2212yes—
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×#638D311yescharacteristic
17C22×C3⋊Dic3D311yescharacteristic, complemented
18C2×C3×#67—31no—
19C2×C3×#67—31no—
20C2×C3×#67—61no—
21C22×C3×Dic3—31no—
22C22×C3×Dic3—31no—
23C22×C3×Dic3—61no—
24#3757D311yescharacteristic, complemented
25C3×#217—31no—
26C3×#217—31no—
27C3×#217—61no—
28#3724—32no—
29Dic3⋊Dic3—32no—
30Dic3⋊Dic3—32no—
31#3722—32no—
32(C2×D3)⋊Dic3—32no—
33(C2×D3)⋊Dic3—32no—
34C22×C33—22no—
35C22×C33D412yes—
36C22×C33C2311yescharacteristic
37C2×C3×C3⋊D3—21no—
38C2×C3×C3⋊D3—22no—
39C3×C3⋊Dic3—22no—
40C3×C3⋊Dic3—21no—
41C3×C3⋊Dic3—22no—
42#935—91no—
43#929—91no—
44#886—91no—
45C23×C32—21no—
46C23×C32—21no—
47C23×C32—21no—
48C23×C32—41no—
49C23×C32C2×D311yescharacteristic
50C23×C32C2×D311yescharacteristic
51C23×C32C2×D311yescharacteristic
52C22×C3⋊D3C2×D311yescharacteristic
53C22×C3×D3—31no—
54C22×C3×D3—31no—
55C22×C3×D3—61no—
56#638—22no—
57C2×C3⋊Dic3—22no—
58C2×C3⋊Dic3C2×D312yes—
59C2×C3⋊Dic3C2×D312yes—
60C2×C3⋊Dic3C2×D311yescharacteristic
61C3×#67—64no—
62C3×#67—62no—
63C3×#67—62no—
64C2×C3×Dic3—32no—
65C2×C3×Dic3—32no—
66C2×C3×Dic3—32no—
67C2×C3×Dic3—32no—
68C2×C3×Dic3—31no—
69C2×C3×Dic3—31no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—64no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—62no—
76C2×C33—22no—
77C2×C33C2×D411yescharacteristic
78C2×C33#4112yes—
79C3×C3⋊D3—41no—
80C2×C3×D4—91no—
81C22×C3×C4—91no—
82C2×#67—31no—
83C2×#67—31no—
84C2×#67—61no—
85C22×Dic3—31no—
86C22×Dic3—31no—
87C22×Dic3—61no—
88C3×C22⋊C4—91no—
89#217—31no—
90#217—31no—
91#217—61no—
92#217—92no—
93(C2×D3)⋊C4—92no—
94(C2×D3)⋊C4—92no—
95C4⋊Dic3—92no—
96Dic3⋊C4—92no—
97Dic3⋊C4—92no—
98C22×C32—22no—
99C22×C32—24no—
100C22×C32—22no—
101C22×C32—22no—
102C22×C32—22no—
103C22×C32—22no—
104C22×C32—22no—
105C22×C32—21no—
106C22×C32—21no—
107C22×C32—21no—
108C22×C32—42no—
109C22×C32—41no—
110C22×C32—42no—
111C22×C32—42no—
112C22×C32—44no—
113C22×C32D1212yes—
114C22×C32#6712yes—
115C22×C32#6712yes—
116C22×C32C22×D311yescharacteristic
117C22×C32C22×D311yescharacteristic
118C22×C32C22×D311yescharacteristic
119C2×C3⋊D3—21no—
120C2×C3⋊D3—22no—
121C2×C3×D3—62no—
122C2×C3×D3—61no—
123C2×C3×D3—62no—
124C2×C3×D3—62no—
125C2×C3×D3—61no—
126C2×C3×D3—122no—
127C3⋊Dic3—22no—
128C3⋊Dic3—21no—
129C3⋊Dic3—22no—
130C3×Dic3—64no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—61no—
134C3×Dic3—62no—
135C3×Dic3—62no—
136C3×Dic3—62no—
137C3×Dic3—61no—
138C3×Dic3—122no—
139#123—271noSylow
140C33#12311yescharacteristic, complemented, Sylow
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—41no—
145C23×C3—91no—
146C23×C3D3211yescharacteristic
147C23×C3D3211yescharacteristic
148C23×C3D3211yescharacteristic
149C22×D3—31no—
150C22×D3—31no—
151C22×D3—61no—
152C3×D4—182no—
153C2×C3×C4—91no—
154C2×C3×C4—92no—
155C2×C3×C4—92no—
156C2×C3×C4—182no—
157#67—64no—
158#67—62no—
159#67—62no—
160C2×Dic3—31no—
161C2×Dic3—32no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—31no—
165C2×Dic3—32no—
166C2×Dic3—62no—
167C2×Dic3—61no—
168C2×Dic3—62no—
169C2×Dic3—64no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×C32—22no—
173C2×C32—24no—
174C2×C32—22no—
175C2×C32—22no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—21no—
180C2×C32—21no—
181C2×C32—21no—
182C2×C32—41no—
183C2×C32—42no—
184C2×C32—42no—
185C2×C32—42no—
186C2×C32—44no—
187C2×C32C2×D1211yescharacteristic
188C2×C32#23512yes—
189C2×C32#23712yes—
190C2×C32#23712yes—
191C2×C32C2×#6711yescharacteristic
192C2×C32C2×#6711yescharacteristic
193C3⋊D3—41no—
194C3×D3—122no—
195C3×D3—121no—
196C3×D3—121no—
197C2×D4—91no—
198C22×C4—91no—
199C4⋊C4—272no—
200C22⋊C4—91no—
201C22⋊C4—272no—
202C22×C3—24no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—44no—
215C22×C3—42no—
216C22×C3—41no—
217C22×C3—182no—
218C22×C3—181no—
219C22×C3#62512yes—
220C22×C3#62612yes—
221C22×C3#62612yes—
222C22×C3C2×D3211yescharacteristic
223C22×C3C2×D3211yescharacteristic
224C22×C3C2×D3211yescharacteristic
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—62no—
228C2×D3—61no—
229C2×D3—61no—
230C2×D3—122no—
231C3×C4—182no—
232C3×C4—182no—
233C3×C4—181no—
234Dic3—64no—
235Dic3—62no—
236Dic3—62no—
237Dic3—62no—
238Dic3—62no—
239Dic3—61no—
240Dic3—62no—
241Dic3—61no—
242Dic3—122no—
243C32—21no—
244C32—21no—
245C32—21no—
246C32—41no—
247C32#88611yescharacteristic, complemented
248C32#92911yescharacteristic, complemented
249C32#93511yescharacteristic, complemented
250C23—91no—
251C23#111311yescharacteristic
252D4—182no—
253C2×C4—92no—
254C2×C4—92no—
255C2×C4—91no—
256C2×C4—182no—
257C2×C3—24no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—22no—
263C2×C3—22no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3—44no—
270C2×C3—42no—
271C2×C3—41no—
272C2×C3—361no—
273C2×C3#380412yes—
274C2×C3#380412yes—
275C2×C3#380512yes—
276C2×C3C2×#62511yescharacteristic
277C2×C3C2×#62611yescharacteristic
278C2×C3C2×#62611yescharacteristic
279D3—122no—
280D3—121no—
281D3—121no—
282C22—22no—
283C22—182no—
284C22—181no—
285C22#629512yes—
286C22C2×#111311yescharacteristic
287C4—182no—
288C4—182no—
289C4—181no—
290C3—21no—
291C3—21no—
292C3—21no—
293C3—41no—
294C3#6396111yescharacteristic, complemented
295C3#6396211yescharacteristic, complemented
296C3#6396611yescharacteristic, complemented
297C2—22no—
298C2—361no—
299C2#8904612yes—
300C2C2×#629511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#1113
commutator, socle, cosocleC22×C33C23
FittingC23×C33C22
prime radicalC2×C3×#638C2
hypercenterC23#1113
hypocenterC33#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×C3⋊Dic3C2semidirect
3. C3×#3757C2semidirect
4. #88815C2semidirect
5. #88814C2semidirect
17. C22×C3⋊Dic3D3semidirect
24. #3757D3semidirect
140. C33#123semidirect
247. C32#886semidirect
248. C32#929semidirect
249. C32#935semidirect
294. C3#63961semidirect
295. C3#63962semidirect
296. C3#63966semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6382
C22×C3×C3⋊Dic32
C3×#37572
#888152
#888142
#639663
#639623
#639613