Finite Group Databasebeta

← C22×D3×SL(2,ℤ/4) #11923622

Subgroups — C22×D3×SL(2,ℤ/4) #11923622

Order 1152.

3364 conjugacy classes of subgroups; only the 311 normal subgroups are shown.

normal subgroups only

This group has 3364 conjugacy classes of subgroups — too many to draw. Only its 311 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
1C23×D3×A4C211yesmaximal, characteristic
2C22×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C22×C3×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C2×D3×SL(2,ℤ/4)C2112yesmaximal, direct product factor
5C23×SL(2,ℤ/4)—31nomaximal
6C22×D3×C22⋊C4—31nomaximal
7C23×C3×A4C2211yescharacteristic
8C22×D3×A4C418yescomplemented
9C22×D3×A4C2216yes—
10C22×D3×Dic3—41nomaximal
11C2×C3⋊SL(2,ℤ/4)C2216yescomplemented
12C2×C3×SL(2,ℤ/4)C2216yescomplemented
13D3×SL(2,ℤ/4)C22116yesdirect product factor
14C25×D3D311yescharacteristic
15C24×A4—31no—
16C23×C4×D3—31no—
17C23×C4×D3—31no—
18C22×SL(2,ℤ/4)—31no—
19C22×SL(2,ℤ/4)—312no—
20C22×SL(2,ℤ/4)—61no—
21C22×SL(2,ℤ/4)D311yescharacteristic, direct product factor
22C22×C3×C22⋊C4—31no—
23C22×#217—31no—
24C22×(C2×D3)⋊C4—31no—
25C22×(C2×D3)⋊C4—31no—
26C2×D3×C22⋊C4—312no—
27C2×D3×C22⋊C4—312no—
28C23×C3×D3—41no—
29C22×C3×A4C2×C414yescomplemented
30C22×C3×A4C2313yes—
31C2×D3×A4C2×C4124yescomplemented
32C2×D3×A4C2314yes—
33C22×C3⋊Dic3—41no—
34C22×C3×Dic3—41no—
35C2×D3×Dic3—412no—
36C3⋊SL(2,ℤ/4)C2314yescomplemented
37C3×SL(2,ℤ/4)C2314yescomplemented
38C23×C22⋊C4—91noSylow
39C25×C3C2×D311yescharacteristic
40C24×D3—31no—
41C24×D3—38no—
42C24×D3—36no—
43C24×D3—31no—
44C24×D3Dic318yescomplemented
45C24×D3C2×D316yes—
46C23×A4—21no—
47C23×A4—38no—
48C23×A4—36no—
49C23×A4C2×D311yescharacteristic
50C23×C3×C4—31no—
51C23×C3×C4—31no—
52C23×Dic3—31no—
53C23×Dic3—31no—
54C23×Dic3—121no—
55C22×C4×D3—312no—
56C22×C4×D3—312no—
57C22×C4×D3—68no—
58C22×C4×D3—61no—
59C22×C4×D3—66no—
60C22×C4×D3—61no—
61C2×SL(2,ℤ/4)—36no—
62C2×SL(2,ℤ/4)—316no—
63C2×SL(2,ℤ/4)—66no—
64C2×SL(2,ℤ/4)C2×D316yesdirect product factor
65C2×C3×C22⋊C4—36no—
66C2×C3×C22⋊C4—36no—
67C2×#217—36no—
68C2×#217—36no—
69C2×(C2×D3)⋊C4—312no—
70C2×(C2×D3)⋊C4—312no—
71D3×C22⋊C4—348no—
72D3×C22⋊C4—316no—
73C23×C32—41no—
74C22×C3×D3—48no—
75C22×C3×D3—46no—
76C2×C3×A4C22×C416yescomplemented
77C2×C3×A4C2411yescharacteristic
78D3×A4C22×C418yescomplemented
79C2×C3⋊Dic3—46no—
80C2×C3×Dic3—46no—
81D3×Dic3—416no—
82C26—31no—
83C24×C4—91no—
84C24×C4—91no—
85C22×C22⋊C4—31no—
86C22×C22⋊C4—912no—
87C22×C22⋊C4—912no—
88C22×C22⋊C4—91no—
89C22×C22⋊C4—91no—
90C22×C22⋊C4—91no—
91C24×C3—31no—
92C24×C3—33no—
93C24×C3—34no—
94C24×C3—121no—
95C24×C3C2×Dic314yescomplemented
96C24×C3C22×D313yes—
97C23×D3—31no—
98C23×D3—38no—
99C23×D3—38no—
100C23×D3—36no—
101C23×D3—36no—
102C23×D3—312no—
103C23×D3—36no—
104C23×D3—38no—
105C23×D3—348no—
106C23×D3—324no—
107C23×D3—624no—
108C23×D3—64no—
109C23×D3—64no—
110C23×D3—63no—
111C23×D3C2×Dic3124yescomplemented
112C23×D3S411yescharacteristic
113C23×D3C22×D314yes—
114C22×A4—24no—
115C22×A4—23no—
116C22×A4—324no—
117C22×A4—34no—
118C22×A4C4×D314yescomplemented
119C22×A4C22×D313yes—
120C22×C3×C4—36no—
121C22×C3×C4—36no—
122C22×C3×C4—64no—
123C22×C3×C4—63no—
124C22×C3×C4—61no—
125C22×Dic3—36no—
126C22×Dic3—36no—
127C22×Dic3—41no—
128C22×Dic3—64no—
129C22×Dic3—63no—
130C22×Dic3—61no—
131C22×Dic3—1212no—
132C22×Dic3—121no—
133C22×Dic3—241no—
134C2×C4×D3—316no—
135C2×C4×D3—316no—
136C2×C4×D3—648no—
137C2×C4×D3—612no—
138C2×C4×D3—624no—
139C2×C4×D3—612no—
140SL(2,ℤ/4)—34no—
141SL(2,ℤ/4)—64no—
142SL(2,ℤ/4)C22×D314yesdirect product factor
143C3×C22⋊C4—312no—
144C3×C22⋊C4—34no—
145#217—312no—
146#217—34no—
147(C2×D3)⋊C4—316no—
148(C2×D3)⋊C4—316no—
149C22×C32—44no—
150C22×C32—43no—
151C2×C3×D3—424no—
152C2×C3×D3—44no—
153C3×A4C23×C411yescharacteristic, complemented
154C3⋊Dic3—44no—
155C3×Dic3—44no—
156C25—38no—
157C25—36no—
158C25—91no—
159C25—91no—
160C25—96no—
161C25—98no—
162C25D3211yescharacteristic
163C23×C4—31no—
164C23×C4—31no—
165C23×C4—912no—
166C23×C4—912no—
167C23×C4—91no—
168C23×C4—91no—
169C23×C4—181no—
170C23×C4—186no—
171C23×C4—188no—
172C23×C4—181no—
173C2×C22⋊C4—36no—
174C2×C22⋊C4—36no—
175C2×C22⋊C4—916no—
176C2×C22⋊C4—948no—
177C2×C22⋊C4—912no—
178C2×C22⋊C4—912no—
179C2×C22⋊C4—96no—
180C2×C22⋊C4—96no—
181C23×C3—34no—
182C23×C3—34no—
183C23×C3—33no—
184C23×C3—33no—
185C23×C3—33no—
186C23×C3—312no—
187C23×C3—36no—
188C23×C3—41no—
189C23×C3—66no—
190C23×C3—61no—
191C23×C3—81no—
192C23×C3—126no—
193C23×C3—128no—
194C23×C3C22×Dic316yescomplemented
195C23×C3C2×S411yescharacteristic
196C23×C3C23×D311yescharacteristic
197C22×D3—38no—
198C22×D3—36no—
199C22×D3—324no—
200C22×D3—324no—
201C22×D3—348no—
202C22×D3—34no—
203C22×D3—312no—
204C22×D3—34no—
205C22×D3—324no—
206C22×D3—348no—
207C22×D3—38no—
208C22×D3—624no—
209C22×D3—632no—
210C22×D3—612no—
211C22×D3—624no—
212C22×D3—66no—
213C22×D3SL(2,ℤ/4)18yesdirect product factor
214C22×D3C22×Dic318yescomplemented
215C22×D3C2×S416yes—
216C2×A4—21no—
217C2×A4—26no—
218C2×A4—38no—
219C2×A4C2×C4×D316yescomplemented
220C2×A4C23×D311yescharacteristic
221C2×C3×C4—34no—
222C2×C3×C4—34no—
223C2×C3×C4—612no—
224C2×C3×C4—66no—
225C2×C3×C4—66no—
226C2×Dic3—34no—
227C2×Dic3—34no—
228C2×Dic3—46no—
229C2×Dic3—612no—
230C2×Dic3—66no—
231C2×Dic3—66no—
232C2×Dic3—1216no—
233C2×Dic3—126no—
234C2×Dic3—246no—
235C4×D3—632no—
236C4×D3—616no—
237C4×D3—616no—
238C2×C32—46no—
239C2×C32—41no—
240C3×D3—48no—
241C24—31no—
242C24—31no—
243C24—33no—
244C24—34no—
245C24—34no—
246C24—324no—
247C24—96no—
248C24—96no—
249C24—98no—
250C24—98no—
251C24—96no—
252C24—912no—
253C24—98no—
254C24—948no—
255C24—924no—
256C24—91no—
257C24—183no—
258C24—184no—
259C24—184no—
260C24—1824no—
261C24D3×Dic314yescomplemented
262C24C2×D3213yes—
263C22×C4—36no—
264C22×C4—36no—
265C22×C4—64no—
266C22×C4—63no—
267C22×C4—61no—
268C22×C4—916no—
269C22×C4—916no—
270C22×C4—96no—
271C22×C4—96no—
272C22×C4—1812no—
273C22×C4—1824no—
274C22×C4—1848no—
275C22×C4—1812no—
276C22×C4—184no—
277C22×C4—183no—
278C22×C4—181no—
279C22⋊C4—312no—
280C22⋊C4—34no—
281C22⋊C4—916no—
282C22⋊C4—916no—
283C22⋊C4—912no—
284C22⋊C4—94no—
285C22×C3—36no—
286C22×C3—31no—
287C22×C3—36no—
288C22×C3—36no—
289C22×C3—312no—
290C22×C3—31no—
291C22×C3—33no—
292C22×C3—43no—
293C22×C3—44no—
294C22×C3—63no—
295C22×C3—64no—
296C22×C3—83no—
297C22×C3—84no—
298C22×C3—124no—
299C22×C3—1224no—
300C22×C3C2×SL(2,ℤ/4)14yescomplemented
301C22×C3C23×Dic311yescharacteristic, complemented
302C22×C3C22×S413yes—
303C2×D3—324no—
304C2×D3—34no—
305C2×D3—38no—
306C2×D3—38no—
307C2×D3—348no—
308C2×D3—38no—
309C2×D3—64no—
310C2×D3—624no—
311C2×D3C2×SL(2,ℤ/4)124yesdirect product factor
312C2×D3C22×S414yes—
313A4—21no—
314A4C22×C4×D311yescharacteristic, complemented
315C3×C4—64no—
316C3×C4—64no—
317Dic3—44no—
318Dic3—64no—
319Dic3—64no—
320Dic3—124no—
321Dic3—244no—
322C32—41noSylow
323C23—33no—
324C23—33no—
325C23—36no—
326C23—34no—
327C23—34no—
328C23—38no—
329C23—33no—
330C23—312no—
331C23—36no—
332C23—38no—
333C23—66no—
334C23—61no—
335C23—94no—
336C23—924no—
337C23—912no—
338C23—924no—
339C23—948no—
340C23—94no—
341C23—98no—
342C23—948no—
343C23—924no—
344C23—96no—
345C23—98no—
346C23—1812no—
347C23—1824no—
348C23—186no—
349C23—1824no—
350C23—1832no—
351C23C2×D3×Dic316yescomplemented
352C23D3×S411yescharacteristic
353C23C22×D3211yescharacteristic
354C2×C4—34no—
355C2×C4—34no—
356C2×C4—612no—
357C2×C4—66no—
358C2×C4—66no—
359C2×C4—94no—
360C2×C4—94no—
361C2×C4—1816no—
362C2×C4—1832no—
363C2×C4—1816no—
364C2×C4—1812no—
365C2×C4—186no—
366C2×C4—186no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—36no—
370C2×C3—46no—
371C2×C3—41no—
372C2×C3—86no—
373C2×C3—81no—
374C2×C3—128no—
375C2×C3C22×SL(2,ℤ/4)16yescomplemented
376C2×C3C23×S411yescharacteristic
377D3—38no—
378D3C22×SL(2,ℤ/4)18yesdirect product factor
379C22—31no—
380C22—36no—
381C22—36no—
382C22—36no—
383C22—312no—
384C22—324no—
385C22—31no—
386C22—33no—
387C22—34no—
388C22—63no—
389C22—64no—
390C22—98no—
391C22—98no—
392C22—948no—
393C22—98no—
394C22—924no—
395C22—94no—
396C22—184no—
397C22—1824no—
398C22D3×SL(2,ℤ/4)14yesdirect product factor
399C22C22×D3×Dic311yescharacteristic, complemented
400C22C2×D3×S413yes—
401C4—64no—
402C4—64no—
403C4—184no—
404C4—184no—
405C3—41no—
406C3—81no—
407C3C23×SL(2,ℤ/4)11yescharacteristic, complemented
408C2—31no—
409C2—31no—
410C2—36no—
411C2—38no—
412C2—98no—
413C2C2×D3×SL(2,ℤ/4)16yesdirect product factor
414C2C22×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, hypocenterC3×A4C23×C4
FrattiniC2C22×D3×S4
Fitting, socleC25×C3C2×D3
prime radicalC23×D3×A4C2
cosocleC2×C3×A4C24
LCSG ▷ C3×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊SL(2,ℤ/4)C2semidirect
3. C22×C3×SL(2,ℤ/4)C2semidirect
4. C2×D3×SL(2,ℤ/4)C2direct (N × H)
8. C22×D3×A4C4semidirect
11. C2×C3⋊SL(2,ℤ/4)C22semidirect
12. C2×C3×SL(2,ℤ/4)C22semidirect
13. D3×SL(2,ℤ/4)C22direct (N × H)
21. C22×SL(2,ℤ/4)D3direct (N × H)
29. C22×C3×A4C2×C4semidirect
31. C2×D3×A4C2×C4semidirect
36. C3⋊SL(2,ℤ/4)C23semidirect
37. C3×SL(2,ℤ/4)C23semidirect
44. C24×D3Dic3semidirect
64. C2×SL(2,ℤ/4)C2×D3direct (N × H)
76. C2×C3×A4C22×C4semidirect
78. D3×A4C22×C4semidirect
95. C24×C3C2×Dic3semidirect
111. C23×D3C2×Dic3semidirect
118. C22×A4C4×D3semidirect
142. SL(2,ℤ/4)C22×D3direct (N × H)
153. C3×A4C23×C4semidirect
194. C23×C3C22×Dic3semidirect
213. C22×D3SL(2,ℤ/4)direct (N × H)
214. C22×D3C22×Dic3semidirect
219. C2×A4C2×C4×D3semidirect
261. C24D3×Dic3semidirect
300. C22×C3C2×SL(2,ℤ/4)semidirect
301. C22×C3C23×Dic3semidirect
311. C2×D3C2×SL(2,ℤ/4)direct (N × H)
314. A4C22×C4×D3semidirect
351. C23C2×D3×Dic3semidirect
375. C2×C3C22×SL(2,ℤ/4)semidirect
378. D3C22×SL(2,ℤ/4)direct (N × H)
398. C22D3×SL(2,ℤ/4)direct (N × H)
399. C22C22×D3×Dic3semidirect
407. C3C23×SL(2,ℤ/4)semidirect
413. C2C2×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
C23×D3×A42
C22×C3⋊SL(2,ℤ/4)2
C22×C3×SL(2,ℤ/4)2
C2×D3×SL(2,ℤ/4)2
C23×SL(2,ℤ/4)3
C22×D3×C22⋊C43
C22×D3×Dic34