Finite Group Databasebeta

← #13179722

Subgroups — #13179722

Order 1728.

4116 conjugacy classes of subgroups; only the 675 normal subgroups are shown.

normal subgroups only

This group has 4116 conjugacy classes of subgroups — too many to draw. Only its 675 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×#89078C212yesmaximal, complemented
2C33×#121C211yesmaximal, characteristic, complemented
3#11723980C212yesmaximal, complemented
4#11723980C212yesmaximal, complemented
5#11723971C211yesmaximal, characteristic, complemented
6C2×#88857C211yesmaximal, characteristic, complemented
7#11723937C212yesmaximal, complemented
8#11723925C212yesmaximal, complemented
9#11723919C212yesmaximal, complemented
10#10596068—313nomaximal
11C2×C33×D4C2212yes—
12C2×C33×D4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C2×#6312C2212yes—
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C22×C3⋊(C3⋊Dic3)C2212yes—
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
20#89078—28no—
21C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
22C2×C4×C3⋊(C3⋊D3)C2212yes—
23C2×#6308C2212yes—
24C33×C4⋊C4C2211yescharacteristic
25C33×C22⋊C4C2212yes—
26#88864C2212yes—
27#88859C2212yes—
28#88859C2212yes—
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
30#88857C2212yes—
31#88857C2212yes—
32#88857C2212yes—
33C2×#3830—326no—
34C32×#121D3113yescomplemented
35#64148—326no—
36#64148—326no—
37#64139—313no—
38C2×#3750—313no—
39#64105—326no—
40#64093—326no—
41#64087—326no—
42C23×C33C2312yes—
43C23×C33C2311yescharacteristic
44C22×C3⋊(C3⋊D3)C2312yes—
45C33×D4—24no—
46C33×D4—41no—
47C2×C33×C4—41no—
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C33×C4C2312yes—
51#6312—24no—
52#6312—24no—
53#6312—42no—
54C2×C3⋊(C3⋊Dic3)—24no—
55C2×C3⋊(C3⋊Dic3)—41no—
56C2×C3⋊(C3⋊Dic3)D414yescomplemented
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3⋊(C3⋊Dic3)C2312yes—
60C4⋊(C3⋊(C3⋊D3))—41no—
61C4×C3⋊(C3⋊D3)—24no—
62#6308—24no—
63#5597—913no—
64C2×C32×D4C2×D3126yes—
65C2×C32×D4C2×D3113yes—
66C22×C32×C4C2×D3113yes—
67C2×#638—326no—
68C2×#638—326no—
69C2×#638—326no—
70C22×C3⋊Dic3—313no—
71C22×C3⋊Dic3—326no—
72C22×C3⋊Dic3—313no—
73#3830—6104no—
74C2×C4⋊(C3⋊D3)—313no—
75C2×C4×C3⋊D3—326no—
76C2×#634—326no—
77C32×C4⋊C4C2×D3113yes—
78C32×C22⋊C4C2×D3126yes—
79#3757—326no—
80#3752—326no—
81#3752—326no—
82C4⋊(C3⋊Dic3)—313no—
83#3750—326no—
84#3750—326no—
85#3750—326no—
86C22×C33—21no—
87C22×C33—21no—
88C22×C33—22no—
89C22×C33—22no—
90C22×C33—22no—
91C22×C33C2×D412yes—
92C22×C33C2411yescharacteristic
93C2×C3⋊(C3⋊D3)—22no—
94C2×C3⋊(C3⋊D3)—22no—
95C2×C3⋊(C3⋊D3)—22no—
96C33×C4—22no—
97C33×C4—21no—
98C33×C4—21no—
99C3⋊(C3⋊Dic3)—22no—
100C3⋊(C3⋊Dic3)—22no—
101C3⋊(C3⋊Dic3)C2×D414yes—
102C2×#237—926no—
103C3×#121C3⋊D3113yescomplemented
104#939—926no—
105#939—926no—
106#930—913no—
107C2×Dic3⋊C4—913no—
108#896—926no—
109#884—926no—
110#878—926no—
111C23×C32C22×D3126yes—
112C23×C32C22×D3113yes—
113C22×C3⋊D3—326no—
114C32×D4—252no—
115C32×D4—413no—
116C2×C32×C4—413no—
117C2×C32×C4C22×D3113yes—
118C2×C32×C4C22×D3113yes—
119C2×C32×C4C22×D3126yes—
120#638—652no—
121#638—652no—
122#638—1226no—
123C2×C3⋊Dic3—352no—
124C2×C3⋊Dic3—326no—
125C2×C3⋊Dic3—326no—
126C2×C3⋊Dic3—326no—
127C2×C3⋊Dic3—652no—
128C2×C3⋊Dic3—1213no—
129C4⋊(C3⋊D3)—1213no—
130C4×C3⋊D3—652no—
131#634—652no—
132#537—271noSylow
133C2×C33—22no—
134C2×C33—42no—
135C2×C33C22×D411yescharacteristic
136C2×C33H(5,𝔽2)11yescharacteristic
137C2×C332−1+411yescharacteristic
138C3⋊(C3⋊D3)—42no—
139C2×C3×D4C2×C3⋊D3126yes—
140C2×C3×D4C2×C3⋊D3113yes—
141C22×C3×C4C2×C3⋊D3113yes—
142C2×#67—926no—
143C2×#67—926no—
144C2×#67—926no—
145C22×Dic3—913no—
146C22×Dic3—926no—
147C22×Dic3—913no—
148#237—18104no—
149C2×D12—913no—
150C2×C4×D3—926no—
151C2×Dic6—926no—
152C3×C4⋊C4C2×C3⋊D3113yes—
153C3×C22⋊C4C2×C3⋊D3126yes—
154#217—926no—
155(C2×D3)⋊C4—926no—
156(C2×D3)⋊C4—926no—
157C4⋊Dic3—913no—
158Dic3⋊C4—926no—
159Dic3⋊C4—926no—
160Dic3⋊C4—926no—
161C22×C32—213no—
162C22×C32—213no—
163C22×C32—226no—
164C22×C32—226no—
165C22×C32—226no—
166C22×C32D3×D4126yes—
167C22×C32C23×D3113yes—
168C2×C3⋊D3—626no—
169C2×C3⋊D3—626no—
170C2×C3⋊D3—626no—
171C32×C4—226no—
172C32×C4—213no—
173C32×C4—213no—
174C3⋊Dic3—352no—
175C3⋊Dic3—626no—
176C3⋊Dic3—626no—
177C2×#41—272no—
178C22⋊Q8—272no—
179C22⋊Q8—272no—
180#121—272no—
181#121—272no—
182#121—271no—
183#121—272no—
184#121C3⋊(C3⋊D3)11yescharacteristic, complemented
185C2×C4⋊C4—271no—
186C33#53711yescharacteristic, complemented, Sylow
187C23×C3C22×C3⋊D3126yes—
188C23×C3C22×C3⋊D3113yes—
189C22×D3—926no—
190C3×D4—252no—
191C3×D4—413no—
192C2×C3×C4—413no—
193C2×C3×C4C22×C3⋊D3113yes—
194C2×C3×C4C22×C3⋊D3113yes—
195C2×C3×C4C22×C3⋊D3126yes—
196#67—1852no—
197#67—1852no—
198#67—3626no—
199C2×Dic3—926no—
200C2×Dic3—952no—
201C2×Dic3—926no—
202C2×Dic3—926no—
203C2×Dic3—1852no—
204C2×Dic3—3613no—
205D12—3613no—
206C4×D3—1852no—
207Dic6—1852no—
208C2×C32—226no—
209C2×C32—426no—
210C2×C32C2×D3×D4113yes—
211C2×C32#1004113yes—
212C2×C32#1010113yes—
213C3⋊D3—1226no—
214#41—548no—
215C2×Q8—272no—
216C2×D4—272no—
217C2×D4—272no—
218C2×D4—272no—
219C2×D4—271no—
220C2×D4C2×C3⋊(C3⋊D3)12yes—
221C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
222C22×C4—271no—
223C22×C4—272no—
224C22×C4—271no—
225C22×C4—272no—
226C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
227C4⋊C4—272no—
228C4⋊C4—272no—
229C4⋊C4—272no—
230C4⋊C4—271no—
231C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
232C22⋊C4—272no—
233C22⋊C4—272no—
234C22⋊C4—272no—
235C22⋊C4C2×C3⋊(C3⋊D3)12yes—
236C22×C3—213no—
237C22×C3—213no—
238C22×C3—226no—
239C22×C3—226no—
240C22×C3—226no—
241C22×C3D4×C3⋊D3126yes—
242C22×C3C23×C3⋊D3113yes—
243C2×D3—1826no—
244C2×D3—1826no—
245C2×D3—1826no—
246C3×C4—226no—
247C3×C4—213no—
248C3×C4—213no—
249Dic3—952no—
250Dic3—1826no—
251Dic3—1826no—
252C32#5597113yescomplemented
253C23—272no—
254C23C22×C3⋊(C3⋊D3)12yes—
255C23C22×C3⋊(C3⋊D3)11yescharacteristic
256Q8—544no—
257D4—24no—
258D4—41no—
259D4—544no—
260D4—544no—
261D4—1081no—
262D4—1082no—
263C2×C4—41no—
264C2×C4—272no—
265C2×C4—274no—
266C2×C4—272no—
267C2×C4—272no—
268C2×C4—544no—
269C2×C4—544no—
270C2×C4—1081no—
271C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
272C2×C4C22×C3⋊(C3⋊D3)12yes—
273C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
274C2×C3—226no—
275C2×C3—426no—
276C2×C3C2×D4×C3⋊D3113yes—
277C2×C3#64361113yes—
278C2×C3#64367113yes—
279D3—3626no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—542no—
286C22—542no—
287C22—542no—
288C22D4×C3⋊(C3⋊D3)12yes—
289C22C23×C3⋊(C3⋊D3)11yescharacteristic
290C4—22no—
291C4—21no—
292C4—21no—
293C4—274no—
294C4—542no—
295C4—542no—
296C3#10596068113yescomplemented
297C2—22no—
298C2—42no—
299C2—1082no—
300C2C2×D4×C3⋊(C3⋊D3)11yescharacteristic
301C2#1172467911yescharacteristic
302C2#1172468511yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×#121C2
hypercenter#121C3⋊(C3⋊D3)
hypocenterC33#537
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#89078C2semidirect
2. C33×#121C2semidirect
3. #11723980C2semidirect
4. #11723980C2semidirect
5. #11723971C2semidirect
6. C2×#88857C2semidirect
7. #11723937C2semidirect
8. #11723925C2semidirect
9. #11723919C2semidirect
34. C32×#121D3semidirect
56. C2×C3⋊(C3⋊Dic3)D4semidirect
103. C3×#121C3⋊D3semidirect
184. #121C3⋊(C3⋊D3)semidirect
186. C33#537semidirect
252. C32#5597semidirect
296. C3#10596068semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C33×#1212
#117239802
#117239712
C2×#888572
#117239372
#117239252
#117239192
#105960683