Finite Group Databasebeta

← C2×#21461569 #30614072

Subgroups — C2×#21461569 #30614072

Order 15616.

1970 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519837C212yesmaximal
2C2×C4×#11758217C212yesmaximal, complemented
3#21461569C218yesmaximal, direct product factor
4C2×#16840642C212yesmaximal, complemented
5C22×#14519737C211yesmaximal, characteristic
6C22×#11758217C2214yes—
7C2×#14519837C4116yescomplemented
8C2×#14519837C2218yes—
9C4×#11758217—24no—
10C4×#11758217C2218yescomplemented
11C2×#14519832—22no—
12C2×C4×#92420—22no—
13#16840642C2218yescomplemented
14C23×C22⋊D61C2211yescharacteristic
15C24×Dic61C411yescharacteristic
16C23×C4×D61C411yescharacteristic
17C2×#14519737—22no—
18C2×#14519737—22no—
19C2×#14519737—24no—
20C2×#14519737C414yes—
21C2×#14519737C2214yes—
22C22×#11758196C2211yescharacteristic
23C22×#11758198C411yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25C22×C22⋊C4×C61C411yescharacteristic
26C2×#11758217—24no—
27C2×#11758217—28no—
28C2×#11758217C2×C4116yescomplemented
29C2×#11758217C2318yes—
30C22×#92420—24no—
31#14519837C2×C4132yescomplemented
32#14519832—28no—
33C4×#92420—28no—
34C24×D61C2×C411yescharacteristic
35C22×C22⋊D61—22no—
36C22×C22⋊D61—22no—
37C22×C22⋊D61—24no—
38C22×C22⋊D61C2×C412yes—
39C22×C22⋊D61C2×C418yes—
40C22×C22⋊D61C2312yes—
41C23×Dic61—22no—
42C23×Dic61—21no—
43C23×Dic61—21no—
44C23×Dic61—44no—
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C414yes—
48C23×Dic61C2311yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—44no—
53C22×C4×D61C2×C414yes—
54#14519737—28no—
55#14519737—28no—
56#14519737—28no—
57#14519737—28no—
58C2×#11758196—22no—
59C2×#11758196—22no—
60C2×#11758196C2×C412yes—
61C2×#11758196C2312yes—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198C2×C414yes—
65C2×C4×Dic61C2×C412yes—
66C2×C4×Dic61D412yes—
67C2×C4×Dic61D412yes—
68C2×C4×Dic61D414yes—
69C2×C4×Dic61C2312yes—
70C25×C61C2×C411yescharacteristic
71C23×C4×C61C2×C411yescharacteristic
72C23×C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61C2×C414yes—
76C24×C61—21no—
77C24×C61—21no—
78C24×C61—22no—
79C24×C61—42no—
80C24×C61—42no—
81C24×C61C4214yes—
82C24×C61C22×C411yescharacteristic
83C24×C61C22×C412yes—
84C23×D61—21no—
85C23×D61—21no—
86C23×D61—22no—
87C23×D61—42no—
88C23×D61—42no—
89C23×D61C4214yes—
90C23×D61C22×C412yes—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61C22⋊C412yes—
96C22×C4×C61C22⋊C412yes—
97C22×C4×C61C22×C412yes—
98C22×C4×C61C22×C412yes—
99C2×C22⋊D61—24no—
100C2×C22⋊D61—24no—
101C2×C22⋊D61—216no—
102C2×C22⋊D61—216no—
103C2×C22⋊D61—24no—
104C2×C22⋊D61—24no—
105C2×C22⋊D61C4218yes—
106C2×C22⋊D61C22×C418yes—
107C22×Dic61—24no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—24no—
111C22×Dic61—24no—
112C22×Dic61—44no—
113C22×Dic61—48no—
114C22×Dic61—48no—
115C22×Dic61—44no—
116C22×Dic61—42no—
117C22×Dic61C22⋊C411yescharacteristic
118C22×Dic61C22⋊C412yes—
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C22×C412yes—
121C22×Dic61C22×C412yes—
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C22×C414yes—
124C22×Dic61C2×D412yes—
125C22×Dic61C2×D412yes—
126C22×Dic61#4114yes—
127C22×Dic61C2411yescharacteristic
128C2×C4×D61—24no—
129C2×C4×D61—24no—
130C2×C4×D61—44no—
131C2×C4×D61—48no—
132C2×C4×D61—44no—
133C2×C4×D61—48no—
134#11758217—216no—
135#11758217—28no—
136#11758217C22×C4116yescomplemented
137C2×#92420—216no—
138C2×#92420—28no—
139C22⋊C4×C61—28no—
140#11758198—28no—
141#11758196—24no—
142#11758196—24no—
143C4×Dic61C22⋊C414yes—
144C4×Dic61C22⋊C414yes—
145C4×Dic61C2×D414yes—
146C4×Dic61C2×D414yes—
147C23×C61—22no—
148C23×C61—22no—
149C23×C61—22no—
150C23×C61—22no—
151C23×C61—24no—
152C23×C61—24no—
153C23×C61—42no—
154C23×C61—42no—
155C23×C61—48no—
156C23×C61—42no—
157C23×C61—48no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61C2×C4212yes—
162C23×C61C2×C4214yes—
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61C2×C22⋊C411yescharacteristic
165C23×C61#11712yes—
166C23×C61C23×C411yescharacteristic
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—24no—
170C22×D61—24no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—48no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—48no—
178C22×D61—42no—
179C22×D61C2×C4214yes—
180C2×C4×C61—24no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61C2×C22⋊C414yes—
188C22⋊D61—216no—
189C22⋊D61—216no—
190C2×Dic61—22no—
191C2×Dic61—24no—
192C2×Dic61—24no—
193C2×Dic61—22no—
194C2×Dic61—48no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—48no—
198C2×Dic61—42no—
199C2×Dic61#10014yescomplemented
200C2×Dic61C2×C4212yes—
201C2×Dic61C2×C22⋊C412yes—
202C2×Dic61C2×C22⋊C412yes—
203C2×Dic61C2×C22⋊C412yes—
204C2×Dic61#11712yes—
205C2×Dic61C4×D418yes—
206C2×Dic61C4×D418yes—
207C2×Dic61C23×C412yes—
208C2×Dic61C22×D412yes—
209C2×Dic61C2×#4112yes—
210C4×D61—48no—
211C4×D61—48no—
212#92420—216no—
213C2×C4×#100—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—24no—
219C22×C61—24no—
220C22×C61—24no—
221C22×C61—44no—
222C22×C61—44no—
223C22×C61—48no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—48no—
227C22×C61C4×C22⋊C412yes—
228C22×C61C4×C22⋊C412yes—
229C22×C61C2×#10012yes—
230C22×C61#41312yes—
231C22×C61C22×C4211yescharacteristic
232C22×C61C22×C22⋊C411yescharacteristic
233C22×C61C2×#11711yescharacteristic
234C2×D61—24no—
235C2×D61—44no—
236C2×D61—48no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—48no—
240C2×D61—44no—
241C4×C61—24no—
242C4×C61—42no—
243Dic61—22no—
244Dic61—42no—
245Dic61C4×C22⋊C414yes—
246Dic61C2×#10014yescomplemented
247Dic61C2×C4×D414yes—
248C22×C4×D4—611no—
249C22×#100—612no—
250C2×C4×M16—612no—
251C4×#100—618no—
252C2×#343—612no—
253C2×C61—21no—
254C2×C61—22no—
255C2×C61—21no—
256C2×C61—44no—
257C2×C61C4×#10014yescomplemented
258C2×C61C2×C4×C22⋊C411yescharacteristic
259C2×C61C22×#10011yescharacteristic
260C2×C61C2×#41311yescharacteristic
261D61—44no—
262C23×D4—611no—
263C24×C4—611no—
264C24×C4—611no—
265C22×M16—614no—
266C2×C4×D4—614no—
267C2×C4×D4—614no—
268C2×C4×D4—1222no—
269C2×C4×D4—1222no—
270C2×C4×D4—1224no—
271C22×C4⋊C4—611no—
272C22×C22⋊C4—611no—
273C22×C22⋊C4#692311yescharacteristic
274C22×C42—611no—
275C2×#100—6116no—
276C2×#100—618no—
277C4×M16—618no—
278C4×M16—1224no—
279C2×#97—1222no—
280C2×C4×C8—1222no—
281#343—618no—
282C61C2×C4×#10011yescharacteristic, complemented, Sylow
283C25—611no—
284C25C2×#692311yescharacteristic
285C22×D4—618no—
286C22×D4—612no—
287C22×D4—612no—
288C22×D4—1222no—
289C22×D4—1222no—
290C22×D4—1224no—
291C23×C4—611no—
292C23×C4—611no—
293C23×C4—611no—
294C23×C4—614no—
295C23×C4—614no—
296C23×C4—1222no—
297C23×C4—1221no—
298C23×C4—1221no—
299C23×C4—1221no—
300C23×C4—1222no—
301C23×C4—1221no—
302C23×C4—2444no—
303C23×C4—2444no—
304C23×C4C2×#692311yescharacteristic
305C23×C4C2×#692311yescharacteristic
306C2×M16—6116no—
307C2×M16—618no—
308C2×M16—1228no—
309C2×M16—1224no—
310C22×C8—1224no—
311C4×D4—1228no—
312C4×D4—1228no—
313C4×D4—1228no—
314C4×D4—1228no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1222no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—614no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4C2×#692314yes—
325C2×C42—612no—
326C2×C42—612no—
327C2×C42—614no—
328C2×C42—612no—
329C2×C42—612no—
330#100—6132no—
331#97—1228no—
332C4×C8—1228no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—42no—
337C24—42no—
338C24—614no—
339C24—612no—
340C24—1221no—
341C24—1221no—
342C24—1222no—
343C24—2442no—
344C24—2442no—
345C24C4×#692314yes—
346C24C22×#692311yescharacteristic
347C24C22×#692312yes—
348C2×D4—618no—
349C2×D4—618no—
350C2×D4—1224no—
351C2×D4—1224no—
352C2×D4—1224no—
353C2×D4—1224no—
354C2×D4—12216no—
355C2×D4—12216no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—614no—
361C22×C4—612no—
362C22×C4—612no—
363C22×C4—611no—
364C22×C4—614no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—612no—
368C22×C4—611no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—1224no—
372C22×C4—1224no—
373C22×C4—1224no—
374C22×C4—1224no—
375C22×C4—1224no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—2444no—
379C22×C4—2448no—
380C22×C4—2444no—
381C22×C4—2448no—
382C22×C4—2444no—
383C22×C4—2448no—
384C22×C4—2444no—
385C22×C4—2448no—
386C22×C4—2442no—
387C22×C4C22⋊#692312yes—
388C22×C4C22⋊#692312yes—
389C22×C4C22×#692312yes—
390C22×C4C22×#692312yes—
391M16—6116no—
392M16—1228no—
393M16—12216no—
394C2×C8—1228no—
395C2×C8—12216no—
396C4⋊C4—1224no—
397C4⋊C4—1224no—
398C22⋊C4—28no—
399C22⋊C4—1228no—
400C42—614no—
401C42—614no—
402C42—614no—
403C42—614no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—42no—
411C23—48no—
412C23—42no—
413C23—42no—
414C23—48no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—614no—
419C23—1222no—
420C23—1222no—
421C23—1224no—
422C23—1224no—
423C23—2442no—
424C23—2448no—
425C23—2442no—
426C23—2442no—
427C23—2448no—
428C23—2442no—
429C23—2442no—
430C23—2442no—
431C23C2×C4×#692312yes—
432C23C2×C4×#692314yes—
433C23C2×C22⋊#692311yescharacteristic
434C23C2×C22⋊#692311yescharacteristic
435C23#1451985612yes—
436C23C23×#692311yescharacteristic
437D4—12216no—
438D4—12216no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—614no—
447C2×C4—612no—
448C2×C4—612no—
449C2×C4—612no—
450C2×C4—612no—
451C2×C4—612no—
452C2×C4—612no—
453C2×C4—612no—
454C2×C4—618no—
455C2×C4—618no—
456C2×C4—612no—
457C2×C4—1224no—
458C2×C4—1224no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—2448no—
462C2×C4—2448no—
463C2×C4—2448no—
464C2×C4—2448no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4C2×C22⋊#692314yes—
469C8—12216no—
470C22—24no—
471C22—24no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—44no—
478C22—44no—
479C22—48no—
480C22—48no—
481C22—42no—
482C22—42no—
483C22—1224no—
484C22—2444no—
485C22—2444no—
486C22—2448no—
487C22—2448no—
488C22—2442no—
489C22—2442no—
490C22C4×C22⋊#692312yes—
491C22C4×C22⋊#692312yes—
492C22D61⋊#10012yes—
493C22#1684074912yes—
494C22C22×C4×#692311yescharacteristic
495C22C22×C22⋊#692311yescharacteristic
496C22C2×#1451985611yescharacteristic
497C4—24no—
498C4—42no—
499C4—614no—
500C4—614no—
501C4—614no—
502C4—1222no—
503C4—2442no—
504C2—21no—
505C2—21no—
506C2—22no—
507C2—44no—
508C2—2444no—
509C2#2146156914yesdirect product factor
510C2C2×C4×C22⋊#692311yescharacteristic
511C2C2×D61⋊#10011yescharacteristic
512C2C2×#1684074911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C61C22×C42
FrattiniC23C23×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C22
cosocleC22×Dic61C24
hypercenterC22×C22⋊C4#6923
hypocenterC61C2×C4×#100
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758217C2semidirect
3. #21461569C2direct (N × H)
4. C2×#16840642C2semidirect
7. C2×#14519837C4semidirect
10. C4×#11758217C22semidirect
13. #16840642C22semidirect
28. C2×#11758217C2×C4semidirect
31. #14519837C2×C4semidirect
136. #11758217C22×C4semidirect
199. C2×Dic61#100semidirect
246. Dic61C2×#100semidirect
257. C2×C61C4×#100semidirect
282. C61C2×C4×#100semidirect
509. C2#21461569direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198372
C2×C4×#117582172
#214615692
C2×#168406422
C22×#145197372
C2×C4×#10061