Finite Group Databasebeta

← C22×#73104 #72269820

Subgroups — C22×#73104 #72269820

Order 1536.

3051 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#73104C2112yesmaximal, direct product factor
2C22×A4⋊C16C212yesmaximal, complemented
3C23×C8×A4C211yesmaximal, characteristic
4C22×#2112—31nomaximal, Sylow
5C23×C4×A4C411yescharacteristic
6C22×C8×A4C416yes—
7C22×C8×A4C411yescharacteristic
8C22×C8×A4C2211yescharacteristic
9C22×C8×A4C2216yes—
10C22×#812—41nomaximal
11#73104C22116yesdirect product factor
12C2×A4⋊C16C22112yescomplemented
13C25×C8D311yescharacteristic
14C23×M32—31no—
15C23×M32—31no—
16C2×#2112—312no—
17C2×#2112—312no—
18C22×C22⋊C16—32no—
19C22×C22⋊C16—32no—
20C24×A4C811yescharacteristic
21C22×C4×A4C816yes—
22C22×C4×A4C811yescharacteristic
23C22×C4×A4C2×C416yes—
24C22×C4×A4C2×C411yescharacteristic
25C23×C3×C8—41no—
26C2×C8×A4C2×C413yes—
27C2×C8×A4C2×C4112yes—
28C2×C8×A4C2×C416yes—
29C2×C8×A4C2313yes—
30C2×C8×A4C2314yes—
31C2×#812—412no—
32C22×C3⋊C16—42no—
33A4⋊C16C2318yescomplemented
34C25×C4Dic311yescharacteristic
35C24×C8—31no—
36C24×C8—31no—
37C24×C8—36no—
38C24×C8—31no—
39C24×C8—31no—
40C24×C8—36no—
41C24×C8Dic311yescharacteristic
42C24×C8Dic316yes—
43C24×C8C2×D311yescharacteristic
44C24×C8C2×D316yes—
45C22×M32—312no—
46C22×M32—312no—
47C22×M32—61no—
48C22×M32—61no—
49C22×M32—66no—
50C22×M32—66no—
51C22×M32—61no—
52C22×M32—61no—
53C23×C16—32no—
54C23×C16—32no—
55#2112—348no—
56#2112—316no—
57C2×C22⋊C16—324no—
58C2×C22⋊C16—312no—
59C2×C22⋊C16—312no—
60C23×A4—24no—
61C23×A4C2×C816yes—
62C23×A4C2×C811yescharacteristic
63C23×C3×C4—41no—
64C2×C4×A4C2×C8112yes—
65C2×C4×A4C2×C813yes—
66C2×C4×A4C2×C816yes—
67C2×C4×A4C22×C414yes—
68C2×C4×A4C22×C413yes—
69C22×C3×C8—41no—
70C22×C3×C8—46no—
71C22×C3×C8—46no—
72C22×C3×C8—41no—
73C8×A4C22×C413yes—
74C8×A4C22×C414yes—
75C8×A4C2411yescharacteristic
76#812—416no—
77C2×C3⋊C16—412no—
78C26C3⋊C811yescharacteristic
79C24×C4—31no—
80C24×C4—31no—
81C24×C4—31no—
82C24×C4—36no—
83C24×C4—31no—
84C24×C4—36no—
85C24×C4C3⋊C816yes—
86C24×C4C3⋊C811yescharacteristic
87C24×C4C2×Dic316yes—
88C24×C4C2×Dic311yescharacteristic
89C23×C8—31no—
90C23×C8—31no—
91C23×C8—36no—
92C23×C8—36no—
93C23×C8—31no—
94C23×C8—36no—
95C23×C8—36no—
96C23×C8—31no—
97C23×C8—31no—
98C23×C8—33no—
99C23×C8—36no—
100C23×C8—312no—
101C23×C8—31no—
102C23×C8—36no—
103C23×C8—36no—
104C23×C8—36no—
105C23×C8—36no—
106C23×C8—324no—
107C23×C8—31no—
108C23×C8—36no—
109C23×C8—33no—
110C23×C8—36no—
111C23×C8—312no—
112C23×C8—63no—
113C23×C8—66no—
114C23×C8—612no—
115C23×C8—63no—
116C23×C8—64no—
117C23×C8—61no—
118C23×C8—66no—
119C23×C8C2×Dic313yes—
120C23×C8C2×Dic316yes—
121C23×C8C2×Dic3112yes—
122C23×C8S411yescharacteristic
123C23×C8C22×D313yes—
124C23×C8C22×D314yes—
125C2×M32—316no—
126C2×M32—316no—
127C2×M32—612no—
128C2×M32—612no—
129C2×M32—624no—
130C2×M32—624no—
131C2×M32—612no—
132C2×M32—612no—
133C22×C16—312no—
134C22×C16—312no—
135C22×C16—62no—
136C22×C16—66no—
137C22×C16—66no—
138C22×C16—62no—
139C22⋊C16—332no—
140C22⋊C16—324no—
141C22⋊C16—38no—
142C24×C3—41no—
143C22×A4—212no—
144C22×A4M3214yescomplemented
145C22×A4C22×C814yes—
146C22×A4C22×C813yes—
147C22×C3×C4—41no—
148C22×C3×C4—46no—
149C22×C3×C4—46no—
150C22×C3×C4—41no—
151C4×A4C22×C813yes—
152C4×A4C22×C814yes—
153C4×A4C23×C411yescharacteristic
154C2×C3×C8—412no—
155C2×C3×C8—46no—
156C2×C3×C8—43no—
157C2×C3×C8—43no—
158C2×C3×C8—44no—
159C3⋊C16—48no—
160C25—24no—
161C25—31no—
162C25—31no—
163C25—36no—
164C25—64no—
165C25C2×C3⋊C816yes—
166C25C2×C3⋊C811yescharacteristic
167C23×C4—31no—
168C23×C4—36no—
169C23×C4—31no—
170C23×C4—31no—
171C23×C4—36no—
172C23×C4—36no—
173C23×C4—31no—
174C23×C4—36no—
175C23×C4—31no—
176C23×C4—33no—
177C23×C4—36no—
178C23×C4—312no—
179C23×C4—31no—
180C23×C4—36no—
181C23×C4—36no—
182C23×C4—36no—
183C23×C4—36no—
184C23×C4—324no—
185C23×C4—31no—
186C23×C4—36no—
187C23×C4—33no—
188C23×C4—36no—
189C23×C4—312no—
190C23×C4—63no—
191C23×C4—64no—
192C23×C4—63no—
193C23×C4—66no—
194C23×C4—612no—
195C23×C4—66no—
196C23×C4—61no—
197C23×C4C2×C3⋊C8112yes—
198C23×C4C2×C3⋊C813yes—
199C23×C4C2×C3⋊C816yes—
200C23×C4SL(2,ℤ/4)11yescharacteristic
201C23×C4C22×Dic314yes—
202C23×C4C22×Dic313yes—
203C22×C8—36no—
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—36no—
207C22×C8—312no—
208C22×C8—36no—
209C22×C8—33no—
210C22×C8—36no—
211C22×C8—324no—
212C22×C8—36no—
213C22×C8—36no—
214C22×C8—33no—
215C22×C8—36no—
216C22×C8—312no—
217C22×C8—33no—
218C22×C8—34no—
219C22×C8—33no—
220C22×C8—312no—
221C22×C8—36no—
222C22×C8—33no—
223C22×C8—36no—
224C22×C8—312no—
225C22×C8—36no—
226C22×C8—312no—
227C22×C8—312no—
228C22×C8—312no—
229C22×C8—33no—
230C22×C8—34no—
231C22×C8—33no—
232C22×C8—34no—
233C22×C8—33no—
234C22×C8—34no—
235C22×C8—63no—
236C22×C8—66no—
237C22×C8—612no—
238C22×C8—612no—
239C22×C8—612no—
240C22×C8—64no—
241C22×C8—61no—
242C22×C8—66no—
243C22×C8—63no—
244C22×C8—66no—
245C22×C8—612no—
246C22×C8—66no—
247C22×C8—63no—
248C22×C8—612no—
249C22×C8SL(2,ℤ/4)11yescharacteristic
250C22×C8SL(2,ℤ/4)16yes—
251C22×C8C22×Dic313yes—
252C22×C8C22×Dic314yes—
253C22×C8C2×S416yes—
254C22×C8C2×S411yescharacteristic
255C22×C8C23×D311yescharacteristic
256M32—616no—
257M32—616no—
258M32—616no—
259M32—616no—
260C2×C16—38no—
261C2×C16—38no—
262C2×C16—612no—
263C2×C16—612no—
264C2×C16—612no—
265C2×C16—612no—
266C23×C3—41no—
267C23×C3—46no—
268C23×C3—84no—
269C2×A4—24no—
270C2×A4C2×M3216yescomplemented
271C2×A4C23×C811yescharacteristic
272C2×C3×C4—46no—
273C2×C3×C4—43no—
274C2×C3×C4—412no—
275C2×C3×C4—43no—
276C2×C3×C4—44no—
277C3×C8—43no—
278C3×C8—44no—
279C3×C8—41no—
280C24—212no—
281C24—31no—
282C24—33no—
283C24—36no—
284C24—34no—
285C24—38no—
286C24—36no—
287C24—36no—
288C24—31no—
289C24—312no—
290C24—64no—
291C24—64no—
292C24—612no—
293C24—63no—
294C24—64no—
295C24—64no—
296C24—624no—
297C24—624no—
298C24#81214yescomplemented
299C24A4⋊C811yescharacteristic
300C24C22×C3⋊C814yes—
301C24C22×C3⋊C813yes—
302C22×C4—36no—
303C22×C4—36no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—36no—
307C22×C4—33no—
308C22×C4—312no—
309C22×C4—33no—
310C22×C4—36no—
311C22×C4—312no—
312C22×C4—36no—
313C22×C4—324no—
314C22×C4—36no—
315C22×C4—36no—
316C22×C4—33no—
317C22×C4—34no—
318C22×C4—33no—
319C22×C4—312no—
320C22×C4—36no—
321C22×C4—33no—
322C22×C4—34no—
323C22×C4—36no—
324C22×C4—33no—
325C22×C4—312no—
326C22×C4—36no—
327C22×C4—312no—
328C22×C4—312no—
329C22×C4—312no—
330C22×C4—33no—
331C22×C4—34no—
332C22×C4—33no—
333C22×C4—34no—
334C22×C4—61no—
335C22×C4—66no—
336C22×C4—63no—
337C22×C4—612no—
338C22×C4—66no—
339C22×C4—64no—
340C22×C4—612no—
341C22×C4—612no—
342C22×C4—63no—
343C22×C4—63no—
344C22×C4—612no—
345C22×C4—66no—
346C22×C4—612no—
347C22×C4—66no—
348C22×C4A4⋊C816yes—
349C22×C4A4⋊C811yescharacteristic
350C22×C4C22×C3⋊C813yes—
351C22×C4C22×C3⋊C814yes—
352C22×C4C2×SL(2,ℤ/4)16yes—
353C22×C4C2×SL(2,ℤ/4)11yescharacteristic
354C22×C4C23×Dic311yescharacteristic
355C2×C8—33no—
356C2×C8—34no—
357C2×C8—33no—
358C2×C8—312no—
359C2×C8—36no—
360C2×C8—33no—
361C2×C8—34no—
362C2×C8—33no—
363C2×C8—34no—
364C2×C8—36no—
365C2×C8—312no—
366C2×C8—312no—
367C2×C8—312no—
368C2×C8—31no—
369C2×C8—33no—
370C2×C8—34no—
371C2×C8—33no—
372C2×C8—34no—
373C2×C8—31no—
374C2×C8—63no—
375C2×C8—64no—
376C2×C8—63no—
377C2×C8—612no—
378C2×C8—66no—
379C2×C8C2×SL(2,ℤ/4)112yes—
380C2×C8C2×SL(2,ℤ/4)16yes—
381C2×C8C2×SL(2,ℤ/4)13yes—
382C2×C8C22×S413yes—
383C2×C8C22×S414yes—
384C16—68no—
385C16—68no—
386C22×C3—44no—
387C22×C3—43no—
388C22×C3—44no—
389C22×C3—812no—
390A4C22×M3211yescharacteristic, complemented
391C3×C4—41no—
392C3×C4—44no—
393C3×C4—43no—
394C23—24no—
395C23—24no—
396C23—34no—
397C23—34no—
398C23—33no—
399C23—33no—
400C23—33no—
401C23—34no—
402C23—312no—
403C23—36no—
404C23—324no—
405C23—34no—
406C23—312no—
407C23—36no—
408C23—612no—
409C23—612no—
410C23—624no—
411C23—64no—
412C23—61no—
413C23—66no—
414C23—66no—
415C23—624no—
416C23—612no—
417C23—624no—
418C23—624no—
419C23—632no—
420C23—64no—
421C23C2×#81216yescomplemented
422C23C2×A4⋊C816yes—
423C23C2×A4⋊C811yescharacteristic
424C23C23×C3⋊C811yescharacteristic
425C2×C4—312no—
426C2×C4—34no—
427C2×C4—36no—
428C2×C4—33no—
429C2×C4—33no—
430C2×C4—31no—
431C2×C4—33no—
432C2×C4—34no—
433C2×C4—34no—
434C2×C4—33no—
435C2×C4—36no—
436C2×C4—312no—
437C2×C4—33no—
438C2×C4—312no—
439C2×C4—312no—
440C2×C4—34no—
441C2×C4—34no—
442C2×C4—33no—
443C2×C4—31no—
444C2×C4—63no—
445C2×C4—63no—
446C2×C4—66no—
447C2×C4—64no—
448C2×C4—612no—
449C2×C4C2×A4⋊C8112yes—
450C2×C4C2×A4⋊C816yes—
451C2×C4C2×A4⋊C813yes—
452C2×C4C22×SL(2,ℤ/4)14yes—
453C2×C4C22×SL(2,ℤ/4)13yes—
454C8—31no—
455C8—33no—
456C8—34no—
457C8C22×SL(2,ℤ/4)13yes—
458C8C22×SL(2,ℤ/4)14yes—
459C8C23×S411yescharacteristic
460C2×C3—41no—
461C2×C3—46no—
462C2×C3—84no—
463C22—212no—
464C22—31no—
465C22—36no—
466C22—31no—
467C22—33no—
468C22—36no—
469C22—36no—
470C22—312no—
471C22—38no—
472C22—34no—
473C22—64no—
474C22—64no—
475C22—624no—
476C22—63no—
477C22—64no—
478C22—64no—
479C22—624no—
480C22—612no—
481C22#7310414yesdirect product factor
482C22C22×#81211yescharacteristic, complemented
483C22C22×A4⋊C814yes—
484C22C22×A4⋊C813yes—
485C4—31no—
486C4—33no—
487C4—34no—
488C4C22×A4⋊C814yes—
489C4C22×A4⋊C813yes—
490C4C23×SL(2,ℤ/4)11yescharacteristic
491C3—41noSylow
492C2—24no—
493C2—31no—
494C2—31no—
495C2—36no—
496C2—64no—
497C2C2×#7310416yesdirect product factor
498C2C23×A4⋊C811yescharacteristic
subgroupG/H
centerC22×C8C2×S4
commutatorC2×A4C23×C8
FrattiniC8C23×S4
FittingC25×C8D3
socleC25C2×C3⋊C8
prime radicalC24×A4C8
cosocleC8×A4C24
hypercenterC23×C8S4
hypocenterA4C22×M32
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C8 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. C2×#73104C2direct (N × H)
2. C22×A4⋊C16C2semidirect
11. #73104C22direct (N × H)
12. C2×A4⋊C16C22semidirect
33. A4⋊C16C23semidirect
144. C22×A4M32semidirect
270. C2×A4C2×M32semidirect
298. C24#812semidirect
390. A4C22×M32semidirect
421. C23C2×#812semidirect
481. C22#73104direct (N × H)
482. C22C22×#812semidirect
497. C2C2×#73104direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#731042
C22×A4⋊C162
C23×C8×A42
C22×#21123
C22×#8124