Finite Group Databasebeta

← C2×#11722390 #13164493

Subgroups — C2×#11722390 #13164493

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
1C2×C3×#41⋊C9C211yesmaximal, characteristic, complemented
2#11722390C214yesmaximal, direct product factor
3C2×C3⋊#3689C211yesmaximal, characteristic, complemented
4C2×C3⋊#3688C211yesmaximal, characteristic, complemented
5C2×#64159—31nomaximal
6C2×#63691—33nomaximal
7C2×C4×C3⋊D9—41nomaximal
8C3×#41⋊C9C2212yescomplemented
9C2×C3×Q8⋊C9C2211yescharacteristic
10C3⋊#3689C2212yescomplemented
11C3⋊#3688C2212yescomplemented
12C2×C32×#41D311yescharacteristic
13C2×#3828—31no—
14#64159—34no—
15#64159—34no—
16C2×#3756—31no—
17C2×#3755—31no—
18C2×#3754—31no—
19C2×#3753—31no—
20C22×C3⋊(C3⋊C8)—31no—
21C2×#41⋊C9D313yescomplemented
22#63691—312no—
23C2×#3689—33no—
24C2×#3688—33no—
25C22×C3⋊D9—41no—
26C2×C3×C4×C9—41no—
27C2×C3⋊Dic9—41no—
28C4×C3⋊D9—44no—
29C3×Q8⋊C9C2311yescharacteristic
30C2×#950—93no—
31C2×#950—91no—
32C32×#41—32no—
33C32×#41C2×D312yes—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C32×D4—31no—
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—32no—
45#3755—32no—
46#3755—32no—
47#3754—34no—
48#3753—34no—
49C2×C3⋊(C3⋊C8)—31no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—34no—
52C2×C4×D9—123no—
53#41⋊C9C2×D316yescomplemented
54C2×Q8⋊C9C2×D313yes—
55#3689—36no—
56#3688—36no—
57C22×C3×C9—41no—
58C2×C3⋊D9—42no—
59C2×C3⋊D9—44no—
60C3×C4×C9—42no—
61C3⋊Dic9—42no—
62C2×C3×#41D913yescomplemented
63C2×C3×#41C3⋊D311yescharacteristic
64C2×#235—93no—
65C2×#235—91no—
66#950—912no—
67#950—94no—
68#950—912no—
69#950—94no—
70C2×#216—93no—
71C2×#216—91no—
72C2×#215—93no—
73C2×#215—91no—
74C2×#214—93no—
75C2×#214—91no—
76C2×#213—93no—
77C2×#213—91no—
78C22×C3⋊C8—93no—
79C22×C3⋊C8—91no—
80C23×C32—31no—
81C22×C3⋊D3—61no—
82C32×Q8—31no—
83C32×Q8C22×D311yescharacteristic
84C32×D4—32no—
85C32×D4—61no—
86C2×C32×C4—31no—
87C2×C32×C4—32no—
88C2×C32×C4—32no—
89C2×C32×C4S411yescharacteristic
90#638—64no—
91C2×C3⋊Dic3—61no—
92C4⋊(C3⋊D3)—32no—
93C4⋊(C3⋊D3)—61no—
94C4×C3⋊D3—64no—
95#634—32no—
96#634—61no—
97C22×D9—123no—
98C3⋊(C3⋊C8)—32no—
99C3⋊(C3⋊C8)—32no—
100C2×C4×C9—43no—
101C2×Dic9—123no—
102C4×D9—1212no—
103Q8⋊C9C22×D313yes—
104C2×#135—271noSylow
105C2×C3×C9—42no—
106C2×C3×C9—41no—
107C3⋊D9—44no—
108C3×#41—36no—
109C3×#41—32no—
110C3×#41C2×D916yescomplemented
111C3×#41C2×C3⋊D312yes—
112C2×C3×Q8C2×D913yes—
113C2×C3×Q8C2×C3⋊D311yescharacteristic
114C2×C3×D4—33no—
115C2×C3×D4—31no—
116C22×C3×C4—33no—
117C22×C3×C4—31no—
118C2×#67—181no—
119C2×#67—183no—
120#235—94no—
121#235—912no—
122#235—182no—
123#235—186no—
124C2×D12—91no—
125C2×D12—93no—
126C2×C4×D3—183no—
127C2×C4×D3—181no—
128C2×Dic6—91no—
129C2×Dic6—93no—
130#216—96no—
131#216—92no—
132#216—96no—
133#216—92no—
134#215—96no—
135#215—92no—
136#215—96no—
137#215—92no—
138#214—912no—
139#214—94no—
140#213—912no—
141#213—94no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—93no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—93no—
146C2×C3⋊C8—94no—
147C2×C3⋊C8—912no—
148C22×C32—32no—
149C22×C32—62no—
150C22×C32C2×S411yescharacteristic
151C2×C3⋊D3—62no—
152C2×C3⋊D3—122no—
153C32×C4—31no—
154C32×C4—31no—
155C32×C4C2×S412yes—
156C3⋊Dic3—62no—
157C22×C9—43no—
158C2×D9—126no—
159C2×D9—1212no—
160C4×C9—46no—
161Dic9—126no—
162C2×#41—271no—
163C2×#41C3⋊D911yescharacteristic, complemented
164#135—274no—
165#135—274no—
166C2×Q16—271no—
167C2×SD16—271no—
168C2×SD16—271no—
169C2×D8—271no—
170C22×C8—271no—
171C3×C9—41noSylow
172C23×C3—31no—
173C23×C3—33no—
174C22×D3—181no—
175C22×D3—183no—
176C3×Q8—33no—
177C3×Q8—31no—
178C3×Q8C22×D913yes—
179C3×Q8C22×C3⋊D311yescharacteristic
180C3×D4—36no—
181C3×D4—32no—
182C3×D4—61no—
183C3×D4—63no—
184C2×C3×C4—31no—
185C2×C3×C4—33no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—36no—
189C2×C3×C4—36no—
190C2×C3×C4(C22⋊C9)⋊C213yes—
191C2×C3×C4C3⋊S411yescharacteristic
192#67—184no—
193#67—1812no—
194C2×Dic3—181no—
195C2×Dic3—183no—
196D12—92no—
197D12—96no—
198D12—181no—
199D12—183no—
200C4×D3—184no—
201C4×D3—1812no—
202Dic6—92no—
203Dic6—96no—
204Dic6—181no—
205Dic6—183no—
206C3⋊C8—92no—
207C3⋊C8—96no—
208C3⋊C8—92no—
209C3⋊C8—96no—
210C2×C32—62no—
211C2×C32#98512yes—
212C2×C32C22×S411yescharacteristic
213C3⋊D3—122no—
214C2×C9—46no—
215C2×C9—43no—
216D9—1212no—
217#41—32no—
218#41—274no—
219#41—542no—
220#41C2×C3⋊D912yescomplemented
221C2×Q8—271no—
222C2×Q8C2×C3⋊D911yescharacteristic
223C2×D4—31no—
224C2×D4—271no—
225C2×D4—541no—
226C22×C4—31no—
227C22×C4—541no—
228Q16—272no—
229Q16—272no—
230SD16—274no—
231SD16—272no—
232SD16—272no—
233D8—274no—
234C2×C8—271no—
235C2×C8—271no—
236C2×C8—274no—
237C22×C3—32no—
238C22×C3—36no—
239C22×C3—62no—
240C22×C3—66no—
241C22×C3C2×(C22⋊C9)⋊C213yes—
242C22×C3C2×C3⋊S411yescharacteristic
243C2×D3—182no—
244C2×D3—186no—
245C2×D3—362no—
246C2×D3—366no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—33no—
250C3×C4—33no—
251C3×C4C2×(C22⋊C9)⋊C216yes—
252C3×C4C2×C3⋊S412yes—
253Dic3—182no—
254Dic3—186no—
255C32C2×#98511yescharacteristic
256C9—43no—
257C23—31no—
258C23—541no—
259Q8—31no—
260Q8—272no—
261Q8—541no—
262Q8C22×C3⋊D911yescharacteristic
263D4—32no—
264D4—61no—
265D4—272no—
266D4—541no—
267D4—544no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—541no—
272C2×C4—544no—
273C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
274C8—272no—
275C8—272no—
276C2×C3—66no—
277C2×C3—62no—
278C2×C3#6369116yescomplemented
279C2×C3C22×(C22⋊C9)⋊C213yes—
280C2×C3#6426612yes—
281C2×C3C22×C3⋊S411yescharacteristic
282D3—366no—
283D3—362no—
284C22—32no—
285C22—62no—
286C22—542no—
287C22—1082no—
288C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
289C4—31no—
290C4—31no—
291C4—542no—
292C4C2×C3⋊((C22⋊C9)⋊C2)12yes—
293C3C2×#6369113yescomplemented
294C3C2×#6426611yescharacteristic
295C2—62no—
296C2—1082no—
297C2#1172239012yesdirect product factor
298C2C22×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×Q8⋊C9C23
FrattiniC2×C3C22×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. #11722390C2direct (N × H)
3. C2×C3⋊#3689C2semidirect
4. C2×C3⋊#3688C2semidirect
8. C3×#41⋊C9C22semidirect
10. C3⋊#3689C22semidirect
11. C3⋊#3688C22semidirect
21. C2×#41⋊C9D3semidirect
53. #41⋊C9C2×D3semidirect
62. C2×C3×#41D9semidirect
110. C3×#41C2×D9semidirect
163. C2×#41C3⋊D9semidirect
220. #41C2×C3⋊D9semidirect
278. C2×C3#63691semidirect
293. C3C2×#63691semidirect
297. C2#11722390direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
#117223902
C2×C3⋊#36892
C2×C3⋊#36882
C2×#641593
C2×#636913
C2×C4×C3⋊D94