Finite Group Databasebeta

← C3×#85492 #11922778

Subgroups — C3×#85492 #11922778

Order 1152.

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

normal subgroups only

This group has 876 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×D4×A4C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#5432C214yesmaximal, complemented
4C3×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5#85492C311yesmaximal, characteristic, direct product factor
6C3×#2435—31nomaximal
7C23×C3×A4C2212yes—
8C3×D4×A4—22no—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3×SL(2,ℤ/4)C2212yes—
11C2×C3×SL(2,ℤ/4)C2212yes—
12C3×A4⋊Q8—22no—
13C3×#936—41nomaximal
14C23×C3×D4D311yescharacteristic
15C2×D4×A4—21no—
16C2×D4×A4C2×C311yescharacteristic
17C2×A4⋊Q8C2×C311yescharacteristic, complemented
18C2×C3×#124—31no—
19C2×C3×#124—31no—
20C2×C3×C22⋊Q8—31no—
21#5432C2×C314yescomplemented
22C4×SL(2,ℤ/4)C2×C311yescharacteristic, complemented
23C3×#394—34no—
24C3×#390—31no—
25C3×#390—31no—
26C3×#379—34no—
27C3×C4×C22⋊C4—31no—
28C22×C3×A4—22no—
29C22×C3×A4—24no—
30C22×C3×A4C2311yescharacteristic
31C2×C32×D4—41no—
32C2×C3×Dic6—41no—
33C3×C4×A4D412yes—
34C3×SL(2,ℤ/4)—22no—
35C3×SL(2,ℤ/4)—22no—
36C3×#217—44no—
37C3×C4×Dic3—41no—
38#2435—31noSylow
39C25×C3C2×D312yes—
40C23×A4—22no—
41C23×A4C22×C312yes—
42C22×C3×Q8—31no—
43C22×C3×D4—22no—
44C22×C3×D4—31no—
45C22×C3×D4—31no—
46C22×C3×D4—32no—
47C22×C3×D4—62no—
48C23×C3×C4C2×D311yescharacteristic
49D4×A4—24no—
50D4×A4—22no—
51C2×C4×A4—21no—
52C2×C4×A4C22×C311yescharacteristic
53C2×SL(2,ℤ/4)C22×C312yes—
54C2×SL(2,ℤ/4)C22×C312yes—
55A4⋊Q8—22no—
56C3×#124—64no—
57C3×#124—62no—
58C3×#124—61no—
59C3×#124—61no—
60C3×C22⋊Q8—62no—
61C3×C22⋊Q8—62no—
62C2×C3×C4⋊C4—31no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C22⋊C4—34no—
65C2×C3×C22⋊C4—34no—
66C2×C3×C22⋊C4—32no—
67C2×C3×C22⋊C4—32no—
68C2×C3×C22⋊C4—34no—
69C2×C3×C22⋊C4—34no—
70C2×C3×C42—31no—
71C2×C3×C42—31no—
72#936—41no—
73C3×#95—32no—
74C23×C32—42no—
75C2×C3×A4—42no—
76C2×C3×A4C2×D411yescharacteristic
77C2×C3×A4#4112yes—
78C32×D4—82no—
79C2×C32×C4—41no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—42no—
82C3×Dic6—82no—
83C23×D4C3×D311yescharacteristic
84C2×#124—31no—
85C2×#124—31no—
86C2×C22⋊Q8—31no—
87#394—34no—
88#390—31no—
89#390—31no—
90#379—34no—
91C4×C22⋊C4—31no—
92C24×C3—24no—
93C24×C3—22no—
94C24×C3—32no—
95C24×C3—32no—
96C24×C3—62no—
97C24×C3—64no—
98C24×C3C22×D311yescharacteristic
99C22×A4—24no—
100C22×A4—22no—
101C22×A4—24no—
102C22×A4—21no—
103C22×A4—44no—
104C22×A4C23×C311yescharacteristic
105C2×C3×Q8—62no—
106C2×C3×Q8—61no—
107C2×C3×Q8—62no—
108C2×C3×Q8—61no—
109C2×C3×D4—34no—
110C2×C3×D4—32no—
111C2×C3×D4—31no—
112C2×C3×D4—41no—
113C2×C3×D4—62no—
114C2×C3×D4—62no—
115C2×C3×D4—61no—
116C2×C3×D4—62no—
117C2×C3×D4—64no—
118C2×C3×D4—64no—
119C2×C3×D4—81no—
120C2×C3×D4S411yescharacteristic
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C22×C3×C4—32no—
124C22×C3×C4—32no—
125C22×C3×C4—32no—
126C22×C3×C4—32no—
127C22×C3×C4—32no—
128C22×C3×C4#6712yes—
129C2×Dic6—41no—
130C4×A4—22no—
131C4×A4C3×D412yes—
132SL(2,ℤ/4)—22no—
133SL(2,ℤ/4)—22no—
134C3×C4⋊C4—62no—
135C3×C4⋊C4—64no—
136C3×C22⋊C4—62no—
137C3×C22⋊C4—62no—
138C3×C22⋊C4—68no—
139C3×C22⋊C4—64no—
140C3×C22⋊C4—62no—
141C3×C22⋊C4—62no—
142C3×C22⋊C4—64no—
143C3×C22⋊C4—124no—
144C3×C22⋊C4—124no—
145C3×C42—61no—
146C3×C42—62no—
147C3×C42—61no—
148#217—44no—
149C4×Dic3—41no—
150C22×C32—41no—
151C22×C32—82no—
152C22×C32—84no—
153C3×A4#12411yescharacteristic, complemented
154C32×C4—42no—
155C3×Dic3—82no—
156C3×Dic3—82no—
157C25C2×C3×D312yes—
158C22×Q8—31no—
159C22×D4—22no—
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—32no—
163C22×D4—62no—
164C23×C4C2×C3×D311yescharacteristic
165#124—64no—
166#124—62no—
167#124—61no—
168#124—61no—
169C22⋊Q8—62no—
170C22⋊Q8—62no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—32no—
173C2×C22⋊C4—34no—
174C2×C22⋊C4—34no—
175C2×C22⋊C4—32no—
176C2×C22⋊C4—32no—
177C2×C22⋊C4—34no—
178C2×C22⋊C4—34no—
179C2×C42—31no—
180C2×C42—31no—
181#95—32no—
182C23×C3—31no—
183C23×C3—31no—
184C23×C3—32no—
185C23×C3—34no—
186C23×C3—32no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—64no—
190C23×C3—62no—
191C23×C3—62no—
192C23×C3—64no—
193C23×C3—64no—
194C23×C3—64no—
195C23×C3—62no—
196C23×C3—82no—
197C23×C3—124no—
198C23×C3—122no—
199C23×C3—124no—
200C23×C3#23712yes—
201C23×C3C2×#6711yescharacteristic
202C23×C3C2×S412yes—
203C2×A4—22no—
204C2×A4—21no—
205C2×A4—44no—
206C2×A4—42no—
207C2×A4C2×C3×D411yescharacteristic
208C2×A4C3×#4112yes—
209C3×Q8—122no—
210C3×Q8—122no—
211C3×D4—22no—
212C3×D4—64no—
213C3×D4—64no—
214C3×D4—62no—
215C3×D4—82no—
216C3×D4—84no—
217C2×C3×C4—31no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—41no—
222C2×C3×C4—61no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—64no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—81no—
235C2×C3×C4C2×S411yescharacteristic
236C2×Dic3—42no—
237C2×Dic3—42no—
238Dic6—82no—
239C2×C32—42no—
240C2×C32—41no—
241C2×C32—162no—
242C24—24no—
243C24—22no—
244C24—32no—
245C24—32no—
246C24—62no—
247C24—64no—
248C24C22×C3×D311yescharacteristic
249C2×Q8—62no—
250C2×Q8—61no—
251C2×Q8—62no—
252C2×Q8—61no—
253C2×D4—34no—
254C2×D4—32no—
255C2×D4—31no—
256C2×D4—62no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—62no—
260C2×D4—64no—
261C2×D4—64no—
262C2×D4C3×S411yescharacteristic
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—32no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—32no—
269C22×C4—32no—
270C22×C4C3×#6712yes—
271C4⋊C4—62no—
272C4⋊C4—64no—
273C22⋊C4—62no—
274C22⋊C4—68no—
275C22⋊C4—64no—
276C22⋊C4—62no—
277C22⋊C4—64no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—124no—
281C22⋊C4—124no—
282C42—61no—
283C42—62no—
284C42—61no—
285C22×C3—24no—
286C22×C3—22no—
287C22×C3—32no—
288C22×C3—31no—
289C22×C3—32no—
290C22×C3—32no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—41no—
294C22×C3—61no—
295C22×C3—68no—
296C22×C3—64no—
297C22×C3—62no—
298C22×C3—82no—
299C22×C3—84no—
300C22×C3—84no—
301C22×C3—81no—
302C22×C3—122no—
303C22×C3—122no—
304C22×C3—124no—
305C22×C3—122no—
306C22×C3—122no—
307C22×C3—164no—
308C22×C3#93611yescharacteristic, complemented
309C22×C3C22×S411yescharacteristic
310A4—21no—
311A4C3×#12411yescharacteristic, complemented
312C3×C4—32no—
313C3×C4—42no—
314C3×C4—82no—
315C3×C4—122no—
316C3×C4—122no—
317C3×C4—122no—
318C3×C4—122no—
319C3×C4GL(2,ℤ/4)12yes—
320Dic3—82no—
321Dic3—82no—
322C32—41noSylow
323C23—31no—
324C23—31no—
325C23—32no—
326C23—34no—
327C23—32no—
328C23—42no—
329C23—64no—
330C23—62no—
331C23—62no—
332C23—64no—
333C23—64no—
334C23—64no—
335C23—62no—
336C23—124no—
337C23—122no—
338C23—124no—
339C23C3×#23712yes—
340C23C2×C3×#6711yescharacteristic
341C23C2×C3×S412yes—
342Q8—122no—
343Q8—122no—
344D4—22no—
345D4—64no—
346D4—64no—
347D4—62no—
348C2×C4—31no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—61no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—64no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4C2×C3×S411yescharacteristic
365C2×C3—31no—
366C2×C3—32no—
367C2×C3—31no—
368C2×C3—42no—
369C2×C3—42no—
370C2×C3—41no—
371C2×C3—82no—
372C2×C3—81no—
373C2×C3—122no—
374C2×C3—162no—
375C2×C3—164no—
376C2×C3#591412yes—
377C2×C3C2×GL(2,ℤ/4)11yescharacteristic
378C22—24no—
379C22—22no—
380C22—32no—
381C22—31no—
382C22—32no—
383C22—32no—
384C22—31no—
385C22—31no—
386C22—61no—
387C22—68no—
388C22—64no—
389C22—62no—
390C22—122no—
391C22—122no—
392C22—124no—
393C22—122no—
394C22—122no—
395C22C3×#93611yescharacteristic, complemented
396C22C22×C3×S411yescharacteristic
397C4—32no—
398C4—122no—
399C4—122no—
400C4—122no—
401C4—122no—
402C4C3×GL(2,ℤ/4)12yes—
403C3—41no—
404C3—81no—
405C3#8549211yescharacteristic, direct product factor
406C2—31no—
407C2—32no—
408C2—31no—
409C2—42no—
410C2—122no—
411C2C3×#591412yes—
412C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×D4D3
socleC24×C3C22×D3
prime radicalC2×C3×D4×A4C2
hypercenterC2×C3×D4S4
hypocenterA4C3×#124
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×A4⋊Q8C2semidirect
3. C3×#5432C2semidirect
4. C3×C4×SL(2,ℤ/4)C2semidirect
5. #85492C3direct (N × H)
17. C2×A4⋊Q8C2×C3semidirect
21. #5432C2×C3semidirect
22. C4×SL(2,ℤ/4)C2×C3semidirect
153. C3×A4#124semidirect
308. C22×C3#936semidirect
311. A4C3×#124semidirect
395. C22C3×#936semidirect
405. C3#85492direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
C2×C3×A4⋊Q82
C3×#54322
C3×C4×SL(2,ℤ/4)2
#854923
C3×#24353
C3×#9364