Finite Group Databasebeta

← C3×#85307 #11922592

Subgroups — C3×#85307 #11922592

Order 1152.

716 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×C3×C4×S4C211yesmaximal, characteristic
2C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×(C2×S4)⋊C4C212yesmaximal
4C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C3×C4×SL(2,ℤ/4)C212yesmaximal, complemented
6#85307C311yesmaximal, characteristic, direct product factor
7C3×#2352—31nomaximal
8C22×C3×S4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)C2212yes—
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C3×C4×S4C414yescomplemented
14C3×#892—41nomaximal
15C2×C4×S4C2×C311yescharacteristic
16C2×C3×C4×D4—31no—
17C2×C3×#117—31no—
18C22×C3×C4⋊C4D311yescharacteristic, complemented
19A4×C4⋊C4—21no—
20A4×C4⋊C4C2×C311yescharacteristic, complemented
21(C2×S4)⋊C4C2×C312yes—
22C4⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
23C4×SL(2,ℤ/4)C2×C312yescomplemented
24C3×#388—32no—
25C3×#387—32no—
26C3×#386—32no—
27C3×#382—31no—
28C3×#380—31no—
29C3×#380—31no—
30C3×C4×C4⋊C4—31no—
31C3×C4×C22⋊C4—32no—
32C22×C3×A4C2311yescharacteristic
33C2×C3×S4—21no—
34C2×C3×S4—21no—
35C2×C3×S4C2×C412yes—
36C2×C3×C4×D3—41no—
37C3×C4×A4—22no—
38C3×C4×A4C2×C412yes—
39C3×SL(2,ℤ/4)—22no—
40C3×SL(2,ℤ/4)C2×C412yes—
41C32×C4⋊C4—41no—
42C3×(C2×D3)⋊C4—42no—
43C3×C4⋊Dic3—41no—
44C3×C4×Dic3—42no—
45#2352—31noSylow
46C22×S4C22×C311yescharacteristic
47C22×C3×D4—31no—
48C23×C3×C4—31no—
49C23×C3×C4C2×D312yes—
50C23×C3×C4C2×D311yescharacteristic
51C2×C4×A4—21no—
52C2×C4×A4—22no—
53C2×C4×A4C22×C312yes—
54C2×C4×A4C22×C311yescharacteristic
55C2×SL(2,ℤ/4)C22×C312yes—
56C2×SL(2,ℤ/4)C22×C311yescharacteristic
57C4×S4C3×C414yescomplemented
58C3×C4×D4—34no—
59C3×C4×D4—34no—
60C3×#117—61no—
61C3×#117—62no—
62C3×#117—61no—
63C2×C3×C4⋊C4—31no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—32no—
67C2×C3×C4⋊C4—31no—
68C2×C3×C4⋊C4—31no—
69C2×C3×C22⋊C4—31no—
70C2×C3×C22⋊C4—31no—
71C2×C3×C22⋊C4—32no—
72C2×C3×C22⋊C4—32no—
73C2×C3×C22⋊C4—32no—
74C2×C3×C42—31no—
75C2×C3×C42—32no—
76C2×C3×C42—32no—
77#892—41no—
78C3×#95—32no—
79C3×#95—32no—
80C3×#95—32no—
81C3×#95—32no—
82C3×#95—32no—
83C22×C3×D3—41no—
84C2×C3×A4C22×C411yescharacteristic
85C2×C3×A4#4111yescharacteristic
86C2×C3×A4#4111yescharacteristic
87C3×S4—22no—
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—42no—
92C3×C4×D3—44no—
93C2×C4×D4—31no—
94C2×#117—31no—
95C22×C4⋊C4C3×D311yescharacteristic, complemented
96#388—32no—
97#387—32no—
98#386—32no—
99#382—31no—
100#380—31no—
101#380—31no—
102C4×C4⋊C4—31no—
103C4×C22⋊C4—32no—
104C24×C3—31no—
105C24×C3C22×D311yescharacteristic
106C22×A4—21no—
107C22×A4C23×C311yescharacteristic
108C2×S4—21no—
109C2×S4—21no—
110C2×S4C2×C3×C412yes—
111C2×C3×D4—32no—
112C2×C3×D4—32no—
113C2×C3×D4—61no—
114C2×C3×D4—61no—
115C2×C3×D4—61no—
116C2×C3×D4—61no—
117C22×C3×C4—22no—
118C22×C3×C4—31no—
119C22×C3×C4—31no—
120C22×C3×C4—32no—
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—34no—
125C22×C3×C4—32no—
126C22×C3×C4—32no—
127C22×C3×C4—32no—
128C22×C3×C4—32no—
129C22×C3×C4—61no—
130C22×C3×C4—62no—
131C22×C3×C4—61no—
132C22×C3×C4—62no—
133C22×C3×C4C4×D312yes—
134C2×C4×D3—41no—
135C4×A4—22no—
136C4×A4—22no—
137C4×A4—42no—
138C4×A4C2×C3×C412yes—
139SL(2,ℤ/4)—22no—
140SL(2,ℤ/4)C2×C3×C412yes—
141C3×C4⋊C4—34no—
142C3×C4⋊C4—32no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4—61no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—81no—
150C3×C4⋊C4S411yescharacteristic, complemented
151C3×C22⋊C4—34no—
152C3×C22⋊C4—32no—
153C3×C22⋊C4—32no—
154C3×C22⋊C4—62no—
155C3×C22⋊C4—62no—
156C3×C22⋊C4—62no—
157C3×C22⋊C4—62no—
158C3×C22⋊C4—122no—
159C3×C42—34no—
160C3×C42—62no—
161C3×C42—62no—
162C3×C42—122no—
163(C2×D3)⋊C4—42no—
164C4⋊Dic3—41no—
165C4×Dic3—42no—
166C22×C32—41no—
167C2×C3×D3—42no—
168C2×C3×D3—81no—
169C2×C3×D3—81no—
170C3×A4#11711yescharacteristic, complemented
171C32×C4—42no—
172C32×C4—82no—
173C3×Dic3—42no—
174C3×Dic3—82no—
175C22×D4—31no—
176C23×C4—31no—
177C23×C4C2×C3×D312yes—
178C23×C4C2×C3×D311yescharacteristic
179C4×D4—34no—
180C4×D4—34no—
181#117—61no—
182#117—62no—
183#117—61no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—32no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—32no—
195C2×C42—31no—
196C2×C42—32no—
197C2×C42—32no—
198#95—32no—
199#95—32no—
200#95—32no—
201#95—32no—
202#95—32no—
203C23×C3—31no—
204C23×C3—31no—
205C23×C3—31no—
206C23×C3—31no—
207C23×C3—32no—
208C23×C3—61no—
209C23×C3—61no—
210C23×C3—61no—
211C23×C3—61no—
212C23×C3—61no—
213C23×C3—61no—
214C23×C3C2×C4×D311yescharacteristic
215C23×C3#23711yescharacteristic
216C23×C3#23911yescharacteristic
217C22×D3—41no—
218C2×A4—21no—
219C2×A4—21no—
220C2×A4—21no—
221C2×A4C22×C3×C411yescharacteristic
222C2×A4C3×#4111yescharacteristic
223C2×A4C3×#4111yescharacteristic
224S4—22no—
225C3×D4—62no—
226C3×D4—62no—
227C3×D4—62no—
228C3×D4—62no—
229C2×C3×C4—31no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—32no—
236C2×C3×C4—32no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—61no—
242C2×C3×C4—64no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—64no—
250C2×C3×C4—62no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—81no—
259C2×C3×C4—82no—
260C2×C3×C4C2×S412yes—
261C2×C3×C4C2×S411yescharacteristic
262C2×Dic3—41no—
263C2×Dic3—42no—
264C4×D3—44no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C3×D3—82no—
269C24—31no—
270C24C22×C3×D311yescharacteristic
271C2×D4—32no—
272C2×D4—32no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C22×C4—22no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—32no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—34no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—61no—
290C22×C4—62no—
291C22×C4—61no—
292C22×C4—62no—
293C22×C4C3×C4×D312yes—
294C4⋊C4—34no—
295C4⋊C4—32no—
296C4⋊C4—31no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—62no—
300C4⋊C4—61no—
301C4⋊C4C3×S411yescharacteristic, complemented
302C22⋊C4—34no—
303C22⋊C4—32no—
304C22⋊C4—32no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—122no—
310C42—34no—
311C42—62no—
312C42—62no—
313C42—122no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—31no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—41no—
324C22×C3—61no—
325C22×C3—62no—
326C22×C3—62no—
327C22×C3—62no—
328C22×C3—81no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3#89211yescharacteristic, complemented
334C22×C3C22×S411yescharacteristic
335C2×D3—42no—
336C2×D3—81no—
337C2×D3—81no—
338A4—21no—
339A4C3×#11711yescharacteristic, complemented
340C3×C4—22no—
341C3×C4—32no—
342C3×C4—42no—
343C3×C4—62no—
344C3×C4—62no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—82no—
348C3×C4—82no—
349C3×C4—122no—
350C3×C4—122no—
351C3×C4—162no—
352C3×C4C4×S412yes—
353Dic3—42no—
354Dic3—82no—
355C32—41noSylow
356C23—31no—
357C23—31no—
358C23—31no—
359C23—31no—
360C23—32no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23C2×C3×C4×D311yescharacteristic
368C23C3×#23711yescharacteristic
369C23C3×#23911yescharacteristic
370D4—62no—
371D4—62no—
372D4—62no—
373D4—62no—
374C2×C4—31no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—61no—
385C2×C4—64no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—64no—
393C2×C4—62no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4C2×C3×S412yes—
402C2×C4C2×C3×S411yescharacteristic
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—81no—
411C2×C3—81no—
412C2×C3—81no—
413C2×C3—122no—
414C2×C3C2×C4×S411yescharacteristic
415C2×C3#591411yescharacteristic
416C2×C3#591911yescharacteristic
417D3—82no—
418C22—31no—
419C22—31no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—61no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—121no—
432C22—121no—
433C22—121no—
434C22—121no—
435C22C3×#89211yescharacteristic, complemented
436C22C22×C3×S411yescharacteristic
437C4—22no—
438C4—32no—
439C4—62no—
440C4—62no—
441C4—62no—
442C4—62no—
443C4—122no—
444C4—122no—
445C4C3×C4×S412yes—
446C3—41no—
447C3—81no—
448C3#8530711yescharacteristic, direct product factor
449C2—31no—
450C2—31no—
451C2—31no—
452C2—31no—
453C2—122no—
454C2C2×C3×C4×S411yescharacteristic
455C2C3×#591411yescharacteristic
456C2C3×#591911yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutatorC2×A4C22×C3×C4
FrattiniC22C22×C3×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
cosocleC22×A4C23×C3
hypercenterC3×C4⋊C4S4
hypocenterA4C3×#117
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×A4×C4⋊C4C2semidirect
4. C3×C4⋊SL(2,ℤ/4)C2semidirect
5. C3×C4×SL(2,ℤ/4)C2semidirect
6. #85307C3direct (N × H)
13. C3×C4×S4C4semidirect
18. C22×C3×C4⋊C4D3semidirect
20. A4×C4⋊C4C2×C3semidirect
22. C4⋊SL(2,ℤ/4)C2×C3semidirect
23. C4×SL(2,ℤ/4)C2×C3semidirect
57. C4×S4C3×C4semidirect
95. C22×C4⋊C4C3×D3semidirect
150. C3×C4⋊C4S4semidirect
170. C3×A4#117semidirect
301. C4⋊C4C3×S4semidirect
333. C22×C3#892semidirect
339. A4C3×#117semidirect
435. C22C3×#892semidirect
448. C3#85307direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
C3×A4×C4⋊C42
C3×(C2×S4)⋊C42
C3×C4⋊SL(2,ℤ/4)2
C3×C4×SL(2,ℤ/4)2
#853073
C3×#23523
C3×#8924