Finite Group Databasebeta

← C2×#21445051 #29655786

Subgroups — C2×#21445051 #29655786

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840415C211yesmaximal, characteristic
2C2×#16840039C211yesmaximal, characteristic, complemented
3C2×#16839964C211yesmaximal, characteristic, complemented
4#21445051C218yesmaximal, direct product factor
5C2×#16839508C211yesmaximal, characteristic, complemented
6C2×#16839491C211yesmaximal, characteristic, complemented
7C2×#16839491C211yesmaximal, characteristic, complemented
8C2×C61×#500C211yesmaximal, characteristic, complemented
9C22×#11758226—21no—
10C22×#11758201—21no—
11#16840415—22no—
12#16840415—22no—
13#16840415—22no—
14#16840415C2214yes—
15C2×#14519762—22no—
16C2×#14519753—21no—
17C2×#14519752—21no—
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840039—22no—
22#16840039C2214yescomplemented
23#16839964—22no—
24#16839964C2214yescomplemented
25C2×#14519710—21no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701—21no—
29C2×#14519697—21no—
30#16839508C2214yescomplemented
31#16839491C2214yescomplemented
32#16839491C2214yescomplemented
33C2×C4⋊Q8×C61C2211yescharacteristic
34C2×C4⋊Q8×C61C2211yescharacteristic
35C61×#500—22no—
36C61×#500C2214yescomplemented
37C2×#107×C61—22no—
38C2×C4×C8×C61C2211yescharacteristic
39C2×#11758226—24no—
40C2×#11758226—24no—
41C2×#11758226—44no—
42C2×#11758226—44no—
43C22×D244—22no—
44C22×D244D411yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61—21no—
48C22×C4×D61—21no—
49C2×#11758201—24no—
50C2×#11758201—44no—
51C22×#92418—21no—
52C22×#92418—21no—
53#14519762—28no—
54#14519762—28no—
55#14519753—24no—
56#14519753—24no—
57#14519752—24no—
58#14519752—24no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D412yes—
64C4×D244D412yes—
65C4×D244C2312yes—
66C4×D244C2312yes—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×#11758196—22no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—24no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75#14519710—24no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2312yes—
80#14519701—24no—
81#14519697—24no—
82C22×Q8×C61—21no—
83C22×Q8×C61—21no—
84C22×C8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61D412yes—
91C4⋊Q8×C61C2312yes—
92C4⋊Q8×C61C2312yes—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D412yes—
97C2×C42×C61C2311yescharacteristic
98#107×C61—28no—
99C4×C8×C61—21no—
100C4×C8×C61C2312yes—
101C23×D61—21no—
102C23×D61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—41no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
111C22×C4×C61—22no—
112C22×C4×C61—41no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—21no—
118#11758226—48no—
119#11758226—416no—
120#11758226—48no—
121C2×D244—21no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—28no—
126C2×D244—21no—
127C2×D244—48no—
128C2×D244D814yes—
129C2×D244C2×D412yes—
130C2×D244C2×D412yes—
131C2×C4×D61—28no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—48no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×C8×C61—24no—
142C2×C8×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61C2×D414yes—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2411yescharacteristic
155#11758201—48no—
156#11758198—28no—
157#11758198—22no—
158#11758198—22no—
159#11758198—22no—
160#11758198—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61C2×D412yes—
164C4⋊Dic61C2×D412yes—
165#11758196—28no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—42no—
169C2×#92418—42no—
170C23×C61#12011yescharacteristic
171C22×D61—22no—
172C22×D61—24no—
173C22×D61—22no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—44no—
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61—82no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—82no—
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61C2×D812yes—
194C2×C4×C61#13512yes—
195C2×C4×C61AGL(1,ℤ/8)12yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204D244—22no—
205D244—22no—
206D244—22no—
207D244—24no—
208D244—416no—
209D244—42no—
210D244C2×D814yes—
211C4×D61—416no—
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—42no—
216C4×D61—44no—
217C4×D61—44no—
218C8×C61—24no—
219#92418—42no—
220#92418—42no—
221C2×#1647—611nomaximal, Sylow
222C22×C61#44712yes—
223C22×C61#45212yes—
224C22×C61#45612yes—
225C22×C61C2×#12011yescharacteristic
226C2×D61—24no—
227C2×D61—48no—
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—42no—
232C2×D61—42no—
233C2×D61—84no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—44no—
237C4×C61—82no—
238C4×C61#44712yes—
239C4×C61#45212yes—
240C4×C61C22×D811yescharacteristic
241C4×C61C2×#13511yescharacteristic
242C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C4×C61C2×#13711yescharacteristic
244Dic61—42no—
245Dic61—42no—
246C2×#547—611no—
247C2×#500D6111yescharacteristic, complemented
248C2×#474—611no—
249C2×#461—611no—
250#1647—618no—
251C2×#332—611no—
252C2×D4⋊C8—611no—
253C2×D4⋊C8—611no—
254C2×C61#164714yescomplemented
255C2×C61C2×#44711yescharacteristic
256C2×C61C2×#45211yescharacteristic
257C2×C61C2×#45611yescharacteristic
258D61—44no—
259D61—82no—
260C22×#41—1221no—
261C22×SD16—1221no—
262#547—614no—
263#547—1222no—
264#547—1222no—
265#547—1222no—
266C2×C4⋊Q8C2×D6111yescharacteristic
267C2×C4⋊Q8C2×D6111yescharacteristic
268C2×#123—1222no—
269C2×C22⋊Q8—1221no—
270C2×#121—1221no—
271C2×C4×D4—611no—
272C2×C4×D4—611no—
273C22×C4⋊C4—1221no—
274#500—22no—
275#500C2×D6114yescomplemented
276#474—614no—
277#474—1222no—
278#461—614no—
279#461—1222no—
280C2×#107—22no—
281C2×#107—1221no—
282C2×C4⋊C8—611no—
283C2×C4⋊C8—611no—
284C2×#103—1221no—
285C2×#102—1221no—
286C2×C4×C8C2×D6111yescharacteristic
287#332—614no—
288D4⋊C8—614no—
289D4⋊C8—614no—
290C61C2×#164711yescharacteristic, complemented, Sylow
291C2×#41—1224no—
292C2×#41—1224no—
293C2×#41—2444no—
294C2×#41—2444no—
295C22×Q8—21no—
296C22×Q8—21no—
297C22×D4—611no—
298C22×D4—611no—
299C22×D4—1222no—
300C23×C4—1222no—
301C23×C4—1221no—
302C23×C4—1221no—
303C2×SD16—1224no—
304C2×SD16—2444no—
305C22×C8—21no—
306C22×C8—1221no—
307C22×C8—1221no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8C22⋊D6112yes—
314C4⋊Q8C22×D6112yes—
315C4⋊Q8C22×D6112yes—
316#123—1228no—
317#123—1228no—
318C22⋊Q8—1224no—
319C22⋊Q8—1224no—
320#121—1224no—
321#121—1224no—
322C4×D4—612no—
323C4×D4—612no—
324C4×D4—612no—
325C4×D4—612no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4—1224no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4C22⋊D6112yes—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1221no—
342C2×C42C22×D6111yescharacteristic
343#107—28no—
344#107—1224no—
345C4⋊C8—612no—
346C4⋊C8—612no—
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349#103—1224no—
350#102—1224no—
351C4×C8—21no—
352C4×C8C22×D6112yes—
353C24—1221no—
354C24—1221no—
355#41—2448no—
356#41—2448no—
357#41—24416no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×D4—612no—
367C2×D4—612no—
368C2×D4—614no—
369C2×D4—1228no—
370C2×D4—1222no—
371C2×D4—1221no—
372C2×D4—1221no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—2448no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—1228no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—2448no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393SD16—2448no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—2442no—
399C2×C8—2442no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—612no—
408C4⋊C4—612no—
409C4⋊C4—1228no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4C2×C22⋊D6114yes—
413C22⋊C4—1228no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C22⋊C4—1222no—
417C22⋊C4—1222no—
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42C2×C22⋊D6111yescharacteristic
421C42C23×D6111yescharacteristic
422C23—1222no—
423C23—1224no—
424C23—1222no—
425C23—2442no—
426C23—2442no—
427C23—2442no—
428C23—2442no—
429C23—2442no—
430C23#1451974511yescharacteristic
431Q8—42no—
432Q8—42no—
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437D4—614no—
438D4—1222no—
439D4—1222no—
440D4—1222no—
441D4—1224no—
442D4—24416no—
443D4—2442no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—82no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—24416no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2444no—
459C2×C4—2444no—
460C2×C4#1451974511yescharacteristic
461C2×C4#1451974511yescharacteristic
462C2×C4#1451974511yescharacteristic
463C2×C4D8×D6112yes—
464C2×C4#1451980012yes—
465C2×C4#1451980112yes—
466C2×C4#1451980912yes—
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C22×C22⋊D6111yescharacteristic
470C8—24no—
471C8—2442no—
472C8—2442no—
473C22—1224no—
474C22—4884no—
475C22—2442no—
476C22—2442no—
477C22—2442no—
478C22—2442no—
479C22—2442no—
480C22—2448no—
481C22#1683989412yes—
482C22#1683992312yes—
483C22#1683994412yes—
484C22C2×#1451974511yescharacteristic
485C4—21no—
486C4—21no—
487C4—44no—
488C4—82no—
489C4—2442no—
490C4—2442no—
491C4#1683989412yes—
492C4#1683992312yes—
493C4C2×D8×D6111yescharacteristic
494C4C2×#1451980011yescharacteristic
495C4C2×#1451980111yescharacteristic
496C4C2×#1451980911yescharacteristic
497C2—4882no—
498C2—2444no—
499C2#2144505114yesdirect product factor
500C2C2×#1683989411yescharacteristic
501C2C2×#1683992311yescharacteristic
502C2C2×#1683994411yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#500C2
socleC23×C61#120
prime radicalC2×#16840415C2
hypercenterC2×#500D61
hypocenterC61C2×#1647
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
2. C2×#16840039C2semidirect
3. C2×#16839964C2semidirect
4. #21445051C2direct (N × H)
5. C2×#16839508C2semidirect
6. C2×#16839491C2semidirect
7. C2×#16839491C2semidirect
8. C2×C61×#500C2semidirect
22. #16840039C22semidirect
24. #16839964C22semidirect
30. #16839508C22semidirect
31. #16839491C22semidirect
32. #16839491C22semidirect
36. C61×#500C22semidirect
247. C2×#500D61semidirect
254. C2×C61#1647semidirect
275. #500C2×D61semidirect
290. C61C2×#1647semidirect
499. C2#21445051direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404152
C2×#168400392
C2×#168399642
#214450512
C2×#168395082
C2×#168394912
C2×C61×#5002
C2×#164761