Finite Group Databasebeta

← C2×#11723341 #13176666

Subgroups — C2×#11723341 #13176666

Order 1728.

1900 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 1900 conjugacy classes of subgroups — too many to draw. Only its 276 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×#89018C212yesmaximal, complemented
2C2×#88852C211yesmaximal, characteristic, complemented
3C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
4C2×C32×Dic12C211yesmaximal, characteristic, complemented
5C2×#88795C212yesmaximal, complemented
6#11723341C218yesmaximal, direct product factor
7C2×#64128—31nomaximal
8C2×#63808—34nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6C2212yes—
12C2×#6292—22no—
13C2×#6289—22no—
14#89018—24no—
15#89018C2218yescomplemented
16C2×C33×C8C2211yescharacteristic
17#88852C2214yescomplemented
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×C8×C3⋊D3C2214yescomplemented
20C32×Dic12C2214yescomplemented
21#88795C2218yescomplemented
22C2×#3832—32no—
23C2×#3799—38no—
24C2×C32×Q16—31no—
25C2×#3755—32no—
26#64128—38no—
27C2×#3745—31no—
28C2×#3745—34no—
29C2×#3745—64no—
30C2×C8×C3⋊D3D311yescharacteristic, complemented
31C2×C3×Dic12D314yescomplemented
32C2×C3×D3×C8—34no—
33C2×#3717—38no—
34#63808—332no—
35C22×C3⋊(C3⋊D3)—22no—
36C22×C3×C3⋊D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C4⋊(C3⋊(C3⋊D3))—22no—
39C4⋊(C3⋊(C3⋊D3))C2314yes—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4×C3⋊D3D412yes—
42C3×C4×C3⋊D3C2312yes—
43C2×C32×Dic3—22no—
44C32×Dic6—22no—
45C32×Dic6C2314yes—
46#6292—28no—
47#6289—28no—
48C33×C8C2312yes—
49C3×C3⋊(C3⋊C8)C2312yes—
50C2×#919—94no—
51C2×#904—91no—
52C2×C32×Q8—32no—
53#3832—38no—
54#3832—64no—
55C2×C4⋊(C3⋊D3)—32no—
56C2×C4⋊(C3⋊D3)—38no—
57C2×C4⋊(C3⋊D3)—62no—
58C2×C4⋊(C3⋊D3)—68no—
59C2×C4×C3⋊D3—62no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×C4×D3—34no—
62C2×C3×Dic6C2×D318yes—
63C2×#626—68no—
64C2×#624—68no—
65#3799—332no—
66#3799—616no—
67C32×Q16—34no—
68C2×C32×C8—24no—
69C2×C32×C8C2×D314yes—
70C2×C32×C8C2×D311yescharacteristic
71#3755—38no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73#3745—34no—
74#3745—316no—
75#3745—616no—
76C8×C3⋊D3C2×D314yescomplemented
77C2×C3×C3⋊C8—34no—
78C3×Dic12C2×D3116yescomplemented
79C3×D3×C8—316no—
80#3717—332no—
81C22×C33C2×D411yescharacteristic
82C2×C3⋊(C3⋊D3)—24no—
83C2×C3⋊(C3⋊D3)—44no—
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3C2×D412yes—
86C33×C4C2×D411yescharacteristic
87C33×C4C2411yescharacteristic
88C3×C3⋊Dic3C2×D412yes—
89C32×Dic3—24no—
90C2×#239—98no—
91C2×#235—92no—
92C2×C3×Q16—34no—
93C22×C3×C8—91no—
94C2×#215—98no—
95#919—932no—
96C2×Dic12C3⋊D311yescharacteristic, complemented
97#904—98no—
98C2×D24—94no—
99C2×D24—91no—
100C2×D24—184no—
101C2×#204—92no—
102C2×D3×C8—34no—
103C22×C3⋊D3—62no—
104C22×C3⋊D3—68no—
105C22×C3⋊D3—128no—
106C22×C3⋊D3D1211yescharacteristic
107C22×C3×D3—34no—
108C32×Q8—34no—
109C32×Q8—62no—
110C2×C32×C4—24no—
111C2×C32×C4—62no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3D1211yescharacteristic
115C4⋊(C3⋊D3)—34no—
116C4⋊(C3⋊D3)—316no—
117C4⋊(C3⋊D3)—616no—
118C4⋊(C3⋊D3)—68no—
119C4⋊(C3⋊D3)—62no—
120C4⋊(C3⋊D3)—68no—
121C4⋊(C3⋊D3)—128no—
122C4×C3⋊D3—68no—
123C4×C3⋊D3D1212yes—
124C4×C3⋊D3C22×D312yes—
125C2×C3×Dic3—28no—
126C2×C3×Dic3—34no—
127C3×C4×D3—38no—
128C3×C4×D3—38no—
129C3×Dic6—28no—
130C3×Dic6C22×D3116yes—
131#626—632no—
132#624—632no—
133C32×C8—28no—
134C32×C8C22×D318yes—
135C32×C8C22×D312yes—
136C3⋊(C3⋊C8)C22×D312yes—
137C3×C3⋊C8—38no—
138C2×#135—271noSylow
139C2×C33#13512yescomplemented
140C2×C33C22×D411yescharacteristic
141C3⋊(C3⋊D3)—44no—
142C3×C3⋊D3—22no—
143C2×C3×Q8—38no—
144C22×C3×C4—91no—
145C2×#67—182no—
146#239—932no—
147#239—1816no—
148#235—98no—
149#235—184no—
150C2×D12—98no—
151C2×D12—92no—
152C2×D12—188no—
153C2×D12—188no—
154C2×C4×D3—34no—
155C2×C4×D3—182no—
156C2×C4×D3—188no—
157C2×Dic6C2×C3⋊D312yes—
158C3×Q16—316no—
159C2×C3×C8—24no—
160C2×C3×C8—91no—
161C2×C3×C8—94no—
162C2×C3×C8D3214yes—
163C2×C3×C8C2×C3⋊D311yescharacteristic
164#215—932no—
165C2×C3⋊C8—34no—
166Dic12C2×C3⋊D314yescomplemented
167D24—916no—
168D24—94no—
169D24—1816no—
170#204—98no—
171D3×C8—316no—
172C22×C32—24no—
173C22×C32C2×D1211yescharacteristic
174C22×C32D3×D414yes—
175C2×C3⋊D3—22no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—616no—
178C2×C3⋊D3—1216no—
179C2×C3⋊D3—124no—
180C2×C3⋊D3—1216no—
181C2×C3⋊D3—2416no—
182C2×C3⋊D3C2×D1212yes—
183C2×C3×D3—38no—
184C2×C3×D3—68no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—64no—
188C32×C4C2×D1211yescharacteristic
189C32×C4D3×D414yes—
190C32×C4C23×D314yes—
191C32×C4C23×D311yescharacteristic
192C3⋊Dic3C2×D1212yes—
193C3×Dic3—216no—
194C3×Dic3—38no—
195C2×#41—272no—
196#135—278no—
197C2×Q16—31no—
198C2×SD16—272no—
199C2×D8—271no—
200C22×C8—91no—
201C33C2×#13511yescharacteristic, complemented, Sylow
202C23×C3—91no—
203C22×D3—34no—
204C22×D3—188no—
205C22×D3—182no—
206C22×D3—368no—
207C3×Q8—316no—
208C3×Q8—68no—
209C2×C3×C4—24no—
210C2×C3×C4—68no—
211C2×C3×C4—91no—
212C2×C3×C4—92no—
213C2×C3×C4—92no—
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216#67—188no—
217C2×Dic3—22no—
218C2×Dic3—34no—
219D12—916no—
220D12—94no—
221D12—1816no—
222D12—1832no—
223D12—188no—
224D12—182no—
225D12—368no—
226C4×D3—38no—
227C4×D3—38no—
228C4×D3—188no—
229C4×D3—1832no—
230Dic6—22no—
231Dic6C22×C3⋊D314yes—
232C3×C8—28no—
233C3×C8—92no—
234C3×C8C2×D3218yes—
235C3×C8C22×C3⋊D312yes—
236C3⋊C8—38no—
237C2×C32—24no—
238C2×C32—28no—
239C2×C32#90412yescomplemented
240C2×C32#91918yescomplemented
241C2×C32C22×D1211yescharacteristic
242C2×C32C2×D3×D414yes—
243C3⋊D3—22no—
244C3⋊D3—124no—
245C3⋊D3—1216no—
246C3⋊D3—2416no—
247C3×D3—68no—
248#41—278no—
249#41—544no—
250C2×Q8—32no—
251C2×D4—272no—
252C2×D4—542no—
253C22×C4—91no—
254C22×C4—542no—
255Q16—34no—
256SD16—278no—
257D8—274no—
258C2×C8—91no—
259C2×C8—94no—
260C2×C8D3×C3⋊D311yescharacteristic
261C22×C3—24no—
262C22×C3—92no—
263C22×C3—182no—
264C22×C3D3×D1214yes—
265C22×C3D4×C3⋊D311yescharacteristic
266C2×D3—38no—
267C2×D3—68no—
268C2×D3—1816no—
269C2×D3—184no—
270C2×D3—3616no—
271C2×D3—3616no—
272C2×D3—364no—
273C2×D3—7216no—
274C3×C4—24no—
275C3×C4—24no—
276C3×C4—616no—
277C3×C4—92no—
278C3×C4D3×D1214yes—
279C3×C4D4×C3⋊D311yescharacteristic
280C3×C4C22×D3214yes—
281C3×C4C23×C3⋊D311yescharacteristic
282Dic3—24no—
283Dic3—38no—
284C32—24no—
285C32C2×#90411yescharacteristic, complemented
286C32C2×#91914yescomplemented
287C23—91no—
288C23—542no—
289Q8—34no—
290Q8—62no—
291D4—274no—
292D4—548no—
293D4—542no—
294C2×C4—62no—
295C2×C4—91no—
296C2×C4—92no—
297C2×C4—92no—
298C2×C4—548no—
299C2×C4C2×D3×C3⋊D311yescharacteristic
300C8—92no—
301C8C2×D3×C3⋊D312yes—
302C2×C3—24no—
303C2×C3—28no—
304C2×C3—182no—
305C2×C3#6380818yescomplemented
306C2×C3#6412812yescomplemented
307C2×C3C2×D3×D1214yes—
308C2×C3C2×D4×C3⋊D311yescharacteristic
309D3—68no—
310D3—3616no—
311D3—364no—
312D3—7216no—
313C22—92no—
314C22—182no—
315C22—544no—
316C22—1084no—
317C22C3⋊D3×D1211yescharacteristic
318C4—64no—
319C4—92no—
320C4C3⋊D3×D1211yescharacteristic
321C4C22×D3×C3⋊D311yescharacteristic
322C3—24no—
323C3C2×#6380814yescomplemented
324C3C2×#6412811yescharacteristic, complemented
325C2—182no—
326C2—1084no—
327C2#1172334112yesdirect product factor
328C2C2×C3⋊D3×D1211yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89018C2semidirect
2. C2×#88852C2semidirect
3. C2×C3×C8×C3⋊D3C2semidirect
4. C2×C32×Dic12C2semidirect
5. C2×#88795C2semidirect
6. #11723341C2direct (N × H)
15. #89018C22semidirect
17. #88852C22semidirect
19. C3×C8×C3⋊D3C22semidirect
20. C32×Dic12C22semidirect
21. #88795C22semidirect
30. C2×C8×C3⋊D3D3semidirect
31. C2×C3×Dic12D3semidirect
76. C8×C3⋊D3C2×D3semidirect
78. C3×Dic12C2×D3semidirect
96. C2×Dic12C3⋊D3semidirect
139. C2×C33#135semidirect
166. Dic12C2×C3⋊D3semidirect
201. C33C2×#135semidirect
239. C2×C32#904semidirect
240. C2×C32#919semidirect
285. C32C2×#904semidirect
286. C32C2×#919semidirect
305. C2×C3#63808semidirect
306. C2×C3#64128semidirect
323. C3C2×#63808semidirect
324. C3C2×#64128semidirect
327. C2#11723341direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890182
C2×#888522
C2×C3×C8×C3⋊D32
C2×C32×Dic122
C2×#887952
#117233412
C2×#641283
C2×#638083