Finite Group Databasebeta

← C2×#21445129 #29661158

Subgroups — C2×#21445129 #29661158

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840516C211yesmaximal, characteristic
2C2×#16840168C211yesmaximal, characteristic, complemented
3C2×#16840154C211yesmaximal, characteristic
4#21445129C218yesmaximal, direct product factor
5C2×#16839504C211yesmaximal, characteristic, complemented
6C2×#16839503C211yesmaximal, characteristic, complemented
7C2×#16839501C211yesmaximal, characteristic
8C2×C61×#501C211yesmaximal, characteristic, complemented
9C22×#11758202—21no—
10C22×#11758201—21no—
11#16840516—22no—
12#16840516—22no—
13#16840516—22no—
14#16840516C2214yes—
15C2×#14519776—21no—
16C2×#14519772—21no—
17C2×#14519772—21no—
18C2×#14519769C2211yescharacteristic
19C2×#14519765C2211yescharacteristic
20C2×#14519762—21no—
21C2×#14519757—21no—
22C2×#14519755—21no—
23#16840168—22no—
24#16840168C2214yescomplemented
25#16840154—22no—
26#16840154C2214yes—
27C2×#14519710—21no—
28C2×#14519708—21no—
29C2×#14519688C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31#16839504C2214yescomplemented
32#16839503C2214yescomplemented
33#16839501C2214yes—
34C2×C4⋊Q8×C61C2211yescharacteristic
35C2×#125×C61C2211yescharacteristic
36C61×#501—22no—
37C61×#501C2214yescomplemented
38C2×#107×C61—21no—
39C2×#106×C61—21no—
40C2×#97×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122D411yescharacteristic
43C22×D244D411yescharacteristic
44C22×C4×D61—21no—
45C2×#11758202—24no—
46C2×#11758202—44no—
47C2×#11758201—24no—
48C2×#11758201—44no—
49C22×#92418—21no—
50C22×#92418—21no—
51#14519776—24no—
52#14519776—24no—
53#14519772—24no—
54#14519772—24no—
55#14519772—24no—
56#14519772—24no—
57#14519769—21no—
58#14519769—21no—
59#14519769D412yes—
60#14519769C2312yes—
61#14519765—21no—
62#14519765—21no—
63#14519765D412yes—
64#14519765C2312yes—
65#14519762—24no—
66#14519762—24no—
67#14519757—24no—
68#14519757—24no—
69#14519755—24no—
70#14519755—24no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61D411yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83#14519710—24no—
84#14519708—24no—
85#14519688—21no—
86#14519688—21no—
87#14519688C2312yes—
88#14519688C2312yes—
89C22×Q8×C61—21no—
90C22×C8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61C2312yes—
95#125×C61—21no—
96#125×C61—21no—
97#125×C61D412yes—
98#125×C61C2312yes—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C42×C61C2311yescharacteristic
106#107×C61—24no—
107#106×C61—24no—
108#97×C61—21no—
109#97×C61C2312yes—
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—42no—
113C2×Q8×C61—42no—
114C2×Q8×C61—41no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—41no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C2×D244—21no—
125C2×D244—42no—
126C2×D244C2×D412yes—
127C2×C4×D61—24no—
128C2×C4×D61—82no—
129C2×Dic122—24no—
130C2×Dic122—21no—
131C2×Dic122—42no—
132C2×Dic122—82no—
133C2×Dic122C2×D412yes—
134C2×C8×C61—24no—
135C2×C8×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D412yes—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2411yescharacteristic
152#11758202—48no—
153#11758201—48no—
154#11758198—24no—
155#11758198—24no—
156#11758198—24no—
157#11758198—22no—
158#11758198—22no—
159#11758198—22no—
160#11758198—22no—
161C4⋊Dic61—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61C2×D412yes—
165C4⋊Dic61C2×D412yes—
166#11758196—24no—
167#11758196—22no—
168#11758196—22no—
169#11758196—22no—
170#11758196—22no—
171C4×Dic61—24no—
172C2×#92418—24no—
173C2×#92418—24no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61#12011yescharacteristic
177C22×D61—22no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181Q8×C61—42no—
182Q8×C61—42no—
183Q8×C61—82no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—82no—
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205D244—42no—
206D244—81no—
207C4×D61—84no—
208Dic122—42no—
209Dic122—81no—
210Dic122—84no—
211C8×C61—24no—
212#92418—24no—
213#92418—24no—
214C2×#1660—611nomaximal, Sylow
215C22×C61#45312yes—
216C22×C61#45612yes—
217C22×C61#45612yes—
218C22×C61C2×#12011yescharacteristic
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—84no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61—82no—
232Dic61—42no—
233Dic61—42no—
234Dic61—42no—
235C2×#563—611no—
236C2×#501—611no—
237C2×#501D6111yescharacteristic, complemented
238C2×#497—611no—
239#1660—618no—
240C2×#330—611no—
241C2×#330—611no—
242C2×#329—611no—
243C2×C61#166014yescomplemented
244C2×C61C2×#45311yescharacteristic
245C2×C61C2×#45611yescharacteristic
246C2×C61C2×#45611yescharacteristic
247D61—82no—
248C22×Q16—1221no—
249C22×SD16—1221no—
250#563—614no—
251#563—1222no—
252#563—1222no—
253#563—1222no—
254C2×C4⋊Q8—1221no—
255C2×C4⋊Q8C2×D6111yescharacteristic
256C2×#126—1221no—
257C2×#126—1221no—
258C2×#125—611no—
259C2×#125C2×D6111yescharacteristic
260C2×#124—611no—
261C2×#123—1221no—
262C2×C22⋊Q8—1221no—
263C2×C22⋊Q8—1221no—
264#501—22no—
265#501—614no—
266#501—1222no—
267#501C2×D6114yescomplemented
268#497—614no—
269#497—1222no—
270C2×#107—21no—
271C2×#107—1221no—
272C2×#106—21no—
273C2×#106—1221no—
274C2×#97—611no—
275C2×#97—611no—
276C2×#97C2×D6111yescharacteristic
277#330—614no—
278#330—614no—
279#329—614no—
280C61C2×#166011yescharacteristic, complemented, Sylow
281C22×Q8—21no—
282C22×Q8—611no—
283C22×Q8—1221no—
284C22×D4—611no—
285C23×C4—1221no—
286C2×Q16—1224no—
287C2×Q16—2444no—
288C2×SD16—1224no—
289C2×SD16—2444no—
290C22×C8—21no—
291C22×C8—1221no—
292C22×C8—1221no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—1224no—
297C4⋊Q8—1224no—
298C4⋊Q8C22×D6112yes—
299#126—1224no—
300#126—1224no—
301#126—1224no—
302#126—1224no—
303#125—21no—
304#125—21no—
305#125—612no—
306#125—612no—
307#125—1221no—
308#125—1221no—
309#125C22⋊D6112yes—
310#125C22×D6112yes—
311#124—612no—
312#124—612no—
313#124—1221no—
314#124—1221no—
315#123—1224no—
316#123—1224no—
317C22⋊Q8—1224no—
318C22⋊Q8—1224no—
319C22⋊Q8—1224no—
320C22⋊Q8—1224no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4C22⋊D6111yescharacteristic
332C2×C4⋊C4C22⋊D6111yescharacteristic
333C2×C22⋊C4—1221no—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1221no—
338C2×C42—1221no—
339C2×C42C22×D6111yescharacteristic
340#107—24no—
341#107—1224no—
342#106—24no—
343#106—1224no—
344#97—21no—
345#97—612no—
346#97—612no—
347#97—1221no—
348#97—1221no—
349#97C22×D6112yes—
350C24—1221no—
351C2×Q8—22no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—41no—
355C2×Q8—612no—
356C2×Q8—1224no—
357C2×Q8—1221no—
358C2×Q8—4882no—
359C2×Q8—2442no—
360C2×D4—612no—
361C2×D4—1221no—
362C2×D4—2442no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—1224no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—4882no—
371C22×C4D4×D6111yescharacteristic
372C22×C4D4×D6111yescharacteristic
373C22×C4C2×C22⋊D6111yescharacteristic
374Q16—2448no—
375SD16—2448no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—1224no—
379C2×C8—1224no—
380C2×C8—1222no—
381C2×C8—1222no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—612no—
393C4⋊C4—612no—
394C4⋊C4—1224no—
395C4⋊C4—1224no—
396C4⋊C4—1222no—
397C4⋊C4—1222no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1222no—
401C4⋊C4—1222no—
402C4⋊C4C2×C22⋊D6112yes—
403C4⋊C4C2×C22⋊D6112yes—
404C22⋊C4—1224no—
405C22⋊C4—1224no—
406C22⋊C4—1224no—
407C22⋊C4—1222no—
408C22⋊C4—1222no—
409C22⋊C4—1222no—
410C22⋊C4—1222no—
411C42—1224no—
412C42D4×D6111yescharacteristic
413C42D4×D6111yescharacteristic
414C42C2×C22⋊D6111yescharacteristic
415C42C23×D6111yescharacteristic
416C23—1222no—
417C23—2442no—
418C23—2442no—
419C23—2442no—
420C23#1451974511yescharacteristic
421Q8—42no—
422Q8—42no—
423Q8—82no—
424Q8—4881no—
425Q8—4884no—
426Q8—2442no—
427D4—4881no—
428D4—2442no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—4884no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4#1451974511yescharacteristic
446C2×C4#1451974511yescharacteristic
447C2×C4#1451974511yescharacteristic
448C2×C4C2×D4×D6111yescharacteristic
449C2×C4C2×D4×D6111yescharacteristic
450C2×C4C22×C22⋊D6111yescharacteristic
451C8—24no—
452C8—1224no—
453C8—1224no—
454C22—4884no—
455C22—2442no—
456C22—2442no—
457C22—2442no—
458C22#1683992812yes—
459C22#1683994312yes—
460C22#1683994412yes—
461C22C2×#1451974511yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—42no—
470C4—82no—
471C4—2442no—
472C4—2442no—
473C4—2442no—
474C2—4882no—
475C2#2144512914yesdirect product factor
476C2C2×#1683992811yescharacteristic
477C2C2×#1683994311yescharacteristic
478C2C2×#1683994411yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#501C2
socleC23×C61#120
prime radicalC22×D244D4
hypercenterC2×#501D61
hypocenterC61C2×#1660
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#16840168C2semidirect
4. #21445129C2direct (N × H)
5. C2×#16839504C2semidirect
6. C2×#16839503C2semidirect
8. C2×C61×#501C2semidirect
24. #16840168C22semidirect
31. #16839504C22semidirect
32. #16839503C22semidirect
37. C61×#501C22semidirect
237. C2×#501D61semidirect
243. C2×C61#1660semidirect
267. #501C2×D61semidirect
280. C61C2×#1660semidirect
475. C2#21445129direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168405162
C2×#168401682
C2×#168401542
#214451292
C2×#168395042
C2×#168395032
C2×#168395012
C2×C61×#5012
C2×#166061