Finite Group Databasebeta

← C2×#21445077 #29655848

Subgroups — C2×#21445077 #29655848

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic
2C2×#16840167C211yesmaximal, characteristic, complemented
3C2×#16840048C211yesmaximal, characteristic, complemented
4C2×#16839967C211yesmaximal, characteristic
5#21445077C218yesmaximal, direct product factor
6C2×#16839507C211yesmaximal, characteristic, complemented
7C2×#16839494C211yesmaximal, characteristic, complemented
8C2×C61×D4⋊C8C211yesmaximal, characteristic
9C22×#11758200—21no—
10#16840482—22no—
11#16840482—22no—
12#16840482—22no—
13#16840482C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519770—22no—
17C2×#14519764—21no—
18C2×#14519756—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742—21no—
21C2×#14519736—21no—
22#16840167—22no—
23#16840167C2214yescomplemented
24#16840048—22no—
25#16840048C2214yescomplemented
26#16839967—22no—
27#16839967C2214yes—
28C2×#14519711—22no—
29C2×#14519709—21no—
30C2×C4⋊#92418C2211yescharacteristic
31C2×#14519702—21no—
32C2×#14519700—21no—
33#16839507C2214yescomplemented
34#16839494C2214yescomplemented
35C2×C4×D4×C61C2211yescharacteristic
36C2×C4⋊C8×C61C2211yescharacteristic
37C2×C4×C8×C61C2211yescharacteristic
38C61×D4⋊C8C2214yes—
39C22×Dic122—21no—
40C22×Dic122—21no—
41C22×Dic122D411yescharacteristic
42C2×#11758200—24no—
43C2×#11758200—44no—
44C22×#92418—21no—
45#14519777—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777D412yes—
51#14519777C2312yes—
52#14519777C2312yes—
53#14519770—28no—
54#14519770—28no—
55#14519764—24no—
56#14519764—24no—
57#14519756—24no—
58#14519756—24no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61C4×Dic122D412yes—
62C4×Dic122C2312yes—
63#14519742—24no—
64#14519742—24no—
65#14519736—24no—
66#14519736—24no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61D412yes—
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78#14519711—28no—
79#14519709—24no—
80C4⋊#92418—21no—
81C4⋊#92418C2312yes—
82#14519702—24no—
83#14519700—24no—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C22×C8×C61—21no—
87C22×C8×C61—21no—
88C4×D4×C61—21no—
89C4×D4×C61—21no—
90C4×D4×C61D412yes—
91C4×D4×C61C2312yes—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—21no—
96C4⋊C8×C61C2312yes—
97C4×C8×C61—21no—
98C4×C8×C61C2312yes—
99C24×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61C2×D412yes—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×Dic61—22no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—41no—
116C2×Dic122—22no—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—21no—
120C2×Dic122—22no—
121C2×Dic122—22no—
122C2×Dic122—42no—
123C2×Dic122—42no—
124C2×Dic122—42no—
125C2×Dic122—42no—
126C2×Dic122—41no—
127C2×Dic122C2×D412yes—
128C2×C8×C61—24no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61—42no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61C2×D412yes—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2411yescharacteristic
140C22⋊Dic61—28no—
141C22⋊Dic61—28no—
142#11758200—48no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4⋊Dic61C2×D412yes—
152C4⋊Dic61C2×D414yes—
153#11758196—28no—
154#11758196—22no—
155#11758196—22no—
156C4×Dic61—28no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159C2×#92418—22no—
160C2×#92418—42no—
161C23×C61—22no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61#12011yescharacteristic
166D4×C61—22no—
167D4×C61—22no—
168D4×C61—42no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—44no—
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#13512yes—
179C2×C4×C61#13512yes—
180C2×C4×C61AGL(1,ℤ/8)12yes—
181C2×C4×C61#13712yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—24no—
188C2×Dic61—44no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—82no—
193Dic122—22no—
194Dic122—22no—
195Dic122—42no—
196Dic122—42no—
197Dic122—44no—
198Dic122—42no—
199Dic122—42no—
200Dic122—42no—
201Dic122—82no—
202C8×C61—24no—
203C8×C61—42no—
204#92418—42no—
205C2×#1650—611nomaximal, Sylow
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—84no—
210C22×C61#44912yes—
211C22×C61#45212yes—
212C22×C61#45612yes—
213C22×C61C2×#12011yescharacteristic
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—42no—
217C4×C61#44912yes—
218C4×C61#45212yes—
219C4×C61C2×#13511yescharacteristic
220C4×C61C2×#13511yescharacteristic
221C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
222C4×C61C2×#13711yescharacteristic
223Dic61—44no—
224Dic61—42no—
225Dic61—42no—
226Dic61—82no—
227C2×#556—611no—
228C2×#500—611no—
229C2×#475—611no—
230C2×#461—611no—
231#1650—618no—
232C2×#332—611no—
233C2×Q8⋊C8—611no—
234C2×D4⋊C8D6111yescharacteristic
235C2×C61—82no—
236C2×C61#165014yescomplemented
237C2×C61C2×#44911yescharacteristic
238C2×C61C2×#45211yescharacteristic
239C2×C61C2×#45611yescharacteristic
240C22×SD16—1221no—
241#556—614no—
242#556—1222no—
243#556—1222no—
244#556—1222no—
245C2×C4⋊Q8—611no—
246C2×C4⋊Q8—611no—
247C2×#126—1222no—
248C2×#124—1221no—
249C2×C22⋊Q8—1221no—
250C2×C4×Q8—611no—
251C2×C4×Q8—1221no—
252C2×C4×D4C2×D6111yescharacteristic
253C2×#117—1221no—
254#500—614no—
255#500—1222no—
256#475—614no—
257#475—1222no—
258#461—614no—
259#461—1222no—
260C2×#107—1222no—
261C2×#106—1221no—
262C2×C4⋊C8—611no—
263C2×C4⋊C8C2×D6111yescharacteristic
264C2×#103—1221no—
265C2×#103—1221no—
266C2×C4×C8C2×D6111yescharacteristic
267#332—614no—
268Q8⋊C8—614no—
269D4⋊C8C2×D6114yes—
270C61C2×#165011yescharacteristic, complemented, Sylow
271C22×Q8—611no—
272C22×Q8—1221no—
273C22×Q8—1221no—
274C22×D4D24411yescharacteristic
275C23×C4—21no—
276C2×SD16—1224no—
277C2×SD16—2444no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—1221no—
281C4⋊Q8—612no—
282C4⋊Q8—612no—
283C4⋊Q8—612no—
284C4⋊Q8—1221no—
285C4⋊Q8—1221no—
286C4⋊Q8—1221no—
287C4⋊Q8—1221no—
288C4⋊Q8—1221no—
289#126—1228no—
290#126—1228no—
291#124—1224no—
292#124—1224no—
293C22⋊Q8—1224no—
294C22⋊Q8—1224no—
295C4×Q8—612no—
296C4×Q8—612no—
297C4×Q8—1224no—
298C4×Q8—1224no—
299C4×Q8—1221no—
300C4×Q8—1221no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4D24412yes—
304C4×D4C22×D6112yes—
305#117—1224no—
306#117—1224no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4D24411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1222no—
318C2×C42—1222no—
319C2×C42—1221no—
320C2×C42C22×D6111yescharacteristic
321#107—1228no—
322#106—1224no—
323C4⋊C8—21no—
324C4⋊C8—612no—
325C4⋊C8—1221no—
326C4⋊C8C22×D6112yes—
327#103—1224no—
328#103—1224no—
329C4×C8—21no—
330C4×C8C22×D6112yes—
331C24—21no—
332C2×Q8—612no—
333C2×Q8—1222no—
334C2×Q8—1221no—
335C2×Q8—1222no—
336C2×Q8—1222no—
337C2×Q8—1222no—
338C2×Q8—1222no—
339C2×Q8—2442no—
340C2×Q8—2442no—
341C2×Q8—2441no—
342C2×Q8—2442no—
343C2×Q8—2442no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4C2×D24412yes—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—1221no—
354C22×C4—1222no—
355C22×C4—1221no—
356C22×C4—2441no—
357C22×C4C2×D24411yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C22×C4D4×D6111yescharacteristic
360SD16—2448no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—42no—
365C2×C8—1222no—
366C2×C8—2442no—
367C4⋊C4—21no—
368C4⋊C4—612no—
369C4⋊C4—614no—
370C4⋊C4—1228no—
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C4⋊C4—1221no—
374C4⋊C4—1222no—
375C4⋊C4—1222no—
376C4⋊C4—1222no—
377C4⋊C4—1222no—
378C4⋊C4—1222no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4C2×D24412yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—1228no—
385C22⋊C4—1228no—
386C42—1228no—
387C42—1222no—
388C42—1222no—
389C42C2×D24411yescharacteristic
390C42D4×D6111yescharacteristic
391C42D4×D6111yescharacteristic
392C42C23×D6111yescharacteristic
393C23—22no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23#1451974711yescharacteristic
398Q8—1222no—
399Q8—1222no—
400Q8—4882no—
401Q8—2442no—
402Q8—2442no—
403Q8—2442no—
404Q8—2442no—
405Q8—2444no—
406Q8—2442no—
407D4—22no—
408D4—22no—
409D4—42no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—1222no—
417C2×C4—1224no—
418C2×C4—1222no—
419C2×C4—4882no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4—2444no—
423C2×C4—2442no—
424C2×C4#1451974711yescharacteristic
425C2×C4#1451974711yescharacteristic
426C2×C4#1451974711yescharacteristic
427C2×C4#1451979612yes—
428C2×C4#1451979912yes—
429C2×C4#1451980512yes—
430C2×C4#1451980612yes—
431C2×C4C2×D4×D6111yescharacteristic
432C2×C4C2×D4×D6111yescharacteristic
433C2×C4C22×D24411yescharacteristic
434C8—24no—
435C8—42no—
436C8—2442no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—84no—
441C22#1683990912yes—
442C22#1683992512yes—
443C22#1683994612yes—
444C22C2×#1451974711yescharacteristic
445C4—21no—
446C4—21no—
447C4—42no—
448C4—4882no—
449C4—2442no—
450C4—2444no—
451C4—2442no—
452C4#1683990912yes—
453C4#1683992512yes—
454C4C2×#1451979611yescharacteristic
455C4C2×#1451979911yescharacteristic
456C4C2×#1451980511yescharacteristic
457C4C2×#1451980611yescharacteristic
458C2—82no—
459C2#2144507714yesdirect product factor
460C2C2×#1683990911yescharacteristic
461C2C2×#1683992511yescharacteristic
462C2C2×#1683994611yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×D4⋊C8C2
socleC23×C61#120
prime radicalC22×D4×C61D4
hypercenterC2×D4⋊C8D61
hypocenterC61C2×#1650
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16840167C2semidirect
3. C2×#16840048C2semidirect
5. #21445077C2direct (N × H)
6. C2×#16839507C2semidirect
7. C2×#16839494C2semidirect
23. #16840167C22semidirect
25. #16840048C22semidirect
33. #16839507C22semidirect
34. #16839494C22semidirect
236. C2×C61#1650semidirect
270. C61C2×#1650semidirect
459. C2#21445077direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404822
C2×#168401672
C2×#168400482
C2×#168399672
#214450772
C2×#168395072
C2×#168394942
C2×C61×D4⋊C82
C2×#165061