Finite Group Databasebeta

← #13175383

Subgroups — #13175383

Order 1728.

1764 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 272 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
1C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C32×#870C211yesmaximal, characteristic, complemented
4#11723536C212yesmaximal, complemented
5#11723442C214yesmaximal
6#11723413C212yesmaximal
7#11723391C214yesmaximal, complemented
8#10596159—31nomaximal
9#10593854—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2212yes—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×#6293C2214yes—
13C33×C42C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2212yes—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C3×#3750C2214yes—
18C32×C4⋊Dic3C2212yes—
19C32×Dic3⋊C4C2214yes—
20#88806C2214yes—
21#88804C2214yes—
22Dic3×C3⋊Dic3C2214yes—
23C32×#125—31no—
24#64105—34no—
25#64101—32no—
26#64097—32no—
27#64096—34no—
28C42×C3⋊D3—31no—
29C42×C3⋊D3—34no—
30C42×C3⋊D3—64no—
31#64079D311yescharacteristic, complemented
32C3×#870—34no—
33C3×#870D314yescomplemented
34#63919—38no—
35#63865—316no—
36#63849—38no—
37#63838—316no—
38C22×C3⋊(C3⋊D3)C2311yescharacteristic
39C2×C33×C4C2312yes—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)C2312yes—
43C4×C3⋊(C3⋊D3)—24no—
44C4×C3⋊(C3⋊D3)Q814yescomplemented
45C2×C3×C3⋊Dic3C2314yes—
46C2×C32×Dic3C2314yes—
47#6293—28no—
48#5688—94no—
49#5688—91no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—34no—
52C2×C4×C3⋊D3—32no—
53C2×C4×C3⋊D3—38no—
54C2×C4×C3⋊D3—64no—
55C2×C4×C3⋊D3—68no—
56C2×#634—34no—
57C2×#627—316no—
58C32×C4⋊C4—34no—
59C32×C4⋊C4—32no—
60C32×C42—24no—
61C32×C42C2×D314yes—
62C32×C42C2×D311yescharacteristic
63#3752—34no—
64C4⋊(C3⋊Dic3)C2×D312yes—
65#3750—34no—
66#3750C2×D314yes—
67C4×C3⋊Dic3—32no—
68C4×C3⋊Dic3—31no—
69C4×C3⋊Dic3—38no—
70C4×C3⋊Dic3—34no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—68no—
73C4×C3⋊Dic3—64no—
74C3×C4⋊Dic3—38no—
75C3×C4⋊Dic3C2×D318yes—
76C3×Dic3⋊C4—316no—
77C3×Dic3⋊C4C2×D3116yes—
78#3724—316no—
79#3722—316no—
80Dic32—316no—
81C22×C33C2411yescharacteristic
82C2×C3⋊(C3⋊D3)—22no—
83C2×C3⋊(C3⋊D3)C2×Q812yes—
84C33×C4—22no—
85C33×C4C2×Q812yes—
86C3⋊(C3⋊Dic3)C2×Q812yes—
87C3⋊(C3⋊Dic3)#4114yes—
88C3×C3⋊Dic3—24no—
89C32×Dic3—24no—
90C3×#125—34no—
91C3×#125—91no—
92#896—916no—
93#896—94no—
94#892—98no—
95#892—92no—
96#888—98no—
97#888—92no—
98#887—916no—
99#887—94no—
100C42×D3—94no—
101C42×D3—91no—
102C42×D3—184no—
103#870—34no—
104#870C3⋊D311yescharacteristic, complemented
105C22×C3⋊D3—31no—
106C22×C3⋊D3—34no—
107C22×C3⋊D3—64no—
108C2×C32×C4—28no—
109C2×C32×C4—24no—
110C2×C32×C4—34no—
111C2×C32×C4C22×D318yes—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D312yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—32no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—38no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—68no—
121C2×C3⋊Dic3C22×D314yes—
122C4×C3⋊D3—34no—
123C4×C3⋊D3—316no—
124C4×C3⋊D3—616no—
125C4×C3⋊D3—64no—
126C4×C3⋊D3—616no—
127C4×C3⋊D3—632no—
128#634—68no—
129C2×C3×Dic3—316no—
130C2×C3×Dic3C22×D3116yes—
131#627—632no—
132#533—271noSylow
133C2×C33C22×Q811yescharacteristic
134C2×C33C2×#4112yes—
135C3⋊(C3⋊D3)—22no—
136C2×C4×D3—94no—
137C2×C4×D3—91no—
138C2×C4×D3—98no—
139C2×C4×D3—92no—
140C2×C4×D3—184no—
141C2×C4×D3—188no—
142C2×Dic6—916no—
143C2×Dic6—94no—
144C3×C4⋊C4—316no—
145C3×C4⋊C4—38no—
146C3×C4⋊C4—94no—
147C3×C4⋊C4—92no—
148C3×C42—24no—
149C3×C42D3214yes—
150C3×C42C2×C3⋊D311yescharacteristic
151(C2×D3)⋊C4—916no—
152(C2×D3)⋊C4—94no—
153C4⋊Dic3—38no—
154C4⋊Dic3C2×C3⋊D312yes—
155Dic3⋊C4—316no—
156Dic3⋊C4—916no—
157Dic3⋊C4—94no—
158Dic3⋊C4C2×C3⋊D314yes—
159C4×Dic3—916no—
160C4×Dic3—98no—
161C4×Dic3—94no—
162C4×Dic3—92no—
163C4×Dic3—91no—
164C4×Dic3—94no—
165C4×Dic3—188no—
166C4×Dic3—184no—
167C22×C32—24no—
168C22×C32C23×D314yes—
169C22×C32C23×D311yescharacteristic
170C2×C3⋊D3—32no—
171C2×C3⋊D3—38no—
172C2×C3⋊D3—68no—
173C2×C3⋊D3—616no—
174C2×C3⋊D3—68no—
175C2×C3⋊D3—62no—
176C32×C4—216no—
177C32×C4—28no—
178C32×C4—22no—
179C32×C4—28no—
180C32×C4—64no—
181C32×C4D3×Q818yes—
182C32×C4D3×Q812yes—
183C3⋊Dic3—24no—
184C3⋊Dic3—32no—
185C3⋊Dic3—34no—
186C3⋊Dic3—38no—
187C3⋊Dic3—316no—
188C3⋊Dic3—68no—
189C3⋊Dic3—616no—
190C3×Dic3—216no—
191C3×Dic3—616no—
192C4⋊Q8—272no—
193#125—31no—
194#125—91no—
195C22⋊Q8—274no—
196C4×Q8—274no—
197#117—272no—
198C2×C42—271no—
199C33#53311yescharacteristic, complemented, Sylow
200C22×D3—94no—
201C22×D3—91no—
202C22×D3—184no—
203C2×C3×C4—28no—
204C2×C3×C4—24no—
205C2×C3×C4—316no—
206C2×C3×C4—94no—
207C2×C3×C4C2×D3218yes—
208C2×C3×C4C2×D3214yes—
209C2×C3×C4C22×C3⋊D312yes—
210C2×C3×C4C22×C3⋊D311yescharacteristic
211C2×Dic3—316no—
212C2×Dic3—94no—
213C2×Dic3—98no—
214C2×Dic3—91no—
215C2×Dic3—92no—
216C2×Dic3—184no—
217C2×Dic3—188no—
218C2×Dic3C22×C3⋊D314yes—
219C4×D3—916no—
220C4×D3—94no—
221C4×D3—1816no—
222C4×D3—1816no—
223C4×D3—184no—
224C4×D3—1832no—
225Dic6—188no—
226Dic6—1832no—
227C2×C32—24no—
228C2×C32—28no—
229C2×C32C2×D3×Q814yes—
230C2×C32C2×D3×Q811yescharacteristic
231C2×C32D3×#4118yes—
232C2×C32D3×#4112yes—
233C3⋊D3—616no—
234C3⋊D3—68no—
235C3⋊D3—62no—
236C2×Q8—274no—
237C22×C4—271no—
238C22×C4—272no—
239C4⋊C4—32no—
240C4⋊C4—34no—
241C4⋊C4—92no—
242C4⋊C4—94no—
243C4⋊C4—274no—
244C22⋊C4—274no—
245C42—274no—
246C42—272no—
247C42—271no—
248C42D3×C3⋊D311yescharacteristic
249C22×C3—24no—
250C22×C3C22×D3214yes—
251C22×C3C23×C3⋊D311yescharacteristic
252C2×D3—98no—
253C2×D3—92no—
254C2×D3—188no—
255C2×D3—1816no—
256C2×D3—188no—
257C2×D3—182no—
258C3×C4—216no—
259C3×C4—28no—
260C3×C4—22no—
261C3×C4—28no—
262C3×C4—616no—
263C3×C4—184no—
264C3×C4#379718yes—
265C3×C4Q8×C3⋊D312yes—
266Dic3—24no—
267Dic3—616no—
268Dic3—98no—
269Dic3—916no—
270Dic3—92no—
271Dic3—94no—
272Dic3—188no—
273Dic3—1816no—
274C32—24no—
275C32#568814yescomplemented
276C32#568811yescharacteristic, complemented
277C23—271no—
278Q8—548no—
279C2×C4—34no—
280C2×C4—94no—
281C2×C4—271no—
282C2×C4—272no—
283C2×C4—274no—
284C2×C4—544no—
285C2×C4C2×D3×C3⋊D312yes—
286C2×C4C2×D3×C3⋊D311yescharacteristic
287C2×C3—24no—
288C2×C3—28no—
289C2×C3C2×#379714yes—
290C2×C3#6430618yes—
291C2×C3C2×Q8×C3⋊D311yescharacteristic
292C2×C3#41×C3⋊D312yes—
293D3—1816no—
294D3—188no—
295D3—182no—
296C22—272no—
297C22—542no—
298C22C22×D3×C3⋊D311yescharacteristic
299C4—22no—
300C4—64no—
301C4—184no—
302C4—272no—
303C4—274no—
304C4#8901912yes—
305C3—24no—
306C3#1059385414yescomplemented
307C3#1059615911yescharacteristic, complemented
308C2—542no—
309C2C2×#8901911yescharacteristic
310C2#1172457812yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC22×C3⋊(C3⋊D3)C23
hypercenterC42D3×C3⋊D3
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64079C2semidirect
3. C32×#870C2semidirect
4. #11723536C2semidirect
7. #11723391C2semidirect
31. #64079D3semidirect
33. C3×#870D3semidirect
44. C4×C3⋊(C3⋊D3)Q8semidirect
104. #870C3⋊D3semidirect
199. C33#533semidirect
275. C32#5688semidirect
276. C32#5688semidirect
306. C3#10593854semidirect
307. C3#10596159semidirect

Maximal subgroups

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

subgroupindex
C42×C3⋊(C3⋊D3)2
C3×#640792
C32×#8702
#117235362
#117234422
#117234132
#117233912
#105961593
#105938543