Finite Group Databasebeta

← C22×#16839532 #29677465

Subgroups — C22×#16839532 #29677465

Order 15616.

2190 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
1C23×#11758194C211yesmaximal, characteristic
2C23×#11758194C211yesmaximal, characteristic
3C2×#16839532C2112yesmaximal, direct product factor
4C22×#117×C61C211yesmaximal, characteristic
5C22×#11758194—21no—
6C22×#11758194—23no—
7C22×#11758194—23no—
8C22×#11758194—21no—
9C22×#11758194—21no—
10C22×#11758194—23no—
11C22×#11758194—23no—
12C22×#11758194C412yescomplemented
13C22×#11758194C416yescomplemented
14C22×#11758194C416yescomplemented
15C22×#11758194C2216yes—
16C22×#11758194C2216yes—
17C23×#92418—21no—
18C23×#92418C412yescomplemented
19#16839532C22116yesdirect product factor
20C24×C4×C61C2211yescharacteristic
21C2×#117×C61—23no—
22C2×#117×C61—23no—
23C2×#117×C61C416yes—
24C2×#117×C61C2216yes—
25C22×C4⋊C4×C61—21no—
26C22×C22⋊C4×C61C412yes—
27C22×C42×C61—21no—
28C2×#11758194—26no—
29C2×#11758194—26no—
30C2×#11758194—212no—
31C2×#11758194—26no—
32C2×#11758194—26no—
33C2×#11758194—212no—
34C2×#11758194—26no—
35C2×#11758194—26no—
36C2×#11758194—26no—
37C2×#11758194—26no—
38C2×#11758194—26no—
39C2×#11758194—26no—
40C2×#11758194—26no—
41C2×#11758194C2×C4112yescomplemented
42C2×#11758194C2×C4112yescomplemented
43C2×#11758194C2×C4112yescomplemented
44C2×#11758194C2314yes—
45C2×#11758194C2314yes—
46C22×#92418—26no—
47C22×#92418—22no—
48C22×#92418—26no—
49C22×#92418—22no—
50C22×#92418—26no—
51C22×#92418C2×C4112yescomplemented
52C25×C61C2×C411yescharacteristic
53C23×C4×C61—22no—
54C23×C4×C61C2×C411yescharacteristic
55C23×C4×C61C2×C411yescharacteristic
56C23×C4×C61C2×C413yes—
57C23×C4×C61C2×C413yes—
58C23×C4×C61C2×C413yes—
59C23×C4×C61D411yescharacteristic
60C23×C4×C61D413yes—
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D413yes—
64C23×C4×C61D413yes—
65C23×C4×C61Q811yescharacteristic
66C23×C4×C61Q813yes—
67C23×C4×C61C2313yes—
68#117×C61—212no—
69#117×C61—26no—
70#117×C61—26no—
71#117×C61C2×C4112yes—
72#117×C61C2314yes—
73C2×C4⋊C4×C61—26no—
74C2×C4⋊C4×C61—26no—
75C2×C22⋊C4×C61—26no—
76C2×C22⋊C4×C61C2×C4112yes—
77C2×C42×C61—26no—
78C2×C42×C61—26no—
79C24×C61—44no—
80C24×C61C4211yescharacteristic
81C24×C61C4213yes—
82C24×C61C22⋊C411yescharacteristic
83C24×C61C22⋊C413yes—
84C24×C61C4⋊C411yescharacteristic
85C24×C61C4⋊C413yes—
86C24×C61C22×C413yes—
87C22×C4×C61—212no—
88C22×C4×C61—48no—
89C22×C4×C61C4213yes—
90C22×C4×C61C4213yes—
91C22×C4×C61C4216yes—
92C22×C4×C61C22⋊C413yes—
93C22×C4×C61C22⋊C413yes—
94C22×C4×C61C22⋊C416yes—
95C22×C4×C61C22⋊C413yes—
96C22×C4×C61C22⋊C413yes—
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22⋊C413yes—
99C22×C4×C61C22⋊C413yes—
100C22×C4×C61C22⋊C413yes—
101C22×C4×C61C22⋊C413yes—
102C22×C4×C61C22⋊C416yes—
103C22×C4×C61C22⋊C416yes—
104C22×C4×C61C22⋊C411yescharacteristic
105C22×C4×C61C4⋊C413yes—
106C22×C4×C61C4⋊C413yes—
107C22×C4×C61C4⋊C416yes—
108C22×C4×C61C4⋊C413yes—
109C22×C4×C61C4⋊C413yes—
110C22×C4×C61C4⋊C413yes—
111C22×C4×C61C4⋊C413yes—
112C22×C4×C61C4⋊C413yes—
113C22×C4×C61C4⋊C413yes—
114C22×C4×C61C4⋊C416yes—
115C22×C4×C61C4⋊C416yes—
116C22×C4×C61C4⋊C411yescharacteristic
117C22×C4×C61C4⋊C411yescharacteristic
118C22×C4×C61C22×C413yes—
119C22×C4×C61C22×C413yes—
120C22×C4×C61C22×C413yes—
121C22×C4×C61C22×C413yes—
122C22×C4×C61C22×C413yes—
123C22×C4×C61C2×D413yes—
124C22×C4×C61C2×D413yes—
125C22×C4×C61C2×D413yes—
126C22×C4×C61C2×D413yes—
127C22×C4×C61C2×D413yes—
128C22×C4×C61C2×D413yes—
129C22×C4×C61C2×Q813yes—
130C22×C4×C61C2×Q813yes—
131C22×C4×C61C2411yescharacteristic
132C4⋊C4×C61—212no—
133C4⋊C4×C61—24no—
134C22⋊C4×C61—212no—
135C22⋊C4×C61C22×C418yes—
136C42×C61—212no—
137C42×C61—24no—
138#11758194—212no—
139#11758194—212no—
140#11758194—212no—
141#11758194—212no—
142#11758194—24no—
143#11758194—24no—
144#11758194—24no—
145#11758194C22×C418yescomplemented
146C2×#92418—212no—
147C2×#92418—212no—
148C2×#92418—212no—
149C2×#92418—212no—
150C2×#92418—24no—
151C2×#92418C22×C418yescomplemented
152C23×C61—24no—
153C23×C61—24no—
154C23×C61—24no—
155C23×C61—46no—
156C23×C61—46no—
157C23×C61—46no—
158C23×C61—46no—
159C23×C61#9513yes—
160C23×C61#9513yes—
161C23×C61#9513yes—
162C23×C61#9516yes—
163C23×C61#9511yescharacteristic
164C23×C61C2×C4213yes—
165C23×C61C2×C4213yes—
166C23×C61C2×C22⋊C413yes—
167C23×C61C2×C22⋊C413yes—
168C23×C61C2×C4⋊C413yes—
169C23×C61C2×C4⋊C413yes—
170C23×C61C23×C411yescharacteristic
171C2×C4×C61—28no—
172C2×C4×C61—424no—
173C2×C4×C61#9516yes—
174C2×C4×C61#9516yes—
175C2×C4×C61#9516yes—
176C2×C4×C61#9516yes—
177C2×C4×C61#9516yes—
178C2×C4×C61#9516yes—
179C2×C4×C61#9513yes—
180C2×C4×C61#9513yes—
181C2×C4×C61#9513yes—
182C2×C4×C61#9513yes—
183C2×C4×C61C2×C4213yes—
184C2×C4×C61C2×C4213yes—
185C2×C4×C61C2×C22⋊C413yes—
186C2×C4×C61C2×C22⋊C413yes—
187C2×C4×C61C2×C22⋊C413yes—
188C2×C4×C61C2×C22⋊C413yes—
189C2×C4×C61C2×C22⋊C413yes—
190C2×C4×C61C2×C22⋊C413yes—
191C2×C4×C61C2×C22⋊C413yes—
192C2×C4×C61C2×C22⋊C413yes—
193C2×C4×C61C2×C22⋊C413yes—
194C2×C4×C61C2×C22⋊C413yes—
195C2×C4×C61C2×C4⋊C413yes—
196C2×C4×C61C2×C4⋊C413yes—
197C2×C4×C61C2×C4⋊C413yes—
198C2×C4×C61C2×C4⋊C413yes—
199C2×C4×C61C2×C4⋊C413yes—
200C2×C4×C61C2×C4⋊C413yes—
201C2×C4×C61C2×C4⋊C413yes—
202C2×C4×C61C2×C4⋊C413yes—
203C2×C4×C61C2×C4⋊C413yes—
204C2×C4×C61C2×C4⋊C413yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C23×C411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×Q811yescharacteristic
211#92418—28no—
212#92418—28no—
213C22×#344—611nomaximal, Sylow
214C22×C61—26no—
215C22×C61—26no—
216C22×C61—26no—
217C22×C61—26no—
218C22×C61—26no—
219C22×C61—26no—
220C22×C61—46no—
221C22×C61—43no—
222C22×C61—43no—
223C22×C61—43no—
224C22×C61—41no—
225C22×C61#34414yescomplemented
226C22×C61C2×#9513yes—
227C22×C61C2×#9513yes—
228C22×C61C2×#9513yes—
229C22×C61C2×#9513yes—
230C22×C61C22×C4211yescharacteristic
231C22×C61C22×C22⋊C411yescharacteristic
232C22×C61C22×C4⋊C411yescharacteristic
233C4×C61—48no—
234C4×C61C2×#9513yes—
235C4×C61C2×#9513yes—
236C4×C61C2×#9513yes—
237C4×C61C2×#9513yes—
238C4×C61C22×C22⋊C411yescharacteristic
239C4×C61C22×C22⋊C411yescharacteristic
240C4×C61C22×C4⋊C411yescharacteristic
241C4×C61C22×C4⋊C411yescharacteristic
242C23×M16—611no—
243C23×M16—611no—
244C22×#117D6111yescharacteristic
245C2×#344—6112no—
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—23no—
249C2×C61—23no—
250C2×C61—23no—
251C2×C61—21no—
252C2×C61C2×#34416yescomplemented
253C2×C61C22×#9511yescharacteristic
254C24×C4C2×D6111yescharacteristic
255C22×M16—616no—
256C22×M16—616no—
257C22×M16—612no—
258C22×M16—616no—
259C22×M16—616no—
260C22×M16—1223no—
261C22×M16—1221no—
262C22×M16—1223no—
263C22×M16—1223no—
264C22×M16—1221no—
265C22×M16—1223no—
266C22×M16—1221no—
267C23×C8—612no—
268C23×C8—1221no—
269C2×#117—23no—
270C2×#117—23no—
271C2×#117Dic6116yes—
272C2×#117C2×D6116yes—
273C22×C4⋊C4—21no—
274C22×C22⋊C4Dic6112yes—
275C22×C42—21no—
276#344—6116no—
277C61C22×#34411yescharacteristic, complemented, Sylow
278C25C2×Dic6111yescharacteristic
279C23×C4—22no—
280C23×C4Dic12211yescharacteristic
281C23×C4Dic12213yes—
282C23×C4C4×D6111yescharacteristic
283C23×C4C4×D6113yes—
284C23×C4C4×D6113yes—
285C23×C4C4×D6111yescharacteristic
286C23×C4D24413yes—
287C23×C4D24411yescharacteristic
288C23×C4C2×Dic6113yes—
289C23×C4C22⋊D6111yescharacteristic
290C23×C4C22⋊D6113yes—
291C23×C4C22⋊D6113yes—
292C23×C4C22⋊D6111yescharacteristic
293C23×C4C22×D6113yes—
294C2×M16—6112no—
295C2×M16—614no—
296C2×M16—6112no—
297C2×M16—6112no—
298C2×M16—614no—
299C2×M16—12212no—
300C2×M16—1226no—
301C2×M16—1226no—
302C2×M16—1226no—
303C2×M16—1226no—
304C2×M16—12212no—
305C2×M16—1226no—
306C2×M16—1226no—
307C2×M16—1226no—
308C2×M16—1226no—
309C2×M16—1226no—
310C2×M16—1226no—
311C2×M16—1226no—
312C22×C8—6112no—
313C22×C8—1226no—
314C22×C8—1222no—
315C22×C8—1226no—
316C22×C8—1222no—
317C22×C8—1226no—
318#117—212no—
319#117—26no—
320#117—26no—
321#117C2×Dic61112yes—
322#117C22×D6114yes—
323C2×C4⋊C4—26no—
324C2×C4⋊C4—26no—
325C2×C22⋊C4—26no—
326C2×C22⋊C4C2×Dic61112yes—
327C2×C42—26no—
328C2×C42—26no—
329C24—44no—
330C24C4×Dic6111yescharacteristic
331C24C4×Dic6113yes—
332C24C4⋊Dic6111yescharacteristic
333C24C4⋊Dic6113yes—
334C24C22⋊Dic6111yescharacteristic
335C24C22⋊Dic6113yes—
336C24C22×Dic6113yes—
337C22×C4—212no—
338C22×C4—48no—
339C22×C4C4×Dic6113yes—
340C22×C4C4×Dic6116yes—
341C22×C4C4×Dic6113yes—
342C22×C4#1175819613yes—
343C22×C4#1175819613yes—
344C22×C4#1175819613yes—
345C22×C4#1175819613yes—
346C22×C4#1175819616yes—
347C22×C4#1175819616yes—
348C22×C4#1175819611yescharacteristic
349C22×C4#1175819611yescharacteristic
350C22×C4#1175819613yes—
351C22×C4#1175819613yes—
352C22×C4C4⋊Dic6113yes—
353C22×C4C4⋊Dic6116yes—
354C22×C4C4⋊Dic6113yes—
355C22×C4#1175819811yescharacteristic
356C22×C4#1175819813yes—
357C22×C4#1175819813yes—
358C22×C4#1175819813yes—
359C22×C4#1175819813yes—
360C22×C4#1175819816yes—
361C22×C4#1175819816yes—
362C22×C4#1175819813yes—
363C22×C4#1175819813yes—
364C22×C4#1175819811yescharacteristic
365C22×C4C22⋊Dic6113yes—
366C22×C4C22⋊Dic6113yes—
367C22×C4C22⋊Dic6116yes—
368C22×C4C2×Dic12213yes—
369C22×C4C2×Dic12213yes—
370C22×C4C2×C4×D6113yes—
371C22×C4C2×C4×D6113yes—
372C22×C4C2×C4×D6113yes—
373C22×C4C2×C4×D6113yes—
374C22×C4C2×D24413yes—
375C22×C4C2×D24413yes—
376C22×C4C22×Dic6113yes—
377C22×C4C2×C22⋊D6113yes—
378C22×C4C2×C22⋊D6113yes—
379C22×C4C2×C22⋊D6113yes—
380C22×C4C2×C22⋊D6113yes—
381C22×C4C23×D6111yescharacteristic
382M16—618no—
383M16—1224no—
384M16—1224no—
385M16—1224no—
386M16—12212no—
387M16—12212no—
388M16—12212no—
389M16—12212no—
390C2×C8—618no—
391C2×C8—1224no—
392C2×C8—12212no—
393C2×C8—12212no—
394C2×C8—12212no—
395C2×C8—12212no—
396C4⋊C4—24no—
397C4⋊C4—212no—
398C22⋊C4—212no—
399C22⋊C4C22×Dic6118yes—
400C42—24no—
401C42—212no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—46no—
406C23—46no—
407C23—46no—
408C23—46no—
409C23#1451968413yes—
410C23#1451968413yes—
411C23#1451968413yes—
412C23#1451968416yes—
413C23#1451968411yescharacteristic
414C23C2×C4×Dic6113yes—
415C23C2×C4×Dic6113yes—
416C23C2×C22⋊Dic6113yes—
417C23C2×C22⋊Dic6113yes—
418C23C2×C4⋊Dic6113yes—
419C23C2×C4⋊Dic6113yes—
420C23C23×Dic6111yescharacteristic
421C2×C4—28no—
422C2×C4—424no—
423C2×C4#1451968416yes—
424C2×C4#1451968416yes—
425C2×C4#1451968416yes—
426C2×C4#1451968416yes—
427C2×C4#1451968416yes—
428C2×C4#1451968416yes—
429C2×C4#1451968413yes—
430C2×C4#1451968413yes—
431C2×C4#1451968413yes—
432C2×C4#1451968413yes—
433C2×C4C2×C4×Dic6113yes—
434C2×C4C2×C4×Dic6113yes—
435C2×C4C2×#1175819813yes—
436C2×C4C2×#1175819813yes—
437C2×C4C2×#1175819813yes—
438C2×C4C2×#1175819813yes—
439C2×C4C2×#1175819813yes—
440C2×C4C2×#1175819813yes—
441C2×C4C2×#1175819813yes—
442C2×C4C2×#1175819813yes—
443C2×C4C2×C22⋊Dic6113yes—
444C2×C4C2×C22⋊Dic6113yes—
445C2×C4C2×#1175819613yes—
446C2×C4C2×#1175819613yes—
447C2×C4C2×#1175819613yes—
448C2×C4C2×#1175819613yes—
449C2×C4C2×#1175819613yes—
450C2×C4C2×#1175819613yes—
451C2×C4C2×#1175819613yes—
452C2×C4C2×#1175819613yes—
453C2×C4C2×C4⋊Dic6113yes—
454C2×C4C2×C4⋊Dic6113yes—
455C2×C4C22×C4×D6111yescharacteristic
456C2×C4C22×C4×D6111yescharacteristic
457C2×C4C22×C22⋊D6111yescharacteristic
458C2×C4C22×C22⋊D6111yescharacteristic
459C2×C4C22×D24411yescharacteristic
460C2×C4C22×Dic12211yescharacteristic
461C8—1228no—
462C8—1228no—
463C22—26no—
464C22—26no—
465C22—26no—
466C22—26no—
467C22—26no—
468C22—26no—
469C22—41no—
470C22—46no—
471C22—43no—
472C22—43no—
473C22—43no—
474C22#1683953214yesdirect product factor
475C22C2×#1451968413yes—
476C22C2×#1451968413yes—
477C22C2×#1451968413yes—
478C22C2×#1451968413yes—
479C22C22×C4×Dic6111yescharacteristic
480C22C22×C22⋊Dic6111yescharacteristic
481C22C22×C4⋊Dic6111yescharacteristic
482C4—48no—
483C4C2×#1451968413yes—
484C4C2×#1451968413yes—
485C4C2×#1451968413yes—
486C4C2×#1451968413yes—
487C4C22×#1175819811yescharacteristic
488C4C22×#1175819811yescharacteristic
489C4C22×#1175819611yescharacteristic
490C4C22×#1175819611yescharacteristic
491C2—21no—
492C2—21no—
493C2—21no—
494C2—23no—
495C2—23no—
496C2—23no—
497C2C2×#1683953216yesdirect product factor
498C2C22×#1451968411yescharacteristic
subgroupG/H
centerC22×C4#11758198
commutatorC22×C61C22×C42
FrattiniC22×C4C23×D61
FittingC22×#117×C61C2
socleC23×C61#95
prime radicalC25×C61C2×C4
cosocleC22×C4×C61C24
hypercenterC22×#117D61
hypocenterC61C22×#344
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#16839532C2direct (N × H)
12. C22×#11758194C4semidirect
13. C22×#11758194C4semidirect
14. C22×#11758194C4semidirect
18. C23×#92418C4semidirect
19. #16839532C22direct (N × H)
41. C2×#11758194C2×C4semidirect
42. C2×#11758194C2×C4semidirect
43. C2×#11758194C2×C4semidirect
51. C22×#92418C2×C4semidirect
145. #11758194C22×C4semidirect
151. C2×#92418C22×C4semidirect
225. C22×C61#344semidirect
252. C2×C61C2×#344semidirect
277. C61C22×#344semidirect
474. C22#16839532direct (N × H)
497. C2C2×#16839532direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117581942
C2×#168395322
C22×#117×C612
C22×#34461