Finite Group Databasebeta

← C3×#73147 #11920925

Subgroups — C3×#73147 #11920925

Order 1152.

456 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 72 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×Q8×A4C211yesmaximal, characteristic
2C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C3×A4⋊C8C211yesmaximal, characteristic
4#73147C311yesmaximal, characteristic, direct product factor
5C3×#1787—31nomaximal
6C3×Q8×A4—21no—
7C3×Q8×A4C412yescomplemented
8C2×C3×C4×A4—21no—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×SL(2,ℤ/4)—21no—
11C3×A4⋊C8—21no—
12C3×#835—41nomaximal
13C23×C3×Q8D311yescharacteristic
14C2×Q8×A4—21no—
15C2×Q8×A4C2×C311yescharacteristic
16C4⋊SL(2,ℤ/4)C2×C311yescharacteristic
17C2×A4⋊C8C2×C311yescharacteristic
18C2×C3×#103—31no—
19C2×C3×#103—31no—
20C2×C3×#103—31no—
21C2×C3×#103—31no—
22C2×C3×C22⋊C8—31no—
23C3×#380—31no—
24C22×C3×A4D411yescharacteristic
25C2×C32×Q8—41no—
26C3×C4×A4—22no—
27C3×C4×A4C2×C411yescharacteristic
28C3×C4×A4D411yescharacteristic
29C3×SL(2,ℤ/4)—41no—
30C3×C4⋊Dic3—41no—
31C2×C3×C3⋊C8—41no—
32#1787—31noSylow
33C22×C3×Q8—21no—
34C22×C3×Q8—31no—
35C22×C3×Q8—31no—
36C22×C3×Q8—32no—
37C22×C3×Q8—61no—
38C22×C3×Q8—61no—
39C22×C3×Q8Dic312yescomplemented
40C23×C3×C4—21no—
41C23×C3×C4C2×D311yescharacteristic
42Q8×A4—22no—
43Q8×A4—21no—
44Q8×A4—22no—
45Q8×A4C3×C412yescomplemented
46C2×C4×A4—22no—
47C2×C4×A4—21no—
48C2×C4×A4—21no—
49C2×C4×A4C22×C311yescharacteristic
50C2×SL(2,ℤ/4)—21no—
51C22×C3×C8—31no—
52C22×C3×C8—31no—
53C2×C3×C4⋊C4—31no—
54C2×C3×C4⋊C4—31no—
55C2×C3×C22⋊C4—61no—
56A4⋊C8—21no—
57C3×#103—62no—
58C3×#103—62no—
59C3×#103—62no—
60C3×#103—61no—
61C3×#103—61no—
62C3×C22⋊C8—61no—
63C3×C22⋊C8—61no—
64C3×#95—61no—
65#835—41no—
66C2×C3×A4C22⋊C411yescharacteristic
67C2×C3×A4SD1611yescharacteristic
68C2×C3×A4Q1611yescharacteristic
69C32×Q8—42no—
70C32×Q8—81no—
71C2×C32×C4—41no—
72C2×C32×C4—81no—
73C2×C3×Dic3—81no—
74C3×C3⋊C8—81no—
75C23×Q8C3×D311yescharacteristic
76C2×#103—31no—
77C2×#103—31no—
78C2×#103—31no—
79C2×#103—31no—
80C2×C22⋊C8—31no—
81#380—31no—
82C24×C3#6711yescharacteristic
83C22×A4—21no—
84C22×A4C3×D411yescharacteristic
85C2×C3×Q8—32no—
86C2×C3×Q8—32no—
87C2×C3×Q8—32no—
88C2×C3×Q8—32no—
89C2×C3×Q8—32no—
90C2×C3×Q8—31no—
91C2×C3×Q8—41no—
92C2×C3×Q8—61no—
93C2×C3×Q8—61no—
94C2×C3×Q8—61no—
95C2×C3×Q8—62no—
96C2×C3×Q8—61no—
97C2×C3×Q8—61no—
98C2×C3×Q8—62no—
99C2×C3×Q8—64no—
100C2×C3×Q8—81no—
101C2×C3×Q8S411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—61no—
108C22×C3×C4—62no—
109C22×C3×C4—61no—
110C22×C3×C4—61no—
111C22×C3×C4—61no—
112C22×C3×C4C2×Dic311yescharacteristic
113C22×C3×C4#6711yescharacteristic
114C4×A4—24no—
115C4×A4—22no—
116C4×A4—21no—
117C4×A4—21no—
118C4×A4C2×C3×C411yescharacteristic
119C4×A4C3×D411yescharacteristic
120SL(2,ℤ/4)—41no—
121C2×C3×C8—61no—
122C2×C3×C8—61no—
123C2×C3×C8—61no—
124C2×C3×C8—61no—
125C2×C3×C8—61no—
126C2×C3×C8—61no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—61no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—61no—
131C3×C22⋊C4—121no—
132C3×C22⋊C4—121no—
133C4⋊Dic3—41no—
134C2×C3⋊C8—41no—
135C22×C32—41no—
136C3×A4#10311yescharacteristic, complemented
137C32×C4—41no—
138C32×C4—41no—
139C32×C4—82no—
140C3×Dic3—161no—
141C22×Q8—21no—
142C22×Q8—31no—
143C22×Q8—31no—
144C22×Q8—32no—
145C22×Q8—61no—
146C22×Q8—61no—
147C22×Q8C3×Dic312yescomplemented
148C23×C4—21no—
149C23×C4C2×C3×D311yescharacteristic
150C22×C8—31no—
151C22×C8—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C22⋊C4—61no—
155#103—62no—
156#103—62no—
157#103—62no—
158#103—61no—
159#103—61no—
160C22⋊C8—61no—
161C22⋊C8—61no—
162#95—61no—
163C23×C3—31no—
164C23×C3—31no—
165C23×C3—31no—
166C23×C3—31no—
167C23×C3#21511yescharacteristic
168C23×C3#21611yescharacteristic
169C23×C3#21711yescharacteristic
170C2×A4—21no—
171C2×A4—21no—
172C2×A4—21no—
173C2×A4C3×C22⋊C411yescharacteristic
174C2×A4C3×SD1611yescharacteristic
175C2×A4C3×Q1611yescharacteristic
176C3×Q8—21no—
177C3×Q8—34no—
178C3×Q8—32no—
179C3×Q8—42no—
180C3×Q8—62no—
181C3×Q8—62no—
182C3×Q8—62no—
183C3×Q8—61no—
184C3×Q8—81no—
185C3×Q8—82no—
186C3×Q8—82no—
187C3×Q8SL(2,ℤ/4)12yescomplemented
188C2×C3×C4—21no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—41no—
197C2×C3×C4—62no—
198C2×C3×C4—62no—
199C2×C3×C4—61no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—81no—
204C2×C3×C4—82no—
205C2×C3×C4—81no—
206C2×C3×C4—121no—
207C2×C3×C4—121no—
208C2×C3×C4—121no—
209C2×C3×C4—121no—
210C2×C3×C4—121no—
211C2×C3×C4—121no—
212C2×C3×C4C2×S411yescharacteristic
213C2×Dic3—81no—
214C3×C8—121no—
215C3×C8—121no—
216C3⋊C8—81no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—41no—
220C24C3×#6711yescharacteristic
221C2×Q8—32no—
222C2×Q8—32no—
223C2×Q8—32no—
224C2×Q8—32no—
225C2×Q8—32no—
226C2×Q8—31no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—62no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8—64no—
235C2×Q8C3×S411yescharacteristic
236C22×C4—22no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—61no—
242C22×C4—62no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4C2×C3×Dic311yescharacteristic
247C22×C4C3×#6711yescharacteristic
248C2×C8—61no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—41no—
270C22×C3—61no—
271C22×C3—81no—
272C22×C3#83511yescharacteristic, complemented
273C22×C3GL(2,ℤ/4)11yescharacteristic
274A4—21no—
275A4C3×#10311yescharacteristic, complemented
276C3×C4—22no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—62no—
282C3×C4—82no—
283C3×C4—84no—
284C3×C4—81no—
285C3×C4—81no—
286C3×C4—241no—
287C3×C4—241no—
288C3×C4C2×SL(2,ℤ/4)11yescharacteristic
289C3×C4GL(2,ℤ/4)11yescharacteristic
290Dic3—161no—
291C32—41noSylow
292C23—31no—
293C23—31no—
294C23—31no—
295C23—31no—
296C23C3×#21511yescharacteristic
297C23C3×#21611yescharacteristic
298C23C3×#21711yescharacteristic
299Q8—21no—
300Q8—34no—
301Q8—32no—
302Q8—62no—
303Q8—62no—
304Q8—62no—
305Q8—61no—
306Q8C3×SL(2,ℤ/4)12yescomplemented
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—61no—
318C2×C4—62no—
319C2×C4—61no—
320C2×C4—62no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4C2×C3×S411yescharacteristic
328C8—121no—
329C8—121no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—81no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3#541611yescharacteristic
341C2×C3#541711yescharacteristic
342C2×C3#543211yescharacteristic
343C22—31no—
344C22—31no—
345C22—31no—
346C22—31no—
347C22—31no—
348C22—31no—
349C22—31no—
350C22—31no—
351C22—31no—
352C22—61no—
353C22C3×#83511yescharacteristic, complemented
354C22C3×GL(2,ℤ/4)11yescharacteristic
355C4—22no—
356C4—31no—
357C4—31no—
358C4—62no—
359C4—241no—
360C4—241no—
361C4C2×C3×SL(2,ℤ/4)11yescharacteristic
362C4C3×GL(2,ℤ/4)11yescharacteristic
363C3—41no—
364C3—81no—
365C3#7314711yescharacteristic, direct product factor
366C2—31no—
367C2—31no—
368C2—31no—
369C2—31no—
370C2C3×#541611yescharacteristic
371C2C3×#541711yescharacteristic
372C2C3×#543211yescharacteristic
subgroupG/H
centerC22×C3GL(2,ℤ/4)
commutatorC4×A4C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC23×C3×Q8D3
socleC24×C3#67
prime radicalC22×C3×A4D4
cosocleC2×C4×A4C22×C3
hypercenterC2×C3×Q8S4
hypocenterA4C3×#103
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
4. #73147C3direct (N × H)
7. C3×Q8×A4C4semidirect
39. C22×C3×Q8Dic3semidirect
45. Q8×A4C3×C4semidirect
136. C3×A4#103semidirect
147. C22×Q8C3×Dic3semidirect
187. C3×Q8SL(2,ℤ/4)semidirect
272. C22×C3#835semidirect
275. A4C3×#103semidirect
306. Q8C3×SL(2,ℤ/4)semidirect
353. C22C3×#835semidirect
365. C3#73147direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C3×C4⋊SL(2,ℤ/4)2
C2×C3×A4⋊C82
#731473
C3×#17873
C3×#8354