Finite Group Databasebeta

← C3×#85302 #11922586

Subgroups — C3×#85302 #11922586

Order 1152.

664 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 108 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×A4⋊Q8C211yesmaximal, characteristic
2C3×A4×C4⋊C4C211yesmaximal, characteristic
3C3×SL(2,ℤ/4)⋊C4C212yesmaximal
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic
5C3×C4×SL(2,ℤ/4)C212yesmaximal
6#85302C311yesmaximal, characteristic, direct product factor
7C3×#2360—31nomaximal
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×SL(2,ℤ/4)C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)C2212yes—
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13C3×A4⋊Q8C414yescomplemented
14C3×#887—41nomaximal
15C2×A4⋊Q8C2×C311yescharacteristic
16C2×C3×C22⋊Q8—31no—
17C2×C3×C4×Q8—31no—
18C22×C3×C4⋊C4D311yescharacteristic
19A4×C4⋊C4—21no—
20A4×C4⋊C4C2×C311yescharacteristic
21SL(2,ℤ/4)⋊C4C2×C312yes—
22C4×SL(2,ℤ/4)C2×C311yescharacteristic
23C4×SL(2,ℤ/4)C2×C312yes—
24C3×#391—31no—
25C3×#389—31no—
26C3×#389—32no—
27C3×#387—32no—
28C3×#385—32no—
29C3×C4×C4⋊C4—31no—
30C3×C4×C22⋊C4—31no—
31C3×C4×C22⋊C4—32no—
32C22×C3×A4C2311yescharacteristic
33C2×C3×Dic6—41no—
34C3×C4×A4—22no—
35C3×C4×A4C2×C412yes—
36C3×SL(2,ℤ/4)—21no—
37C3×SL(2,ℤ/4)C2×C414yes—
38C3×SL(2,ℤ/4)Q812yescomplemented
39C32×C4⋊C4—41no—
40C3×Dic3⋊C4—42no—
41C3×C4×Dic3—42no—
42C3×C4×Dic3—41no—
43#2360—31noSylow
44C22×C3×Q8—31no—
45C23×C3×C4C2×D312yes—
46C23×C3×C4C2×D311yescharacteristic
47C2×C4×A4—21no—
48C2×C4×A4—22no—
49C2×C4×A4C22×C312yes—
50C2×C4×A4C22×C311yescharacteristic
51C2×SL(2,ℤ/4)C22×C311yescharacteristic
52C2×SL(2,ℤ/4)C22×C312yes—
53C2×SL(2,ℤ/4)C22×C311yescharacteristic
54A4⋊Q8C3×C414yescomplemented
55C3×C22⋊Q8—34no—
56C3×C22⋊Q8—34no—
57C3×C4×Q8—61no—
58C3×C4×Q8—62no—
59C3×C4×Q8—61no—
60C2×C3×C4⋊C4—31no—
61C2×C3×C4⋊C4—32no—
62C2×C3×C4⋊C4—31no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C4⋊C4—32no—
66C2×C3×C4⋊C4—32no—
67C2×C3×C22⋊C4—31no—
68C2×C3×C22⋊C4—32no—
69C2×C3×C22⋊C4—31no—
70C2×C3×C42—31no—
71C2×C3×C42—32no—
72C2×C3×C42—31no—
73C2×C3×C42—32no—
74#887—41no—
75C3×#95—31no—
76C3×#95—32no—
77C3×#95—31no—
78C3×#95—32no—
79C3×#95—32no—
80C3×#95—32no—
81C2×C3×A4C22×C411yescharacteristic
82C2×C3×A4C2×Q811yescharacteristic
83C2×C3×A4#4111yescharacteristic
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C3×Dic3—41no—
87C2×C3×Dic3—42no—
88C2×C3×Dic3—41no—
89C3×Dic6—44no—
90C2×C22⋊Q8—31no—
91C2×C4×Q8—31no—
92C22×C4⋊C4C3×D311yescharacteristic
93#391—31no—
94#389—31no—
95#389—32no—
96#387—32no—
97#385—32no—
98C4×C4⋊C4—31no—
99C4×C22⋊C4—31no—
100C4×C22⋊C4—32no—
101C24×C3C22×D311yescharacteristic
102C22×A4—21no—
103C22×A4C23×C311yescharacteristic
104C2×C3×Q8—34no—
105C2×C3×Q8—61no—
106C2×C3×Q8—62no—
107C2×C3×Q8—61no—
108C22×C3×C4—22no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C22×C3×C4—32no—
118C22×C3×C4—31no—
119C22×C3×C4—31no—
120C22×C3×C4—62no—
121C22×C3×C4C4×D312yes—
122C2×Dic6—41no—
123C4×A4—22no—
124C4×A4—22no—
125C4×A4—42no—
126C4×A4C2×C3×C412yes—
127SL(2,ℤ/4)—21no—
128SL(2,ℤ/4)C2×C3×C414yes—
129SL(2,ℤ/4)C3×Q812yescomplemented
130C3×C4⋊C4—34no—
131C3×C4⋊C4—38no—
132C3×C4⋊C4—32no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—62no—
137C3×C4⋊C4—62no—
138C3×C4⋊C4—81no—
139C3×C4⋊C4—122no—
140C3×C4⋊C4S411yescharacteristic
141C3×C22⋊C4—32no—
142C3×C22⋊C4—34no—
143C3×C22⋊C4—34no—
144C3×C22⋊C4—32no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—61no—
147C3×C42—62no—
148C3×C42—61no—
149C3×C42—62no—
150C3×C42—62no—
151C3×C42—61no—
152C3×C42—122no—
153Dic3⋊C4—42no—
154C4×Dic3—42no—
155C4×Dic3—41no—
156C22×C32—41no—
157C3×A4C4×Q811yescharacteristic, complemented
158C32×C4—42no—
159C32×C4—82no—
160C3×Dic3—44no—
161C3×Dic3—42no—
162C3×Dic3—81no—
163C22×Q8—31no—
164C23×C4C2×C3×D312yes—
165C23×C4C2×C3×D311yescharacteristic
166C22⋊Q8—34no—
167C22⋊Q8—34no—
168C4×Q8—61no—
169C4×Q8—62no—
170C4×Q8—61no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—32no—
178C2×C22⋊C4—32no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C42—32no—
182C2×C42—31no—
183C2×C42—31no—
184C2×C42—32no—
185#95—31no—
186#95—32no—
187#95—31no—
188#95—32no—
189#95—32no—
190#95—32no—
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—31no—
195C23×C3C2×C4×D311yescharacteristic
196C23×C3#23711yescharacteristic
197C23×C3D3×Q811yescharacteristic
198C2×A4—21no—
199C2×A4—21no—
200C2×A4—21no—
201C2×A4C22×C3×C411yescharacteristic
202C2×A4C2×C3×Q811yescharacteristic
203C2×A4C3×#4111yescharacteristic
204C3×Q8—64no—
205C3×Q8—64no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—34no—
211C2×C3×C4—34no—
212C2×C3×C4—32no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—81no—
237C2×C3×C4—82no—
238C2×C3×C4C2×S412yes—
239C2×C3×C4C2×S411yescharacteristic
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243Dic6—44no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C24C22×C3×D311yescharacteristic
248C2×Q8—34no—
249C2×Q8—61no—
250C2×Q8—62no—
251C2×Q8—61no—
252C22×C4—22no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—62no—
265C22×C4C3×C4×D312yes—
266C4⋊C4—34no—
267C4⋊C4—38no—
268C4⋊C4—32no—
269C4⋊C4—31no—
270C4⋊C4—61no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—122no—
274C4⋊C4C3×S411yescharacteristic
275C22⋊C4—34no—
276C22⋊C4—32no—
277C22⋊C4—34no—
278C22⋊C4—32no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C42—61no—
282C42—62no—
283C42—62no—
284C42—62no—
285C42—61no—
286C42—122no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—41no—
297C22×C3—61no—
298C22×C3—81no—
299C22×C3#88711yescharacteristic, complemented
300C22×C3C22×S411yescharacteristic
301A4—21no—
302A4C3×C4×Q811yescharacteristic, complemented
303C3×C4—22no—
304C3×C4—32no—
305C3×C4—42no—
306C3×C4—64no—
307C3×C4—62no—
308C3×C4—64no—
309C3×C4—62no—
310C3×C4—62no—
311C3×C4—82no—
312C3×C4—82no—
313C3×C4—121no—
314C3×C4—121no—
315C3×C4—162no—
316C3×C4C4×S412yes—
317Dic3—44no—
318Dic3—42no—
319Dic3—81no—
320C32—41noSylow
321C23—31no—
322C23—31no—
323C23—31no—
324C23—31no—
325C23C2×C3×C4×D311yescharacteristic
326C23C3×#23711yescharacteristic
327C23C3×D3×Q811yescharacteristic
328Q8—64no—
329Q8—64no—
330C2×C4—31no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—34no—
335C2×C4—34no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4C2×C3×S412yes—
359C2×C4C2×C3×S411yescharacteristic
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—81no—
368C2×C3—81no—
369C2×C3—81no—
370C2×C3C2×C4×S411yescharacteristic
371C2×C3#591411yescharacteristic
372C2×C3Q8×S411yescharacteristic
373C22—31no—
374C22—31no—
375C22—31no—
376C22—31no—
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—61no—
383C22C3×#88711yescharacteristic, complemented
384C22C22×C3×S411yescharacteristic
385C4—22no—
386C4—32no—
387C4—64no—
388C4—62no—
389C4—64no—
390C4—62no—
391C4—62no—
392C4—121no—
393C4—121no—
394C4C3×C4×S412yes—
395C3—41no—
396C3—81no—
397C3#8530211yescharacteristic, direct product factor
398C2—31no—
399C2—31no—
400C2—31no—
401C2—31no—
402C2C2×C3×C4×S411yescharacteristic
403C2C3×#591411yescharacteristic
404C2C3×Q8×S411yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutatorC2×A4C22×C3×C4
FrattiniC22C22×C3×S4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3×A4C23
cosocleC22×A4C23×C3
hypercenterC3×C4⋊C4S4
hypocenterA4C3×C4×Q8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #85302C3direct (N × H)
13. C3×A4⋊Q8C4semidirect
38. C3×SL(2,ℤ/4)Q8semidirect
54. A4⋊Q8C3×C4semidirect
129. SL(2,ℤ/4)C3×Q8semidirect
157. C3×A4C4×Q8semidirect
299. C22×C3#887semidirect
302. A4C3×C4×Q8semidirect
383. C22C3×#887semidirect
397. C3#85302direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×A4⋊Q82
C3×A4×C4⋊C42
C3×SL(2,ℤ/4)⋊C42
C3×C4×SL(2,ℤ/4)2
#853023
C3×#23603
C3×#8874