Finite Group Databasebeta

← C2×#21444986 #29655737

Subgroups — C2×#21444986 #29655737

Order 15616.

1174 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×Q8×D244C211yesmaximal, characteristic
2C2×#16840113C211yesmaximal, characteristic, complemented
3C2×#16840057C211yesmaximal, characteristic, complemented
4C2×#16840053C211yesmaximal, characteristic
5#21444986C218yesmaximal, direct product factor
6C2×#16839507C211yesmaximal, characteristic, complemented
7C2×#16839491C211yesmaximal, characteristic
8C2×C61×Q8⋊C8C211yesmaximal, characteristic, complemented
9C22×Q8×D61—21no—
10Q8×D244—22no—
11Q8×D244—22no—
12Q8×D244—22no—
13Q8×D244C2214yes—
14C2×#14519777—21no—
15C2×#14519777C2211yescharacteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519757—22no—
18C2×#14519757—21no—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21#16840113—22no—
22#16840113C2214yescomplemented
23#16840057—22no—
24#16840057C2214yescomplemented
25#16840053—22no—
26#16840053C2214yes—
27C2×#14519710—21no—
28C2×#14519709—21no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519702—22no—
31C2×#14519701—21no—
32C2×#14519699—21no—
33#16839507C2214yescomplemented
34#16839491C2214yes—
35C2×C4×Q8×C61C2211yescharacteristic
36C2×C4⋊C8×C61C2211yescharacteristic
37C2×C4×C8×C61C2211yescharacteristic
38C61×Q8⋊C8C2214yescomplemented
39C22×Dic122—22no—
40C22×Dic122—21no—
41C22×Dic122—21no—
42C2×Q8×D61—24no—
43C2×Q8×D61—24no—
44C2×Q8×D61—44no—
45C2×Q8×D61—44no—
46C22×D244D411yescharacteristic
47C22×C4×D61—22no—
48C22×C4×D61—21no—
49C22×#92418—21no—
50#14519777—24no—
51#14519777—21no—
52#14519777—21no—
53#14519777—24no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777D412yes—
58#14519777C2312yes—
59#14519777C2312yes—
60#14519757—24no—
61#14519757—28no—
62#14519757—24no—
63#14519757—28no—
64C4×D244—24no—
65C4×D244—24no—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244D412yes—
69C4×D244C2312yes—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61D412yes—
76C2×#11758198—22no—
77C2×#11758198—21no—
78#14519710—24no—
79#14519709—24no—
80C4⋊#92418—21no—
81C4⋊#92418C2312yes—
82#14519702—28no—
83#14519701—24no—
84#14519699—24no—
85C22×Q8×C61D411yescharacteristic
86C22×C8×C61—21no—
87C22×C8×C61—21no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61D412yes—
91C4×Q8×C61C2312yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2312yes—
98C4×C8×C61—21no—
99C4×C8×C61C2312yes—
100C23×D61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61SD1614yes—
104C2×Q8×C61C2×D412yes—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—41no—
113Q8×D61—48no—
114Q8×D61—416no—
115Q8×D61—48no—
116C2×D244—21no—
117C2×D244—22no—
118C2×D244Q1614yes—
119C2×D244C2×D412yes—
120C2×C4×D61—28no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—48no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×Dic122—28no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—48no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×Dic122—44no—
134C2×Dic122—41no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—42no—
138C2×C8×C61—42no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61C2×D412yes—
143C42×C61—22no—
144C42×C61—22no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2411yescharacteristic
149#11758198—28no—
150#11758198—22no—
151#11758198—22no—
152C4⋊Dic61—28no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161C4⋊Dic61C2×D414yes—
162C4⋊Dic61C2×D412yes—
163C2×#92418—22no—
164C2×#92418—42no—
165C23×C61#12011yescharacteristic
166C22×D61—22no—
167C22×D61—24no—
168C22×D61—42no—
169C22×D61—42no—
170Q8×C61—22no—
171Q8×C61—24no—
172Q8×C61C2×SD1614yes—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—42no—
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12011yescharacteristic
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61C2×SD1612yes—
182C2×C4×C61C2×Q1612yes—
183C2×C4×C61AGL(1,ℤ/8)12yes—
184C2×C4×C61#13712yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×Dic61—22no—
189C2×Dic61—24no—
190C2×Dic61—42no—
191C2×Dic61—44no—
192C2×Dic61—42no—
193C2×Dic61—82no—
194D244—22no—
195D244—24no—
196D244C2×Q1614yes—
197C4×D61—416no—
198C4×D61—42no—
199C4×D61—42no—
200C4×D61—44no—
201Dic122—416no—
202Dic122—42no—
203Dic122—42no—
204Dic122—44no—
205Dic122—44no—
206Dic122—82no—
207C8×C61—42no—
208C8×C61—42no—
209#92418—42no—
210C2×#1637—611nomaximal, Sylow
211C22×C61#45012yes—
212C22×C61#45112yes—
213C22×C61#45412yes—
214C22×C61C2×#12011yescharacteristic
215C2×D61—24no—
216C2×D61—48no—
217C2×D61—42no—
218C2×D61—42no—
219C4×C61—24no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—42no—
223C4×C61#45012yes—
224C4×C61#45112yes—
225C4×C61C22×SD1611yescharacteristic
226C4×C61C22×Q1611yescharacteristic
227C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
228C4×C61C2×#13711yescharacteristic
229Dic61—42no—
230Dic61—44no—
231Dic61—82no—
232C2×D4×Q8—611no—
233C2×#487—611no—
234C2×#477—611no—
235C2×#476—611no—
236#1637—618no—
237C2×#332—611no—
238C2×Q8⋊C8D6111yescharacteristic, complemented
239C2×D4⋊C8—611no—
240C2×C61#163714yescomplemented
241C2×C61C2×#45011yescharacteristic
242C2×C61C2×#45111yescharacteristic
243C2×C61C2×#45411yescharacteristic
244D61—44no—
245C23×Q8—1221no—
246D4×Q8—614no—
247D4×Q8—1222no—
248D4×Q8—1222no—
249D4×Q8—1222no—
250C2×C4⋊Q8—611no—
251C2×C4⋊Q8—611no—
252C2×C4⋊Q8—1221no—
253C2×C22⋊Q8—1222no—
254C2×C22⋊Q8—1221no—
255C2×C4×Q8C2×D6111yescharacteristic
256C2×C4×D4—611no—
257C2×C4×D4—1221no—
258#487—614no—
259#487—1222no—
260#477—614no—
261#477—1222no—
262#476—614no—
263#476—1222no—
264C2×#107—1221no—
265C2×#106—1221no—
266C2×C4⋊C8—611no—
267C2×C4⋊C8C2×D6111yescharacteristic
268C2×#103—1222no—
269C2×#103—1221no—
270C2×#102—1221no—
271C2×C4×C8C2×D6111yescharacteristic
272#332—614no—
273Q8⋊C8C2×D6114yescomplemented
274D4⋊C8—614no—
275C61C2×#163711yescharacteristic, complemented, Sylow
276C22×Q8—1222no—
277C22×Q8—1224no—
278C22×Q8—1224no—
279C22×Q8—1221no—
280C22×Q8—1221no—
281C22×Q8—2444no—
282C22×Q8—2444no—
283C22×Q8D24411yescharacteristic
284C22×D4—611no—
285C23×C4—1222no—
286C23×C4—1221no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—1221no—
290C4⋊Q8—612no—
291C4⋊Q8—612no—
292C4⋊Q8—612no—
293C4⋊Q8—1224no—
294C4⋊Q8—1224no—
295C4⋊Q8—1221no—
296C4⋊Q8—1221no—
297C4⋊Q8—1221no—
298C4⋊Q8—1221no—
299C4⋊Q8—1221no—
300C22⋊Q8—1228no—
301C22⋊Q8—1224no—
302C22⋊Q8—1228no—
303C22⋊Q8—1224no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8D24412yes—
307C4×Q8C22×D6112yes—
308C4×D4—612no—
309C4×D4—612no—
310C4×D4—1224no—
311C4×D4—1224no—
312C4×D4—1221no—
313C4×D4—1221no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4D24411yescharacteristic
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1221no—
324C2×C42—21no—
325C2×C42C22×D6111yescharacteristic
326#107—1224no—
327#106—1224no—
328C4⋊C8—21no—
329C4⋊C8—612no—
330C4⋊C8—1221no—
331C4⋊C8C22×D6112yes—
332#103—1228no—
333#103—1224no—
334#102—1224no—
335C4×C8—21no—
336C4×C8C22×D6112yes—
337C24—1221no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—1228no—
341C2×Q8—1222no—
342C2×Q8—1222no—
343C2×Q8—1222no—
344C2×Q8—2444no—
345C2×Q8—2448no—
346C2×Q8—2448no—
347C2×Q8—24416no—
348C2×Q8—2448no—
349C2×Q8—2441no—
350C2×Q8—2442no—
351C2×Q8—2442no—
352C2×Q8#1175819014yes—
353C2×Q8C2×D24412yes—
354C2×D4—612no—
355C2×D4—614no—
356C2×D4—1221no—
357C2×D4—1222no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—1228no—
361C22×C4—1222no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4—1222no—
365C22×C4—2448no—
366C22×C4—2441no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4C2×D24411yescharacteristic
370C22×C4D4×D6111yescharacteristic
371C22×C4D4×D6111yescharacteristic
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—1222no—
377C2×C8—2442no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—614no—
382C4⋊C4—612no—
383C4⋊C4—1228no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—1222no—
390C4⋊C4—2441no—
391C4⋊C4—2441no—
392C4⋊C4C2×D24412yes—
393C22⋊C4—1228no—
394C22⋊C4—1222no—
395C22⋊C4—1222no—
396C42—22no—
397C42—22no—
398C42C2×D24411yescharacteristic
399C42D4×D6111yescharacteristic
400C42D4×D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—1222no—
403C23—1224no—
404C23—2442no—
405C23—2442no—
406C23#1451974711yescharacteristic
407Q8—22no—
408Q8—24no—
409Q8—4882no—
410Q8—2444no—
411Q8—24416no—
412Q8—2442no—
413Q8—2442no—
414Q8—2444no—
415Q8C2×#1175819014yes—
416D4—614no—
417D4—1222no—
418D4—1224no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—1224no—
425C2×C4—1222no—
426C2×C4—4882no—
427C2×C4—24416no—
428C2×C4—2442no—
429C2×C4—2444no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2444no—
434C2×C4#1451974711yescharacteristic
435C2×C4#1451974711yescharacteristic
436C2×C4#1451974711yescharacteristic
437C2×C4C2×#1175819012yes—
438C2×C4Q16×D6112yes—
439C2×C4#1451980512yes—
440C2×C4#1451980612yes—
441C2×C4C2×D4×D6111yescharacteristic
442C2×C4C2×D4×D6111yescharacteristic
443C2×C4C22×D24411yescharacteristic
444C8—42no—
445C8—42no—
446C8—2442no—
447C22—1224no—
448C22—2442no—
449C22—2448no—
450C22—2442no—
451C22#1683991512yes—
452C22#1683991912yes—
453C22#1683993612yes—
454C22C2×#1451974711yescharacteristic
455C4—21no—
456C4—21no—
457C4—24no—
458C4—42no—
459C4—4882no—
460C4—2444no—
461C4—2442no—
462C4#1683991512yes—
463C4#1683991912yes—
464C4C22×#1175819011yescharacteristic
465C4C2×Q16×D6111yescharacteristic
466C4C2×#1451980511yescharacteristic
467C4C2×#1451980611yescharacteristic
468C2—2444no—
469C2#2144498614yesdirect product factor
470C2C2×#1683991511yescharacteristic
471C2C2×#1683991911yescharacteristic
472C2C2×#1683993611yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×Q8⋊C8C2
socleC23×C61#120
prime radicalC22×D244D4
hypercenterC2×Q8⋊C8D61
hypocenterC61C2×#1637
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16840113C2semidirect
3. C2×#16840057C2semidirect
5. #21444986C2direct (N × H)
6. C2×#16839507C2semidirect
8. C2×C61×Q8⋊C8C2semidirect
22. #16840113C22semidirect
24. #16840057C22semidirect
33. #16839507C22semidirect
38. C61×Q8⋊C8C22semidirect
238. C2×Q8⋊C8D61semidirect
240. C2×C61#1637semidirect
273. Q8⋊C8C2×D61semidirect
275. C61C2×#1637semidirect
469. C2#21444986direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D2442
C2×#168401132
C2×#168400572
C2×#168400532
#214449862
C2×#168395072
C2×#168394912
C2×C61×Q8⋊C82
C2×#163761