Finite Group Databasebeta

← C2×C3×#5417 #11922782

Subgroups — C2×C3×#5417 #11922782

Order 1152.

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

normal subgroups only

This group has 708 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×#5417C214yesmaximal, direct product factor
4C2×C3×A4⋊C8C211yesmaximal, characteristic, complemented
5C2×#5417C311yesmaximal, characteristic, direct product factor
6C2×C3×#448—31nomaximal
7C22×C3×S4—21no—
8C3×Q8×A4—21no—
9C3×Q8×A4C2212yescomplemented
10C2×C3×C4×A4—21no—
11C2×C3×C4×A4C2211yescharacteristic
12C3×C4⋊S4—21no—
13C3×C4⋊S4C2212yes—
14C2×C3×#215—41nomaximal
15C3×A4⋊C8C2212yescomplemented
16C23×C3×Q8D311yescharacteristic, complemented
17C2×Q8×A4—21no—
18C2×Q8×A4C2×C311yescharacteristic, complemented
19C2×C4⋊S4C2×C311yescharacteristic
20C22×C3×SD16—31no—
21C22×C3×SD16—31no—
22C2×C3×#121—31no—
23#5417C2×C314yesdirect product factor
24C2×A4⋊C8C2×C311yescharacteristic, complemented
25C3×#448—34no—
26C3×#448—34no—
27C2×C3×#103—31no—
28C2×C3×#103—31no—
29C2×C3×C22⋊C8—31no—
30C22×C3×A4D411yescharacteristic
31C2×C3×S4—22no—
32C2×C3×S4—42no—
33C2×C32×Q8—41no—
34C2×C3×D12—41no—
35C3×C4×A4—22no—
36C3×C4×A4D411yescharacteristic
37C3×C4×A4C2311yescharacteristic
38C3×#215—44no—
39C2×C3×C3⋊C8—41no—
40C2×#448—31noSylow
41C22×S4—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—31no—
44C22×C3×Q8—31no—
45C22×C3×Q8—32no—
46C22×C3×Q8—61no—
47C22×C3×Q8—61no—
48C22×C3×Q8C2×D312yescomplemented
49C22×C3×D4—31no—
50C22×C3×D4—61no—
51C23×C3×C4—21no—
52C23×C3×C4C2×D311yescharacteristic
53Q8×A4—22no—
54Q8×A4—21no—
55Q8×A4—22no—
56Q8×A4C22×C312yescomplemented
57C2×C4×A4—22no—
58C2×C4×A4—21no—
59C2×C4×A4—21no—
60C2×C4×A4C22×C311yescharacteristic
61C4⋊S4—21no—
62C4⋊S4C22×C312yes—
63C2×C3×SD16—34no—
64C2×C3×SD16—34no—
65C2×C3×SD16—62no—
66C2×C3×SD16—62no—
67C2×C3×SD16—62no—
68C2×C3×SD16—61no—
69C2×C3×SD16—61no—
70C22×C3×C8—31no—
71C22×C3×C8—31no—
72C3×#121—32no—
73C3×#121—32no—
74C3×#121—61no—
75C3×#121—61no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C22⋊C4—61no—
78C2×#215—41no—
79A4⋊C8C22×C312yescomplemented
80C3×#103—34no—
81C3×#103—34no—
82C3×C22⋊C8—32no—
83C3×C22⋊C8—32no—
84C22×C3×D3—81no—
85C2×C3×A4SD1612yescomplemented
86C2×C3×A4C2×D411yescharacteristic
87C3×S4—42no—
88C32×Q8—42no—
89C32×Q8—81no—
90C2×C32×C4—41no—
91C2×C32×C4—81no—
92C3×D12—42no—
93C3×D12—81no—
94C3×C3⋊C8—42no—
95C23×Q8C3×D311yescharacteristic, complemented
96C22×SD16—31no—
97C22×SD16—31no—
98C2×#121—31no—
99#448—34no—
100#448—34no—
101C2×#103—31no—
102C2×#103—31no—
103C2×C22⋊C8—31no—
104C24×C3—61no—
105C24×C3#6711yescharacteristic
106C22×A4—21no—
107C22×A4C3×D411yescharacteristic
108C2×S4—22no—
109C2×S4—42no—
110C2×C3×Q8—32no—
111C2×C3×Q8—32no—
112C2×C3×Q8—32no—
113C2×C3×Q8—32no—
114C2×C3×Q8—32no—
115C2×C3×Q8—31no—
116C2×C3×Q8—41no—
117C2×C3×Q8—61no—
118C2×C3×Q8—61no—
119C2×C3×Q8—61no—
120C2×C3×Q8—62no—
121C2×C3×Q8—61no—
122C2×C3×Q8—61no—
123C2×C3×Q8—62no—
124C2×C3×Q8—64no—
125C2×C3×Q8—81no—
126C2×C3×Q8S411yescharacteristic, complemented
127C2×C3×D4—32no—
128C2×C3×D4—61no—
129C2×C3×D4—61no—
130C2×C3×D4—61no—
131C2×C3×D4—61no—
132C2×C3×D4—62no—
133C2×C3×D4—62no—
134C2×C3×D4—61no—
135C2×C3×D4—122no—
136C2×C3×D4—122no—
137C22×C3×C4—22no—
138C22×C3×C4—31no—
139C22×C3×C4—31no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—61no—
143C22×C3×C4—62no—
144C22×C3×C4—61no—
145C22×C3×C4—61no—
146C22×C3×C4#6711yescharacteristic
147C22×C3×C4C22×D311yescharacteristic
148C2×D12—41no—
149C4×A4—24no—
150C4×A4—22no—
151C4×A4—21no—
152C4×A4—21no—
153C4×A4C3×D411yescharacteristic
154C4×A4C23×C311yescharacteristic
155C3×SD16—68no—
156C3×SD16—64no—
157C3×SD16—64no—
158C2×C3×C8—32no—
159C2×C3×C8—32no—
160C2×C3×C8—61no—
161C2×C3×C8—61no—
162C2×C3×C8—61no—
163C2×C3×C8—61no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—61no—
166C3×C22⋊C4—62no—
167C3×C22⋊C4—62no—
168#215—44no—
169C2×C3⋊C8—41no—
170C22×C32—41no—
171C2×C3×D3—82no—
172C2×C3×D3—162no—
173C3×A4C2×SD1611yescharacteristic, complemented
174C32×C4—41no—
175C32×C4—41no—
176C32×C4—82no—
177C22×Q8—21no—
178C22×Q8—31no—
179C22×Q8—31no—
180C22×Q8—32no—
181C22×Q8—61no—
182C22×Q8—61no—
183C22×Q8C2×C3×D312yescomplemented
184C22×D4—31no—
185C22×D4—61no—
186C23×C4—21no—
187C23×C4C2×C3×D311yescharacteristic
188C2×SD16—34no—
189C2×SD16—34no—
190C2×SD16—62no—
191C2×SD16—62no—
192C2×SD16—62no—
193C2×SD16—61no—
194C2×SD16—61no—
195C22×C8—31no—
196C22×C8—31no—
197#121—32no—
198#121—32no—
199#121—61no—
200#121—61no—
201C2×C4⋊C4—31no—
202C2×C22⋊C4—61no—
203#103—34no—
204#103—34no—
205C22⋊C8—32no—
206C22⋊C8—32no—
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—32no—
210C23×C3—62no—
211C23×C3—121no—
212C23×C3—122no—
213C23×C3—122no—
214C23×C3—121no—
215C23×C3#21512yescomplemented
216C23×C3C2×#6711yescharacteristic
217C22×D3—81no—
218C2×A4—22no—
219C2×A4—21no—
220C2×A4C3×SD1612yescomplemented
221C2×A4C2×C3×D411yescharacteristic
222S4—42no—
223C3×Q8—21no—
224C3×Q8—34no—
225C3×Q8—32no—
226C3×Q8—42no—
227C3×Q8—62no—
228C3×Q8—62no—
229C3×Q8—62no—
230C3×Q8—61no—
231C3×Q8—81no—
232C3×Q8—82no—
233C3×Q8—82no—
234C3×Q8C2×S412yescomplemented
235C3×D4—62no—
236C3×D4—62no—
237C3×D4—121no—
238C3×D4—124no—
239C3×D4—122no—
240C3×D4—122no—
241C3×D4—121no—
242C2×C3×C4—21no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—41no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—61no—
254C2×C3×C4—62no—
255C2×C3×C4—61no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—81no—
259C2×C3×C4—82no—
260C2×C3×C4—81no—
261C2×C3×C4—122no—
262C2×C3×C4C2×S411yescharacteristic
263D12—42no—
264D12—81no—
265C3×C8—62no—
266C3×C8—62no—
267C3⋊C8—42no—
268C2×C32—42no—
269C2×C32—41no—
270C3×D3—162no—
271C24—61no—
272C24C3×#6711yescharacteristic
273C2×Q8—32no—
274C2×Q8—32no—
275C2×Q8—32no—
276C2×Q8—32no—
277C2×Q8—32no—
278C2×Q8—31no—
279C2×Q8—61no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×Q8—62no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×Q8—62no—
286C2×Q8—64no—
287C2×Q8C3×S411yescharacteristic, complemented
288C2×D4—32no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—61no—
296C2×D4—122no—
297C2×D4—122no—
298C22×C4—22no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—61no—
304C22×C4—62no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4C3×#6711yescharacteristic
308C22×C4C22×C3×D311yescharacteristic
309SD16—68no—
310SD16—64no—
311SD16—64no—
312C2×C8—32no—
313C2×C8—32no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—61no—
318C4⋊C4—32no—
319C4⋊C4—61no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22×C3—32no—
323C22×C3—31no—
324C22×C3—32no—
325C22×C3—32no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—41no—
329C22×C3—61no—
330C22×C3—81no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3—122no—
334C22×C3—242no—
335C22×C3—242no—
336C22×C3C2×#21511yescharacteristic, complemented
337C22×C3GL(2,ℤ/4)11yescharacteristic
338C2×D3—82no—
339C2×D3—162no—
340A4—21no—
341A4C2×C3×SD1611yescharacteristic, complemented
342C3×C4—22no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—62no—
348C3×C4—82no—
349C3×C4—84no—
350C3×C4—81no—
351C3×C4—81no—
352C3×C4—122no—
353C3×C4GL(2,ℤ/4)11yescharacteristic
354C3×C4C22×S411yescharacteristic
355C32—41noSylow
356C23—31no—
357C23—31no—
358C23—32no—
359C23—62no—
360C23—121no—
361C23—122no—
362C23—122no—
363C23—121no—
364C23C3×#21512yescomplemented
365C23C2×C3×#6711yescharacteristic
366Q8—21no—
367Q8—34no—
368Q8—32no—
369Q8—62no—
370Q8—62no—
371Q8—62no—
372Q8—61no—
373Q8C2×C3×S412yescomplemented
374D4—62no—
375D4—62no—
376D4—121no—
377D4—124no—
378D4—122no—
379D4—122no—
380D4—121no—
381C2×C4—21no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—61no—
392C2×C4—62no—
393C2×C4—61no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—122no—
397C2×C4C2×C3×S411yescharacteristic
398C8—62no—
399C8—62no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—32no—
403C2×C3—42no—
404C2×C3—41no—
405C2×C3—82no—
406C2×C3—81no—
407C2×C3—242no—
408C2×C3#541712yesdirect product factor
409C2×C3C2×GL(2,ℤ/4)11yescharacteristic
410D3—162no—
411C22—32no—
412C22—31no—
413C22—32no—
414C22—32no—
415C22—31no—
416C22—31no—
417C22—61no—
418C22—122no—
419C22—122no—
420C22—122no—
421C22—242no—
422C22—242no—
423C22C2×C3×#21511yescharacteristic, complemented
424C22C3×GL(2,ℤ/4)11yescharacteristic
425C4—22no—
426C4—31no—
427C4—31no—
428C4—62no—
429C4—122no—
430C4C3×GL(2,ℤ/4)11yescharacteristic
431C4C22×C3×S411yescharacteristic
432C3—41no—
433C3—81no—
434C3C2×#541711yescharacteristic, direct product factor
435C2—31no—
436C2—31no—
437C2—32no—
438C2—242no—
439C2C3×#541712yesdirect product factor
440C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3GL(2,ℤ/4)
commutator, cosocleC4×A4C23×C3
FrattiniC4C22×C3×S4
FittingC23×C3×Q8D3
socleC24×C3#67
prime radicalC2×C3×C4⋊S4C2
hypercenterC2×C3×Q8S4
hypocenterA4C2×C3×SD16
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
1. C2×C3×Q8×A4C2semidirect
3. C3×#5417C2direct (N × H)
4. C2×C3×A4⋊C8C2semidirect
5. C2×#5417C3direct (N × H)
9. C3×Q8×A4C22semidirect
15. C3×A4⋊C8C22semidirect
16. C23×C3×Q8D3semidirect
18. C2×Q8×A4C2×C3semidirect
23. #5417C2×C3direct (N × H)
24. C2×A4⋊C8C2×C3semidirect
48. C22×C3×Q8C2×D3semidirect
56. Q8×A4C22×C3semidirect
79. A4⋊C8C22×C3semidirect
85. C2×C3×A4SD16semidirect
95. C23×Q8C3×D3semidirect
126. C2×C3×Q8S4semidirect
173. C3×A4C2×SD16semidirect
183. C22×Q8C2×C3×D3semidirect
215. C23×C3#215semidirect
220. C2×A4C3×SD16semidirect
234. C3×Q8C2×S4semidirect
287. C2×Q8C3×S4semidirect
336. C22×C3C2×#215semidirect
341. A4C2×C3×SD16semidirect
364. C23C3×#215semidirect
373. Q8C2×C3×S4semidirect
408. C2×C3#5417direct (N × H)
423. C22C2×C3×#215semidirect
434. C3C2×#5417direct (N × H)
439. C2C3×#5417direct (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×#54172
C2×C3×A4⋊C82
C2×#54173
C2×C3×#4483
C2×C3×#2154