Finite Group Databasebeta

← C2×#11724186 #13180778

Subgroups — C2×#11724186 #13180778

Order 1728.

1654 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 133 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×C3⋊(C3⋊S4)C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C32×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4#11724186C214yesmaximal, direct product factor
5C2×(C3⋊S4)⋊C4—34nomaximal
6C2×#64090—31nomaximal
7C22×C32×A4C2211yescharacteristic
8C2×C3⋊(C3⋊S4)C414yescomplemented
9C2×C3⋊(C3⋊S4)C2212yes—
10C2×#6289—41nomaximal
11A4×C3⋊Dic3C2212yescomplemented
12C32×SL(2,ℤ/4)C2212yescomplemented
13C22×C3⋊S4—34no—
14C22×C3⋊S4—64no—
15C22×#638—31no—
16C23×C3⋊Dic3D311yescharacteristic, complemented
17C22×C4×C3⋊D3—31no—
18C2×Dic3×A4—34no—
19C2×C3×SL(2,ℤ/4)D314yescomplemented
20(C3⋊S4)⋊C4—316no—
21C2×C32×C22⋊C4—31no—
22C2×#3752—31no—
23C2×#3750—31no—
24C2×C4×C3⋊Dic3—31no—
25#64090—34no—
26#64090—34no—
27C22×C3⋊(C3⋊D3)—41no—
28C2×C32×A4C2×C412yescomplemented
29C2×C32×A4C2311yescharacteristic
30C3⋊(C3⋊S4)C2×C414yescomplemented
31C2×C3×C3⋊Dic3—41no—
32C2×C32×Dic3—41no—
33#6289—44no—
34C2×C4×S4—91no—
35C2×#67⋊C4—94no—
36C24×C32C2×D311yescharacteristic
37C23×C3⋊D3—31no—
38C22×C3×A4—24no—
39C22×C3×A4C2×D314yes—
40C2×C3⋊S4—316no—
41C2×C3⋊S4—38no—
42C2×C3⋊S4—616no—
43C2×C3⋊S4—68no—
44C22×C32×C4—31no—
45C22×C32×C4—31no—
46C2×#638—32no—
47C2×#638—34no—
48C2×#638—34no—
49C2×#638—32no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—31no—
52C22×C3⋊Dic3—32no—
53C22×C3⋊Dic3—31no—
54C22×C3⋊Dic3C2×D312yescomplemented
55C2×C4×C3⋊D3—34no—
56C2×C4×C3⋊D3—61no—
57C2×C4×C3⋊D3—62no—
58C2×C4×C3⋊D3—61no—
59C2×#624—124no—
60Dic3×A4—38no—
61C3×SL(2,ℤ/4)C2×D318yescomplemented
62C32×C22⋊C4—32no—
63C32×C22⋊C4—32no—
64#3752—34no—
65#3750—34no—
66C4×C3⋊Dic3—34no—
67C22×C33—41no—
68C2×C3⋊(C3⋊D3)—44no—
69C2×C3⋊(C3⋊D3)—42no—
70C32×A4C22×C411yescharacteristic, complemented
71C3×C3⋊Dic3—42no—
72C32×Dic3—42no—
73C22×S4—91no—
74C22×S4—184no—
75C22×#67—94no—
76C23×Dic3—34no—
77C22×C4×D3—94no—
78C2×C4×A4—91no—
79C2×SL(2,ℤ/4)C3⋊D311yescharacteristic, complemented
80C4×S4—94no—
81C2×C3×C22⋊C4—34no—
82C2×(C2×D3)⋊C4—94no—
83C2×Dic3⋊C4—94no—
84C2×C4×Dic3—94no—
85#67⋊C4—916no—
86#67⋊C4—916no—
87C23×C32—31no—
88C23×C32—31no—
89C23×C32—32no—
90C23×C32C4×D312yescomplemented
91C23×C32C22×D311yescharacteristic
92C22×C3⋊D3—32no—
93C22×C3⋊D3—34no—
94C22×C3⋊D3—62no—
95C22×C3⋊D3—61no—
96C22×C3⋊D3—61no—
97C22×C3⋊D3—124no—
98C22×C3⋊D3—244no—
99C2×C3×A4—28no—
100C2×C3×A4—24no—
101C2×C3×A4C4×D318yescomplemented
102C2×C3×A4C22×D314yes—
103C3⋊S4—316no—
104C3⋊S4—616no—
105C2×C32×C4—32no—
106C2×C32×C4—32no—
107C2×C32×C4—61no—
108C2×C32×C4—62no—
109C2×C32×C4—61no—
110#638—38no—
111#638—34no—
112#638—34no—
113C2×C3⋊Dic3—32no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—32no—
117C2×C3⋊Dic3—32no—
118C2×C3⋊Dic3—32no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3S411yescharacteristic, complemented
121C4×C3⋊D3—64no—
122C4×C3⋊D3—64no—
123C2×C3×Dic3—44no—
124C2×C3×Dic3—124no—
125#624—1216no—
126C2×C4×D4—271noSylow
127C2×C33—42no—
128C2×C33—41no—
129C3⋊(C3⋊D3)—44no—
130C24×C3D3214yes—
131C23×D3—94no—
132C22×A4—24no—
133C22×A4C2×C3⋊D311yescharacteristic
134C2×S4—94no—
135C2×S4—92no—
136C2×S4—1816no—
137C2×S4—188no—
138C22×C3×C4—34no—
139C22×C3×C4—34no—
140C2×#67—98no—
141C2×#67—916no—
142C2×#67—916no—
143C2×#67—98no—
144C22×Dic3—38no—
145C22×Dic3—94no—
146C22×Dic3—94no—
147C22×Dic3—98no—
148C22×Dic3—94no—
149C2×C4×D3—916no—
150C2×C4×D3—184no—
151C2×C4×D3—188no—
152C2×C4×D3—184no—
153C2×C4×D3—361no—
154C4×A4—92no—
155SL(2,ℤ/4)C2×C3⋊D312yescomplemented
156C3×C22⋊C4—38no—
157C3×C22⋊C4—38no—
158(C2×D3)⋊C4—916no—
159Dic3⋊C4—916no—
160C4×Dic3—916no—
161C22×C32—32no—
162C22×C32—31no—
163C22×C32—32no—
164C22×C32—32no—
165C22×C32—31no—
166C22×C32—31no—
167C22×C32—44no—
168C22×C32—61no—
169C22×C32—84no—
170C22×C32C2×C4×D311yescharacteristic, complemented
171C22×C32C2×S411yescharacteristic
172C2×C3⋊D3—34no—
173C2×C3⋊D3—62no—
174C2×C3⋊D3—64no—
175C2×C3⋊D3—64no—
176C2×C3⋊D3—62no—
177C2×C3⋊D3—128no—
178C2×C3⋊D3—1216no—
179C2×C3⋊D3—248no—
180C2×C3⋊D3—2416no—
181C3×A4—24no—
182C3×A4C2×C4×D314yescomplemented
183C32×C4—62no—
184C32×C4—62no—
185C3⋊Dic3—34no—
186C3⋊Dic3—32no—
187C3⋊Dic3C2×S412yescomplemented
188C3×Dic3—48no—
189C3×Dic3—128no—
190C22×D4—271no—
191C23×C4—91no—
192C23×C4—271no—
193C4×D4—274no—
194C4×D4—274no—
195C2×C4⋊C4—271no—
196C2×C22⋊C4—31no—
197C2×C22⋊C4—271no—
198C2×C42—271no—
199C33—41noSylow
200C23×C3—34no—
201C23×C3—34no—
202C23×C3—38no—
203C23×C3#62418yescomplemented
204C23×C3C2×D3214yes—
205C22×D3—98no—
206C22×D3—916no—
207C22×D3—184no—
208C22×D3—188no—
209C22×D3—184no—
210C22×D3—361no—
211C22×D3—724no—
212C2×A4—28no—
213C2×A4—24no—
214C2×A4C4×C3⋊D312yescomplemented
215C2×A4C22×C3⋊D311yescharacteristic
216S4—94no—
217S4—1816no—
218C2×C3×C4—38no—
219C2×C3×C4—38no—
220C2×C3×C4—64no—
221C2×C3×C4—68no—
222C2×C3×C4—64no—
223C2×C3×C4—361no—
224#67—932no—
225#67—916no—
226#67—916no—
227C2×Dic3—34no—
228C2×Dic3—41no—
229C2×Dic3—98no—
230C2×Dic3—98no—
231C2×Dic3—98no—
232C2×Dic3—916no—
233C2×Dic3—94no—
234C2×Dic3—98no—
235C2×Dic3—184no—
236C4×D3—1816no—
237C4×D3—1816no—
238C4×D3—364no—
239C2×C32—31no—
240C2×C32—31no—
241C2×C32—32no—
242C2×C32—44no—
243C2×C32—48no—
244C2×C32—84no—
245C2×C32—88no—
246C2×C32C4×S412yescomplemented
247C2×C32C22×S411yescharacteristic
248C3⋊D3—64no—
249C3⋊D3—1216no—
250C3⋊D3—2416no—
251C24—271no—
252C24D3×C3⋊D311yescharacteristic
253C2×D4—272no—
254C2×D4—274no—
255C2×D4—274no—
256C2×D4—272no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—92no—
260C22×C4—274no—
261C22×C4—271no—
262C22×C4—271no—
263C22×C4—271no—
264C22×C4—272no—
265C22×C4—541no—
266C22×C4—542no—
267C22×C4—541no—
268C4⋊C4—274no—
269C22⋊C4—32no—
270C22⋊C4—32no—
271C22⋊C4—274no—
272C42—274no—
273C22×C3—38no—
274C22×C3—34no—
275C22×C3—34no—
276C22×C3—34no—
277C22×C3—38no—
278C22×C3—38no—
279C22×C3—41no—
280C22×C3—64no—
281C22×C3—84no—
282C22×C3C2×#62414yescomplemented
283C22×C3D3×S414yes—
284C2×D3—916no—
285C2×D3—188no—
286C2×D3—1816no—
287C2×D3—188no—
288C2×D3—1816no—
289C2×D3—362no—
290C2×D3—364no—
291C2×D3—7216no—
292C2×D3—728no—
293A4—24no—
294A4C2×C4×C3⋊D311yescharacteristic, complemented
295C3×C4—68no—
296C3×C4—68no—
297C3×C4—362no—
298Dic3—38no—
299Dic3—42no—
300Dic3—916no—
301Dic3—98no—
302C32—44no—
303C32—84no—
304C32C2×C4×S411yescharacteristic, complemented
305C23—31no—
306C23—31no—
307C23—32no—
308C23—272no—
309C23—274no—
310C23—541no—
311C23—542no—
312C23—541no—
313C23#628912yescomplemented
314C23C2×D3×C3⋊D311yescharacteristic
315D4—278no—
316D4—274no—
317D4—274no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4—61no—
323C2×C4—91no—
324C2×C4—272no—
325C2×C4—272no—
326C2×C4—272no—
327C2×C4—274no—
328C2×C4—271no—
329C2×C4—272no—
330C2×C4—544no—
331C2×C4—544no—
332C2×C4—541no—
333C2×C3—34no—
334C2×C3—34no—
335C2×C3—38no—
336C2×C3—42no—
337C2×C3—41no—
338C2×C3—88no—
339C2×C3—84no—
340C2×C3(C3⋊S4)⋊C418yescomplemented
341C2×C3C2×D3×S414yes—
342D3—1816no—
343D3—364no—
344D3—7216no—
345C22—32no—
346C22—31no—
347C22—32no—
348C22—32no—
349C22—31no—
350C22—31no—
351C22—61no—
352C22—274no—
353C22—542no—
354C22—544no—
355C22—544no—
356C22—542no—
357C22C2×#628911yescharacteristic, complemented
358C22C3⋊D3×S411yescharacteristic
359C4—62no—
360C4—62no—
361C4—92no—
362C4—274no—
363C4—272no—
364C3—41no—
365C3—84no—
366C3C2×(C3⋊S4)⋊C414yescomplemented
367C2—31no—
368C2—31no—
369C2—32no—
370C2—544no—
371C2#1172418612yesdirect product factor
372C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×C3⋊D3×S4
Fitting, socleC24×C32C2×D3
prime radicalC22×C3⋊(C3⋊S4)C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×C3⋊Dic3C2semidirect
3. C2×C32×SL(2,ℤ/4)C2semidirect
4. #11724186C2direct (N × H)
8. C2×C3⋊(C3⋊S4)C4semidirect
11. A4×C3⋊Dic3C22semidirect
12. C32×SL(2,ℤ/4)C22semidirect
16. C23×C3⋊Dic3D3semidirect
19. C2×C3×SL(2,ℤ/4)D3semidirect
28. C2×C32×A4C2×C4semidirect
30. C3⋊(C3⋊S4)C2×C4semidirect
54. C22×C3⋊Dic3C2×D3semidirect
61. C3×SL(2,ℤ/4)C2×D3semidirect
70. C32×A4C22×C4semidirect
79. C2×SL(2,ℤ/4)C3⋊D3semidirect
90. C23×C32C4×D3semidirect
101. C2×C3×A4C4×D3semidirect
120. C2×C3⋊Dic3S4semidirect
155. SL(2,ℤ/4)C2×C3⋊D3semidirect
170. C22×C32C2×C4×D3semidirect
182. C3×A4C2×C4×D3semidirect
187. C3⋊Dic3C2×S4semidirect
203. C23×C3#624semidirect
214. C2×A4C4×C3⋊D3semidirect
246. C2×C32C4×S4semidirect
282. C22×C3C2×#624semidirect
294. A4C2×C4×C3⋊D3semidirect
304. C32C2×C4×S4semidirect
313. C23#6289semidirect
340. C2×C3(C3⋊S4)⋊C4semidirect
357. C22C2×#6289semidirect
366. C3C2×(C3⋊S4)⋊C4semidirect
371. C2#11724186direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊S4)2
C2×A4×C3⋊Dic32
C2×C32×SL(2,ℤ/4)2
#117241862
C2×(C3⋊S4)⋊C43
C2×#640903
C2×#62894