Finite Group Databasebeta

← C22×#64207 #11923621

Subgroups — C22×#64207 #11923621

Order 1152.

2716 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 2716 conjugacy classes of subgroups — too many to draw. Only its 191 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×C3⋊S4C211yesmaximal, characteristic, complemented
2C23×C3×S4C211yesmaximal, characteristic, complemented
3C22×Dic3×A4C211yesmaximal, characteristic, complemented
4C2×#64207C2112yesmaximal, direct product factor
5C22×C4⋊S4—31nomaximal
6C22×#939—31nomaximal
7C23×C3×A4C2211yescharacteristic
8C22×C3⋊S4—24no—
9C22×C3⋊S4C2216yescomplemented
10C22×C3×S4—24no—
11C22×C3×S4C2216yescomplemented
12C22×#626—41nomaximal
13C2×Dic3×A4C2216yescomplemented
14#64207C22116yesdirect product factor
15C23×S4—31no—
16C23×S4—61no—
17C23×S4D311yescharacteristic, complemented
18C23×C3×D4—31no—
19C23×#67—31no—
20C23×#67—31no—
21C24×Dic3D311yescharacteristic, complemented
22C22×C4×A4—31no—
23C2×C4⋊S4—312no—
24C22×#217—31no—
25C2×#939—312no—
26C2×#939—312no—
27C22×(C2×D3)⋊C4—31no—
28C22×Dic3⋊C4—31no—
29C23×C3⋊D3—41no—
30C23×C3×D3—41no—
31C22×C3×A4D414yescomplemented
32C22×C3×A4C2313yes—
33C2×C3⋊S4—212no—
34C2×C3⋊S4C2314yescomplemented
35C2×C3×S4—212no—
36C2×C3×S4C2314yescomplemented
37C22×C3×Dic3—41no—
38C2×#626—412no—
39Dic3×A4C2314yescomplemented
40C22×#121—91noSylow
41C25×C3—31no—
42C25×C3C2×D311yescharacteristic
43C24×D3—31no—
44C23×A4—21no—
45C23×A4C2×D311yescharacteristic
46C22×S4—24no—
47C22×S4—36no—
48C22×S4—66no—
49C22×S4—68no—
50C22×S4—64no—
51C22×S4C2×D316yescomplemented
52C22×C3×D4—36no—
53C22×C3×D4—36no—
54C22×C3×D4—64no—
55C22×C3×D4—64no—
56C23×C3×C4—31no—
57C22×#67—312no—
58C22×#67—36no—
59C22×#67—36no—
60C22×#67—61no—
61C22×#67—66no—
62C22×#67—64no—
63C22×#67—64no—
64C22×#67—61no—
65C23×Dic3—31no—
66C23×Dic3—31no—
67C23×Dic3—36no—
68C23×Dic3—31no—
69C23×Dic3C2×D316yescomplemented
70C22×D12—121no—
71C2×C4×A4—36no—
72C4⋊S4—316no—
73C2×#217—312no—
74#939—348no—
75#939—316no—
76C2×(C2×D3)⋊C4—312no—
77C2×Dic3⋊C4—312no—
78C23×C32—41no—
79C22×C3⋊D3—46no—
80C22×C3⋊D3—84no—
81C22×C3×D3—46no—
82C22×C3×D3—84no—
83C2×C3×A4C2×D416yescomplemented
84C2×C3×A4C2411yescharacteristic
85C3⋊S4—24no—
86C3×S4—24no—
87C2×C3×Dic3—46no—
88#626—416no—
89C23×D4—31no—
90C23×D4—91no—
91C23×D4—91no—
92C24×C4—31no—
93C2×#121—912no—
94C2×#121—912no—
95C22×C4⋊C4—91no—
96C22×C22⋊C4—91no—
97C22×C22⋊C4—91no—
98C24×C3—31no—
99C24×C3—33no—
100C24×C3—34no—
101C24×C3—36no—
102C24×C3—64no—
103C24×C3—63no—
104C24×C3—64no—
105C24×C3—61no—
106C24×C3D1214yescomplemented
107C24×C3C22×D313yes—
108C23×D3—36no—
109C23×D3—41no—
110C23×D3—64no—
111C23×D3—63no—
112C23×D3—64no—
113C23×D3—61no—
114C23×D3—121no—
115C23×D3—241no—
116C22×A4—24no—
117C22×A4—23no—
118C22×A4#6714yescomplemented
119C22×A4C22×D313yes—
120C2×S4—212no—
121C2×S4—34no—
122C2×S4—64no—
123C2×S4—624no—
124C2×S4—612no—
125C2×S4C22×D314yescomplemented
126C2×C3×D4—312no—
127C2×C3×D4—34no—
128C2×C3×D4—624no—
129C2×C3×D4—612no—
130C2×C3×D4—612no—
131C22×C3×C4—36no—
132C22×C3×C4—64no—
133C22×C3×C4—121no—
134C2×#67—316no—
135C2×#67—312no—
136C2×#67—34no—
137C2×#67—612no—
138C2×#67—624no—
139C2×#67—624no—
140C2×#67—612no—
141C2×#67—612no—
142C2×#67—612no—
143C22×Dic3—31no—
144C22×Dic3—36no—
145C22×Dic3—36no—
146C22×Dic3—36no—
147C22×Dic3—312no—
148C22×Dic3—36no—
149C22×Dic3—64no—
150C22×Dic3—63no—
151C22×Dic3—64no—
152C22×Dic3S411yescharacteristic, complemented
153C22×Dic3C22×D314yescomplemented
154C2×D12—1212no—
155C4×A4—34no—
156#217—316no—
157(C2×D3)⋊C4—316no—
158Dic3⋊C4—316no—
159C22×C32—44no—
160C22×C32—43no—
161C2×C3⋊D3—44no—
162C2×C3⋊D3—812no—
163C2×C3×D3—44no—
164C2×C3×D3—812no—
165C3×A4C22×D411yescharacteristic, complemented
166C3×Dic3—44no—
167C25—31no—
168C25—91no—
169C25D3211yescharacteristic
170C22×D4—36no—
171C22×D4—36no—
172C22×D4—64no—
173C22×D4—64no—
174C22×D4—912no—
175C22×D4—96no—
176C22×D4—96no—
177C22×D4—181no—
178C22×D4—186no—
179C22×D4—181no—
180C22×D4—184no—
181C22×D4—184no—
182C23×C4—31no—
183C23×C4—36no—
184C23×C4—91no—
185C23×C4—91no—
186C23×C4—96no—
187C23×C4—91no—
188#121—916no—
189#121—948no—
190C2×C4⋊C4—912no—
191C2×C22⋊C4—912no—
192C2×C22⋊C4—912no—
193C23×C3—34no—
194C23×C3—34no—
195C23×C3—33no—
196C23×C3—33no—
197C23×C3—33no—
198C23×C3—312no—
199C23×C3—36no—
200C23×C3—34no—
201C23×C3—41no—
202C23×C3—61no—
203C23×C3—66no—
204C23×C3—66no—
205C23×C3—612no—
206C23×C3—612no—
207C23×C3—66no—
208C23×C3—81no—
209C23×C3—1212no—
210C23×C3—124no—
211C23×C3C2×D1216yescomplemented
212C23×C3C2×S411yescharacteristic
213C23×C3C23×D311yescharacteristic
214C22×D3—34no—
215C22×D3—46no—
216C22×D3—612no—
217C22×D3—66no—
218C22×D3—612no—
219C22×D3—66no—
220C22×D3—84no—
221C22×D3—126no—
222C22×D3—1212no—
223C22×D3—124no—
224C22×D3—246no—
225C22×D3—244no—
226C22×D3—248no—
227C2×A4—21no—
228C2×A4—26no—
229C2×A4C2×#6716yescomplemented
230C2×A4C23×D311yescharacteristic
231S4—24no—
232S4—68no—
233S4—64no—
234C3×D4—624no—
235C3×D4—64no—
236C3×D4—64no—
237C2×C3×C4—34no—
238C2×C3×C4—612no—
239C2×C3×C4—126no—
240#67—616no—
241#67—624no—
242#67—64no—
243#67—64no—
244#67—616no—
245C2×Dic3—36no—
246C2×Dic3—34no—
247C2×Dic3—312no—
248C2×Dic3—34no—
249C2×Dic3—34no—
250C2×Dic3—66no—
251C2×Dic3—612no—
252C2×Dic3C2×S416yescomplemented
253D12—1216no—
254C2×C32—46no—
255C2×C32—41no—
256C3⋊D3—84no—
257C3×D3—84no—
258C24—31no—
259C24—33no—
260C24—34no—
261C24—36no—
262C24—64no—
263C24—63no—
264C24—64no—
265C24—61no—
266C24—96no—
267C24—183no—
268C24—184no—
269C24—184no—
270C24—181no—
271C24#62614yescomplemented
272C24C2×D3213yes—
273C2×D4—312no—
274C2×D4—34no—
275C2×D4—624no—
276C2×D4—612no—
277C2×D4—612no—
278C2×D4—916no—
279C2×D4—912no—
280C2×D4—94no—
281C2×D4—1812no—
282C2×D4—1824no—
283C2×D4—1812no—
284C2×D4—1824no—
285C2×D4—1812no—
286C2×D4—1812no—
287C22×C4—31no—
288C22×C4—36no—
289C22×C4—34no—
290C22×C4—64no—
291C22×C4—96no—
292C22×C4—96no—
293C22×C4—91no—
294C22×C4—96no—
295C22×C4—912no—
296C22×C4—96no—
297C22×C4—184no—
298C22×C4—183no—
299C22×C4—184no—
300C4⋊C4—916no—
301C22⋊C4—916no—
302C22⋊C4—916no—
303C22×C3—36no—
304C22×C3—31no—
305C22×C3—36no—
306C22×C3—36no—
307C22×C3—312no—
308C22×C3—31no—
309C22×C3—33no—
310C22×C3—43no—
311C22×C3—44no—
312C22×C3—63no—
313C22×C3—64no—
314C22×C3—64no—
315C22×C3—64no—
316C22×C3—64no—
317C22×C3—83no—
318C22×C3—84no—
319C22×C3—1212no—
320C22×C3—1212no—
321C22×C3C4⋊S414yescomplemented
322C22×C3C22×D1211yescharacteristic, complemented
323C22×C3C22×S413yes—
324C2×D3—44no—
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—64no—
328C2×D3—812no—
329C2×D3—124no—
330C2×D3—1212no—
331C2×D3—1212no—
332C2×D3—244no—
333C2×D3—2412no—
334C2×D3—2424no—
335A4—21no—
336A4C22×#6711yescharacteristic, complemented
337C3×C4—64no—
338C3×C4—124no—
339Dic3—34no—
340Dic3—64no—
341Dic3C22×S414yescomplemented
342C32—41noSylow
343C23—33no—
344C23—33no—
345C23—34no—
346C23—34no—
347C23—33no—
348C23—312no—
349C23—36no—
350C23—34no—
351C23—61no—
352C23—66no—
353C23—612no—
354C23—66no—
355C23—612no—
356C23—66no—
357C23—94no—
358C23—1212no—
359C23—124no—
360C23—1812no—
361C23—186no—
362C23—1812no—
363C23—186no—
364C23—3612no—
365C23—364no—
366C23C2×#62616yescomplemented
367C23D3×S411yescharacteristic
368C23C22×D3211yescharacteristic
369D4—64no—
370D4—624no—
371D4—64no—
372D4—1816no—
373D4—1816no—
374D4—184no—
375D4—1824no—
376D4—184no—
377C2×C4—36no—
378C2×C4—34no—
379C2×C4—612no—
380C2×C4—94no—
381C2×C4—912no—
382C2×C4—96no—
383C2×C4—94no—
384C2×C4—94no—
385C2×C4—186no—
386C2×C4—1812no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—36no—
390C2×C3—46no—
391C2×C3—41no—
392C2×C3—86no—
393C2×C3—81no—
394C2×C3—124no—
395C2×C3C2×C4⋊S416yescomplemented
396C2×C3C23×S411yescharacteristic
397D3—84no—
398D3—124no—
399D3—244no—
400D3—248no—
401C22—31no—
402C22—36no—
403C22—36no—
404C22—36no—
405C22—312no—
406C22—31no—
407C22—33no—
408C22—63no—
409C22—64no—
410C22—64no—
411C22—64no—
412C22—64no—
413C22—1212no—
414C22—1212no—
415C22—184no—
416C22—184no—
417C22—184no—
418C22—3612no—
419C22—3612no—
420C22#6420714yesdirect product factor
421C22C22×#62611yescharacteristic, complemented
422C22C2×D3×S413yes—
423C4—34no—
424C4—64no—
425C4—94no—
426C4—184no—
427C3—41no—
428C3—81no—
429C3C22×C4⋊S411yescharacteristic, complemented
430C2—31no—
431C2—31no—
432C2—36no—
433C2—124no—
434C2—364no—
435C2C2×#6420716yesdirect product factor
436C2C22×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, cosocleC2×C3×A4C24
FrattiniC2C22×D3×S4
Fitting, socleC25×C3C2×D3
hypocenterC3×A4C22×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊S4C2semidirect
2. C23×C3×S4C2semidirect
3. C22×Dic3×A4C2semidirect
4. C2×#64207C2direct (N × H)
9. C22×C3⋊S4C22semidirect
11. C22×C3×S4C22semidirect
13. C2×Dic3×A4C22semidirect
14. #64207C22direct (N × H)
17. C23×S4D3semidirect
21. C24×Dic3D3semidirect
31. C22×C3×A4D4semidirect
34. C2×C3⋊S4C23semidirect
36. C2×C3×S4C23semidirect
39. Dic3×A4C23semidirect
51. C22×S4C2×D3semidirect
69. C23×Dic3C2×D3semidirect
83. C2×C3×A4C2×D4semidirect
106. C24×C3D12semidirect
118. C22×A4#67semidirect
125. C2×S4C22×D3semidirect
152. C22×Dic3S4semidirect
153. C22×Dic3C22×D3semidirect
165. C3×A4C22×D4semidirect
211. C23×C3C2×D12semidirect
229. C2×A4C2×#67semidirect
252. C2×Dic3C2×S4semidirect
271. C24#626semidirect
321. C22×C3C4⋊S4semidirect
322. C22×C3C22×D12semidirect
336. A4C22×#67semidirect
341. Dic3C22×S4semidirect
366. C23C2×#626semidirect
395. C2×C3C2×C4⋊S4semidirect
420. C22#64207direct (N × H)
421. C22C22×#626semidirect
429. C3C22×C4⋊S4semidirect
435. C2C2×#64207direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊S42
C23×C3×S42
C22×Dic3×A42
C2×#642072
C22×C4⋊S43
C22×#9393
C22×#6264