Finite Group Databasebeta

← C22×#16839533 #29677466

Subgroups — C22×#16839533 #29677466

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
2C22×#14519736C211yesmaximal, characteristic
3C2×#16839533C2112yesmaximal, direct product factor
4C23×M16×C61C211yesmaximal, characteristic
5C22×#11758194—21no—
6C22×#11758194—23no—
7C22×#11758194—23no—
8C22×#11758194C412yescomplemented
9C22×#11758194C416yescomplemented
10C22×#11758194C2216yes—
11C23×#92418—21no—
12C2×#14519736—23no—
13C2×#14519736—23no—
14C2×#14519736C416yes—
15C2×#14519736C2216yes—
16C22×C4⋊Dic61—21no—
17C22×C22⋊Dic61C412yes—
18C22×C4×Dic61—21no—
19#16839533C22116yesdirect product factor
20C24×C4×C61C2211yescharacteristic
21C22×M16×C61—21no—
22C22×M16×C61—23no—
23C22×M16×C61—21no—
24C22×M16×C61—23no—
25C22×M16×C61C416yescomplemented
26C22×M16×C61C2216yes—
27C23×C8×C61C412yescomplemented
28C23×Dic61—22no—
29C2×#11758194—26no—
30C2×#11758194—26no—
31C2×#11758194—212no—
32C2×#11758194—26no—
33C2×#11758194—26no—
34C2×#11758194—26no—
35C2×#11758194C2×C4112yescomplemented
36C2×#11758194C2×C4112yescomplemented
37C2×#11758194C2314yes—
38C22×#92418—26no—
39C22×#92418—22no—
40C22×#92418—26no—
41#14519736—212no—
42#14519736—26no—
43#14519736—26no—
44#14519736C2×C4112yes—
45#14519736C2314yes—
46C2×C4⋊Dic61—26no—
47C2×C4⋊Dic61—26no—
48C2×C22⋊Dic61—26no—
49C2×C22⋊Dic61C2×C4112yes—
50C2×C4×Dic61—26no—
51C2×C4×Dic61—26no—
52C25×C61C2×C411yescharacteristic
53C23×C4×C61C2×C413yes—
54C23×C4×C61C2×C411yescharacteristic
55C23×C4×C61C2×C413yes—
56C23×C4×C61C2×C411yescharacteristic
57C23×C4×C61C2×C413yes—
58C23×C4×C61D411yescharacteristic
59C23×C4×C61D413yes—
60C23×C4×C61D411yescharacteristic
61C23×C4×C61D413yes—
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D413yes—
64C23×C4×C61Q811yescharacteristic
65C23×C4×C61Q813yes—
66C23×C4×C61C2313yes—
67C2×M16×C61—26no—
68C2×M16×C61—26no—
69C2×M16×C61—212no—
70C2×M16×C61—26no—
71C2×M16×C61—26no—
72C2×M16×C61—26no—
73C2×M16×C61—26no—
74C2×M16×C61C2×C4112yescomplemented
75C2×M16×C61C2314yes—
76C22×C8×C61—26no—
77C22×C8×C61—22no—
78C22×C8×C61C2×C4112yescomplemented
79C24×C61—44no—
80C24×C61C4211yescharacteristic
81C24×C61C4213yes—
82C24×C61C22⋊C411yescharacteristic
83C24×C61C22⋊C413yes—
84C24×C61C4⋊C411yescharacteristic
85C24×C61C4⋊C413yes—
86C24×C61C22×C413yes—
87C22×C4×C61C4213yes—
88C22×C4×C61C4213yes—
89C22×C4×C61C4216yes—
90C22×C4×C61C22⋊C413yes—
91C22×C4×C61C22⋊C413yes—
92C22×C4×C61C22⋊C416yes—
93C22×C4×C61C22⋊C413yes—
94C22×C4×C61C22⋊C411yescharacteristic
95C22×C4×C61C22⋊C413yes—
96C22×C4×C61C22⋊C413yes—
97C22×C4×C61C22⋊C416yes—
98C22×C4×C61C22⋊C413yes—
99C22×C4×C61C22⋊C413yes—
100C22×C4×C61C22⋊C413yes—
101C22×C4×C61C22⋊C416yes—
102C22×C4×C61C22⋊C411yescharacteristic
103C22×C4×C61C4⋊C413yes—
104C22×C4×C61C4⋊C413yes—
105C22×C4×C61C4⋊C416yes—
106C22×C4×C61C4⋊C413yes—
107C22×C4×C61C4⋊C413yes—
108C22×C4×C61C4⋊C413yes—
109C22×C4×C61C4⋊C416yes—
110C22×C4×C61C4⋊C411yescharacteristic
111C22×C4×C61C4⋊C413yes—
112C22×C4×C61C4⋊C413yes—
113C22×C4×C61C4⋊C413yes—
114C22×C4×C61C4⋊C416yes—
115C22×C4×C61C4⋊C411yescharacteristic
116C22×C4×C61C22×C413yes—
117C22×C4×C61C22×C413yes—
118C22×C4×C61C22×C413yes—
119C22×C4×C61C22×C413yes—
120C22×C4×C61C22×C413yes—
121C22×C4×C61C2×D413yes—
122C22×C4×C61C2×D413yes—
123C22×C4×C61C2×D413yes—
124C22×C4×C61C2×D413yes—
125C22×C4×C61C2×D413yes—
126C22×C4×C61C2×D413yes—
127C22×C4×C61C2×Q813yes—
128C22×C4×C61C2×Q813yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—212no—
131C22×Dic61—48no—
132M16×C61—212no—
133M16×C61—212no—
134M16×C61—24no—
135M16×C61—24no—
136C2×C8×C61—212no—
137C2×C8×C61—212no—
138C2×C8×C61C22×C418yescomplemented
139C22⋊Dic61—212no—
140C22⋊Dic61C22×C418yes—
141C4⋊Dic61—212no—
142C4⋊Dic61—24no—
143C4×Dic61—212no—
144C4×Dic61—24no—
145#11758194—212no—
146#11758194—212no—
147#11758194—24no—
148#11758194C22×C418yescomplemented
149C2×#92418—212no—
150C2×#92418—212no—
151C2×#92418—24no—
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#9516yes—
172C2×C4×C61#9516yes—
173C2×C4×C61#9516yes—
174C2×C4×C61#9516yes—
175C2×C4×C61#9516yes—
176C2×C4×C61#9516yes—
177C2×C4×C61#9513yes—
178C2×C4×C61#9513yes—
179C2×C4×C61#9513yes—
180C2×C4×C61#9513yes—
181C2×C4×C61C2×C4213yes—
182C2×C4×C61C2×C4213yes—
183C2×C4×C61C2×C22⋊C413yes—
184C2×C4×C61C2×C22⋊C413yes—
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×C4⋊C413yes—
194C2×C4×C61C2×C4⋊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×C61C23×C411yescharacteristic
204C2×C4×C61C23×C411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×Q811yescharacteristic
209C2×Dic61—28no—
210C2×Dic61—424no—
211C8×C61—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×C61C2×#9513yes—
234C4×C61C2×#9513yes—
235C4×C61C2×#9513yes—
236C4×C61C2×#9513yes—
237C4×C61C22×C22⋊C411yescharacteristic
238C4×C61C22×C22⋊C411yescharacteristic
239C4×C61C22×C4⋊C411yescharacteristic
240C4×C61C22×C4⋊C411yescharacteristic
241Dic61—48no—
242C23×M16—611no—
243C23×M16D6111yescharacteristic
244C22×#117—611no—
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—23no—
256C22×M16—23no—
257C22×M16—21no—
258C22×M16—21no—
259C22×M16—612no—
260C22×M16—616no—
261C22×M16—616no—
262C22×M16—1221no—
263C22×M16—1223no—
264C22×M16—1223no—
265C22×M16Dic6116yescomplemented
266C22×M16C2×D6116yes—
267C23×C8—1221no—
268C23×C8Dic6112yescomplemented
269C2×#117—616no—
270C2×#117—616no—
271C2×#117—1223no—
272C2×#117—1223no—
273C22×C4⋊C4—1221no—
274C22×C22⋊C4—612no—
275C22×C42—1221no—
276#344—6116no—
277C61C22×#34411yescharacteristic, complemented, Sylow
278C25C4×D6111yescharacteristic
279C23×C4—1222no—
280C23×C4Dic12211yescharacteristic
281C23×C4Dic12213yes—
282C23×C4C4×D6113yes—
283C23×C4C4×D6111yescharacteristic
284C23×C4C4×D6113yes—
285C23×C4D24411yescharacteristic
286C23×C4D24413yes—
287C23×C4C2×Dic6113yes—
288C23×C4C2×Dic6111yescharacteristic
289C23×C4C22⋊D6113yes—
290C23×C4C22⋊D6111yescharacteristic
291C23×C4C22⋊D6113yes—
292C23×C4C22⋊D6111yescharacteristic
293C23×C4C22×D6113yes—
294C2×M16—212no—
295C2×M16—26no—
296C2×M16—26no—
297C2×M16—26no—
298C2×M16—26no—
299C2×M16—26no—
300C2×M16—26no—
301C2×M16—6112no—
302C2×M16—6112no—
303C2×M16—614no—
304C2×M16—1226no—
305C2×M16—1226no—
306C2×M16—12212no—
307C2×M16—1226no—
308C2×M16—1226no—
309C2×M16—1226no—
310C2×M16C2×Dic61112yescomplemented
311C2×M16C22×D6114yes—
312C22×C8—26no—
313C22×C8—22no—
314C22×C8—1222no—
315C22×C8—1226no—
316C22×C8—1226no—
317C22×C8C2×Dic61112yescomplemented
318#117—6112no—
319#117—614no—
320#117—12212no—
321#117—1226no—
322#117—1226no—
323C2×C4⋊C4—1226no—
324C2×C4⋊C4—1226no—
325C2×C22⋊C4—6112no—
326C2×C22⋊C4—1226no—
327C2×C42—1226no—
328C2×C42—1226no—
329C24—44no—
330C24C4×Dic6111yescharacteristic
331C24C4×Dic6113yes—
332C24#1175819611yescharacteristic
333C24#1175819613yes—
334C24#1175819811yescharacteristic
335C24#1175819813yes—
336C24C2×C4×D6113yes—
337C22×C4—12212no—
338C22×C4—2448no—
339C22×C4C4×Dic6113yes—
340C22×C4C4×Dic6116yes—
341C22×C4C4×Dic6113yes—
342C22×C4#1175819613yes—
343C22×C4#1175819616yes—
344C22×C4#1175819613yes—
345C22×C4#1175819613yes—
346C22×C4#1175819613yes—
347C22×C4#1175819616yes—
348C22×C4#1175819611yescharacteristic
349C22×C4#1175819613yes—
350C22×C4C4⋊Dic6113yes—
351C22×C4C4⋊Dic6113yes—
352C22×C4C4⋊Dic6116yes—
353C22×C4C4⋊Dic6111yescharacteristic
354C22×C4C4⋊Dic6113yes—
355C22×C4#1175819813yes—
356C22×C4#1175819813yes—
357C22×C4#1175819816yes—
358C22×C4#1175819811yescharacteristic
359C22×C4#1175819813yes—
360C22×C4#1175819813yes—
361C22×C4#1175819813yes—
362C22×C4#1175819816yes—
363C22×C4C22⋊Dic6113yes—
364C22×C4C22⋊Dic6116yes—
365C22×C4C22⋊Dic6113yes—
366C22×C4C22⋊Dic6113yes—
367C22×C4C22⋊Dic6111yescharacteristic
368C22×C4C2×Dic12213yes—
369C22×C4C2×Dic12213yes—
370C22×C4C2×C4×D6113yes—
371C22×C4C2×C4×D6113yes—
372C22×C4C2×C4×D6113yes—
373C22×C4C2×D24413yes—
374C22×C4C2×D24413yes—
375C22×C4C22×Dic6113yes—
376C22×C4C22×Dic6113yes—
377C22×C4C2×C22⋊D6113yes—
378C22×C4C2×C22⋊D6113yes—
379C22×C4C2×C22⋊D6113yes—
380C22×C4C2×C22⋊D6113yes—
381C22×C4C23×D6111yescharacteristic
382M16—24no—
383M16—24no—
384M16—212no—
385M16—212no—
386M16—618no—
387M16—1224no—
388M16—12212no—
389M16—12212no—
390C2×C8—212no—
391C2×C8—212no—
392C2×C8—1224no—
393C2×C8—12212no—
394C2×C8—12212no—
395C2×C8C22×Dic6118yescomplemented
396C4⋊C4—1224no—
397C4⋊C4—12212no—
398C22⋊C4—618no—
399C22⋊C4—12212no—
400C42—1224no—
401C42—12212no—
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×#1175819813yes—
417C23C2×#1175819813yes—
418C23C2×#1175819613yes—
419C23C2×#1175819613yes—
420C23C22×C4×D6111yescharacteristic
421C2×C4—1228no—
422C2×C4—24424no—
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×C22⋊Dic6113yes—
442C2×C4C2×C22⋊Dic6113yes—
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×C4⋊Dic6113yes—
452C2×C4C2×C4⋊Dic6113yes—
453C2×C4C2×C4⋊Dic6113yes—
454C2×C4C2×C4⋊Dic6113yes—
455C2×C4C22×C4×D6111yescharacteristic
456C2×C4C23×Dic6111yescharacteristic
457C2×C4C22×C22⋊D6111yescharacteristic
458C2×C4C22×C22⋊D6111yescharacteristic
459C2×C4C22×D24411yescharacteristic
460C2×C4C22×Dic12211yescharacteristic
461C8—28no—
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#1683953314yesdirect product factor
475C22C2×#1451968413yes—
476C22C2×#1451968413yes—
477C22C2×#1451968413yes—
478C22C2×#1451968413yes—
479C22C22×C4×Dic6111yescharacteristic
480C22C22×#1175819811yescharacteristic
481C22C22×#1175819611yescharacteristic
482C4—2448no—
483C4C2×#1451968413yes—
484C4C2×#1451968413yes—
485C4C2×#1451968413yes—
486C4C2×#1451968413yes—
487C4C22×#1175819811yescharacteristic
488C4C22×C22⋊Dic6111yescharacteristic
489C4C22×#1175819611yescharacteristic
490C4C22×C4⋊Dic6111yescharacteristic
491C2—21no—
492C2—21no—
493C2—21no—
494C2—23no—
495C2—23no—
496C2—23no—
497C2C2×#1683953316yesdirect product factor
498C2C22×#1451968411yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic61
commutatorC22×C61C22×C42
FrattiniC22×C4C23×D61
FittingC23×M16×C61C2
socleC23×C61#95
prime radicalC25×C61C2×C4
cosocleC22×C4×C61C24
hypercenterC23×M16D61
hypocenterC61C22×#344
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16839533C2direct (N × H)
8. C22×#11758194C4semidirect
9. C22×#11758194C4semidirect
19. #16839533C22direct (N × H)
25. C22×M16×C61C4semidirect
27. C23×C8×C61C4semidirect
35. C2×#11758194C2×C4semidirect
36. C2×#11758194C2×C4semidirect
74. C2×M16×C61C2×C4semidirect
78. C22×C8×C61C2×C4semidirect
138. C2×C8×C61C22×C4semidirect
148. #11758194C22×C4semidirect
225. C22×C61#344semidirect
252. C2×C61C2×#344semidirect
265. C22×M16Dic61semidirect
268. C23×C8Dic61semidirect
277. C61C22×#344semidirect
310. C2×M16C2×Dic61semidirect
317. C22×C8C2×Dic61semidirect
395. C2×C8C22×Dic61semidirect
474. C22#16839533direct (N × H)
497. C2C2×#16839533direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117581942
C22×#145197362
C2×#168395332
C23×M16×C612
C22×#34461