Finite Group Databasebeta

← C2×#21444970 #29655718

Subgroups — C2×#21444970 #29655718

Order 15616.

1206 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840114C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3C2×#16840052C211yesmaximal, characteristic, complemented
4#21444970C218yesmaximal, direct product factor
5C2×#16839508C211yesmaximal, characteristic, complemented
6C2×D4⋊#92418C211yesmaximal, characteristic, complemented
7C2×D4⋊#92418C211yesmaximal, characteristic, complemented
8C2×C61×#547C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic
10#16840114—22no—
11#16840114C2214yescomplemented
12#16840052—22no—
13#16840052—22no—
14#16840052C2214yescomplemented
15#16840052C2214yescomplemented
16C2×#14519708—21no—
17C2×#14519708—21no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519700—22no—
21C2×#14519698—21no—
22C2×#14519698—21no—
23C2×C4×#92418C2211yescharacteristic
24#16839508C2214yescomplemented
25D4⋊#92418C2214yescomplemented
26D4⋊#92418C2214yescomplemented
27C22×#41×C61—21no—
28C61×#547—22no—
29C61×#547—22no—
30C61×#547—22no—
31C61×#547C2214yes—
32C2×C4⋊Q8×C61C2211yescharacteristic
33C2×#123×C61—22no—
34C2×C22⋊Q8×C61—21no—
35C2×#121×C61—21no—
36C2×C4×D4×C61C2211yescharacteristic
37C2×C4×D4×C61C2211yescharacteristic
38C22×C4⋊C4×C61—21no—
39C22×Dic122—21no—
40C22×#92418—21no—
41C22×#92418—21no—
42C22×#92418—21no—
43#14519777—21no—
44#14519777—21no—
45#14519777—21no—
46#14519777C2312yes—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49#14519708—24no—
50#14519708—24no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519700—28no—
56#14519698—24no—
57#14519698—24no—
58C4×#92418—21no—
59C4×#92418C2312yes—
60C2×#41×C61—24no—
61C2×#41×C61—24no—
62C2×#41×C61—44no—
63C2×#41×C61—44no—
64C22×Q8×C61—21no—
65C22×D4×C61—22no—
66C22×D4×C61D411yescharacteristic
67C22×D4×C61D411yescharacteristic
68C23×C4×C61—22no—
69C23×C4×C61—21no—
70C23×C4×C61—21no—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61D412yes—
74C4⋊Q8×C61C2312yes—
75#123×C61—28no—
76#123×C61—28no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—24no—
79#121×C61—24no—
80#121×C61—24no—
81C4×D4×C61—21no—
82C4×D4×C61—21no—
83C4×D4×C61—21no—
84C4×D4×C61—21no—
85C4×D4×C61D412yes—
86C4×D4×C61D412yes—
87C4×D4×C61C2312yes—
88C4×D4×C61C2312yes—
89C2×C4⋊C4×C61—24no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—24no—
93C2×C4⋊C4×C61D412yes—
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C24×C61—21no—
101C24×C61—21no—
102#41×C61—48no—
103#41×C61—416no—
104#41×C61—48no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C2×D4×C61—28no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—48no—
116C2×D4×C61SD1614yes—
117C2×D4×C61C2×D412yes—
118C2×D4×C61C2×D412yes—
119C22×C4×C61—28no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—48no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×Dic61—41no—
136C2×Dic122—22no—
137C2×Dic122—44no—
138C2×Dic122—41no—
139C4⋊C4×C61—28no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61C2×D412yes—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D414yes—
148C22⋊C4×C61—28no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2411yescharacteristic
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—42no—
165C2×#92418—42no—
166C2×#92418—42no—
167C23×C61—22no—
168C23×C61—24no—
169C23×C61—22no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61#12011yescharacteristic
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—44no—
179D4×C61—22no—
180D4×C61—22no—
181D4×C61—22no—
182D4×C61—24no—
183D4×C61—416no—
184D4×C61—42no—
185D4×C61C2×SD1614yes—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—416no—
191C2×C4×C61—42no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—44no—
199C2×C4×C61—44no—
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61C2×SD1612yes—
204C2×C4×C61#13512yes—
205C2×C4×C61AGL(1,ℤ/8)12yes—
206C2×C4×C61#13712yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×Dic61—42no—
211C2×Dic61—82no—
212Dic122—44no—
213Dic122—82no—
214#92418—42no—
215#92418—42no—
216#92418—42no—
217C2×#1635—611nomaximal, Sylow
218C22×C61—24no—
219C22×C61—48no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—84no—
226C22×C61#45012yes—
227C22×C61#45212yes—
228C22×C61#45412yes—
229C22×C61C2×#12011yescharacteristic
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—44no—
234C4×C61—42no—
235C4×C61#45012yes—
236C4×C61#45212yes—
237C4×C61C22×SD1611yescharacteristic
238C4×C61C2×#13511yescharacteristic
239C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C61C2×#13711yescharacteristic
241Dic61—82no—
242C2×#547D6111yescharacteristic
243C2×#487—611no—
244C2×#476—611no—
245C2×#476—611no—
246#1635—618no—
247C2×#332—611no—
248C2×D4⋊C8—611no—
249C2×D4⋊C8—611no—
250C2×C61—44no—
251C2×C61—82no—
252C2×C61#163514yescomplemented
253C2×C61C2×#45011yescharacteristic
254C2×C61C2×#45211yescharacteristic
255C2×C61C2×#45411yescharacteristic
256C22×#41—21no—
257#547—22no—
258#547—22no—
259#547—22no—
260#547C2×D6114yes—
261C2×C4⋊Q8—611no—
262C2×C4⋊Q8C2×D6111yescharacteristic
263C2×#123—22no—
264C2×C22⋊Q8—21no—
265C2×#121—21no—
266C2×C4×D4C2×D6111yescharacteristic
267C2×C4×D4C2×D6111yescharacteristic
268C22×C4⋊C4—21no—
269#487—614no—
270#487—1222no—
271#476—614no—
272#476—614no—
273#476—1222no—
274#476—1222no—
275C2×#106—1221no—
276C2×#106—1221no—
277C2×C4⋊C8—611no—
278C2×C4⋊C8—611no—
279C2×#103—1222no—
280C2×#102—1221no—
281C2×#102—1221no—
282C2×C4×C8—611no—
283#332—614no—
284D4⋊C8—614no—
285D4⋊C8—614no—
286C61C2×#163511yescharacteristic, complemented, Sylow
287C2×#41—24no—
288C2×#41—24no—
289C2×#41—44no—
290C2×#41—44no—
291C22×Q8—21no—
292C22×Q8—1221no—
293C22×D4—22no—
294C22×D4C22⋊D6111yescharacteristic
295C22×D4C22⋊D6111yescharacteristic
296C23×C4—22no—
297C23×C4—21no—
298C23×C4—21no—
299C22×C8—1221no—
300C22×C8—1221no—
301C22×C8—1221no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—612no—
305C4⋊Q8—1221no—
306C4⋊Q8—1221no—
307C4⋊Q8—1221no—
308C4⋊Q8C22⋊D6112yes—
309C4⋊Q8C22×D6112yes—
310#123—28no—
311#123—28no—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314#121—24no—
315#121—24no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4C22⋊D6112yes—
321C4×D4C22⋊D6112yes—
322C4×D4C22×D6112yes—
323C4×D4C22×D6112yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4C22⋊D6111yescharacteristic
331C2×C4⋊C4C22⋊D6111yescharacteristic
332C2×C4⋊C4C22⋊D6112yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C42C22×D6111yescharacteristic
337#106—1224no—
338#106—1224no—
339C4⋊C8—612no—
340C4⋊C8—612no—
341C4⋊C8—1221no—
342C4⋊C8—1221no—
343#103—1228no—
344#102—1224no—
345#102—1224no—
346C4×C8—612no—
347C4×C8—1221no—
348C24—21no—
349C24—21no—
350#41—48no—
351#41—416no—
352#41—48no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—1222no—
358C2×Q8—2444no—
359C2×Q8—2441no—
360C2×D4—28no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—48no—
367C2×D4#1175820014yes—
368C2×D4C2×C22⋊D6112yes—
369C2×D4C2×C22⋊D6112yes—
370C22×C4—28no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—2441no—
384C22×C4C2×C22⋊D6111yescharacteristic
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8—1222no—
390C2×C8—2442no—
391C2×C8—2442no—
392C2×C8—2442no—
393C4⋊C4—28no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—1222no—
400C4⋊C4—1222no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4C2×C22⋊D6114yes—
404C4⋊C4C2×C22⋊D6112yes—
405C4⋊C4C2×C22⋊D6112yes—
406C22⋊C4—28no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C42C2×C22⋊D6111yescharacteristic
412C42C2×C22⋊D6111yescharacteristic
413C42C2×C22⋊D6111yescharacteristic
414C42C23×D6111yescharacteristic
415C23—22no—
416C23—24no—
417C23—22no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23#1451974611yescharacteristic
424Q8—42no—
425Q8—42no—
426Q8—44no—
427Q8—4882no—
428Q8—2444no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4—24no—
433D4—416no—
434D4—42no—
435D4C2×#1175820014yes—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—416no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—4882no—
451C2×C4—2442no—
452C2×C4#1451974611yescharacteristic
453C2×C4#1451974611yescharacteristic
454C2×C4#1451974611yescharacteristic
455C2×C4C2×#1175820012yes—
456C2×C4#1451979712yes—
457C2×C4#1451980312yes—
458C2×C4#1451980712yes—
459C2×C4C22×C22⋊D6111yescharacteristic
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C8—2442no—
463C8—2442no—
464C8—2442no—
465C22—24no—
466C22—42no—
467C22—48no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—84no—
473C22#1683991312yes—
474C22#1683992412yes—
475C22#1683993512yes—
476C22C2×#1451974611yescharacteristic
477C4—21no—
478C4—21no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—4882no—
483C4#1683991312yes—
484C4#1683992412yes—
485C4C22×#1175820011yescharacteristic
486C4C2×#1451979711yescharacteristic
487C4C2×#1451980311yescharacteristic
488C4C2×#1451980711yescharacteristic
489C2—44no—
490C2—82no—
491C2#2144497014yesdirect product factor
492C2C2×#1683991311yescharacteristic
493C2C2×#1683992411yescharacteristic
494C2C2×#1683993511yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#547C2
socleC23×C61#120
hypercenterC2×#547D61
hypocenterC61C2×#1635
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16840114C2semidirect
2. C2×#16840052C2semidirect
3. C2×#16840052C2semidirect
4. #21444970C2direct (N × H)
5. C2×#16839508C2semidirect
6. C2×D4⋊#92418C2semidirect
7. C2×D4⋊#92418C2semidirect
11. #16840114C22semidirect
14. #16840052C22semidirect
15. #16840052C22semidirect
24. #16839508C22semidirect
25. D4⋊#92418C22semidirect
26. D4⋊#92418C22semidirect
252. C2×C61#1635semidirect
286. C61C2×#1635semidirect
491. C2#21444970direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168401142
C2×#168400522
#214449702
C2×#168395082
C2×D4⋊#924182
C2×C61×#5472
C2×#163561