Finite Group Databasebeta

← C2×C3⋊D3×SL(2,ℤ/4) #13180780

Subgroups — C2×C3⋊D3×SL(2,ℤ/4) #13180780

Order 1728.

1790 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1790 conjugacy classes of subgroups — too many to draw. Only its 149 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×A4×C3⋊D3C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
3C2×C32×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C3⋊D3×SL(2,ℤ/4)C214yesmaximal, direct product factor
5C2×D3×SL(2,ℤ/4)—34nomaximal
6C2×C22⋊C4×C3⋊D3—31nomaximal
7C22×C32×A4C2211yescharacteristic
8C2×A4×C3⋊D3C414yescomplemented
9C2×A4×C3⋊D3C2212yes—
10C2×Dic3×C3⋊D3—41nomaximal
11C3⋊(C3⋊SL(2,ℤ/4))C2212yescomplemented
12C32×SL(2,ℤ/4)C2212yescomplemented
13C24×C3⋊D3D311yescharacteristic
14C22×D3×A4—34no—
15C22×C4×C3⋊D3—31no—
16C22×C4×C3⋊D3—31no—
17C2×C3⋊SL(2,ℤ/4)—34no—
18C2×C3⋊SL(2,ℤ/4)—64no—
19C2×C3×SL(2,ℤ/4)D314yescomplemented
20D3×SL(2,ℤ/4)—316no—
21C2×C32×C22⋊C4—31no—
22C2×#3757—31no—
23C2×#3752—31no—
24C2×#3752—31no—
25C22⋊C4×C3⋊D3—34no—
26C22⋊C4×C3⋊D3—34no—
27C22×C3×C3⋊D3—41no—
28C2×C32×A4C2×C412yescomplemented
29C2×C32×A4C2311yescharacteristic
30A4×C3⋊D3C2×C414yescomplemented
31C2×C3⋊(C3⋊Dic3)—41no—
32C2×C32×Dic3—41no—
33Dic3×C3⋊D3—44no—
34C22×SL(2,ℤ/4)—91no—
35C2×D3×C22⋊C4—94no—
36C24×C32C2×D311yescharacteristic
37C23×C3⋊D3—31no—
38C23×C3⋊D3—32no—
39C23×C3⋊D3—31no—
40C23×C3⋊D3—34no—
41C23×C3⋊D3Dic314yescomplemented
42C23×C3⋊D3C2×D312yes—
43C22×C3×A4—24no—
44C22×C3×A4C2×D314yes—
45C2×D3×A4—316no—
46C2×D3×A4—38no—
47C22×C32×C4—31no—
48C22×C32×C4—31no—
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—31no—
51C2×C4×C3⋊D3—34no—
52C2×C4×C3⋊D3—34no—
53C2×C4×C3⋊D3—64no—
54C2×C4×C3⋊D3—61no—
55C2×C4×C3⋊D3—62no—
56C2×C4×C3⋊D3—61no—
57C2×D3×Dic3—124no—
58C3⋊SL(2,ℤ/4)—38no—
59C3⋊SL(2,ℤ/4)—68no—
60C3×SL(2,ℤ/4)C2×D318yescomplemented
61C32×C22⋊C4—32no—
62C32×C22⋊C4—32no—
63#3757—32no—
64#3757—32no—
65#3752—34no—
66#3752—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—
73C24×D3—34no—
74C23×A4—91no—
75C22×C4×D3—94no—
76C22×C4×D3—94no—
77C2×SL(2,ℤ/4)—91no—
78C2×SL(2,ℤ/4)—94no—
79C2×SL(2,ℤ/4)—184no—
80C2×SL(2,ℤ/4)C3⋊D311yescharacteristic, direct product factor
81C2×C3×C22⋊C4—34no—
82C2×#217—94no—
83C2×(C2×D3)⋊C4—94no—
84C2×(C2×D3)⋊C4—94no—
85D3×C22⋊C4—916no—
86D3×C22⋊C4—916no—
87C23×C32—31no—
88C23×C32—32no—
89C23×C32—31no—
90C23×C32C2×Dic312yescomplemented
91C23×C32C22×D311yescharacteristic
92C22×C3⋊D3—31no—
93C22×C3⋊D3—34no—
94C22×C3⋊D3—34no—
95C22×C3⋊D3—32no—
96C22×C3⋊D3—32no—
97C22×C3⋊D3—34no—
98C22×C3⋊D3—38no—
99C22×C3⋊D3—34no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3—64no—
102C22×C3⋊D3—62no—
103C22×C3⋊D3—61no—
104C22×C3⋊D3C2×Dic314yescomplemented
105C22×C3⋊D3S411yescharacteristic
106C22×C3×D3—124no—
107C2×C3×A4—28no—
108C2×C3×A4—24no—
109C2×C3×A4C4×D318yescomplemented
110C2×C3×A4C22×D314yes—
111D3×A4—316no—
112C2×C32×C4—32no—
113C2×C32×C4—32no—
114C2×C32×C4—62no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C3⋊Dic3—32no—
118C2×C3⋊Dic3—32no—
119C2×C3⋊Dic3—62no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—124no—
123C2×C3⋊Dic3—244no—
124C4×C3⋊D3—68no—
125C4×C3⋊D3—64no—
126C4×C3⋊D3—64no—
127C2×C3×Dic3—44no—
128D3×Dic3—1216no—
129C22×C22⋊C4—271noSylow
130C2×C33—42no—
131C2×C33—41no—
132C3×C3⋊D3—44no—
133C24×C3D3214yes—
134C23×D3—316no—
135C23×D3—38no—
136C23×D3—94no—
137C23×D3—98no—
138C23×D3—94no—
139C23×D3—916no—
140C22×A4—24no—
141C22×A4—94no—
142C22×A4—92no—
143C22×A4C2×C3⋊D311yescharacteristic
144C22×C3×C4—34no—
145C22×C3×C4—34no—
146C22×Dic3—94no—
147C22×Dic3—94no—
148C22×Dic3—361no—
149C2×C4×D3—916no—
150C2×C4×D3—916no—
151C2×C4×D3—184no—
152C2×C4×D3—188no—
153C2×C4×D3—1816no—
154C2×C4×D3—184no—
155SL(2,ℤ/4)—92no—
156SL(2,ℤ/4)—188no—
157SL(2,ℤ/4)C2×C3⋊D312yesdirect product factor
158C3×C22⋊C4—38no—
159C3×C22⋊C4—38no—
160#217—98no—
161#217—98no—
162(C2×D3)⋊C4—916no—
163(C2×D3)⋊C4—916no—
164C22×C32—32no—
165C22×C32—31no—
166C22×C32—32no—
167C22×C32—32no—
168C22×C32—31no—
169C22×C32—31no—
170C22×C32—44no—
171C22×C32—61no—
172C22×C32—84no—
173C22×C32C22×Dic311yescharacteristic, complemented
174C22×C32C2×S411yescharacteristic
175C2×C3⋊D3—34no—
176C2×C3⋊D3—32no—
177C2×C3⋊D3—34no—
178C2×C3⋊D3—34no—
179C2×C3⋊D3—38no—
180C2×C3⋊D3—34no—
181C2×C3⋊D3—62no—
182C2×C3⋊D3—64no—
183C2×C3⋊D3SL(2,ℤ/4)14yesdirect product factor
184C2×C3⋊D3C2×S412yes—
185C2×C3×D3—128no—
186C2×C3×D3—1216no—
187C3×A4—24no—
188C3×A4C2×C4×D314yescomplemented
189C32×C4—62no—
190C32×C4—62no—
191C3⋊Dic3—62no—
192C3⋊Dic3—62no—
193C3⋊Dic3—128no—
194C3⋊Dic3—248no—
195C3×Dic3—48no—
196C25—91no—
197C23×C4—271no—
198C23×C4—271no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—274no—
201C2×C22⋊C4—274no—
202C2×C22⋊C4—271no—
203C2×C22⋊C4—271no—
204C2×C22⋊C4—271no—
205C33—41noSylow
206C23×C3—34no—
207C23×C3—34no—
208C23×C3—38no—
209C23×C3—361no—
210C23×C3D3×Dic318yescomplemented
211C23×C3C2×D3214yes—
212C22×D3—34no—
213C22×D3—316no—
214C22×D3—98no—
215C22×D3—98no—
216C22×D3—916no—
217C22×D3—916no—
218C22×D3—94no—
219C22×D3—98no—
220C22×D3—916no—
221C22×D3—932no—
222C22×D3—916no—
223C22×D3—1816no—
224C22×D3—184no—
225C22×D3—188no—
226C2×A4—28no—
227C2×A4—24no—
228C2×A4—94no—
229C2×A4C4×C3⋊D312yescomplemented
230C2×A4C22×C3⋊D311yescharacteristic
231C2×C3×C4—38no—
232C2×C3×C4—38no—
233C2×C3×C4—68no—
234C2×C3×C4—64no—
235C2×C3×C4—64no—
236C2×Dic3—41no—
237C2×Dic3—98no—
238C2×Dic3—98no—
239C2×Dic3—188no—
240C2×Dic3—184no—
241C2×Dic3—184no—
242C2×Dic3—364no—
243C2×Dic3—361no—
244C2×Dic3—724no—
245C4×D3—1816no—
246C4×D3—1832no—
247C4×D3—1816no—
248C2×C32—31no—
249C2×C32—31no—
250C2×C32—32no—
251C2×C32—44no—
252C2×C32—48no—
253C2×C32—84no—
254C2×C32—88no—
255C2×C32C2×SL(2,ℤ/4)12yescomplemented
256C2×C32C22×S411yescharacteristic
257C3⋊D3—34no—
258C3⋊D3C2×SL(2,ℤ/4)14yesdirect product factor
259C3×D3—1216no—
260C24—94no—
261C24—92no—
262C24—271no—
263C24—271no—
264C24—272no—
265C24—274no—
266C24D3×C3⋊D311yescharacteristic
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—274no—
270C22×C4—274no—
271C22×C4—271no—
272C22×C4—271no—
273C22×C4—541no—
274C22×C4—542no—
275C22×C4—544no—
276C22×C4—541no—
277C22⋊C4—32no—
278C22⋊C4—32no—
279C22⋊C4—274no—
280C22⋊C4—274no—
281C22⋊C4—272no—
282C22⋊C4—272no—
283C22×C3—38no—
284C22×C3—34no—
285C22×C3—34no—
286C22×C3—34no—
287C22×C3—38no—
288C22×C3—38no—
289C22×C3—41no—
290C22×C3—64no—
291C22×C3—84no—
292C22×C3—362no—
293C22×C3—364no—
294C22×C3C2×D3×Dic314yescomplemented
295C22×C3D3×S414yes—
296C2×D3—38no—
297C2×D3—316no—
298C2×D3—916no—
299C2×D3—916no—
300C2×D3—932no—
301C2×D3—98no—
302C2×D3—916no—
303C2×D3—916no—
304C2×D3—188no—
305C2×D3—1816no—
306A4—24no—
307A4C2×C4×C3⋊D311yescharacteristic, complemented
308C3×C4—68no—
309C3×C4—68no—
310Dic3—42no—
311Dic3—188no—
312Dic3—188no—
313Dic3—362no—
314Dic3—728no—
315C32—44no—
316C32—84no—
317C32C22×SL(2,ℤ/4)11yescharacteristic, complemented
318C23—31no—
319C23—31no—
320C23—32no—
321C23—91no—
322C23—94no—
323C23—272no—
324C23—274no—
325C23—272no—
326C23—274no—
327C23—272no—
328C23—278no—
329C23—274no—
330C23—274no—
331C23—271no—
332C23—542no—
333C23—541no—
334C23—544no—
335C23Dic3×C3⋊D312yescomplemented
336C23C2×D3×C3⋊D311yescharacteristic
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—272no—
343C2×C4—272no—
344C2×C4—544no—
345C2×C4—548no—
346C2×C4—544no—
347C2×C4—542no—
348C2×C4—541no—
349C2×C4—541no—
350C2×C3—34no—
351C2×C3—34no—
352C2×C3—38no—
353C2×C3—42no—
354C2×C3—41no—
355C2×C3—88no—
356C2×C3—84no—
357C2×C3—364no—
358C2×C3D3×SL(2,ℤ/4)18yescomplemented
359C2×C3C2×D3×S414yes—
360D3—316no—
361D3—916no—
362C22—32no—
363C22—31no—
364C22—32no—
365C22—32no—
366C22—31no—
367C22—31no—
368C22—61no—
369C22—94no—
370C22—92no—
371C22—274no—
372C22—274no—
373C22—278no—
374C22—274no—
375C22—274no—
376C22—272no—
377C22—542no—
378C22—544no—
379C22C2×Dic3×C3⋊D311yescharacteristic, complemented
380C22C3⋊D3×S411yescharacteristic
381C4—62no—
382C4—62no—
383C4—542no—
384C4—542no—
385C3—41no—
386C3—84no—
387C3C2×D3×SL(2,ℤ/4)14yescomplemented
388C2—31no—
389C2—31no—
390C2—32no—
391C2—94no—
392C2—274no—
393C2C3⋊D3×SL(2,ℤ/4)12yesdirect product factor
394C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×C3⋊D3×S4
Fitting, socleC24×C32C2×D3
prime radicalC22×A4×C3⋊D3C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
3. C2×C32×SL(2,ℤ/4)C2semidirect
4. C3⋊D3×SL(2,ℤ/4)C2direct (N × H)
8. C2×A4×C3⋊D3C4semidirect
11. C3⋊(C3⋊SL(2,ℤ/4))C22semidirect
12. C32×SL(2,ℤ/4)C22semidirect
19. C2×C3×SL(2,ℤ/4)D3semidirect
28. C2×C32×A4C2×C4semidirect
30. A4×C3⋊D3C2×C4semidirect
41. C23×C3⋊D3Dic3semidirect
60. C3×SL(2,ℤ/4)C2×D3semidirect
70. C32×A4C22×C4semidirect
80. C2×SL(2,ℤ/4)C3⋊D3direct (N × H)
90. C23×C32C2×Dic3semidirect
104. C22×C3⋊D3C2×Dic3semidirect
109. C2×C3×A4C4×D3semidirect
157. SL(2,ℤ/4)C2×C3⋊D3direct (N × H)
173. C22×C32C22×Dic3semidirect
183. C2×C3⋊D3SL(2,ℤ/4)direct (N × H)
188. C3×A4C2×C4×D3semidirect
210. C23×C3D3×Dic3semidirect
229. C2×A4C4×C3⋊D3semidirect
255. C2×C32C2×SL(2,ℤ/4)semidirect
258. C3⋊D3C2×SL(2,ℤ/4)direct (N × H)
294. C22×C3C2×D3×Dic3semidirect
307. A4C2×C4×C3⋊D3semidirect
317. C32C22×SL(2,ℤ/4)semidirect
335. C23Dic3×C3⋊D3semidirect
358. C2×C3D3×SL(2,ℤ/4)semidirect
379. C22C2×Dic3×C3⋊D3semidirect
387. C3C2×D3×SL(2,ℤ/4)semidirect
393. C2C3⋊D3×SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×A4×C3⋊D32
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C2×C32×SL(2,ℤ/4)2
C3⋊D3×SL(2,ℤ/4)2
C2×D3×SL(2,ℤ/4)3
C2×C22⋊C4×C3⋊D33
C2×Dic3×C3⋊D34