Finite Group Databasebeta

← C2×#21458232 #30558497

Subgroups — C2×#21458232 #30558497

Order 15616.

1918 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic, complemented
2#21458232C2116yesmaximal, direct product factor
3C2×#16840158C211yesmaximal, characteristic, complemented
4C2×#16840103C212yesmaximal, complemented
5C2×#16840083C212yesmaximal, complemented
6C2×#16840054C211yesmaximal, characteristic, complemented
7C2×#16840042C211yesmaximal, characteristic, complemented
8C2×#16839989C212yesmaximal, complemented
9C2×C4×#11758200C211yesmaximal, characteristic, complemented
10C2×D4×#92418C211yesmaximal, characteristic, complemented
11C2×#16839798C212yesmaximal, complemented
12C2×C61×#555C211yesmaximal, characteristic
13C22×#11758200C2211yescharacteristic
14C23×#92418C2212yes—
15#16840482—24no—
16#16840482C2218yescomplemented
17C2×#14519777C2211yescharacteristic
18C2×#14519770—22no—
19C2×#14519764—21no—
20C2×#14519756C2212yes—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742—21no—
23C2×#14519736C2212yes—
24#16840158C2218yescomplemented
25#16840103C22116yescomplemented
26#16840083C22116yescomplemented
27#16840054C2218yescomplemented
28#16840042C2218yescomplemented
29#16839989C22116yescomplemented
30C4×#11758200C2218yescomplemented
31D4×#92418C2218yescomplemented
32#16839798C22116yescomplemented
33C2×#14519710C2212yes—
34C2×#14519710C2212yes—
35C2×#14519708C2212yes—
36C2×#14519708C2211yescharacteristic
37C2×#14519708C2212yes—
38C2×C4⋊#92418C2211yescharacteristic
39C2×#14519700C2212yes—
40C2×#14519700C2211yescharacteristic
41C2×#14519698C2212yes—
42C2×#14519698C2211yescharacteristic
43C2×C22⋊#92418C2212yes—
44C2×C4×#92418C2211yescharacteristic
45C61×#555—24no—
46C61×#555C2218yes—
47C2×C4⋊Q8×C61C2211yescharacteristic
48C2×#124×C61—22no—
49C2×#121×C61—21no—
50C2×#121×C61C2212yes—
51C2×C4×D4×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#117×C61—21no—
54C2×#117×C61C2212yes—
55C22×Dic122—21no—
56C22×Dic122C2311yescharacteristic
57C2×#11758200—24no—
58C2×#11758200C2314yes—
59C22×#92418—28no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2312yes—
62C22×#92418C2318yes—
63C22×#92418C2311yescharacteristic
64#14519777—22no—
65#14519777C2314yes—
66#14519770—216no—
67#14519764—28no—
68#14519756—24no—
69#14519756C2318yes—
70C4×Dic122—22no—
71C4×Dic122C2314yes—
72#14519742—28no—
73#14519736—24no—
74#14519736C2318yes—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2312yes—
78C2×#11758196—22no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—22no—
81C2×C22⋊Dic61—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84#14519710C2318yes—
85#14519710C2318yes—
86#14519708C2318yes—
87#14519708C2314yes—
88#14519708C2318yes—
89C4⋊#92418C2314yes—
90#14519700C2318yes—
91#14519700C2314yes—
92#14519698C2318yes—
93#14519698C2314yes—
94C22⋊#92418C2318yes—
95C4×#92418C2314yes—
96C22×Q8×C61—21no—
97C22×D4×C61—22no—
98C22×D4×C61D411yescharacteristic
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61C2312yes—
102C4⋊Q8×C61—22no—
103C4⋊Q8×C61C2314yes—
104#124×C61—216no—
105#121×C61—28no—
106#121×C61—24no—
107#121×C61C2318yes—
108C4×D4×C61—22no—
109C4×D4×C61D414yes—
110C4×D4×C61C2314yes—
111C4×D4×C61C2314yes—
112#117×C61—24no—
113#117×C61—28no—
114#117×C61C2318yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61C2312yes—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—22no—
120C2×C22⋊C4×C61—22no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61D412yes—
123C2×C42×C61—22no—
124C2×C42×C61C2311yescharacteristic
125C24×C61—21no—
126C24×C61C2×D412yes—
127C2×Q8×C61—24no—
128C2×Q8×C61—82no—
129C2×D4×C61—28no—
130C2×D4×C61—24no—
131C2×D4×C61—21no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—84no—
135C2×D4×C61C2×D414yes—
136C2×D4×C61C2412yes—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—24no—
142C22×C4×C61—44no—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D414yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2414yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—24no—
155C2×Dic122—21no—
156C2×Dic122—44no—
157C2×Dic122C2412yes—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61C2×D414yes—
162C4⋊C4×C61C2412yes—
163C4⋊C4×C61C2414yes—
164C22⋊C4×C61—28no—
165C22⋊C4×C61—28no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61C2×D418yes—
168C42×C61—28no—
169C42×C61C2×D412yes—
170C42×C61C2412yes—
171C22⋊Dic61—28no—
172C22⋊Dic61—28no—
173#11758200—28no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2414yes—
178C4⋊Dic61C2412yes—
179#11758196—28no—
180#11758196—24no—
181C4×Dic61—28no—
182C4×Dic61—24no—
183C2×#92418—24no—
184C2×#92418—22no—
185C2×#92418—216no—
186C2×#92418#4114yes—
187C2×#92418C2412yes—
188C2×#92418C2414yes—
189C2×#92418C2412yes—
190C23×C61—24no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61—22no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61C22×D414yes—
198C23×C61C22×D411yescharacteristic
199Q8×C61—84no—
200D4×C61—28no—
201D4×C61—22no—
202D4×C61—22no—
203D4×C61—42no—
204D4×C61—88no—
205C2×C4×C61—24no—
206C2×C4×C61—22no—
207C2×C4×C61—28no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—42no—
213C2×C4×C61—44no—
214C2×C4×C61—48no—
215C2×C4×C61#13514yes—
216C2×C4×C61C22×D414yes—
217C2×C4×C61C22×D412yes—
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D412yes—
220C2×C4×C61C2×#4112yes—
221C2×C4×C612−1+412yes—
222C2×C4×C61C2511yescharacteristic
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—42no—
227C2×Dic61—44no—
228C2×Dic61—42no—
229Dic122—22no—
230Dic122—22no—
231Dic122—42no—
232Dic122—48no—
233#92418—24no—
234#92418—22no—
235#92418C2×#4114yes—
236C2×#3340—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—24no—
239C22×C61—24no—
240C22×C61—48no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—84no—
245C22×C61#54712yes—
246C22×C61C2×#13512yes—
247C22×C61#57712yes—
248C22×C61C23×D411yescharacteristic
249C4×C61—24no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—44no—
253C4×C61#54712yes—
254C4×C61C2×#13514yes—
255C4×C61C22×#4111yescharacteristic
256C4×C61C2×2−1+411yescharacteristic
257Dic61—42no—
258Dic61—44no—
259Dic61—42no—
260C2×#556—611no—
261C2×#555D6111yescharacteristic
262#3340—6116no—
263C2×#498—611no—
264C2×#485—612no—
265C2×#482—612no—
266C2×#477—611no—
267C2×#474—611no—
268C2×#465—612no—
269C2×C4×SD16—611no—
270C2×C8×D4—611no—
271C2×#427—612no—
272C2×C61—44no—
273C2×C61—82no—
274C2×C61#334014yescomplemented
275C2×C61C2×#54711yescharacteristic
276C2×C61C22×#13511yescharacteristic
277C2×C61C2×#57711yescharacteristic
278C22×SD16—611no—
279C23×C8—612no—
280#556—618no—
281#556—1224no—
282#555—24no—
283#555C2×D6118yes—
284C2×C4⋊Q8—611no—
285C2×C4⋊Q8C2×D6111yescharacteristic
286C2×#126—1222no—
287C2×#124—22no—
288C2×#124—1221no—
289C2×C22⋊Q8—612no—
290C2×#121—21no—
291C2×#121C2×D6112yes—
292C2×C4×Q8—611no—
293C2×C4×Q8—1221no—
294C2×C4×D4C2×D6111yescharacteristic
295C2×C4×D4C2×D6111yescharacteristic
296C2×#117—21no—
297C2×#117—612no—
298C2×#117C2×D6112yes—
299#498—618no—
300#485—6116no—
301#482—6116no—
302#477—618no—
303#474—618no—
304#465—6116no—
305C4×SD16—618no—
306C8×D4—618no—
307#427—6116no—
308C2×#107—612no—
309C2×#107—612no—
310C2×#106—612no—
311C2×#106—612no—
312C2×#106—611no—
313C2×C4⋊C8—611no—
314C2×#103—612no—
315C2×#103—611no—
316C2×#102—611no—
317C2×#102—612no—
318C2×C22⋊C8—612no—
319C2×C4×C8—611no—
320C61C2×#334011yescharacteristic, complemented, Sylow
321C22×Q8—21no—
322C22×Q8—611no—
323C22×Q8—1221no—
324C22×D4—22no—
325C22×D4C22⋊D6111yescharacteristic
326C22×D4C22×D6111yescharacteristic
327C23×C4—21no—
328C23×C4C22×D6112yes—
329C2×SD16—614no—
330C2×SD16—1224no—
331C22×C8—611no—
332C22×C8—612no—
333C22×C8—618no—
334C22×C8—611no—
335C22×C8—1228no—
336C4⋊Q8—22no—
337C4⋊Q8—614no—
338C4⋊Q8—1222no—
339C4⋊Q8C22×D6114yes—
340#126—12216no—
341#124—216no—
342#124—1228no—
343C22⋊Q8—618no—
344C22⋊Q8—1224no—
345#121—28no—
346#121—24no—
347#121C22×D6118yes—
348C4×Q8—614no—
349C4×Q8—1228no—
350C4×Q8—1222no—
351C4×D4—22no—
352C4×D4C22⋊D6114yes—
353C4×D4C22×D6114yes—
354C4×D4C22×D6114yes—
355#117—24no—
356#117—28no—
357#117—618no—
358#117—1224no—
359#117C22×D6118yes—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4C22⋊D6111yescharacteristic
367C2×C4⋊C4C22×D6111yescharacteristic
368C2×C4⋊C4C22×D6112yes—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4C22⋊D6112yes—
375C2×C42—22no—
376C2×C42—1222no—
377C2×C42—1221no—
378C2×C42C22×D6111yescharacteristic
379#107—618no—
380#107—618no—
381#106—614no—
382#106—618no—
383#106—618no—
384C4⋊C8—614no—
385#103—614no—
386#103—618no—
387#102—614no—
388#102—618no—
389C22⋊C8—618no—
390C4×C8—614no—
391C24—21no—
392C24C2×C22⋊D6112yes—
393C2×Q8—24no—
394C2×Q8—82no—
395C2×Q8—612no—
396C2×Q8—1221no—
397C2×Q8—1224no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—2444no—
401C2×D4—28no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—84no—
407C2×D4C2×C22⋊D6114yes—
408C2×D4C23×D6112yes—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—1221no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4C2×C22⋊D6114yes—
419C22×C4C2×C22⋊D6112yes—
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C23×D6114yes—
423C22×C4C23×D6111yescharacteristic
424SD16—1228no—
425C2×C8—614no—
426C2×C8—612no—
427C2×C8—612no—
428C2×C8—614no—
429C2×C8—12216no—
430C2×C8—1224no—
431C2×C8—1222no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—614no—
436C4⋊C4—612no—
437C4⋊C4—1221no—
438C4⋊C4—1222no—
439C4⋊C4—1224no—
440C4⋊C4—1228no—
441C4⋊C4—1224no—
442C4⋊C4C2×C22⋊D6114yes—
443C4⋊C4C23×D6114yes—
444C4⋊C4C23×D6112yes—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—1228no—
449C22⋊C4—1228no—
450C22⋊C4C2×C22⋊D6118yes—
451C42—28no—
452C42—1228no—
453C42—1224no—
454C42C2×C22⋊D6112yes—
455C42C23×D6112yes—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C22×C22⋊D6114yes—
464C23C22×C22⋊D6111yescharacteristic
465Q8—84no—
466Q8—1222no—
467Q8—1222no—
468Q8—2448no—
469Q8—2442no—
470D4—28no—
471D4—22no—
472D4—22no—
473D4—42no—
474D4—88no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—1222no—
486C2×C4—1224no—
487C2×C4—1222no—
488C2×C4—2444no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4#1451979714yes—
492C2×C4C22×C22⋊D6114yes—
493C2×C4C22×C22⋊D6112yes—
494C2×C4C22×C22⋊D6112yes—
495C2×C4C22×C22⋊D6111yescharacteristic
496C2×C4#41×D6112yes—
497C2×C4#1451982412yes—
498C2×C4C24×D6111yescharacteristic
499C8—614no—
500C8—1224no—
501C8—1222no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—48no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22#1684041112yes—
511C22C2×#1451979712yes—
512C22#1684057312yes—
513C22C23×C22⋊D6111yescharacteristic
514C4—24no—
515C4—22no—
516C4—42no—
517C4—44no—
518C4—2444no—
519C4—2442no—
520C4—2442no—
521C4#1684041112yes—
522C4C2×#1451979714yes—
523C4C2×#41×D6111yescharacteristic
524C4C2×#1451982411yescharacteristic
525C2—44no—
526C2—82no—
527C2#2145823214yesdirect product factor
528C2C2×#1684041111yescharacteristic
529C2C22×#1451979711yescharacteristic
530C2C2×#1684057311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
Fitting, prime radicalC2×C61×#555C2
socleC23×C61C22×D4
hypercenterC2×#555D61
hypocenterC61C2×#3340
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#16840482C2semidirect
2. #21458232C2direct (N × H)
3. C2×#16840158C2semidirect
4. C2×#16840103C2semidirect
5. C2×#16840083C2semidirect
6. C2×#16840054C2semidirect
7. C2×#16840042C2semidirect
8. C2×#16839989C2semidirect
9. C2×C4×#11758200C2semidirect
10. C2×D4×#92418C2semidirect
11. C2×#16839798C2semidirect
16. #16840482C22semidirect
24. #16840158C22semidirect
25. #16840103C22semidirect
26. #16840083C22semidirect
27. #16840054C22semidirect
28. #16840042C22semidirect
29. #16839989C22semidirect
30. C4×#11758200C22semidirect
31. D4×#92418C22semidirect
32. #16839798C22semidirect
274. C2×C61#3340semidirect
320. C61C2×#3340semidirect
527. C2#21458232direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404822
#214582322
C2×#168401582
C2×#168401032
C2×#168400832
C2×#168400542
C2×#168400422
C2×#168399892
C2×C4×#117582002
C2×D4×#924182
C2×#168397982
C2×C61×#5552
C2×#334061