Finite Group Databasebeta

← C3×A4×C22⋊Q8 #11919696

Subgroups — C3×A4×C22⋊Q8 #11919696

Order 1152.

1008 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×A4C211yesmaximal, characteristic, complemented
2C22×C3×C4×A4C211yesmaximal, characteristic
3C3×A4×C4⋊C4C212yesmaximal, complemented
4C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×A4×C22⋊C4C212yesmaximal
6C22×C3×C22⋊Q8C311yesmaximal, characteristic, complemented
7A4×C22⋊Q8C313yesmaximal, direct product factor
8C23×C3×A4C2211yescharacteristic
9C3×Q8×A4—22no—
10C2×C3×C4×A4—22no—
11C2×C3×C4×A4C2211yescharacteristic
12C2×C3×C4×A4C2212yes—
13C2×C3×C4×A4C2212yes—
14C2×C3×C4×A4C2211yescharacteristic
15C32×C22⋊Q8—41nomaximal
16C23×C3×Q8C2×C311yescharacteristic, complemented
17C24×C3×C4C2×C311yescharacteristic
18C2×Q8×A4C2×C313yescomplemented
19C22×C4×A4C2×C313yes—
20C2×C3×C22⋊Q8—31no—
21C2×C3×C22⋊Q8—32no—
22C2×C3×C22⋊Q8—31no—
23C2×C3×C22⋊Q8—32no—
24C2×C3×C22⋊Q8—31no—
25C2×C3×C22⋊Q8—31no—
26C22×C3×C4⋊C4C2×C312yescomplemented
27C22×C3×C4⋊C4C2×C311yescharacteristic, complemented
28C22×C3×C22⋊C4C2×C312yes—
29A4×C4⋊C4C2×C316yescomplemented
30A4×C4⋊C4C2×C313yescomplemented
31A4×C22⋊C4C2×C316yes—
32C22×C3×A4—21no—
33C22×C3×A4—21no—
34C22×C3×A4Q812yescomplemented
35C22×C3×A4C2311yescharacteristic
36C2×C32×Q8—41no—
37C22×C32×C4—41no—
38C3×C4×A4—21no—
39C3×C4×A4—22no—
40C3×C4×A4—22no—
41C3×C4×A4D412yes—
42C32×C4⋊C4—41no—
43C32×C4⋊C4—42no—
44C32×C22⋊C4—42no—
45C22×C22⋊Q8C3211yescharacteristic, complemented, Sylow
46C25×C3C22×C311yescharacteristic
47C23×A4C22×C313yes—
48C22×C3×Q8—22no—
49C22×C3×Q8—31no—
50C22×C3×Q8—32no—
51C22×C3×Q8—31no—
52C22×C3×Q8—62no—
53C23×C3×C4—22no—
54C23×C3×C4—31no—
55C23×C3×C4—31no—
56C23×C3×C4—31no—
57C23×C3×C4—31no—
58C23×C3×C4—62no—
59C23×C3×C4C22×C311yescharacteristic
60C23×C3×C4C22×C312yes—
61C23×C3×C4C22×C312yes—
62C23×C3×C4C22×C311yescharacteristic
63Q8×A4—26no—
64C2×C4×A4—26no—
65C2×C4×A4C22×C313yes—
66C2×C4×A4C22×C316yes—
67C2×C4×A4C22×C316yes—
68C2×C4×A4C22×C313yes—
69C3×C22⋊Q8—32no—
70C3×C22⋊Q8—32no—
71C3×C22⋊Q8—34no—
72C3×C22⋊Q8—34no—
73C3×C22⋊Q8—32no—
74C3×C22⋊Q8—31no—
75C3×C22⋊Q8—31no—
76C3×C22⋊Q8—32no—
77C3×C22⋊Q8—31no—
78C3×C22⋊Q8—32no—
79C3×C22⋊Q8—43no—
80C3×C22⋊Q8A411yescharacteristic, direct product factor
81C2×C3×C4⋊C4—31no—
82C2×C3×C4⋊C4—32no—
83C2×C3×C4⋊C4—31no—
84C2×C3×C4⋊C4—32no—
85C2×C3×C4⋊C4—32no—
86C2×C3×C4⋊C4—32no—
87C2×C3×C4⋊C4—32no—
88C2×C3×C22⋊C4—32no—
89C2×C3×C22⋊C4—32no—
90C2×C3×C22⋊C4—32no—
91C2×C3×C22⋊C4—32no—
92C23×C32—41no—
93C2×C3×A4—22no—
94C2×C3×A4C2×D411yescharacteristic
95C2×C3×A4C2×Q811yescharacteristic
96C2×C3×A4#4111yescharacteristic
97C32×Q8—82no—
98C2×C32×C4—41no—
99C2×C32×C4—42no—
100C2×C32×C4—42no—
101C2×C32×C4—41no—
102C2×C32×C4—82no—
103C23×Q8C2×C3211yescharacteristic, complemented
104C24×C4C2×C3211yescharacteristic
105C2×C22⋊Q8—31no—
106C2×C22⋊Q8—32no—
107C2×C22⋊Q8—31no—
108C2×C22⋊Q8—32no—
109C2×C22⋊Q8—31no—
110C2×C22⋊Q8—31no—
111C22×C4⋊C4C2×C3211yescharacteristic, complemented
112C22×C4⋊C4C2×C3212yescomplemented
113C22×C22⋊C4C2×C3212yes—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—32no—
117C24×C3—31no—
118C24×C3—31no—
119C24×C3—61no—
120C24×C3—61no—
121C24×C3C3×Q812yescomplemented
122C24×C3C23×C311yescharacteristic
123C22×A4—23no—
124C22×A4—23no—
125C22×A4C3×Q816yescomplemented
126C22×A4C23×C313yes—
127C2×C3×Q8—32no—
128C2×C3×Q8—32no—
129C2×C3×Q8—31no—
130C2×C3×Q8—43no—
131C2×C3×Q8—62no—
132C2×C3×Q8—62no—
133C2×C3×Q8—62no—
134C2×C3×Q8—64no—
135C2×C3×Q8—62no—
136C2×C3×Q8—64no—
137C2×C3×Q8C2×A411yescharacteristic, complemented
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—32no—
144C22×C3×C4—32no—
145C22×C3×C4—31no—
146C22×C3×C4—32no—
147C22×C3×C4—32no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—32no—
152C22×C3×C4—32no—
153C22×C3×C4—31no—
154C22×C3×C4—43no—
155C22×C3×C4—62no—
156C22×C3×C4—62no—
157C22×C3×C4—62no—
158C22×C3×C4—62no—
159C22×C3×C4—62no—
160C22×C3×C4—62no—
161C22×C3×C4—62no—
162C22×C3×C4—62no—
163C22×C3×C4—61no—
164C22×C3×C4—62no—
165C22×C3×C4—62no—
166C22×C3×C4C3×D412yes—
167C22×C3×C4C2×A411yescharacteristic
168C4×A4—23no—
169C4×A4—26no—
170C4×A4—26no—
171C4×A4C3×D416yes—
172C3×C4⋊C4—32no—
173C3×C4⋊C4—34no—
174C3×C4⋊C4—32no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—32no—
177C3×C4⋊C4—32no—
178C3×C4⋊C4—32no—
179C3×C4⋊C4—43no—
180C3×C4⋊C4—46no—
181C3×C4⋊C4C2×A411yescharacteristic, complemented
182C3×C4⋊C4C2×A412yescomplemented
183C3×C22⋊C4—34no—
184C3×C22⋊C4—32no—
185C3×C22⋊C4—32no—
186C3×C22⋊C4—32no—
187C3×C22⋊C4—46no—
188C3×C22⋊C4C2×A412yes—
189C22×C32—42no—
190C22×C32—41no—
191C22×C32—81no—
192C22×C32—81no—
193C3×A4C22⋊Q811yescharacteristic, direct product factor
194C32×C4—42no—
195C32×C4—81no—
196C32×C4—82no—
197C32×C4—82no—
198C25C22×C3211yescharacteristic
199C22×Q8—22no—
200C22×Q8—31no—
201C22×Q8—32no—
202C22×Q8—31no—
203C22×Q8—62no—
204C23×C4—22no—
205C23×C4—31no—
206C23×C4—31no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—62no—
210C23×C4C22×C3212yes—
211C23×C4C22×C3212yes—
212C23×C4C22×C3211yescharacteristic
213C23×C4C22×C3211yescharacteristic
214C22⋊Q8—32no—
215C22⋊Q8—32no—
216C22⋊Q8—34no—
217C22⋊Q8—34no—
218C22⋊Q8—32no—
219C22⋊Q8—31no—
220C22⋊Q8—31no—
221C22⋊Q8—32no—
222C22⋊Q8—31no—
223C22⋊Q8—32no—
224C22⋊Q8C3×A411yescharacteristic, direct product factor
225C2×C4⋊C4—31no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C22⋊C4—32no—
233C2×C22⋊C4—32no—
234C2×C22⋊C4—32no—
235C2×C22⋊C4—32no—
236C23×C3—22no—
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—31no—
240C23×C3—31no—
241C23×C3—32no—
242C23×C3—32no—
243C23×C3—32no—
244C23×C3—32no—
245C23×C3—31no—
246C23×C3—43no—
247C23×C3—61no—
248C23×C3—61no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—62no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—62no—
255C23×C3—62no—
256C23×C3—61no—
257C23×C3—61no—
258C23×C3—64no—
259C23×C3C2×C3×D411yescharacteristic
260C23×C3C2×C3×Q811yescharacteristic
261C23×C3C3×#4111yescharacteristic
262C23×C3C22×A411yescharacteristic
263C2×A4—26no—
264C2×A4C2×C3×D413yes—
265C2×A4C2×C3×Q813yes—
266C2×A4C3×#4113yes—
267C3×Q8—22no—
268C3×Q8—64no—
269C3×Q8—64no—
270C3×Q8—62no—
271C3×Q8—86no—
272C2×C3×C4—22no—
273C2×C3×C4—32no—
274C2×C3×C4—31no—
275C2×C3×C4—32no—
276C2×C3×C4—32no—
277C2×C3×C4—32no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—31no—
281C2×C3×C4—43no—
282C2×C3×C4—46no—
283C2×C3×C4—46no—
284C2×C3×C4—43no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—61no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—61no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—64no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—61no—
297C2×C3×C4—62no—
298C2×C3×C4—62no—
299C2×C3×C4—61no—
300C2×C3×C4—62no—
301C2×C3×C4—86no—
302C2×C3×C4C22×A412yes—
303C2×C3×C4C22×A412yes—
304C2×C3×C4C22×A411yescharacteristic
305C2×C3×C4C22×A411yescharacteristic
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—82no—
310C24—21no—
311C24—21no—
312C24—31no—
313C24—32no—
314C24—31no—
315C24—61no—
316C24—61no—
317C24C32×Q812yescomplemented
318C24C23×C3211yescharacteristic
319C2×Q8—32no—
320C2×Q8—32no—
321C2×Q8—31no—
322C2×Q8—62no—
323C2×Q8—62no—
324C2×Q8—62no—
325C2×Q8—64no—
326C2×Q8—62no—
327C2×Q8—64no—
328C2×Q8C2×C3×A411yescharacteristic, complemented
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—31no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—31no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4C32×D412yes—
357C22×C4C2×C3×A411yescharacteristic
358C4⋊C4—32no—
359C4⋊C4—34no—
360C4⋊C4—32no—
361C4⋊C4—31no—
362C4⋊C4—32no—
363C4⋊C4—32no—
364C4⋊C4—32no—
365C4⋊C4C2×C3×A411yescharacteristic, complemented
366C4⋊C4C2×C3×A412yescomplemented
367C22⋊C4—34no—
368C22⋊C4—32no—
369C22⋊C4—32no—
370C22⋊C4—32no—
371C22⋊C4C2×C3×A412yes—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—32no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—31no—
381C22×C3—31no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—32no—
385C22×C3—46no—
386C22×C3—43no—
387C22×C3—62no—
388C22×C3—62no—
389C22×C3—62no—
390C22×C3—62no—
391C22×C3—62no—
392C22×C3—62no—
393C22×C3—62no—
394C22×C3—62no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—83no—
398C22×C3—83no—
399C22×C3C3×C22⋊Q811yescharacteristic, complemented
400C22×C3Q8×A412yescomplemented
401C22×C3C23×A411yescharacteristic
402A4C3×C22⋊Q813yesdirect product factor
403C3×C4—21no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—32no—
407C3×C4—46no—
408C3×C4—61no—
409C3×C4—62no—
410C3×C4—62no—
411C3×C4—83no—
412C3×C4—86no—
413C3×C4—86no—
414C3×C4D4×A412yes—
415C32—41noSylow
416C23—22no—
417C23—31no—
418C23—31no—
419C23—31no—
420C23—31no—
421C23—32no—
422C23—32no—
423C23—32no—
424C23—32no—
425C23—31no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—62no—
431C23—61no—
432C23—61no—
433C23—62no—
434C23—62no—
435C23—61no—
436C23—61no—
437C23—64no—
438C23C2×C32×D411yescharacteristic
439C23C2×C32×Q811yescharacteristic
440C23C32×#4111yescharacteristic
441C23C22×C3×A411yescharacteristic
442Q8—22no—
443Q8—64no—
444Q8—64no—
445Q8—62no—
446C2×C4—22no—
447C2×C4—32no—
448C2×C4—31no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—32no—
454C2×C4—31no—
455C2×C4—62no—
456C2×C4—61no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—61no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—64no—
464C2×C4—62no—
465C2×C4—61no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—61no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4C22×C3×A412yes—
472C2×C4C22×C3×A412yes—
473C2×C4C22×C3×A411yescharacteristic
474C2×C4C22×C3×A411yescharacteristic
475C2×C3—22no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—31no—
479C2×C3—31no—
480C2×C3—43no—
481C2×C3—43no—
482C2×C3—43no—
483C2×C3—62no—
484C2×C3—86no—
485C2×C3C2×D4×A411yescharacteristic
486C2×C3C2×Q8×A411yescharacteristic
487C2×C3A4×#4111yescharacteristic
488C22—21no—
489C22—21no—
490C22—31no—
491C22—31no—
492C22—31no—
493C22—32no—
494C22—31no—
495C22—31no—
496C22—31no—
497C22—31no—
498C22—31no—
499C22—31no—
500C22—32no—
501C22—62no—
502C22—62no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22—61no—
510C22—61no—
511C22C32×C22⋊Q811yescharacteristic, complemented
512C22C3×Q8×A412yescomplemented
513C22C23×C3×A411yescharacteristic
514C4—21no—
515C4—22no—
516C4—22no—
517C4—32no—
518C4—61no—
519C4—62no—
520C4—62no—
521C4C3×D4×A412yes—
522C3—43no—
523C3A4×C22⋊Q811yescharacteristic, direct product factor
524C2—22no—
525C2—31no—
526C2—31no—
527C2—31no—
528C2—31no—
529C2—62no—
530C2C2×C3×D4×A411yescharacteristic
531C2C2×C3×Q8×A411yescharacteristic
532C2C3×A4×#4111yescharacteristic
subgroupG/H
centerC22×C3C23×A4
commutator, cosocleC24C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×C22⋊Q8C3
socleC24×C3C23×C3
prime radicalC23×C3×A4C22
hypercenterC3×C22⋊Q8A4
hypocenterC22C32×C22⋊Q8
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
3. C3×A4×C4⋊C4C2semidirect
4. C3×A4×C4⋊C4C2semidirect
6. C22×C3×C22⋊Q8C3semidirect
7. A4×C22⋊Q8C3direct (N × H)
16. C23×C3×Q8C2×C3semidirect
18. C2×Q8×A4C2×C3semidirect
26. C22×C3×C4⋊C4C2×C3semidirect
27. C22×C3×C4⋊C4C2×C3semidirect
29. A4×C4⋊C4C2×C3semidirect
30. A4×C4⋊C4C2×C3semidirect
34. C22×C3×A4Q8semidirect
45. C22×C22⋊Q8C32semidirect
80. C3×C22⋊Q8A4direct (N × H)
103. C23×Q8C2×C32semidirect
111. C22×C4⋊C4C2×C32semidirect
112. C22×C4⋊C4C2×C32semidirect
121. C24×C3C3×Q8semidirect
125. C22×A4C3×Q8semidirect
137. C2×C3×Q8C2×A4semidirect
181. C3×C4⋊C4C2×A4semidirect
182. C3×C4⋊C4C2×A4semidirect
193. C3×A4C22⋊Q8direct (N × H)
224. C22⋊Q8C3×A4direct (N × H)
317. C24C32×Q8semidirect
328. C2×Q8C2×C3×A4semidirect
365. C4⋊C4C2×C3×A4semidirect
366. C4⋊C4C2×C3×A4semidirect
399. C22×C3C3×C22⋊Q8semidirect
400. C22×C3Q8×A4semidirect
402. A4C3×C22⋊Q8direct (N × H)
511. C22C32×C22⋊Q8semidirect
512. C22C3×Q8×A4semidirect
523. C3A4×C22⋊Q8direct (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
C22×C3×C4×A42
C3×A4×C4⋊C42
C3×A4×C22⋊C42
C22×C3×C22⋊Q83
A4×C22⋊Q83
C32×C22⋊Q84