Finite Group Databasebeta

← C3×#85502 #11922786

Subgroups — C3×#85502 #11922786

Order 1152.

716 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 96 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
2C2×C3×A4⋊Q8C211yesmaximal, characteristic
3C3×SL(2,ℤ/4)⋊C4C214yesmaximal
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic
5#85502C311yesmaximal, characteristic, direct product factor
6C3×#2434—31nomaximal
7C3×Q8×A4—22no—
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×SL(2,ℤ/4)C2212yes—
11C2×C3×SL(2,ℤ/4)C2212yes—
12C3×A4⋊Q8—22no—
13C3×#944—41nomaximal
14C23×C3×Q8D311yescharacteristic
15C2×Q8×A4—21no—
16C2×Q8×A4C2×C311yescharacteristic
17C2×A4⋊Q8C2×C311yescharacteristic
18C2×C3×C4⋊Q8—31no—
19C2×C3×C4⋊Q8—31no—
20C2×C3×C22⋊Q8—31no—
21SL(2,ℤ/4)⋊C4C2×C314yes—
22C4×SL(2,ℤ/4)C2×C311yescharacteristic
23C3×#395—34no—
24C3×#391—31no—
25C3×#391—31no—
26C3×#385—34no—
27C3×C4×C22⋊C4—31no—
28C22×C3×A4C2311yescharacteristic
29C2×C32×Q8—41no—
30C2×C3×Dic6—41no—
31C3×C4×A4—22no—
32C3×C4×A4D412yes—
33C3×SL(2,ℤ/4)—22no—
34C3×SL(2,ℤ/4)Q814yescomplemented
35C3×Dic3⋊C4—44no—
36C3×C4×Dic3—41no—
37#2434—31noSylow
38C22×C3×Q8—22no—
39C22×C3×Q8—31no—
40C22×C3×Q8—31no—
41C22×C3×Q8—32no—
42C22×C3×Q8—31no—
43C22×C3×Q8—62no—
44C23×C3×C4C2×D312yes—
45C23×C3×C4C2×D311yescharacteristic
46Q8×A4—24no—
47Q8×A4—22no—
48C2×C4×A4—22no—
49C2×C4×A4—21no—
50C2×C4×A4C22×C312yes—
51C2×C4×A4C22×C311yescharacteristic
52C2×SL(2,ℤ/4)C22×C312yes—
53C2×SL(2,ℤ/4)C22×C312yes—
54A4⋊Q8—22no—
55C3×C4⋊Q8—64no—
56C3×C4⋊Q8—62no—
57C3×C4⋊Q8—61no—
58C3×C4⋊Q8—61no—
59C3×C22⋊Q8—62no—
60C3×C22⋊Q8—62no—
61C2×C3×C4⋊C4—34no—
62C2×C3×C4⋊C4—34no—
63C2×C3×C4⋊C4—31no—
64C2×C3×C4⋊C4—32no—
65C2×C3×C22⋊C4—32no—
66C2×C3×C22⋊C4—32no—
67C2×C3×C42—31no—
68C2×C3×C42—31no—
69#944—41no—
70C3×#95—32no—
71C3×#95—34no—
72C3×#95—34no—
73C2×C3×A4C2×D411yescharacteristic
74C2×C3×A4C2×Q812yes—
75C32×Q8—82no—
76C2×C32×C4—41no—
77C2×C32×C4—42no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—42no—
80C3×Dic6—82no—
81C23×Q8C3×D311yescharacteristic
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8—31no—
84C2×C22⋊Q8—31no—
85#395—34no—
86#391—31no—
87#391—31no—
88#385—34no—
89C4×C22⋊C4—31no—
90C24×C3C22×D311yescharacteristic
91C22×A4—21no—
92C22×A4C23×C311yescharacteristic
93C2×C3×Q8—34no—
94C2×C3×Q8—32no—
95C2×C3×Q8—31no—
96C2×C3×Q8—41no—
97C2×C3×Q8—61no—
98C2×C3×Q8—62no—
99C2×C3×Q8—62no—
100C2×C3×Q8—62no—
101C2×C3×Q8—64no—
102C2×C3×Q8—64no—
103C2×C3×Q8—62no—
104C2×C3×Q8—61no—
105C2×C3×Q8—62no—
106C2×C3×Q8—61no—
107C2×C3×Q8—81no—
108C2×C3×Q8S411yescharacteristic
109C22×C3×C4—22no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—32no—
113C22×C3×C4—32no—
114C22×C3×C4—32no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C22×C3×C4—32no—
118C22×C3×C4—32no—
119C22×C3×C4—62no—
120C22×C3×C4#6712yes—
121C2×Dic6—41no—
122C4×A4—24no—
123C4×A4—22no—
124C4×A4—22no—
125C4×A4C3×D412yes—
126SL(2,ℤ/4)—22no—
127SL(2,ℤ/4)C3×Q814yescomplemented
128C3×C4⋊C4—62no—
129C3×C4⋊C4—64no—
130C3×C4⋊C4—68no—
131C3×C4⋊C4—64no—
132C3×C4⋊C4—64no—
133C3×C22⋊C4—34no—
134C3×C22⋊C4—34no—
135C3×C22⋊C4—62no—
136C3×C22⋊C4—62no—
137C3×C42—61no—
138C3×C42—62no—
139C3×C42—61no—
140Dic3⋊C4—44no—
141C4×Dic3—41no—
142C22×C32—41no—
143C3×A4C4⋊Q811yescharacteristic, complemented
144C32×C4—42no—
145C32×C4—82no—
146C3×Dic3—44no—
147C3×Dic3—82no—
148C22×Q8—22no—
149C22×Q8—31no—
150C22×Q8—31no—
151C22×Q8—32no—
152C22×Q8—31no—
153C22×Q8—62no—
154C23×C4C2×C3×D312yes—
155C23×C4C2×C3×D311yescharacteristic
156C4⋊Q8—64no—
157C4⋊Q8—62no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C22⋊Q8—62no—
161C22⋊Q8—62no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—32no—
164C2×C4⋊C4—34no—
165C2×C4⋊C4—34no—
166C2×C22⋊C4—32no—
167C2×C22⋊C4—32no—
168C2×C42—31no—
169C2×C42—31no—
170#95—32no—
171#95—34no—
172#95—34no—
173C23×C3—31no—
174C23×C3—32no—
175C23×C3—31no—
176C23×C3D3×Q812yes—
177C23×C3C2×#6711yescharacteristic
178C2×A4—22no—
179C2×A4—21no—
180C2×A4C2×C3×D411yescharacteristic
181C2×A4C2×C3×Q812yes—
182C3×Q8—22no—
183C3×Q8—64no—
184C3×Q8—64no—
185C3×Q8—62no—
186C3×Q8—82no—
187C3×Q8—84no—
188C3×Q8—122no—
189C3×Q8—122no—
190C2×C3×C4—31no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—34no—
195C2×C3×C4—34no—
196C2×C3×C4—34no—
197C2×C3×C4—32no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—62no—
201C2×C3×C4—62no—
202C2×C3×C4—61no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—64no—
212C2×C3×C4—62no—
213C2×C3×C4—82no—
214C2×C3×C4—81no—
215C2×C3×C4C2×S412yes—
216C2×C3×C4C2×S411yescharacteristic
217C2×Dic3—42no—
218C2×Dic3—42no—
219Dic6—82no—
220C2×C32—42no—
221C2×C32—41no—
222C24C22×C3×D311yescharacteristic
223C2×Q8—34no—
224C2×Q8—32no—
225C2×Q8—31no—
226C2×Q8—61no—
227C2×Q8—62no—
228C2×Q8—62no—
229C2×Q8—62no—
230C2×Q8—64no—
231C2×Q8—64no—
232C2×Q8—62no—
233C2×Q8—61no—
234C2×Q8—62no—
235C2×Q8—61no—
236C2×Q8C3×S411yescharacteristic
237C22×C4—22no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—32no—
241C22×C4—32no—
242C22×C4—32no—
243C22×C4—32no—
244C22×C4—32no—
245C22×C4—32no—
246C22×C4—32no—
247C22×C4—62no—
248C22×C4C3×#6712yes—
249C4⋊C4—62no—
250C4⋊C4—64no—
251C4⋊C4—68no—
252C4⋊C4—64no—
253C4⋊C4—64no—
254C22⋊C4—34no—
255C22⋊C4—34no—
256C22⋊C4—62no—
257C22⋊C4—62no—
258C42—61no—
259C42—62no—
260C42—61no—
261C22×C3—32no—
262C22×C3—31no—
263C22×C3—32no—
264C22×C3—32no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—41no—
268C22×C3—61no—
269C22×C3—81no—
270C22×C3#94411yescharacteristic, complemented
271C22×C3C22×S411yescharacteristic
272A4—21no—
273A4C3×C4⋊Q811yescharacteristic, complemented
274C3×C4—22no—
275C3×C4—32no—
276C3×C4—42no—
277C3×C4—64no—
278C3×C4—64no—
279C3×C4—62no—
280C3×C4—82no—
281C3×C4—84no—
282C3×C4—82no—
283C3×C4—122no—
284C3×C4—122no—
285C3×C4GL(2,ℤ/4)12yes—
286Dic3—44no—
287Dic3—82no—
288C32—41noSylow
289C23—31no—
290C23—32no—
291C23—31no—
292C23C3×D3×Q812yes—
293C23C2×C3×#6711yescharacteristic
294Q8—22no—
295Q8—64no—
296Q8—64no—
297Q8—62no—
298Q8—122no—
299Q8—122no—
300C2×C4—31no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—34no—
305C2×C4—34no—
306C2×C4—34no—
307C2×C4—32no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—64no—
320C2×C4—62no—
321C2×C4C2×C3×S412yes—
322C2×C4C2×C3×S411yescharacteristic
323C2×C3—32no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—42no—
327C2×C3—41no—
328C2×C3—82no—
329C2×C3—81no—
330C2×C3Q8×S412yes—
331C2×C3C2×GL(2,ℤ/4)11yescharacteristic
332C22—32no—
333C22—31no—
334C22—32no—
335C22—32no—
336C22—31no—
337C22—31no—
338C22—61no—
339C22C3×#94411yescharacteristic, complemented
340C22C22×C3×S411yescharacteristic
341C4—22no—
342C4—32no—
343C4—64no—
344C4—64no—
345C4—62no—
346C4—122no—
347C4—122no—
348C4C3×GL(2,ℤ/4)12yes—
349C3—41no—
350C3—81no—
351C3#8550211yescharacteristic, direct product factor
352C2—32no—
353C2—31no—
354C2—31no—
355C2C3×Q8×S412yes—
356C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
socleC24×C3C22×D3
prime radicalC22×C3×A4C23
hypercenterC2×C3×Q8S4
hypocenterA4C3×C4⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
5. #85502C3direct (N × H)
34. C3×SL(2,ℤ/4)Q8semidirect
127. SL(2,ℤ/4)C3×Q8semidirect
143. C3×A4C4⋊Q8semidirect
270. C22×C3#944semidirect
273. A4C3×C4⋊Q8semidirect
339. C22C3×#944semidirect
351. C3#85502direct (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
C2×C3×A4⋊Q82
C3×SL(2,ℤ/4)⋊C42
C3×C4×SL(2,ℤ/4)2
#855023
C3×#24343
C3×#9444