Finite Group Databasebeta

← C2×#11721136 #13150709

Subgroups — C2×#11721136 #13150709

Order 1728.

1599 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C22×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
3C2×C4×#1084C211yesmaximal, characteristic, complemented
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×#88450C212yesmaximal, complemented
6#11721136C218yesmaximal, direct product factor
7C2×#63894—31nomaximal
8C2×Dic3×D12—31nomaximal
9C23×(C3⋊D3)⋊C3C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C418yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
16C2×((C3⋊D3)⋊C3)⋊C4—28no—
17C2×((C3⋊D3)⋊C3)⋊C4C2218yescomplemented
18C4×#1084C2214yescomplemented
19C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
20#88450C2218yescomplemented
21C22×C4⋊(C3⋊D3)D311yescharacteristic
22C22×C3×D12—31no—
23C22×#624—32no—
24C22×D3×Dic3—32no—
25C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
26C2×C3×C4⋊Dic3—31no—
27C2×C3×C4×Dic3—31no—
28C2×C3×C4×Dic3—31no—
29C2×C3×C4×Dic3—61no—
30C2×#3722—32no—
31C2×(C2×D3)⋊Dic3—32no—
32#63894—38no—
33Dic3×D12—38no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×(C3⋊D3)⋊C3—24no—
36C22×(C3⋊D3)⋊C3—24no—
37C22×(C3⋊D3)⋊C3C2×C418yescomplemented
38C22×(C3⋊D3)⋊C3C2314yes—
39C2×C4×H(3,𝔽3)C2×C414yescomplemented
40C2×C4×H(3,𝔽3)C2312yes—
41C2×#1084—22no—
42C2×#1084D414yescomplemented
43C2×#1084C2312yes—
44C2×#1084C2312yes—
45C2×(C3⋊Dic3)⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C4⋊((C3⋊D3)⋊C3)C2×C4116yescomplemented
48((C3⋊D3)⋊C3)⋊C4—216no—
49C2×D4×Dic3—91no—
50C2×#893—91no—
51C2×C4×D12—91no—
52C23×C3⋊D3C2×D312yes—
53C23×C3×D3—32no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C3⋊Dic3C2×D312yes—
58C2×C4⋊(C3⋊D3)Dic318yescomplemented
59C2×C4⋊(C3⋊D3)C2×D314yes—
60C22×C3×Dic3—31no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—31no—
63C22×C3×Dic3—31no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—61no—
67C2×C3×D12—38no—
68C2×C3×D12—34no—
69C2×#624—38no—
70C2×#624—68no—
71C2×D3×Dic3—38no—
72C2×D3×Dic3—68no—
73C4⋊(C3⋊Dic3)C2×D314yescomplemented
74C3×C4⋊Dic3—34no—
75C3×C4×Dic3—34no—
76C3×C4×Dic3—34no—
77C3×C4×Dic3—64no—
78#3722—38no—
79(C2×D3)⋊Dic3—38no—
80C22×H(3,𝔽3)C22×C412yes—
81C22×H(3,𝔽3)C2×D412yes—
82C22×H(3,𝔽3)#4112yes—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×(C3⋊D3)⋊C3—216no—
85C2×(C3⋊D3)⋊C3—24no—
86C2×(C3⋊D3)⋊C3—24no—
87C2×(C3⋊D3)⋊C3C22×C418yescomplemented
88C4×H(3,𝔽3)C22×C414yescomplemented
89#1084—22no—
90#1084C2×D414yescomplemented
91(C3⋊Dic3)⋊C3—24no—
92C22×C3×D4—91no—
93C23×Dic3—92no—
94C22×D12—31no—
95C22×D12—91no—
96C22×C4×D3—92no—
97C22×C4×D3—92no—
98C2×C3×C4⋊C4—91no—
99C2×C3×C42—91no—
100C2×#217—92no—
101D4×Dic3—98no—
102C2×(C2×D3)⋊C4—92no—
103C2×(C2×D3)⋊C4—92no—
104C2×C4⋊Dic3—31no—
105C2×C4⋊Dic3—91no—
106C2×C4×Dic3—31no—
107C2×C4×Dic3—31no—
108C2×C4×Dic3—61no—
109#893—98no—
110C4×D12—98no—
111C23×C32—21no—
112C23×C32C22×D311yescharacteristic
113C23×C32C22×D311yescharacteristic
114C22×C3⋊D3—24no—
115C22×C3⋊D3—24no—
116C22×C3⋊D3C2×Dic318yescomplemented
117C22×C3⋊D3C22×D314yes—
118C22×C3×D3—38no—
119C22×C3×D3—34no—
120C22×C3×D3—64no—
121C22×C3×D3—64no—
122C2×C32×C4—24no—
123C2×C32×C4—22no—
124C2×C32×C4C4×D314yescomplemented
125C2×C32×C4C2×Dic314yescomplemented
126C2×C32×C4C22×D312yes—
127C2×C32×C4C22×D312yes—
128C2×C3⋊Dic3—24no—
129C2×C3⋊Dic3C22×D314yes—
130C4⋊(C3⋊D3)C2×Dic3116yescomplemented
131C2×C3×Dic3—34no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C3×D12—316no—
146#624—616no—
147D3×Dic3—616no—
148C2×C4×D4—271noSylow
149C2×H(3,𝔽3)C4×D414yescomplemented
150C2×H(3,𝔽3)C23×C411yescharacteristic
151C2×H(3,𝔽3)C22×D411yescharacteristic
152C2×H(3,𝔽3)C2×#4111yescharacteristic
153(C3⋊D3)⋊C3—28no—
154C24×C3—92no—
155C23×D3—32no—
156C23×D3—92no—
157C2×C3×D4—94no—
158C2×C3×D4—98no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—61no—
162C22×C3×C4—91no—
163C22×C3×C4—92no—
164C22×C3×C4—91no—
165C22×C3×C4D3211yescharacteristic
166C22×Dic3—31no—
167C22×Dic3—31no—
168C22×Dic3—31no—
169C22×Dic3—31no—
170C22×Dic3—32no—
171C22×Dic3—61no—
172C22×Dic3—61no—
173C22×Dic3—92no—
174C22×Dic3—98no—
175C22×Dic3—188no—
176C2×D12—38no—
177C2×D12—34no—
178C2×D12—94no—
179C2×D12—98no—
180C2×C4×D3—98no—
181C2×C4×D3—98no—
182C2×C4×D3—188no—
183C2×C4×D3—188no—
184C3×C4⋊C4—94no—
185C3×C42—94no—
186#217—98no—
187(C2×D3)⋊C4—98no—
188(C2×D3)⋊C4—98no—
189C4⋊Dic3—34no—
190C4⋊Dic3—94no—
191C4×Dic3—34no—
192C4×Dic3—34no—
193C4×Dic3—64no—
194C22×C32—21no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32C2×C4×D312yes—
199C22×C32D3×D412yes—
200C22×C32D3×D412yes—
201C22×C32#23712yes—
202C22×C32#23912yes—
203C22×C32C22×Dic312yes—
204C22×C32C23×D311yescharacteristic
205C22×C32C23×D311yescharacteristic
206C2×C3⋊D3—216no—
207C2×C3⋊D3—24no—
208C2×C3⋊D3—24no—
209C2×C3⋊D3C22×Dic318yescomplemented
210C2×C3×D3—38no—
211C2×C3×D3—616no—
212C2×C3×D3—64no—
213C2×C3×D3—64no—
214C32×C4—24no—
215C32×C4C2×C4×D314yescomplemented
216C32×C4C22×Dic314yescomplemented
217C3⋊Dic3—24no—
218C3×Dic3—34no—
219C3×Dic3—34no—
220C3×Dic3—64no—
221C3×Dic3—64no—
222C3×Dic3—64no—
223C3×Dic3—62no—
224C3×Dic3—62no—
225C22×D4—91no—
226C23×C4—272no—
227C4×D4—278no—
228C2×C4⋊C4—91no—
229C2×C22⋊C4—272no—
230C2×C42—91no—
231H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
232C23×C3—31no—
233C23×C3—31no—
234C23×C3—61no—
235C23×C3—94no—
236C23×C3—98no—
237C23×C3—184no—
238C23×C3—184no—
239C23×C3C2×D3211yescharacteristic
240C22×D3—34no—
241C22×D3—38no—
242C22×D3—64no—
243C22×D3—64no—
244C22×D3—98no—
245C22×D3—94no—
246C22×D3—184no—
247C22×D3—184no—
248C3×D4—916no—
249C2×C3×C4—34no—
250C2×C3×C4—34no—
251C2×C3×C4—32no—
252C2×C3×C4—32no—
253C2×C3×C4—64no—
254C2×C3×C4—62no—
255C2×C3×C4—92no—
256C2×C3×C4—94no—
257C2×C3×C4—94no—
258C2×C3×C4—92no—
259C2×C3×C4—184no—
260C2×C3×C4—182no—
261C2×C3×C4D3×Dic314yes—
262C2×C3×C4C2×D3212yes—
263C2×Dic3—32no—
264C2×Dic3—32no—
265C2×Dic3—34no—
266C2×Dic3—34no—
267C2×Dic3—32no—
268C2×Dic3—32no—
269C2×Dic3—34no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—64no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—64no—
276C2×Dic3—64no—
277C2×Dic3—94no—
278C2×Dic3—184no—
279C2×Dic3—1816no—
280D12—316no—
281D12—916no—
282C4×D3—1816no—
283C4×D3—1816no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—24no—
288C2×C32#89314yescomplemented
289C2×C32D4×Dic314yescomplemented
290C2×C32C22×C4×D311yescharacteristic
291C2×C32C2×D3×D411yescharacteristic
292C2×C32C2×D3×D411yescharacteristic
293C2×C32C2×#23711yescharacteristic
294C2×C32C2×#23911yescharacteristic
295C2×C32C23×Dic311yescharacteristic
296C3⋊D3—28no—
297C3×D3—68no—
298C24—92no—
299C2×D4—98no—
300C2×D4—94no—
301C22×C4—91no—
302C22×C4—91no—
303C22×C4—92no—
304C22×C4—278no—
305C22×C4—548no—
306C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C4⋊C4—94no—
308C22⋊C4—278no—
309C42—94no—
310C22×C3—31no—
311C22×C3—31no—
312C22×C3—32no—
313C22×C3—32no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—32no—
317C22×C3—32no—
318C22×C3—61no—
319C22×C3—62no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—98no—
323C22×C3—184no—
324C22×C3—1816no—
325C22×C3—184no—
326C22×C3#379612yes—
327C22×C3#380212yes—
328C22×C3C2×D3×Dic312yes—
329C22×C3C22×D3211yescharacteristic
330C2×D3—38no—
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—616no—
334C2×D3—98no—
335C2×D3—184no—
336C2×D3—1816no—
337C2×D3—184no—
338C3×C4—34no—
339C3×C4—34no—
340C3×C4—64no—
341C3×C4—94no—
342C3×C4—184no—
343C3×C4—182no—
344C3×C4C2×D3×Dic314yes—
345Dic3—34no—
346Dic3—34no—
347Dic3—64no—
348Dic3—62no—
349Dic3—62no—
350Dic3—64no—
351Dic3—64no—
352Dic3—184no—
353C32—21no—
354C32C2×#89311yescharacteristic, complemented
355C32C2×D4×Dic311yescharacteristic, complemented
356C23—94no—
357C23—98no—
358C23—184no—
359C23—184no—
360C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361D4—916no—
362C2×C4—92no—
363C2×C4—94no—
364C2×C4—92no—
365C2×C4—94no—
366C2×C4—182no—
367C2×C4—184no—
368C2×C4—5416no—
369C2×C4((C3⋊D3)⋊C3)⋊C414yescomplemented
370C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—34no—
374C2×C3—34no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—64no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—188no—
384C2×C3#6389814yes—
385C2×C3C2×#379611yescharacteristic
386C2×C3C2×#380211yescharacteristic
387C2×C3C22×D3×Dic311yescharacteristic
388D3—68no—
389D3—188no—
390C22—98no—
391C22—1816no—
392C22—184no—
393C22—184no—
394C22#8864912yes—
395C22#8865512yes—
396C22C2×((C3⋊D3)⋊C3)⋊C412yes—
397C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C4—94no—
399C4—182no—
400C4—184no—
401C4C2×((C3⋊D3)⋊C3)⋊C414yescomplemented
402C3—31no—
403C3—31no—
404C3—61no—
405C3C2×#6389811yescharacteristic
406C2—188no—
407C2#1172113614yesdirect product factor
408C2C2×#8864911yescharacteristic
409C2C2×#8865511yescharacteristic
410C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
3. C2×C4×#1084C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88450C2semidirect
6. #11721136C2direct (N × H)
14. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
17. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
18. C4×#1084C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. #88450C22semidirect
25. C2×C4⋊(C3⋊Dic3)D3semidirect
37. C22×(C3⋊D3)⋊C3C2×C4semidirect
39. C2×C4×H(3,𝔽3)C2×C4semidirect
42. C2×#1084D4semidirect
47. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
58. C2×C4⋊(C3⋊D3)Dic3semidirect
73. C4⋊(C3⋊Dic3)C2×D3semidirect
87. C2×(C3⋊D3)⋊C3C22×C4semidirect
88. C4×H(3,𝔽3)C22×C4semidirect
90. #1084C2×D4semidirect
116. C22×C3⋊D3C2×Dic3semidirect
124. C2×C32×C4C4×D3semidirect
125. C2×C32×C4C2×Dic3semidirect
130. C4⋊(C3⋊D3)C2×Dic3semidirect
149. C2×H(3,𝔽3)C4×D4semidirect
209. C2×C3⋊D3C22×Dic3semidirect
215. C32×C4C2×C4×D3semidirect
216. C32×C4C22×Dic3semidirect
231. H(3,𝔽3)C2×C4×D4semidirect
288. C2×C32#893semidirect
289. C2×C32D4×Dic3semidirect
354. C32C2×#893semidirect
355. C32C2×D4×Dic3semidirect
369. C2×C4((C3⋊D3)⋊C3)⋊C4semidirect
401. C4C2×((C3⋊D3)⋊C3)⋊C4semidirect
407. C2#11721136direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C22×((C3⋊D3)⋊C3)⋊C42
C2×C4×#10842
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884502
#117211362
C2×#638943
C2×Dic3×D123