Finite Group Databasebeta

← C2×#11722046 #13160372

Subgroups — C2×#11722046 #13160372

Order 1728.

1292 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×#88855C211yesmaximal, characteristic, complemented
3C2×C32×C4×Dic3C211yesmaximal, characteristic, complemented
4#11722046C214yesmaximal, direct product factor
5C2×#63632—31nomaximal
6C2×#63569—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×C3×#638—21no—
9C3×#3828—21no—
10C3×#3828—21no—
11C3×#3828C412yescomplemented
12C3×#3828C2212yes—
13C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×#634C411yescharacteristic, complemented
16C22×C32×Dic3—21no—
17#88855—21no—
18#88855C2212yescomplemented
19C2×C3⋊(C3⋊(C3⋊C8))—21no—
20C32×C4×Dic3—21no—
21C32×C4×Dic3C2212yescomplemented
22C2×#3828D311yescharacteristic
23C2×C3×#235—34no—
24C2×C32×C42—31no—
25C2×#3748—31no—
26C2×#3748—34no—
27C2×#3748—64no—
28C2×C3×C4×Dic3D314yescomplemented
29#63632—34no—
30#63569—316no—
31C23×C33D411yescharacteristic
32C22×C3×C3⋊D3—21no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C3×#638—22no—
40C3×#638—22no—
41C2×C3×C3⋊Dic3—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)C2×C412yescomplemented
44C3×C4×C3⋊D3—22no—
45C3×C4×C3⋊D3—22no—
46C3×#634—21no—
47C3×#634C2×C412yescomplemented
48C2×C32×Dic3—24no—
49C2×C32×Dic3—22no—
50C3⋊(C3⋊(C3⋊C8))—22no—
51C2×#837—91no—
52C2×#805—94no—
53C22×C32×C4—24no—
54C22×C32×C4—61no—
55C22×C32×C4C2×D314yes—
56C22×C32×C4C2×D311yescharacteristic
57C2×#638—21no—
58#3828—21no—
59#3828—21no—
60#3828Dic312yescomplemented
61#3828C2×D312yes—
62C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
63C2×C4×C3⋊D3—21no—
64C2×#634Dic311yescharacteristic, complemented
65C2×C3×#67—64no—
66C22×C3×Dic3—24no—
67C3×#235—38no—
68C3×#235—38no—
69C3×#235—64no—
70C3×#235—64no—
71C2×C3×D12—34no—
72C2×C3×C4×D3—64no—
73C2×C3×Dic6—34no—
74C32×C42—32no—
75C32×C42—61no—
76#3748—32no—
77#3748—38no—
78#3748—68no—
79#3748—68no—
80#3748—64no—
81#3748—61no—
82C2×C3⋊(C3⋊C8)—68no—
83C2×C3⋊(C3⋊C8)—64no—
84C2×C3⋊(C3⋊C8)—61no—
85C3×C4×Dic3—24no—
86C3×C4×Dic3C2×D318yescomplemented
87C22×C33—22no—
88C22×C33C22⋊C411yescharacteristic
89C22×C33C22⋊C411yescharacteristic
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—42no—
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—22no—
98C32×Dic3—24no—
99C2×C3×#41—91no—
100C2×#235—34no—
101C2×C3×C42—34no—
102C2×C4×Dic3C3⋊D311yescharacteristic, complemented
103C2×#208—94no—
104C2×#208—91no—
105C2×#208—184no—
106#837—94no—
107#805—916no—
108C23×C32—24no—
109C23×C32#6714yes—
110C23×C32#6711yescharacteristic
111C22×C3⋊D3—21no—
112C22×C3×D3—64no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—64no—
120C2×C32×C4—62no—
121C2×C32×C4C4×D314yes—
122C2×C32×C4C4×D314yes—
123C2×C32×C4D1214yes—
124C2×C32×C4D1214yes—
125C2×C32×C4C2×Dic311yescharacteristic
126C2×C32×C4C2×Dic311yescharacteristic
127C2×C32×C4#6714yes—
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133#638—22no—
134#638—22no—
135C2×C3⋊Dic3—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)C2×Dic312yescomplemented
138C4×C3⋊D3—22no—
139C4×C3⋊D3—22no—
140#634—21no—
141#634C2×Dic312yescomplemented
142C3×#67—68no—
143C3×#67—68no—
144C2×C3×Dic3—216no—
145C2×C3×Dic3—28no—
146C2×C3×Dic3—64no—
147C3×D12—38no—
148C3×D12—64no—
149C3×C4×D3—68no—
150C3×C4×D3—68no—
151C3×Dic6—38no—
152C3×Dic6—64no—
153C3⋊(C3⋊C8)—616no—
154C3⋊(C3⋊C8)—68no—
155C3⋊(C3⋊C8)—62no—
156C2×#104—271noSylow
157C2×C33—21no—
158C2×C33—21no—
159C2×C33#10412yescomplemented
160C2×C33C2×C22⋊C411yescharacteristic
161C3×C3⋊D3—42no—
162C3×#41—92no—
163C3×#41—92no—
164C3×#41—181no—
165C3×#41—181no—
166C2×C3×Q8—91no—
167C2×C3×D4—91no—
168C2×C3×D4—181no—
169C22×C3×C4—24no—
170C22×C3×C4—64no—
171C22×C3×C4—181no—
172C22×C3×C4D3214yes—
173C22×C3×C4C2×C3⋊D311yescharacteristic
174C2×#67—64no—
175C22×Dic3—21no—
176#235—38no—
177#235—38no—
178#235—64no—
179#235—64no—
180C2×D12—34no—
181C2×C4×D3—64no—
182C2×Dic6—34no—
183C3×C42—38no—
184C3×C42—64no—
185C4×Dic3—21no—
186C4×Dic3C2×C3⋊D312yescomplemented
187#208—98no—
188#208—92no—
189#208—188no—
190#208—188no—
191#208—184no—
192#208—181no—
193C2×C3⋊C8—188no—
194C2×C3⋊C8—184no—
195C2×C3⋊C8—181no—
196C22×C32—22no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—28no—
201C22×C32—48no—
202C22×C32(C2×D3)⋊C414yes—
203C22×C32(C2×D3)⋊C414yes—
204C22×C32#21711yescharacteristic
205C22×C32#21711yescharacteristic
206C22×C32C2×#6714yes—
207C22×C32C2×#6711yescharacteristic
208C2×C3⋊D3—22no—
209C2×C3⋊D3—42no—
210C2×C3×D3—68no—
211C2×C3×D3—128no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—64no—
217C32×C4(C2×D3)⋊C414yes—
218C32×C4(C2×D3)⋊C414yes—
219C32×C4#21711yescharacteristic
220C32×C4#21711yescharacteristic
221C32×C4C2×C4×D314yes—
222C32×C4C2×D1214yes—
223C32×C4C22×Dic311yescharacteristic
224C32×C4C2×#6711yescharacteristic
225C3⋊Dic3—22no—
226C3×Dic3—216no—
227C3×Dic3—68no—
228C2×#41—91no—
229C2×M16—271no—
230C2×C42—31no—
231#104—274no—
232C33C2×#10411yescharacteristic, complemented, Sylow
233C23×C3—24no—
234C23×C3—181no—
235C23×C3#62514yes—
236C23×C3#63811yescharacteristic
237C22×D3—64no—
238C3×Q8—92no—
239C3×Q8—181no—
240C3×D4—92no—
241C3×D4—182no—
242C3×D4—182no—
243C3×D4—181no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—616no—
251C2×C3×C4—68no—
252C2×C3×C4—181no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×C3×C4D3×Dic314yes—
256C2×C3×C4D3×Dic314yes—
257C2×C3×C4#62514yes—
258C2×C3×C4#62614yes—
259C2×C3×C4#62614yes—
260C2×C3×C4C4×C3⋊D311yescharacteristic
261C2×C3×C4C4×C3⋊D311yescharacteristic
262C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
263C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
264C2×C3×C4#63811yescharacteristic
265C2×C3×C4C2×D3214yes—
266C2×C3×C4C22×C3⋊D311yescharacteristic
267#67—68no—
268#67—68no—
269C2×Dic3—24no—
270C2×Dic3—22no—
271C2×Dic3—64no—
272D12—38no—
273D12—64no—
274C4×D3—68no—
275C4×D3—68no—
276Dic6—38no—
277Dic6—64no—
278C3⋊C8—1816no—
279C3⋊C8—188no—
280C3⋊C8—182no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—28no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—44no—
288C2×C32—44no—
289C2×C32#80518yescomplemented
290C2×C32#83712yescomplemented
291C2×C32C2×(C2×D3)⋊C414yes—
292C2×C32C2×#21711yescharacteristic
293C3⋊D3—42no—
294C3×D3—128no—
295#41—92no—
296#41—92no—
297#41—181no—
298#41—181no—
299C2×Q8—91no—
300C2×D4—91no—
301C2×D4—181no—
302C22×C4—61no—
303C22×C4—181no—
304C22×C4D3×C3⋊D311yescharacteristic
305M16—272no—
306M16—541no—
307C2×C8—541no—
308C42—32no—
309C42—61no—
310C22×C3—28no—
311C22×C3—22no—
312C22×C3—24no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—48no—
316C22×C3—182no—
317C22×C3—362no—
318C22×C3(C2×D3)⋊Dic314yes—
319C22×C3(C2×D3)⋊Dic314yes—
320C22×C3#375211yescharacteristic
321C22×C3#375211yescharacteristic
322C22×C3C2×#62514yes—
323C22×C3C2×#63811yescharacteristic
324C2×D3—68no—
325C2×D3—128no—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—24no—
329C3×C4—24no—
330C3×C4—616no—
331C3×C4—182no—
332C3×C4(C2×D3)⋊Dic314yes—
333C3×C4(C2×D3)⋊Dic314yes—
334C3×C4#375211yescharacteristic
335C3×C4#375211yescharacteristic
336C3×C4C2×D3×Dic314yes—
337C3×C4C2×#62614yes—
338C3×C4C2×C4×C3⋊D311yescharacteristic
339C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
340Dic3—24no—
341Dic3—68no—
342C32—24no—
343C32C2×#80514yescomplemented
344C32C2×#83711yescharacteristic, complemented
345C23—181no—
346C23#629011yescharacteristic
347Q8—92no—
348Q8—181no—
349D4—92no—
350D4—182no—
351D4—182no—
352D4—181no—
353C2×C4—64no—
354C2×C4—62no—
355C2×C4—181no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4Dic3×C3⋊D311yescharacteristic
359C2×C4Dic3×C3⋊D311yescharacteristic
360C2×C4#629011yescharacteristic
361C2×C4#629211yescharacteristic
362C2×C4#629211yescharacteristic
363C2×C4C2×D3×C3⋊D311yescharacteristic
364C8—542no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—28no—
370C2×C3—24no—
371C2×C3—44no—
372C2×C3—44no—
373C2×C3—362no—
374C2×C3#6356918yescomplemented
375C2×C3#6363212yescomplemented
376C2×C3C2×(C2×D3)⋊Dic314yes—
377C2×C3C2×#375211yescharacteristic
378D3—128no—
379C22—22no—
380C22—182no—
381C22—362no—
382C22#8880311yescharacteristic
383C22#8880311yescharacteristic
384C22C2×#629011yescharacteristic
385C4—64no—
386C4—182no—
387C4#8880311yescharacteristic
388C4#8880311yescharacteristic
389C4C2×Dic3×C3⋊D311yescharacteristic
390C4C2×#629211yescharacteristic
391C3—24no—
392C3C2×#6356914yescomplemented
393C3C2×#6363211yescharacteristic, complemented
394C2—21no—
395C2—21no—
396C2—362no—
397C2#1172204612yesdirect product factor
398C2C2×#8880311yescharacteristic
subgroupG/H
centerC2×C4#6292
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88855C2semidirect
3. C2×C32×C4×Dic3C2semidirect
4. #11722046C2direct (N × H)
11. C3×#3828C4semidirect
13. C2×C3×C4⋊(C3⋊D3)C4semidirect
15. C2×C3×#634C4semidirect
18. #88855C22semidirect
21. C32×C4×Dic3C22semidirect
28. C2×C3×C4×Dic3D3semidirect
43. C3×C4⋊(C3⋊D3)C2×C4semidirect
47. C3×#634C2×C4semidirect
60. #3828Dic3semidirect
62. C2×C4⋊(C3⋊D3)Dic3semidirect
64. C2×#634Dic3semidirect
86. C3×C4×Dic3C2×D3semidirect
102. C2×C4×Dic3C3⋊D3semidirect
137. C4⋊(C3⋊D3)C2×Dic3semidirect
141. #634C2×Dic3semidirect
159. C2×C33#104semidirect
186. C4×Dic3C2×C3⋊D3semidirect
232. C33C2×#104semidirect
289. C2×C32#805semidirect
290. C2×C32#837semidirect
343. C32C2×#805semidirect
344. C32C2×#837semidirect
374. C2×C3#63569semidirect
375. C2×C3#63632semidirect
392. C3C2×#63569semidirect
393. C3C2×#63632semidirect
397. C2#11722046direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×#888552
C2×C32×C4×Dic32
#117220462
C2×#636323
C2×#635693