Finite Group Databasebeta

← C2×D3×#3777 #13180630

Subgroups — C2×D3×#3777 #13180630

Order 1728.

1002 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 228 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×D3×#157C212yesmaximal
2C22×D3×#157C211yesmaximal, characteristic
3C2×#88946C211yesmaximal, characteristic, complemented
4C2×#88945C212yesmaximal, complemented
5C2×C3×#3777C211yesmaximal, characteristic, complemented
6D3×#3777C218yesmaximal, direct product factor
7C22×#3777—31nomaximal
8C22×D3×C3⋊D3C2211yescharacteristic
9C22×C3⋊#157C2212yes—
10C22×C3⋊#157C2211yescharacteristic
11C22×C3×#157C2212yes—
12C22×C3×#157C2211yescharacteristic
13C2×D3×#157—28no—
14C2×D3×#157C418yescomplemented
15C2×D3×#157C2218yes—
16C2×D3×#157C2214yes—
17#88946C2214yescomplemented
18#88945C2218yescomplemented
19C3×#3777C2214yescomplemented
20C23×#157—31no—
21C23×#157—32no—
22C2×#3777—38no—
23C2×#3777—32no—
24C2×#3777—31no—
25C2×#3777D311yescharacteristic, direct product factor
26C22×C3⋊(C3⋊D3)Q811yescharacteristic
27C22×C3×C3⋊D3C2311yescharacteristic
28C22×C32×D3Q811yescharacteristic
29C2×D3×C3⋊D3C2×C414yes—
30C2×D3×C3⋊D3D414yes—
31C2×D3×C3⋊D3Q812yes—
32C2×D3×C3⋊D3C2312yes—
33C2×C3⋊#157—24no—
34C2×C3⋊#157C2×C414yescomplemented
35C2×C3⋊#157C2314yes—
36C2×C3⋊#157C2312yes—
37C2×C3×#157—24no—
38C2×C3×#157C2×C414yescomplemented
39C2×C3×#157C2314yes—
40C2×C3×#157C2312yes—
41D3×#157—216no—
42D3×#157C2×C4116yescomplemented
43C2×D3×C4⋊C4—91nomaximal
44C23×C3⋊D3—31no—
45C22×D32—121no—
46C22×#157—31no—
47C22×#157—38no—
48C22×#157—34no—
49C22×#157—32no—
50C22×#157—38no—
51C22×#157—68no—
52C22×#157C2×D312yes—
53C22×#157C2×D311yescharacteristic
54#3777—38no—
55#3777—34no—
56#3777C2×D314yesdirect product factor
57C22×C33C2×Q811yescharacteristic
58C2×C3⋊(C3⋊D3)C4⋊C414yescomplemented
59C2×C3⋊(C3⋊D3)C2×Q812yes—
60C2×C3×C3⋊D3C22×C412yes—
61C2×C3×C3⋊D3C2×D412yes—
62C2×C3×C3⋊D3C2×Q811yescharacteristic
63C2×C3×C3⋊D3C2411yescharacteristic
64C2×C32×D3C4⋊C414yescomplemented
65C2×C32×D3C2×Q812yes—
66D3×C3⋊D3C4⋊C418yescomplemented
67D3×C3⋊D3C22×C414yes—
68D3×C3⋊D3C2×D414yes—
69C3⋊#157—24no—
70C3⋊#157C22×C414yescomplemented
71C3×#157—24no—
72C3×#157C22×C414yescomplemented
73C22×C4×D3—91no—
74C22×C4×D3—92no—
75C2×C3×C4⋊C4—91no—
76C2×C4⋊Dic3—91no—
77C2×Dic3⋊C4—92no—
78D3×C4⋊C4—98no—
79C23×C32—31no—
80C22×C3⋊D3—31no—
81C22×C3⋊D3—34no—
82C22×C3⋊D3—34no—
83C22×C3⋊D3—32no—
84C22×C3⋊D3—32no—
85C22×C3⋊D3—121no—
86C22×C3⋊D3—241no—
87C22×C3⋊D3C22×D311yescharacteristic
88C22×C3×D3—41no—
89C22×C3×D3—121no—
90C2×D32—124no—
91C2×D32—124no—
92C2×D32—122no—
93C2×D32—122no—
94C2×#157—24no—
95C2×#157—34no—
96C2×#157—32no—
97C2×#157—316no—
98C2×#157—34no—
99C2×#157—616no—
100C2×#157—64no—
101C2×#157C4×D314yescomplemented
102C2×#157C22×D314yes—
103C2×#157C22×D312yes—
104C22×C4⋊C4—271noSylow
105C2×C33C2×C4⋊C412yescomplemented
106C2×C33C22×Q811yescharacteristic
107C3⋊(C3⋊D3)C2×C4⋊C414yescomplemented
108C3×C3⋊D3C2×C4⋊C412yescomplemented
109C3×C3⋊D3C23×C411yescharacteristic
110C3×C3⋊D3C22×D411yescharacteristic
111C32×D3C2×C4⋊C414yescomplemented
112C23×D3—91no—
113C23×D3—361no—
114C22×C3×C4—91no—
115C22×C3×C4—92no—
116C22×Dic3—91no—
117C22×Dic3—92no—
118C2×C4×D3—98no—
119C2×C4×D3—94no—
120C2×C4×D3—98no—
121C2×C4×D3—188no—
122C3×C4⋊C4—94no—
123C4⋊Dic3—94no—
124Dic3⋊C4—98no—
125C22×C32—34no—
126C22×C32—32no—
127C22×C32—41no—
128C22×C32—81no—
129C22×C32D3×Q811yescharacteristic
130C2×C3⋊D3—34no—
131C2×C3⋊D3—32no—
132C2×C3⋊D3—38no—
133C2×C3⋊D3—34no—
134C2×C3⋊D3—34no—
135C2×C3⋊D3—124no—
136C2×C3⋊D3—122no—
137C2×C3⋊D3—244no—
138C2×C3⋊D3—242no—
139C2×C3⋊D3C2×C4×D312yes—
140C2×C3⋊D3D3×D412yes—
141C2×C3⋊D3D3×Q811yescharacteristic
142C2×C3⋊D3C23×D311yescharacteristic
143C2×C3×D3—42no—
144C2×C3×D3—44no—
145C2×C3×D3—122no—
146C2×C3×D3—122no—
147C2×C3×D3—121no—
148C2×C3×D3—121no—
149D32—128no—
150D32—124no—
151D32—124no—
152#157—24no—
153#157—34no—
154#157—64no—
155#157C2×C4×D314yescomplemented
156C23×C4—271no—
157C23×C4—272no—
158C2×C4⋊C4—91no—
159C2×C4⋊C4—278no—
160C2×C4⋊C4—272no—
161C2×C4⋊C4—271no—
162C33C22×C4⋊C411yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C23×C3—121no—
165C22×D3—91no—
166C22×D3—92no—
167C22×D3—92no—
168C22×D3—94no—
169C22×D3—94no—
170C22×D3—121no—
171C22×D3—362no—
172C22×D3—364no—
173C22×D3—364no—
174C22×D3—362no—
175C22×D3—361no—
176C22×D3—721no—
177C22×D3PSU(3,𝔽2)11yescharacteristic
178C2×C3×C4—94no—
179C2×C3×C4—92no—
180C2×C3×C4—94no—
181C2×C3×C4—184no—
182C2×Dic3—94no—
183C2×Dic3—92no—
184C2×Dic3—94no—
185C2×Dic3—184no—
186C4×D3—916no—
187C4×D3—1816no—
188C2×C32—34no—
189C2×C32—41no—
190C2×C32—42no—
191C2×C32—82no—
192C2×C32—81no—
193C2×C32D3×C4⋊C412yescomplemented
194C2×C32C2×D3×Q811yescharacteristic
195C3⋊D3—34no—
196C3⋊D3—124no—
197C3⋊D3—244no—
198C3⋊D3D3×C4⋊C412yescomplemented
199C3⋊D3C22×C4×D311yescharacteristic
200C3⋊D3C2×D3×D411yescharacteristic
201C3×D3—44no—
202C3×D3—122no—
203C3×D3—121no—
204C3×D3—121no—
205C24—271no—
206C22×C4—91no—
207C22×C4—92no—
208C22×C4—274no—
209C22×C4—278no—
210C22×C4—278no—
211C22×C4—271no—
212C22×C4—272no—
213C22×C4—548no—
214C4⋊C4—94no—
215C4⋊C4—278no—
216C4⋊C4—274no—
217C22×C3—41no—
218C22×C3—81no—
219C22×C3—91no—
220C22×C3—91no—
221C22×C3—92no—
222C22×C3—92no—
223C22×C3—124no—
224C22×C3—122no—
225C22×C3C2×PSU(3,𝔽2)11yescharacteristic
226C2×D3—92no—
227C2×D3—94no—
228C2×D3—94no—
229C2×D3—94no—
230C2×D3—98no—
231C2×D3—122no—
232C2×D3—122no—
233C2×D3—121no—
234C2×D3—121no—
235C2×D3—364no—
236C2×D3—364no—
237C2×D3—368no—
238C2×D3—364no—
239C2×D3—362no—
240C2×D3—724no—
241C2×D3—722no—
242C2×D3#377714yesdirect product factor
243C2×D3C2×PSU(3,𝔽2)12yes—
244C3×C4—94no—
245C3×C4—184no—
246Dic3—94no—
247Dic3—184no—
248C32—41no—
249C32—81no—
250C32C2×D3×C4⋊C411yescharacteristic, complemented
251C23—31no—
252C23—91no—
253C23—272no—
254C23—274no—
255C23—274no—
256C23—272no—
257C23—271no—
258C2×C4—92no—
259C2×C4—94no—
260C2×C4—94no—
261C2×C4—184no—
262C2×C4—2716no—
263C2×C4—272no—
264C2×C4—274no—
265C2×C4—274no—
266C2×C4—5416no—
267C2×C4—544no—
268C2×C3—42no—
269C2×C3—41no—
270C2×C3—82no—
271C2×C3—81no—
272C2×C3—91no—
273C2×C3—91no—
274C2×C3—92no—
275C2×C3—124no—
276C2×C3C2×#377712yescomplemented
277C2×C3C22×PSU(3,𝔽2)11yescharacteristic
278D3—94no—
279D3—122no—
280D3—121no—
281D3—121no—
282D3—364no—
283D3—724no—
284D3C2×#377714yesdirect product factor
285C22—34no—
286C22—32no—
287C22—92no—
288C22—92no—
289C22—91no—
290C22—91no—
291C22—274no—
292C22—274no—
293C22—278no—
294C22—274no—
295C22—272no—
296C22D3×PSU(3,𝔽2)11yescharacteristic
297C4—94no—
298C4—184no—
299C4—274no—
300C4—544no—
301C3—41no—
302C3—81no—
303C3C22×#377711yescharacteristic, complemented
304C2—34no—
305C2—91no—
306C2—91no—
307C2—92no—
308C2—274no—
309C2D3×#377712yesdirect product factor
310C2C2×D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×PSU(3,𝔽2)
commutatorC3×C3⋊D3C23×C4
FrattiniC2C2×D3×PSU(3,𝔽2)
Fitting, socleC22×C33C2×Q8
prime radicalC22×D3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypocenterC33C22×C4⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. C2×#88946C2semidirect
4. C2×#88945C2semidirect
5. C2×C3×#3777C2semidirect
6. D3×#3777C2direct (N × H)
14. C2×D3×#157C4semidirect
17. #88946C22semidirect
18. #88945C22semidirect
19. C3×#3777C22semidirect
25. C2×#3777D3direct (N × H)
34. C2×C3⋊#157C2×C4semidirect
38. C2×C3×#157C2×C4semidirect
42. D3×#157C2×C4semidirect
56. #3777C2×D3direct (N × H)
58. C2×C3⋊(C3⋊D3)C4⋊C4semidirect
64. C2×C32×D3C4⋊C4semidirect
66. D3×C3⋊D3C4⋊C4semidirect
70. C3⋊#157C22×C4semidirect
72. C3×#157C22×C4semidirect
101. C2×#157C4×D3semidirect
105. C2×C33C2×C4⋊C4semidirect
107. C3⋊(C3⋊D3)C2×C4⋊C4semidirect
108. C3×C3⋊D3C2×C4⋊C4semidirect
111. C32×D3C2×C4⋊C4semidirect
155. #157C2×C4×D3semidirect
162. C33C22×C4⋊C4semidirect
193. C2×C32D3×C4⋊C4semidirect
198. C3⋊D3D3×C4⋊C4semidirect
242. C2×D3#3777direct (N × H)
250. C32C2×D3×C4⋊C4semidirect
276. C2×C3C2×#3777semidirect
284. D3C2×#3777direct (N × H)
303. C3C22×#3777semidirect
309. C2D3×#3777direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×#889462
C2×#889452
C2×C3×#37772
D3×#37772
C22×#37773
C2×D3×C4⋊C49