Finite Group Databasebeta

← C2×#11721129 #13150692

Subgroups — C2×#11721129 #13150692

Order 1728.

1555 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22×((C3⋊D3)⋊C3)⋊C4C212yesmaximal
3C2×C4⋊#1084C211yesmaximal, characteristic, complemented
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×#88448C212yesmaximal, complemented
6#11721129C218yesmaximal, direct product factor
7C2×#63900—31nomaximal
8C2×D3×C4⋊Dic3—31nomaximal
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C418yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C2×((C3⋊D3)⋊C3)⋊C4—28no—
17C2×((C3⋊D3)⋊C3)⋊C4C2218yes—
18C4⋊#1084C2214yescomplemented
19C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
20#88448C2218yescomplemented
21C22×C4×C3⋊D3D311yescharacteristic
22C22×C3×C4×D3—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×#3724—32no—
31C2×Dic3⋊Dic3—32no—
32#63900—38no—
33D3×C4⋊Dic3—38no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×(C3⋊D3)⋊C3C2×C414yes—
36C22×(C3⋊D3)⋊C3D414yes—
37C22×(C3⋊D3)⋊C3Q814yes—
38C22×(C3⋊D3)⋊C3C2312yes—
39C2×C4×H(3,𝔽3)C2×C414yescomplemented
40C2×C4×H(3,𝔽3)C2312yes—
41C2×#1084—24no—
42C2×#1084C2314yes—
43C2×(C3⋊Dic3)⋊C3—24no—
44C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C4×(C3⋊D3)⋊C3C2×C4116yescomplemented
48((C3⋊D3)⋊C3)⋊C4—216no—
49C22×C4⋊Dic3—91no—
50C2×D3×C4⋊C4—91no—
51C2×D3×C4⋊C4—91no—
52C23×C3⋊D3C2×D311yescharacteristic
53C23×C3×D3—31no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C3⋊Dic3C2×D312yes—
58C22×C3⋊Dic3C2×D311yescharacteristic
59C2×C4×C3⋊D3Dic318yescomplemented
60C2×C4×C3⋊D3C2×D314yes—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—62no—
66C2×C3×C4×D3—38no—
67C2×C3×C4×D3—34no—
68C2×#624—38no—
69C2×#624—68no—
70C2×D3×Dic3—38no—
71C2×D3×Dic3—68no—
72C4⋊(C3⋊Dic3)C2×D314yescomplemented
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3—64no—
77#3724—38no—
78Dic3⋊Dic3—38no—
79C22×H(3,𝔽3)C22×C412yes—
80C22×H(3,𝔽3)C2×D412yes—
81C22×H(3,𝔽3)C2×Q812yes—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×(C3⋊D3)⋊C3C4⋊C4116yescomplemented
84C2×(C3⋊D3)⋊C3C22×C414yes—
85C2×(C3⋊D3)⋊C3C2×D414yes—
86C2×(C3⋊D3)⋊C3C2×Q814yes—
87C4×H(3,𝔽3)C22×C414yescomplemented
88#1084—24no—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3C22×C414yescomplemented
91C23×C3×C4—91no—
92C23×Dic3—92no—
93C22×C4×D3—31no—
94C22×C4×D3—91no—
95C22×C4×D3—92no—
96C22×C4×D3—92no—
97C2×C3×C4⋊C4—91no—
98C2×C3×C4⋊C4—91no—
99C2×C4⋊Dic3—31no—
100C2×C4⋊Dic3—31no—
101C2×C4⋊Dic3—31no—
102C2×C4⋊Dic3—61no—
103C2×C4⋊Dic3—91no—
104C2×C4⋊Dic3—98no—
105C2×C4⋊Dic3—92no—
106C2×Dic3⋊C4—92no—
107C2×Dic3⋊C4—92no—
108D3×C4⋊C4—98no—
109D3×C4⋊C4—98no—
110C23×C32—21no—
111C23×C32C22×D311yescharacteristic
112C23×C32C22×D311yescharacteristic
113C22×C3⋊D3Dic614yes—
114C22×C3⋊D3D1214yes—
115C22×C3⋊D3C2×Dic314yes—
116C22×C3⋊D3C22×D312yes—
117C22×C3×D3—34no—
118C22×C3×D3—34no—
119C22×C3×D3—34no—
120C22×C3×D3—32no—
121C2×C32×C4—24no—
122C2×C32×C4—22no—
123C2×C32×C4C4×D314yescomplemented
124C2×C32×C4C2×Dic314yescomplemented
125C2×C32×C4C22×D312yes—
126C2×C32×C4C22×D312yes—
127C2×C3⋊Dic3—24no—
128C2×C3⋊Dic3C2×Dic314yescomplemented
129C2×C3⋊Dic3C22×D314yes—
130C2×C3⋊Dic3C22×D312yes—
131C4×C3⋊D3C2×Dic3116yescomplemented
132C2×C3×Dic3—34no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—124no—
142C3×C4×D3—316no—
143#624—616no—
144D3×Dic3—616no—
145C22×C4⋊C4—271noSylow
146C2×H(3,𝔽3)C2×C4⋊C414yescomplemented
147C2×H(3,𝔽3)C23×C411yescharacteristic
148C2×H(3,𝔽3)C22×D411yescharacteristic
149C2×H(3,𝔽3)C22×Q811yescharacteristic
150(C3⋊D3)⋊C3C2×C4⋊C418yescomplemented
151C24×C3—91no—
152C23×D3—31no—
153C23×D3—91no—
154C22×C3×C4—31no—
155C22×C3×C4—31no—
156C22×C3×C4—61no—
157C22×C3×C4—94no—
158C22×C3×C4—98no—
159C22×C3×C4—91no—
160C22×C3×C4—92no—
161C22×C3×C4—92no—
162C22×C3×C4D3211yescharacteristic
163C22×Dic3—31no—
164C22×Dic3—32no—
165C22×Dic3—32no—
166C22×Dic3—32no—
167C22×Dic3—62no—
168C22×Dic3—91no—
169C22×Dic3—92no—
170C22×Dic3—98no—
171C22×Dic3—188no—
172C2×C4×D3—38no—
173C2×C4×D3—34no—
174C2×C4×D3—94no—
175C2×C4×D3—98no—
176C2×C4×D3—98no—
177C2×C4×D3—98no—
178C2×C4×D3—188no—
179C2×C4×D3—188no—
180C3×C4⋊C4—94no—
181C3×C4⋊C4—94no—
182C4⋊Dic3—34no—
183C4⋊Dic3—34no—
184C4⋊Dic3—34no—
185C4⋊Dic3—64no—
186C4⋊Dic3—94no—
187C4⋊Dic3—98no—
188Dic3⋊C4—98no—
189Dic3⋊C4—98no—
190C22×C32—21no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32C2×Dic612yes—
195C22×C32C2×C4×D312yes—
196C22×C32C2×D1212yes—
197C22×C32D3×D412yes—
198C22×C32D3×Q812yes—
199C22×C32C22×Dic312yes—
200C22×C32C23×D311yescharacteristic
201C22×C32C23×D311yescharacteristic
202C2×C3⋊D3C4⋊Dic3116yescomplemented
203C2×C3⋊D3C2×Dic614yes—
204C2×C3⋊D3C2×D1214yes—
205C2×C3⋊D3C22×Dic314yes—
206C2×C3×D3—316no—
207C2×C3×D3—34no—
208C2×C3×D3—34no—
209C2×C3×D3—34no—
210C32×C4—24no—
211C32×C4C2×C4×D314yescomplemented
212C32×C4C22×Dic314yescomplemented
213C3⋊Dic3—24no—
214C3⋊Dic3C22×Dic314yescomplemented
215C3×Dic3—34no—
216C3×Dic3—64no—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—124no—
220C23×C4—91no—
221C23×C4—272no—
222C2×C4⋊C4—91no—
223C2×C4⋊C4—91no—
224C2×C4⋊C4—278no—
225C2×C4⋊C4—272no—
226H(3,𝔽3)C22×C4⋊C411yescharacteristic, complemented, Sylow
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—61no—
230C23×C3—92no—
231C23×C3—94no—
232C23×C3—94no—
233C23×C3—94no—
234C23×C3C2×D3211yescharacteristic
235C22×D3—32no—
236C22×D3—34no—
237C22×D3—34no—
238C22×D3—34no—
239C22×D3—94no—
240C22×D3—94no—
241C22×D3—94no—
242C22×D3—92no—
243C2×C3×C4—34no—
244C2×C3×C4—34no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—64no—
248C2×C3×C4—62no—
249C2×C3×C4—916no—
250C2×C3×C4—92no—
251C2×C3×C4—94no—
252C2×C3×C4—94no—
253C2×C3×C4—94no—
254C2×C3×C4—184no—
255C2×C3×C4—184no—
256C2×C3×C4D3×Dic314yes—
257C2×C3×C4C2×D3212yes—
258C2×Dic3—32no—
259C2×Dic3—34no—
260C2×Dic3—34no—
261C2×Dic3—34no—
262C2×Dic3—34no—
263C2×Dic3—64no—
264C2×Dic3—64no—
265C2×Dic3—64no—
266C2×Dic3—64no—
267C2×Dic3—94no—
268C2×Dic3—92no—
269C2×Dic3—94no—
270C2×Dic3—124no—
271C2×Dic3—184no—
272C2×Dic3—1816no—
273C4×D3—316no—
274C4×D3—916no—
275C4×D3—1816no—
276C4×D3—1816no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—24no—
281C2×C32D3×C4⋊C414yescomplemented
282C2×C32C2×C4⋊Dic314yescomplemented
283C2×C32C22×Dic611yescharacteristic
284C2×C32C22×C4×D311yescharacteristic
285C2×C32C22×D1211yescharacteristic
286C2×C32C2×D3×D411yescharacteristic
287C2×C32C2×D3×Q811yescharacteristic
288C2×C32C23×Dic311yescharacteristic
289C3⋊D3C2×C4⋊Dic318yescomplemented
290C3×D3—38no—
291C24—91no—
292C22×C4—91no—
293C22×C4—98no—
294C22×C4—94no—
295C22×C4—92no—
296C22×C4—92no—
297C22×C4—278no—
298C22×C4—548no—
299C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C4⋊C4—94no—
301C4⋊C4—94no—
302C4⋊C4—278no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—32no—
306C22×C3—32no—
307C22×C3—32no—
308C22×C3—32no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—61no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—94no—
316C22×C3—94no—
317C22×C3—916no—
318C22×C3—94no—
319C22×C3D3×Dic612yes—
320C22×C3D3×D1212yes—
321C22×C3C2×D3×Dic312yes—
322C22×C3C22×D3211yescharacteristic
323C2×D3—34no—
324C2×D3—34no—
325C2×D3—34no—
326C2×D3—316no—
327C2×D3—94no—
328C2×D3—94no—
329C2×D3—916no—
330C2×D3—94no—
331C3×C4—34no—
332C3×C4—34no—
333C3×C4—64no—
334C3×C4—94no—
335C3×C4—184no—
336C3×C4—184no—
337C3×C4C2×D3×Dic314yes—
338Dic3—34no—
339Dic3—64no—
340Dic3—64no—
341Dic3—64no—
342Dic3—94no—
343Dic3—124no—
344Dic3—184no—
345C32—21no—
346C32C2×D3×C4⋊C411yescharacteristic, complemented
347C32C22×C4⋊Dic311yescharacteristic, complemented
348C23—92no—
349C23—94no—
350C23—94no—
351C23—94no—
352C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C2×C4—92no—
354C2×C4—94no—
355C2×C4—916no—
356C2×C4—94no—
357C2×C4—94no—
358C2×C4—184no—
359C2×C4—184no—
360C2×C4—5416no—
361C2×C4((C3⋊D3)⋊C3)⋊C414yescomplemented
362C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—34no—
366C2×C3—34no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—61no—
372C2×C3—64no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—98no—
376C2×C3D3×C4⋊Dic314yes—
377C2×C3C2×D3×Dic611yescharacteristic
378C2×C3C2×D3×D1211yescharacteristic
379C2×C3C22×D3×Dic311yescharacteristic
380D3—38no—
381D3—98no—
382C22—94no—
383C22—916no—
384C22—94no—
385C22—94no—
386C22((C3⋊D3)⋊C3)⋊Q812yes—
387C22#8865412yes—
388C22C2×((C3⋊D3)⋊C3)⋊C412yes—
389C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
390C4—94no—
391C4—184no—
392C4—184no—
393C4C2×((C3⋊D3)⋊C3)⋊C414yescomplemented
394C3—31no—
395C3—31no—
396C3—61no—
397C3C2×D3×C4⋊Dic311yescharacteristic
398C2—98no—
399C2#1172112914yesdirect product factor
400C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
401C2C2×#8865411yescharacteristic
402C2C22×((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 radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×C4⋊C4
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
3. C2×C4⋊#1084C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88448C2semidirect
6. #11721129C2direct (N × H)
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
18. C4⋊#1084C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. #88448C22semidirect
25. C2×C4⋊(C3⋊Dic3)D3semidirect
39. C2×C4×H(3,𝔽3)C2×C4semidirect
44. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
47. C4×(C3⋊D3)⋊C3C2×C4semidirect
59. C2×C4×C3⋊D3Dic3semidirect
72. C4⋊(C3⋊Dic3)C2×D3semidirect
83. C2×(C3⋊D3)⋊C3C4⋊C4semidirect
87. C4×H(3,𝔽3)C22×C4semidirect
90. (C3⋊Dic3)⋊C3C22×C4semidirect
123. C2×C32×C4C4×D3semidirect
124. C2×C32×C4C2×Dic3semidirect
128. C2×C3⋊Dic3C2×Dic3semidirect
131. C4×C3⋊D3C2×Dic3semidirect
146. C2×H(3,𝔽3)C2×C4⋊C4semidirect
150. (C3⋊D3)⋊C3C2×C4⋊C4semidirect
202. C2×C3⋊D3C4⋊Dic3semidirect
211. C32×C4C2×C4×D3semidirect
212. C32×C4C22×Dic3semidirect
214. C3⋊Dic3C22×Dic3semidirect
226. H(3,𝔽3)C22×C4⋊C4semidirect
281. C2×C32D3×C4⋊C4semidirect
282. C2×C32C2×C4⋊Dic3semidirect
289. C3⋊D3C2×C4⋊Dic3semidirect
346. C32C2×D3×C4⋊C4semidirect
347. C32C22×C4⋊Dic3semidirect
361. C2×C4((C3⋊D3)⋊C3)⋊C4semidirect
393. C4C2×((C3⋊D3)⋊C3)⋊C4semidirect
399. C2#11721129direct (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)⋊C32
C22×((C3⋊D3)⋊C3)⋊C42
C2×C4⋊#10842
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884482
#117211292
C2×#639003
C2×D3×C4⋊Dic33