Finite Group Databasebeta

← C3×#85508 #11922792

Subgroups — C3×#85508 #11922792

Order 1152.

784 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 88 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, complemented
2C2×C3×C4⋊S4C211yesmaximal, characteristic
3C3×(C2×S4)⋊C4C214yesmaximal, complemented
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5#85508C311yesmaximal, characteristic, direct product factor
6C3×#2436—31nomaximal
7C22×C3×S4C2212yes—
8C3×Q8×A4—22no—
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)C2212yes—
12C3×C4⋊S4—22no—
13C3×#947—41nomaximal
14C23×C3×Q8D311yescharacteristic, complemented
15C2×Q8×A4—21no—
16C2×Q8×A4C2×C311yescharacteristic, complemented
17C2×C4⋊S4C2×C311yescharacteristic
18C2×C3×#124—31no—
19C2×C3×#124—31no—
20C2×C3×#121—31no—
21(C2×S4)⋊C4C2×C314yescomplemented
22C4×SL(2,ℤ/4)C2×C311yescharacteristic, complemented
23C3×#393—34no—
24C3×#391—31no—
25C3×#391—31no—
26C3×#386—34no—
27C3×C4×C22⋊C4—31no—
28C22×C3×A4C2311yescharacteristic
29C2×C3×S4—22no—
30C2×C3×S4—24no—
31C2×C32×Q8—41no—
32C2×C3×D12—41no—
33C3×C4×A4—22no—
34C3×C4×A4D412yes—
35C3×SL(2,ℤ/4)—22no—
36C3×(C2×D3)⋊C4—44no—
37C3×C4×Dic3—41no—
38#2436—31noSylow
39C22×S4C22×C312yes—
40C22×C3×Q8—22no—
41C22×C3×Q8—31no—
42C22×C3×Q8—31no—
43C22×C3×Q8—32no—
44C22×C3×Q8—62no—
45C22×C3×D4—31no—
46C22×C3×D4—32no—
47C23×C3×C4C2×D312yes—
48C23×C3×C4C2×D311yescharacteristic
49Q8×A4—24no—
50Q8×A4—22no—
51C2×C4×A4—22no—
52C2×C4×A4—21no—
53C2×C4×A4C22×C312yes—
54C2×C4×A4C22×C311yescharacteristic
55C2×SL(2,ℤ/4)C22×C312yes—
56C4⋊S4—22no—
57C3×#124—64no—
58C3×#124—62no—
59C3×#124—61no—
60C3×#124—61no—
61C3×#121—62no—
62C3×#121—62no—
63C2×C3×C4⋊C4—31no—
64C2×C3×C22⋊C4—34no—
65C2×C3×C22⋊C4—34no—
66C2×C3×C22⋊C4—32no—
67C2×C3×C22⋊C4—32no—
68C2×C3×C42—31no—
69C2×C3×C42—31no—
70#947—41no—
71C3×#95—32no—
72C3×#95—34no—
73C3×#95—34no—
74C22×C3×D3—42no—
75C2×C3×A4C2×D411yescharacteristic
76C2×C3×A4#4112yes—
77C3×S4—42no—
78C32×Q8—82no—
79C2×C32×C4—41no—
80C2×C32×C4—42no—
81C2×C3×Dic3—42no—
82C3×D12—82no—
83C23×Q8C3×D311yescharacteristic, complemented
84C2×#124—31no—
85C2×#124—31no—
86C2×#121—31no—
87#393—34no—
88#391—31no—
89#391—31no—
90#386—34no—
91C4×C22⋊C4—31no—
92C24×C3—32no—
93C24×C3C22×D311yescharacteristic
94C22×A4—21no—
95C22×A4C23×C311yescharacteristic
96C2×S4—22no—
97C2×S4—24no—
98C2×C3×Q8—34no—
99C2×C3×Q8—32no—
100C2×C3×Q8—31no—
101C2×C3×Q8—41no—
102C2×C3×Q8—61no—
103C2×C3×Q8—62no—
104C2×C3×Q8—62no—
105C2×C3×Q8—62no—
106C2×C3×Q8—64no—
107C2×C3×Q8—64no—
108C2×C3×Q8—81no—
109C2×C3×Q8S411yescharacteristic, complemented
110C2×C3×D4—62no—
111C2×C3×D4—61no—
112C2×C3×D4—62no—
113C2×C3×D4—62no—
114C2×C3×D4—64no—
115C2×C3×D4—64no—
116C2×C3×D4—62no—
117C2×C3×D4—61no—
118C22×C3×C4—22no—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C22×C3×C4—32no—
122C22×C3×C4—32no—
123C22×C3×C4—32no—
124C22×C3×C4—32no—
125C22×C3×C4—32no—
126C22×C3×C4—32no—
127C22×C3×C4—62no—
128C22×C3×C4#6712yes—
129C2×D12—41no—
130C4×A4—24no—
131C4×A4—22no—
132C4×A4—22no—
133C4×A4C3×D412yes—
134SL(2,ℤ/4)—22no—
135C3×C4⋊C4—62no—
136C3×C22⋊C4—64no—
137C3×C22⋊C4—68no—
138C3×C22⋊C4—64no—
139C3×C22⋊C4—62no—
140C3×C22⋊C4—62no—
141C3×C22⋊C4—64no—
142C3×C42—61no—
143C3×C42—62no—
144C3×C42—61no—
145(C2×D3)⋊C4—44no—
146C4×Dic3—41no—
147C22×C32—41no—
148C2×C3×D3—82no—
149C2×C3×D3—84no—
150C3×A4#12411yescharacteristic, complemented
151C32×C4—42no—
152C32×C4—82no—
153C3×Dic3—82no—
154C22×Q8—22no—
155C22×Q8—31no—
156C22×Q8—31no—
157C22×Q8—32no—
158C22×Q8—62no—
159C22×D4—31no—
160C22×D4—32no—
161C23×C4C2×C3×D312yes—
162C23×C4C2×C3×D311yescharacteristic
163#124—64no—
164#124—62no—
165#124—61no—
166#124—61no—
167#121—62no—
168#121—62no—
169C2×C4⋊C4—31no—
170C2×C22⋊C4—32no—
171C2×C22⋊C4—34no—
172C2×C22⋊C4—34no—
173C2×C22⋊C4—32no—
174C2×C42—31no—
175C2×C42—31no—
176#95—32no—
177#95—34no—
178#95—34no—
179C23×C3—31no—
180C23×C3—32no—
181C23×C3—31no—
182C23×C3—62no—
183C23×C3—62no—
184C23×C3—64no—
185C23×C3—64no—
186C23×C3—62no—
187C23×C3#23912yes—
188C23×C3C2×#6711yescharacteristic
189C22×D3—42no—
190C2×A4—22no—
191C2×A4—21no—
192C2×A4C2×C3×D411yescharacteristic
193C2×A4C3×#4112yes—
194S4—42no—
195C3×Q8—22no—
196C3×Q8—64no—
197C3×Q8—64no—
198C3×Q8—62no—
199C3×Q8—82no—
200C3×Q8—84no—
201C3×D4—122no—
202C3×D4—124no—
203C3×D4—122no—
204C3×D4—122no—
205C3×D4—122no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—34no—
211C2×C3×C4—32no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—61no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—64no—
225C2×C3×C4—62no—
226C2×C3×C4—82no—
227C2×C3×C4—81no—
228C2×C3×C4C2×S412yes—
229C2×C3×C4C2×S411yescharacteristic
230C2×Dic3—42no—
231D12—82no—
232C2×C32—42no—
233C2×C32—41no—
234C3×D3—162no—
235C24—32no—
236C24C22×C3×D311yescharacteristic
237C2×Q8—34no—
238C2×Q8—32no—
239C2×Q8—31no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—62no—
243C2×Q8—62no—
244C2×Q8—64no—
245C2×Q8—64no—
246C2×Q8C3×S411yescharacteristic, complemented
247C2×D4—62no—
248C2×D4—61no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—64no—
252C2×D4—64no—
253C2×D4—62no—
254C2×D4—61no—
255C22×C4—22no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—32no—
264C22×C4—62no—
265C22×C4C3×#6712yes—
266C4⋊C4—62no—
267C22⋊C4—64no—
268C22⋊C4—68no—
269C22⋊C4—64no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—64no—
273C42—61no—
274C42—62no—
275C42—61no—
276C22×C3—32no—
277C22×C3—31no—
278C22×C3—32no—
279C22×C3—32no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—41no—
283C22×C3—61no—
284C22×C3—81no—
285C22×C3—122no—
286C22×C3—124no—
287C22×C3—122no—
288C22×C3—124no—
289C22×C3—122no—
290C22×C3#94711yescharacteristic, complemented
291C22×C3C22×S411yescharacteristic
292C2×D3—82no—
293C2×D3—84no—
294A4—21no—
295A4C3×#12411yescharacteristic, complemented
296C3×C4—22no—
297C3×C4—32no—
298C3×C4—42no—
299C3×C4—62no—
300C3×C4—82no—
301C3×C4—84no—
302C3×C4—82no—
303C3×C4—122no—
304C3×C4—122no—
305C3×C4—122no—
306C3×C4GL(2,ℤ/4)12yes—
307Dic3—82no—
308C32—41noSylow
309C23—31no—
310C23—32no—
311C23—31no—
312C23—62no—
313C23—62no—
314C23—64no—
315C23—64no—
316C23—62no—
317C23C3×#23912yes—
318C23C2×C3×#6711yescharacteristic
319Q8—22no—
320Q8—64no—
321Q8—64no—
322Q8—62no—
323D4—122no—
324D4—124no—
325D4—122no—
326D4—122no—
327D4—122no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—34no—
333C2×C4—32no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—61no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—64no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4C2×C3×S412yes—
347C2×C4C2×C3×S411yescharacteristic
348C2×C3—31no—
349C2×C3—32no—
350C2×C3—31no—
351C2×C3—42no—
352C2×C3—41no—
353C2×C3—82no—
354C2×C3—81no—
355C2×C3—242no—
356C2×C3#591912yes—
357C2×C3C2×GL(2,ℤ/4)11yescharacteristic
358D3—162no—
359C22—32no—
360C22—31no—
361C22—32no—
362C22—32no—
363C22—31no—
364C22—31no—
365C22—61no—
366C22—122no—
367C22—124no—
368C22—122no—
369C22—124no—
370C22—122no—
371C22C3×#94711yescharacteristic, complemented
372C22C22×C3×S411yescharacteristic
373C4—22no—
374C4—32no—
375C4—62no—
376C4—122no—
377C4—122no—
378C4—122no—
379C4C3×GL(2,ℤ/4)12yes—
380C3—41no—
381C3—81no—
382C3#8550811yescharacteristic, direct product factor
383C2—31no—
384C2—32no—
385C2—31no—
386C2—242no—
387C2C3×#591912yes—
388C2C2×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 radicalC2×C3×C4⋊S4C2
hypercenterC2×C3×Q8S4
hypocenterA4C3×#124
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
3. C3×(C2×S4)⋊C4C2semidirect
4. C3×C4×SL(2,ℤ/4)C2semidirect
5. #85508C3direct (N × H)
14. C23×C3×Q8D3semidirect
16. C2×Q8×A4C2×C3semidirect
21. (C2×S4)⋊C4C2×C3semidirect
22. C4×SL(2,ℤ/4)C2×C3semidirect
83. C23×Q8C3×D3semidirect
109. C2×C3×Q8S4semidirect
150. C3×A4#124semidirect
246. C2×Q8C3×S4semidirect
290. C22×C3#947semidirect
295. A4C3×#124semidirect
371. C22C3×#947semidirect
382. C3#85508direct (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×C4⋊S42
C3×(C2×S4)⋊C42
C3×C4×SL(2,ℤ/4)2
#855083
C3×#24363
C3×#9474