Finite Group Databasebeta

← C2×#11722385 #13164484

Subgroups — C2×#11722385 #13164484

Order 1728.

832 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C22×C3×Q8⋊C9C211yesmaximal, characteristic, complemented
2#11722385C214yesmaximal, direct product factor
3C2×C3⋊#3689C211yesmaximal, characteristic, complemented
4C2×C3⋊#3688C211yesmaximal, characteristic, complemented
5C2×#64151—31nomaximal
6C2×#63689—33nomaximal
7C2×#6233—41nomaximal
8C2×C3×Q8⋊C9C2211yescharacteristic
9C2×C3×Q8⋊C9C2212yescomplemented
10C3⋊#3689C2212yescomplemented
11C3⋊#3688C2212yescomplemented
12C22×C32×Q8D311yescharacteristic
13C2×#3828—31no—
14C2×#3756—31no—
15C2×#3756—31no—
16#64151—34no—
17#64151—34no—
18C2×#3755—31no—
19C2×#3755—31no—
20C2×#3748—31no—
21C22×Q8⋊C9D313yescomplemented
22#63689—312no—
23C2×#3689—33no—
24C2×#3688—33no—
25C23×C3×C9—41no—
26C22×C3⋊D9—41no—
27#6233—44no—
28C2×C3⋊Dic9—41no—
29C3×Q8⋊C9C2311yescharacteristic
30C2×#942—93no—
31C2×#942—91no—
32C2×C32×Q8—31no—
33C2×C32×Q8—32no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C32×Q8C2×D312yes—
36C22×C32×C4—31no—
37C2×#638—61no—
38#3828—34no—
39#3828—62no—
40C2×C4⋊(C3⋊D3)—31no—
41C2×C4×C3⋊D3—61no—
42C2×#634—31no—
43#3756—32no—
44#3756—34no—
45#3756—32no—
46#3755—34no—
47#3755—32no—
48#3755—32no—
49#3748—34no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×#611—123no—
53C2×Q8⋊C9C2×D313yes—
54C2×Q8⋊C9C2×D316yescomplemented
55#3689—36no—
56#3688—36no—
57C22×C3×C9—41no—
58C22×C3×C9—42no—
59C22×C3×C9—82no—
60C2×C3⋊D9—42no—
61C2×C3⋊D9—82no—
62C3⋊Dic9—42no—
63C22×C3×Q8D913yescomplemented
64C22×C3×Q8C3⋊D311yescharacteristic
65C2×#235—93no—
66C2×#235—91no—
67C2×#216—93no—
68C2×#216—91no—
69C2×#216—93no—
70C2×#216—91no—
71#942—912no—
72#942—94no—
73#942—912no—
74#942—94no—
75C2×#215—93no—
76C2×#215—91no—
77C2×#215—93no—
78C2×#215—91no—
79C2×#208—93no—
80C2×#208—91no—
81C23×C32S411yescharacteristic
82C22×C3⋊D3—61no—
83C32×Q8—31no—
84C32×Q8—32no—
85C32×Q8—61no—
86C32×Q8C22×D311yescharacteristic
87C2×C32×C4—31no—
88C2×C32×C4—32no—
89C2×C32×C4—31no—
90C2×C32×C4—32no—
91#638—64no—
92C2×C3⋊Dic3—61no—
93C4⋊(C3⋊D3)—32no—
94C4⋊(C3⋊D3)—61no—
95C4×C3⋊D3—64no—
96#634—32no—
97#634—61no—
98C23×C9—43no—
99C22×D9—123no—
100C3⋊(C3⋊C8)—32no—
101C3⋊(C3⋊C8)—32no—
102#611—1212no—
103C2×Dic9—123no—
104Q8⋊C9C22×D313yes—
105C2×#137—271noSylow
106C2×C3×C9—42no—
107C2×C3×C9—41no—
108C2×C3×C9—82no—
109C3⋊D9—82no—
110C2×C3×Q8—33no—
111C2×C3×Q8—31no—
112C2×C3×Q8—36no—
113C2×C3×Q8—32no—
114C2×C3×Q8C2×D913yes—
115C2×C3×Q8C2×D916yescomplemented
116C2×C3×Q8C2×C3⋊D311yescharacteristic
117C2×C3×Q8C2×C3⋊D312yes—
118C22×C3×C4—33no—
119C22×C3×C4—31no—
120C2×#67—183no—
121C2×#67—181no—
122#235—94no—
123#235—912no—
124#235—182no—
125#235—186no—
126C2×D12—91no—
127C2×D12—93no—
128C2×C4×D3—181no—
129C2×C4×D3—183no—
130C2×Dic6—91no—
131C2×Dic6—93no—
132#216—96no—
133#216—912no—
134#216—92no—
135#216—94no—
136#216—96no—
137#216—92no—
138#215—96no—
139#215—912no—
140#215—94no—
141#215—92no—
142#215—96no—
143#215—92no—
144#208—94no—
145#208—912no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—93no—
149C2×C3⋊C8—93no—
150C22×C32—22no—
151C22×C32C2×S412yes—
152C22×C32C2×S411yescharacteristic
153C2×C3⋊D3—62no—
154C2×C3⋊D3—122no—
155C32×C4—31no—
156C32×C4—31no—
157C32×C4—32no—
158C3⋊Dic3—62no—
159C22×C9—43no—
160C22×C9—46no—
161C22×C9—86no—
162C2×D9—126no—
163C2×D9—246no—
164Dic9—126no—
165C2×#41—271no—
166C22×Q8C3⋊D911yescharacteristic, complemented
167#137—274no—
168#137—274no—
169C2×Q16—271no—
170C2×Q16—271no—
171C2×SD16—271no—
172C2×SD16—271no—
173C2×M16—271no—
174C3×C9—41noSylow
175C23×C3(C22⋊C9)⋊C213yes—
176C23×C3C3⋊S411yescharacteristic
177C22×D3—181no—
178C22×D3—183no—
179C3×Q8—33no—
180C3×Q8—36no—
181C3×Q8—31no—
182C3×Q8—32no—
183C3×Q8—61no—
184C3×Q8—63no—
185C3×Q8C22×D913yes—
186C3×Q8C22×C3⋊D311yescharacteristic
187C2×C3×C4—31no—
188C2×C3×C4—32no—
189C2×C3×C4—31no—
190C2×C3×C4—32no—
191C2×C3×C4—33no—
192C2×C3×C4—36no—
193C2×C3×C4—33no—
194C2×C3×C4—36no—
195#67—184no—
196#67—1812no—
197C2×Dic3—181no—
198C2×Dic3—183no—
199D12—92no—
200D12—96no—
201D12—181no—
202D12—183no—
203C4×D3—184no—
204C4×D3—1812no—
205Dic6—92no—
206Dic6—96no—
207Dic6—181no—
208Dic6—183no—
209C3⋊C8—92no—
210C3⋊C8—92no—
211C3⋊C8—96no—
212C3⋊C8—96no—
213C2×C32—22no—
214C2×C32#98312yes—
215C2×C32C22×S411yescharacteristic
216C3⋊D3—122no—
217C2×C9—46no—
218C2×C9—43no—
219C2×C9—86no—
220D9—246no—
221#41—274no—
222#41—542no—
223C2×Q8—31no—
224C2×Q8—32no—
225C2×Q8—271no—
226C2×Q8C2×C3⋊D911yescharacteristic
227C2×Q8C2×C3⋊D912yescomplemented
228C2×D4—271no—
229C2×D4—541no—
230C22×C4—31no—
231C22×C4—541no—
232Q16—272no—
233Q16—274no—
234Q16—272no—
235SD16—272no—
236SD16—274no—
237SD16—272no—
238M16—274no—
239C2×C8—271no—
240C2×C8—271no—
241C22×C3—22no—
242C22×C3—26no—
243C22×C3C2×(C22⋊C9)⋊C216yes—
244C22×C3C2×(C22⋊C9)⋊C213yes—
245C22×C3C2×C3⋊S412yes—
246C22×C3C2×C3⋊S411yescharacteristic
247C2×D3—182no—
248C2×D3—186no—
249C2×D3—362no—
250C2×D3—366no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—32no—
254C3×C4—33no—
255C3×C4—33no—
256C3×C4—36no—
257Dic3—182no—
258Dic3—186no—
259C32C2×#98311yescharacteristic
260C9—43no—
261C23—541no—
262C23C3⋊((C22⋊C9)⋊C2)11yescharacteristic
263Q8—31no—
264Q8—32no—
265Q8—61no—
266Q8—272no—
267Q8—541no—
268Q8C22×C3⋊D911yescharacteristic
269D4—272no—
270D4—541no—
271D4—544no—
272C2×C4—32no—
273C2×C4—32no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—544no—
277C2×C4—541no—
278C8—272no—
279C8—272no—
280C2×C3—26no—
281C2×C3—22no—
282C2×C3#6368916yescomplemented
283C2×C3C22×(C22⋊C9)⋊C213yes—
284C2×C3#6426412yes—
285C2×C3C22×C3⋊S411yescharacteristic
286D3—366no—
287D3—362no—
288C22—22no—
289C22—542no—
290C22—1082no—
291C22C2×C3⋊((C22⋊C9)⋊C2)12yes—
292C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
293C4—31no—
294C4—32no—
295C4—31no—
296C4—542no—
297C3C2×#6368913yescomplemented
298C3C2×#6426411yescharacteristic
299C2—22no—
300C2—1082no—
301C2#1172238512yesdirect product factor
302C2C22×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×Q8⋊C9C23
FrattiniC2×C3C22×C3⋊S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
hypercenterC23C3⋊((C22⋊C9)⋊C2)
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8⋊C9C2semidirect
2. #11722385C2direct (N × H)
3. C2×C3⋊#3689C2semidirect
4. C2×C3⋊#3688C2semidirect
9. C2×C3×Q8⋊C9C22semidirect
10. C3⋊#3689C22semidirect
11. C3⋊#3688C22semidirect
21. C22×Q8⋊C9D3semidirect
54. C2×Q8⋊C9C2×D3semidirect
63. C22×C3×Q8D9semidirect
115. C2×C3×Q8C2×D9semidirect
166. C22×Q8C3⋊D9semidirect
227. C2×Q8C2×C3⋊D9semidirect
282. C2×C3#63689semidirect
297. C3C2×#63689semidirect
301. C2#11722385direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
#117223852
C2×C3⋊#36892
C2×C3⋊#36882
C2×#641513
C2×#636893
C2×#62334