Finite Group Databasebeta

← C2×A4×#215 #11923366

Subgroups — C2×A4×#215 #11923366

Order 1152.

838 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 99 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×A4×D12C211yesmaximal, characteristic
3A4×#215C214yesmaximal, direct product factor
4C2×A4×C3⋊C8C211yesmaximal, characteristic, complemented
5C23×#215C311yesmaximal, characteristic, complemented
6C2×A4×SD16—31nomaximal
7C22×D3×A4—21no—
8C3×Q8×A4—21no—
9C3×Q8×A4C2212yescomplemented
10C2×C3×C4×A4—21no—
11C2×C3×C4×A4C2211yescharacteristic
12A4×D12—21no—
13A4×D12C2212yes—
14C2×C3×#215—41nomaximal
15A4×C3⋊C8C2212yescomplemented
16C23×C3×Q8C2×C311yescharacteristic, complemented
17C23×D12C2×C311yescharacteristic
18C2×Q8×A4—21no—
19C2×Q8×A4D311yescharacteristic, complemented
20C2×D4×A4—31no—
21C22×#215—31no—
22C22×#215—31no—
23C22×#215—31no—
24C22×#215—31no—
25C22×#215—34no—
26C22×#215C2×C314yescomplemented
27C23×C3⋊C8C2×C311yescharacteristic, complemented
28A4×SD16—34no—
29C2×C8×A4—31no—
30C22×C3×A4D411yescharacteristic
31C2×D3×A4—22no—
32C2×D3×A4—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—
40C23×SD16—31noSylow
41C24×D3—21no—
42C23×A4—61no—
43C22×C3×Q8—21no—
44C22×C3×Q8—31no—
45C22×C3×Q8—32no—
46C22×C3×Q8—31no—
47C22×C3×Q8—61no—
48C22×C3×Q8—61no—
49C22×C3×Q8C22×C312yescomplemented
50C23×C3×C4—21no—
51C23×C3×C4C22×C311yescharacteristic
52C22×D12—21no—
53C22×D12—31no—
54C22×D12—31no—
55C22×D12—32no—
56C22×D12—61no—
57C22×D12—61no—
58C22×D12C22×C312yes—
59Q8×A4—22no—
60Q8×A4—21no—
61Q8×A4—22no—
62Q8×A4C2×D312yescomplemented
63D4×A4—32no—
64D4×A4—61no—
65C2×C4×A4—22no—
66C2×C4×A4—21no—
67C2×C4×A4—21no—
68C2×C4×A4C2×D311yescharacteristic
69C2×C3×SD16—121no—
70C2×#215—34no—
71C2×#215—34no—
72C2×#215—32no—
73C2×#215—34no—
74C2×#215—31no—
75C2×#215—31no—
76C2×#215—34no—
77C2×#215—34no—
78C2×#215—34no—
79C2×#215—34no—
80C2×#215—34no—
81C2×#215—31no—
82C2×#215A411yescharacteristic, direct product factor
83C22×C3⋊C8—31no—
84C22×C3⋊C8—31no—
85C22×C3⋊C8—32no—
86C22×C3⋊C8C22×C312yescomplemented
87C8×A4—32no—
88C22×C3×D3—81no—
89C2×C3×A4SD1612yescomplemented
90C2×C3×A4C2×D411yescharacteristic
91D3×A4—42no—
92C32×Q8—42no—
93C32×Q8—81no—
94C2×C32×C4—41no—
95C2×C32×C4—81no—
96C3×D12—42no—
97C3×D12—81no—
98C3×C3⋊C8—42no—
99C23×Q8C3×D311yescharacteristic, complemented
100C23×D4—31no—
101C22×SD16—34no—
102C22×SD16—91no—
103C22×SD16—91no—
104C22×SD16—94no—
105C22×SD16—91no—
106C22×SD16—91no—
107C23×C8—31no—
108C24×C3C3×D411yescharacteristic
109C23×D3—22no—
110C23×D3—42no—
111C23×D3—61no—
112C23×D3—62no—
113C23×D3—61no—
114C23×D3—122no—
115C22×A4—21no—
116C22×A4—62no—
117C22×A4—122no—
118C22×A4#6711yescharacteristic
119C2×C3×Q8—32no—
120C2×C3×Q8—32no—
121C2×C3×Q8—32no—
122C2×C3×Q8—32no—
123C2×C3×Q8—31no—
124C2×C3×Q8—41no—
125C2×C3×Q8—61no—
126C2×C3×Q8—61no—
127C2×C3×Q8—61no—
128C2×C3×Q8—62no—
129C2×C3×Q8—61no—
130C2×C3×Q8—62no—
131C2×C3×Q8—61no—
132C2×C3×Q8—64no—
133C2×C3×Q8—61no—
134C2×C3×Q8—81no—
135C2×C3×Q8C2×A411yescharacteristic, complemented
136C2×C3×D4—121no—
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×C4C3×D411yescharacteristic
146C22×C3×C4C23×C311yescharacteristic
147C2×D12—32no—
148C2×D12—31no—
149C2×D12—32no—
150C2×D12—32no—
151C2×D12—32no—
152C2×D12—61no—
153C2×D12—61no—
154C2×D12—61no—
155C2×D12—61no—
156C2×D12—64no—
157C2×D12—62no—
158C2×D12—61no—
159C2×D12—61no—
160C2×D12—62no—
161C2×D12C2×A411yescharacteristic
162C4×A4—24no—
163C4×A4—22no—
164C4×A4—21no—
165C4×A4—21no—
166C4×A4#6711yescharacteristic
167C4×A4C22×D311yescharacteristic
168C3×SD16—124no—
169C2×C3×C8—121no—
170#215—38no—
171#215—34no—
172#215—34no—
173#215—34no—
174#215C2×A414yesdirect product factor
175C2×C3⋊C8—32no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—32no—
178C2×C3⋊C8—32no—
179C2×C3⋊C8—32no—
180C2×C3⋊C8C2×A411yescharacteristic, complemented
181C22×C32—41no—
182C2×C3×D3—82no—
183C2×C3×D3—162no—
184C3×A4C2×SD1611yescharacteristic, complemented
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—82no—
188C25—61no—
189C22×Q8—21no—
190C22×Q8—31no—
191C22×Q8—32no—
192C22×Q8—31no—
193C22×Q8—61no—
194C22×Q8—61no—
195C22×Q8C2×C3×D312yescomplemented
196C22×D4—32no—
197C22×D4—61no—
198C22×D4—91no—
199C22×D4—91no—
200C22×D4—92no—
201C22×D4—181no—
202C22×D4—181no—
203C23×C4—21no—
204C23×C4C2×C3×D311yescharacteristic
205C2×SD16—31no—
206C2×SD16—91no—
207C2×SD16—94no—
208C2×SD16—94no—
209C2×SD16—91no—
210C2×SD16—94no—
211C2×SD16—94no—
212C2×SD16—94no—
213C2×SD16—92no—
214C2×SD16—91no—
215C2×SD16—94no—
216C2×SD16—94no—
217C2×SD16—94no—
218C22×C8—32no—
219C22×C8—91no—
220C22×C8—92no—
221C22×C8—91no—
222C23×C3—31no—
223C23×C3—32no—
224C23×C3—31no—
225C23×C3—241no—
226C23×C3C3×SD1612yescomplemented
227C23×C3C2×C3×D411yescharacteristic
228C22×D3—21no—
229C22×D3—42no—
230C22×D3—62no—
231C22×D3—62no—
232C22×D3—62no—
233C22×D3—61no—
234C22×D3—62no—
235C22×D3—122no—
236C22×D3—122no—
237C22×D3—124no—
238C22×D3—124no—
239C22×D3—122no—
240C22×D3—122no—
241C22×D3—122no—
242C2×A4—22no—
243C2×A4—21no—
244C2×A4—122no—
245C2×A4#21512yesdirect product factor
246C2×A4C2×#6711yescharacteristic
247C3×Q8—21no—
248C3×Q8—32no—
249C3×Q8—42no—
250C3×Q8—62no—
251C3×Q8—62no—
252C3×Q8—62no—
253C3×Q8—62no—
254C3×Q8—61no—
255C3×Q8—81no—
256C3×Q8—82no—
257C3×Q8—82no—
258C3×Q8C22×A412yescomplemented
259C3×D4—122no—
260C3×D4—241no—
261C2×C3×C4—21no—
262C2×C3×C4—32no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—41no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—62no—
274C2×C3×C4—61no—
275C2×C3×C4—62no—
276C2×C3×C4—81no—
277C2×C3×C4—82no—
278C2×C3×C4—81no—
279C2×C3×C4C22×A411yescharacteristic
280D12—21no—
281D12—32no—
282D12—62no—
283D12—62no—
284D12—62no—
285D12—62no—
286D12—61no—
287D12C22×A412yes—
288C3×C8—122no—
289C3⋊C8—32no—
290C3⋊C8C22×A412yescomplemented
291C2×C32—41no—
292C2×C32—42no—
293C3×D3—162no—
294C24—62no—
295C24—122no—
296C24—181no—
297C24—182no—
298C24—181no—
299C24—362no—
300C24C3×#6711yescharacteristic
301C2×Q8—32no—
302C2×Q8—32no—
303C2×Q8—32no—
304C2×Q8—32no—
305C2×Q8—31no—
306C2×Q8—61no—
307C2×Q8—61no—
308C2×Q8—61no—
309C2×Q8—62no—
310C2×Q8—61no—
311C2×Q8—62no—
312C2×Q8—61no—
313C2×Q8—64no—
314C2×Q8—61no—
315C2×Q8D3×A411yescharacteristic, complemented
316C2×D4—31no—
317C2×D4—92no—
318C2×D4—92no—
319C2×D4—92no—
320C2×D4—92no—
321C2×D4—91no—
322C2×D4—181no—
323C2×D4—181no—
324C2×D4—181no—
325C2×D4—181no—
326C2×D4—184no—
327C2×D4—181no—
328C2×D4—182no—
329C2×D4—181no—
330C2×D4—182no—
331C22×C4—22no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—61no—
337C22×C4—62no—
338C22×C4—61no—
339C22×C4C3×#6711yescharacteristic
340C22×C4C22×C3×D311yescharacteristic
341SD16—34no—
342SD16—94no—
343SD16—94no—
344SD16—98no—
345SD16—94no—
346C2×C8—31no—
347C2×C8—92no—
348C2×C8—92no—
349C2×C8—92no—
350C2×C8—91no—
351C2×C8—92no—
352C22×C3—32no—
353C22×C3—31no—
354C22×C3—32no—
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—32no—
358C22×C3—32no—
359C22×C3—41no—
360C22×C3—81no—
361C22×C3—242no—
362C22×C3—482no—
363C22×C3C2×C3×SD1611yescharacteristic, complemented
364C22×C3D4×A411yescharacteristic
365C2×D3—22no—
366C2×D3—42no—
367C2×D3—62no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—124no—
371C2×D3—124no—
372C2×D3—122no—
373C2×D3—122no—
374C2×D3—122no—
375A4—21no—
376A4C2×#21511yescharacteristic, direct product factor
377C3×C4—22no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—62no—
383C3×C4—82no—
384C3×C4—84no—
385C3×C4—81no—
386C3×C4—81no—
387C3×C4D4×A411yescharacteristic
388C3×C4C23×A411yescharacteristic
389C32—41noSylow
390C23—31no—
391C23—32no—
392C23—31no—
393C23—61no—
394C23—122no—
395C23—182no—
396C23—182no—
397C23—182no—
398C23—182no—
399C23—181no—
400C23—362no—
401C23—362no—
402C23—362no—
403C23—364no—
404C23—362no—
405C23—362no—
406C23—364no—
407C23C3×#21512yescomplemented
408C23C2×C3×#6711yescharacteristic
409Q8—21no—
410Q8—32no—
411Q8—62no—
412Q8—62no—
413Q8—62no—
414Q8—62no—
415Q8—61no—
416Q8C2×D3×A412yescomplemented
417D4—32no—
418D4—61no—
419D4—92no—
420D4—182no—
421D4—182no—
422D4—182no—
423D4—182no—
424D4—181no—
425C2×C4—21no—
426C2×C4—32no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—61no—
438C2×C4—62no—
439C2×C4C2×D3×A411yescharacteristic
440C8—32no—
441C8—92no—
442C2×C3—32no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—41no—
446C2×C3—42no—
447C2×C3—81no—
448C2×C3—82no—
449C2×C3—482no—
450C2×C3A4×SD1612yescomplemented
451C2×C3C2×D4×A411yescharacteristic
452D3—42no—
453D3—122no—
454C22—32no—
455C22—31no—
456C22—32no—
457C22—31no—
458C22—31no—
459C22—32no—
460C22—32no—
461C22—62no—
462C22—122no—
463C22—182no—
464C22—362no—
465C22—362no—
466C22—364no—
467C22—362no—
468C22—362no—
469C22—364no—
470C22—362no—
471C22C2×C3×#21511yescharacteristic, complemented
472C22A4×#6711yescharacteristic
473C4—22no—
474C4—31no—
475C4—31no—
476C4—62no—
477C4A4×#6711yescharacteristic
478C4C22×D3×A411yescharacteristic
479C3—41no—
480C3—81no—
481C3C2×A4×SD1611yescharacteristic, complemented
482C2—31no—
483C2—32no—
484C2—31no—
485C2—122no—
486C2—362no—
487C2A4×#21512yesdirect product factor
488C2C2×A4×#6711yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC22×C3×C4C23×C3
FrattiniC4C22×D3×A4
FittingC23×C3×Q8C2×C3
socleC24×C3C3×D4
prime radicalC2×A4×D12C2
hypercenterC2×Q8D3×A4
hypocenterC22×C3C2×C3×SD16
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
3. A4×#215C2direct (N × H)
4. C2×A4×C3⋊C8C2semidirect
5. C23×#215C3semidirect
9. C3×Q8×A4C22semidirect
15. A4×C3⋊C8C22semidirect
16. C23×C3×Q8C2×C3semidirect
19. C2×Q8×A4D3semidirect
26. C22×#215C2×C3semidirect
27. C23×C3⋊C8C2×C3semidirect
49. C22×C3×Q8C22×C3semidirect
62. Q8×A4C2×D3semidirect
82. C2×#215A4direct (N × H)
86. C22×C3⋊C8C22×C3semidirect
89. C2×C3×A4SD16semidirect
99. C23×Q8C3×D3semidirect
135. C2×C3×Q8C2×A4semidirect
174. #215C2×A4direct (N × H)
180. C2×C3⋊C8C2×A4semidirect
184. C3×A4C2×SD16semidirect
195. C22×Q8C2×C3×D3semidirect
226. C23×C3C3×SD16semidirect
245. C2×A4#215direct (N × H)
258. C3×Q8C22×A4semidirect
290. C3⋊C8C22×A4semidirect
315. C2×Q8D3×A4semidirect
363. C22×C3C2×C3×SD16semidirect
376. A4C2×#215direct (N × H)
407. C23C3×#215semidirect
416. Q8C2×D3×A4semidirect
450. C2×C3A4×SD16semidirect
471. C22C2×C3×#215semidirect
481. C3C2×A4×SD16semidirect
487. C2A4×#215direct (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×A4×D122
A4×#2152
C2×A4×C3⋊C82
C23×#2153
C2×A4×SD163
C2×C3×#2154