Finite Group Databasebeta

← C2×#21445002 #29655690

Subgroups — C2×#21445002 #29655690

Order 15616.

1042 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840498C211yesmaximal, characteristic
2C2×#16840044C211yesmaximal, characteristic, complemented
3C2×#16839969C211yesmaximal, characteristic, complemented
4#21445002C218yesmaximal, direct product factor
5C2×#16839506C211yesmaximal, characteristic
6C2×#16839494C211yesmaximal, characteristic, complemented
7C2×#16839494C211yesmaximal, characteristic, complemented
8C2×C61×#486C211yesmaximal, characteristic
9C22×#11758202—21no—
10#16840498—22no—
11#16840498—22no—
12#16840498—22no—
13#16840498C2214yes—
14C2×#14519764—21no—
15C2×#14519764—21no—
16C2×#14519758—22no—
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61—21no—
20C2×D4×Dic61—21no—
21#16840044—22no—
22#16840044C2214yescomplemented
23#16839969—22no—
24#16839969C2214yescomplemented
25C2×#14519708—21no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701—21no—
29C2×#14519700—21no—
30#16839506C2214yes—
31#16839494C2214yescomplemented
32#16839494C2214yescomplemented
33C2×C4⋊Q8×C61C2211yescharacteristic
34C2×#127×C61C2211yescharacteristic
35C61×#486—22no—
36C61×#486C2214yes—
37C2×#102×C61—22no—
38C2×C4×C8×C61C2211yescharacteristic
39C22×Dic122D411yescharacteristic
40C22×Dic122D411yescharacteristic
41C23×Dic61—22no—
42C2×#11758202—24no—
43C2×#11758202—44no—
44C22×#92418—21no—
45C22×#92418—21no—
46#14519764—24no—
47#14519764—24no—
48#14519764—24no—
49#14519764—24no—
50#14519758—28no—
51#14519758—28no—
52C4×Dic122—21no—
53C4×Dic122—21no—
54C4×Dic122—21no—
55C4×Dic122—21no—
56C4×Dic122D412yes—
57C4×Dic122D412yes—
58C4×Dic122C2312yes—
59C4×Dic122C2312yes—
60D4×Dic61—24no—
61D4×Dic61—24no—
62D4×Dic61—24no—
63D4×Dic61—24no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61D411yescharacteristic
66C2×#11758196—21no—
67C2×#11758196—21no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61—21no—
73#14519708—24no—
74C4⋊#92418—21no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77C4⋊#92418C2312yes—
78#14519701—24no—
79#14519700—24no—
80C22×Q8×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61D412yes—
84C22×C8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61C2312yes—
89#127×C61—21no—
90#127×C61—21no—
91#127×C61D412yes—
92#127×C61C2312yes—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C42×C61C2311yescharacteristic
96#102×C61—28no—
97C4×C8×C61—21no—
98C4×C8×C61C2312yes—
99C24×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61—41no—
102C2×Q8×C61—42no—
103C2×Q8×C61—42no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—44no—
110C2×D4×C61—42no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C2×D4×C61C2×D414yes—
115C22×C4×C61—41no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—28no—
124C22×Dic61—84no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122C2×D412yes—
132C2×Dic122C2×D412yes—
133C2×C8×C61—22no—
134C2×C8×C61—42no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61C2411yescharacteristic
143C22⋊Dic61—28no—
144C22⋊Dic61—28no—
145C22⋊Dic61—28no—
146#11758202—48no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61C2×D412yes—
150C4⋊Dic61C2×D412yes—
151#11758196—22no—
152#11758196—22no—
153#11758196—22no—
154#11758196—22no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—42no—
162C2×#92418—42no—
163C23×C61—24no—
164C23×C61—44no—
165C23×C61—44no—
166C23×C61—44no—
167C23×C61#12011yescharacteristic
168Q8×C61—42no—
169Q8×C61—42no—
170Q8×C61—82no—
171D4×C61—44no—
172D4×C61—42no—
173D4×C61—42no—
174D4×C61—42no—
175D4×C61—42no—
176D4×C61—44no—
177D4×C61—44no—
178D4×C61—82no—
179C2×C4×C61—22no—
180C2×C4×C61—42no—
181C2×C4×C61—82no—
182C2×C4×C61#12011yescharacteristic
183C2×C4×C61#12011yescharacteristic
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#13512yes—
186C2×C4×C61#13512yes—
187C2×C4×C61AGL(1,ℤ/8)12yes—
188C2×C4×C61#13712yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—88no—
201Dic122—22no—
202Dic122—22no—
203Dic122—22no—
204Dic122—22no—
205Dic122—42no—
206Dic122—42no—
207C8×C61—42no—
208#92418—42no—
209#92418—42no—
210C2×#1640—611nomaximal, Sylow
211C22×C61—44no—
212C22×C61—44no—
213C22×C61—44no—
214C22×C61—88no—
215C22×C61#44912yes—
216C22×C61#45212yes—
217C22×C61#45512yes—
218C22×C61C2×#12011yescharacteristic
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—82no—
222C4×C61#44912yes—
223C4×C61#45212yes—
224C4×C61C2×#13511yescharacteristic
225C4×C61C2×#13511yescharacteristic
226C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C61C2×#13711yescharacteristic
228Dic61—42no—
229Dic61—42no—
230Dic61—42no—
231Dic61—42no—
232C2×#559—611no—
233C2×#486D6111yescharacteristic
234C2×#475—611no—
235C2×#462—611no—
236#1640—618no—
237C2×#331—611no—
238C2×Q8⋊C8—611no—
239C2×Q8⋊C8—611no—
240C2×C61—84no—
241C2×C61#164014yescomplemented
242C2×C61C2×#44911yescharacteristic
243C2×C61C2×#45211yescharacteristic
244C2×C61C2×#45511yescharacteristic
245C22×Q16—1221no—
246#559—614no—
247#559—1222no—
248#559—1222no—
249#559—1222no—
250C2×C4⋊Q8C2×D6111yescharacteristic
251C2×#127C2×D6111yescharacteristic
252C2×#124—1221no—
253C2×#124—1221no—
254C2×#123—1222no—
255C2×C4×Q8—611no—
256C2×C4×Q8—611no—
257C2×C4×D4—1221no—
258C2×C4×D4—1221no—
259#486—22no—
260#486C2×D6114yes—
261#475—614no—
262#475—1222no—
263#462—614no—
264#462—1222no—
265C2×#106—1221no—
266C2×C4⋊C8—611no—
267C2×C4⋊C8—611no—
268C2×#103—1221no—
269C2×#103—1221no—
270C2×#102—22no—
271C2×C4×C8C2×D6111yescharacteristic
272#331—614no—
273Q8⋊C8—614no—
274Q8⋊C8—614no—
275C61C2×#164011yescharacteristic, complemented, Sylow
276C22×Q8—21no—
277C22×Q8—611no—
278C22×Q8—611no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C22⋊D6112yes—
282C23×C4—1222no—
283C2×Q16—1224no—
284C2×Q16—2444no—
285C22×C8—21no—
286C22×C8—1221no—
287C22×C8—1221no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8C22×D6112yes—
292#127—21no—
293#127—21no—
294#127C22⋊D6112yes—
295#127C22×D6112yes—
296#124—1224no—
297#124—1224no—
298#124—1224no—
299#124—1224no—
300#123—1228no—
301#123—1228no—
302C4×Q8—612no—
303C4×Q8—612no—
304C4×Q8—612no—
305C4×Q8—612no—
306C4×Q8—1221no—
307C4×Q8—1221no—
308C4×Q8—1221no—
309C4×Q8—1221no—
310C4×D4—1224no—
311C4×D4—1224no—
312C4×D4—1224no—
313C4×D4—1224no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C42—1221no—
324C2×C42—1221no—
325C2×C42C22×D6111yescharacteristic
326#106—1224no—
327C4⋊C8—612no—
328C4⋊C8—612no—
329C4⋊C8—1221no—
330C4⋊C8—1221no—
331#103—1224no—
332#103—1224no—
333#102—28no—
334C4×C8—21no—
335C4×C8C22×D6112yes—
336C24—22no—
337C2×Q8—22no—
338C2×Q8—41no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—612no—
342C2×Q8—612no—
343C2×Q8—1221no—
344C2×Q8—1221no—
345C2×Q8—1222no—
346C2×Q8—1222no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—44no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C2×C22⋊D6114yes—
360C22×C4—41no—
361C22×C4—1228no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—4884no—
367C22×C4D4×D6111yescharacteristic
368C22×C4D4×D6111yescharacteristic
369C22×C4C2×C22⋊D6111yescharacteristic
370Q16—2448no—
371C2×C8—22no—
372C2×C8—42no—
373C2×C8—1222no—
374C2×C8—1222no—
375C2×C8—2442no—
376C2×C8—2442no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—612no—
382C4⋊C4—612no—
383C4⋊C4—1222no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C22⋊C4—1228no—
390C22⋊C4—1228no—
391C22⋊C4—1228no—
392C42—1222no—
393C42—1222no—
394C42—1222no—
395C42—1222no—
396C42D4×D6111yescharacteristic
397C42D4×D6111yescharacteristic
398C42C2×C22⋊D6111yescharacteristic
399C42C23×D6111yescharacteristic
400C23—24no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23#1451974511yescharacteristic
405Q8—42no—
406Q8—42no—
407Q8—82no—
408Q8—1222no—
409Q8—1222no—
410Q8—1222no—
411Q8—1222no—
412Q8—2442no—
413Q8—2442no—
414D4—44no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—42no—
419D4—44no—
420D4—44no—
421D4—82no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—4888no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4#1451974511yescharacteristic
435C2×C4#1451974511yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4#1451979612yes—
438C2×C4#1451979812yes—
439C2×C4#1451980112yes—
440C2×C4#1451980612yes—
441C2×C4C2×D4×D6111yescharacteristic
442C2×C4C2×D4×D6111yescharacteristic
443C2×C4C22×C22⋊D6111yescharacteristic
444C8—42no—
445C8—2442no—
446C8—2442no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—88no—
451C22#1683990412yes—
452C22#1683992112yes—
453C22#1683993912yes—
454C22C2×#1451974511yescharacteristic
455C4—21no—
456C4—21no—
457C4—82no—
458C4—2442no—
459C4—2442no—
460C4—2442no—
461C4—2442no—
462C4#1683990412yes—
463C4#1683992112yes—
464C4C2×#1451979611yescharacteristic
465C4C2×#1451979811yescharacteristic
466C4C2×#1451980111yescharacteristic
467C4C2×#1451980611yescharacteristic
468C2—84no—
469C2#2144500214yesdirect product factor
470C2C2×#1683990411yescharacteristic
471C2C2×#1683992111yescharacteristic
472C2C2×#1683993911yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#486C2
socleC23×C61#120
prime radicalC2×#127×C61C22
hypercenterC2×#486D61
hypocenterC61C2×#1640
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
2. C2×#16840044C2semidirect
3. C2×#16839969C2semidirect
4. #21445002C2direct (N × H)
6. C2×#16839494C2semidirect
7. C2×#16839494C2semidirect
22. #16840044C22semidirect
24. #16839969C22semidirect
31. #16839494C22semidirect
32. #16839494C22semidirect
241. C2×C61#1640semidirect
275. C61C2×#1640semidirect
469. C2#21445002direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404982
C2×#168400442
C2×#168399692
#214450022
C2×#168395062
C2×#168394942
C2×C61×#4862
C2×#164061