Finite Group Databasebeta

← C2×#21458729 #30560502

Subgroups — C2×#21458729 #30560502

Order 15616.

1854 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic122C211yesmaximal, characteristic, complemented
2#21458729C2116yesmaximal, direct product factor
3C2×#16840172C212yesmaximal, complemented
4C2×#16840165C211yesmaximal, characteristic, complemented
5C2×#16840066C212yesmaximal, complemented
6C2×#16840054C211yesmaximal, characteristic, complemented
7C2×#16840052C211yesmaximal, characteristic
8C2×#16840042C212yesmaximal
9C2×#16839893C211yesmaximal, characteristic, complemented
10C2×#16839862C212yesmaximal
11C2×C4×#11758200C211yesmaximal, characteristic
12C2×C61×#554C211yesmaximal, characteristic
13C22×#11758200C2211yescharacteristic
14Q8×Dic122—24no—
15Q8×Dic122C2218yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519777C2212yes—
18C2×#14519776—21no—
19C2×#14519776—22no—
20C2×Q8×Dic61—21no—
21C2×C4×Dic122C2212yes—
22C2×C4×Dic122C2211yescharacteristic
23#16840172C22116yescomplemented
24#16840165C2218yescomplemented
25#16840066C22116yescomplemented
26#16840054C2218yescomplemented
27#16840052C2218yes—
28#16840042C22116yes—
29#16839893C2218yescomplemented
30#16839862C22116yes—
31C4×#11758200C2218yes—
32C2×#14519710C2212yes—
33C2×#14519710C2212yes—
34C2×#14519710C2212yes—
35C2×#14519708C2212yes—
36C2×#14519708C2211yescharacteristic
37C2×C4⋊#92418C2212yes—
38C2×C4⋊#92418C2211yescharacteristic
39C2×#14519700C2212yes—
40C2×#14519700C2211yescharacteristic
41C2×#14519698C2212yes—
42C2×#14519698C2211yescharacteristic
43C2×#14519688C2212yes—
44C2×C4×#92418C2211yescharacteristic
45C61×#554—24no—
46C61×#554C2218yes—
47C2×C4⋊Q8×C61C2211yescharacteristic
48C2×#125×C61C2212yes—
49C2×C22⋊Q8×C61—21no—
50C2×C22⋊Q8×C61—22no—
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×D4×C61C2212yes—
53C2×C4×D4×C61C2211yescharacteristic
54C22×C4⋊C4×C61—21no—
55C22×Dic122—21no—
56C22×Dic122—22no—
57C22×Dic122C2311yescharacteristic
58C2×#11758200Q818yescomplemented
59C2×#11758200C2314yes—
60C22×#92418C2311yescharacteristic
61C22×#92418C2312yes—
62C22×#92418C2311yescharacteristic
63#14519777—24no—
64#14519777—22no—
65#14519777C2318yes—
66#14519777C2314yes—
67#14519776—28no—
68#14519776—216no—
69Q8×Dic61—28no—
70C4×Dic122—24no—
71C4×Dic122—22no—
72C4×Dic122C2314yes—
73C4×Dic122C2318yes—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—21no—
82#14519710C2318yes—
83#14519710C2318yes—
84#14519710C2318yes—
85#14519708C2318yes—
86#14519708C2314yes—
87C4⋊#92418C2318yes—
88C4⋊#92418C2314yes—
89#14519700C2318yes—
90#14519700C2314yes—
91#14519698C2314yes—
92#14519698C2318yes—
93#14519688C2318yes—
94C4×#92418C2314yes—
95C22×Q8×C61—21no—
96C22×Q8×C61D411yescharacteristic
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61—22no—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61C2314yes—
102#125×C61—24no—
103#125×C61C2318yes—
104C22⋊Q8×C61—28no—
105C22⋊Q8×C61—216no—
106C4×Q8×C61D414yes—
107C4×Q8×C61C2314yes—
108C4×D4×C61—24no—
109C4×D4×C61—22no—
110C4×D4×C61C2318yes—
111C4×D4×C61C2314yes—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—22no—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61—28no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D412yes—
118C2×C4⋊C4×C61C2312yes—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—22no—
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2312yes—
124C24×C61—21no—
125C2×Q8×C61—24no—
126C2×Q8×C61—24no—
127C2×Q8×C61—82no—
128C2×Q8×C61C2×D414yes—
129C2×D4×C61—21no—
130C2×D4×C61—22no—
131C2×D4×C61C2×Q814yes—
132C2×D4×C61C2412yes—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—28no—
137C22×C4×C61—44no—
138C22×C4×C61—84no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C2×Dic122—21no—
148C2×Dic122—22no—
149C2×Dic122—24no—
150C2×Dic122—28no—
151C2×Dic122—84no—
152C2×Dic122—82no—
153C2×Dic122C2×Q814yes—
154C2×Dic122C2412yes—
155C4⋊C4×C61—28no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—28no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D414yes—
161C4⋊C4×C61C2×D418yes—
162C4⋊C4×C61C2412yes—
163C4⋊C4×C61C2414yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—28no—
166C42×C61C2×D414yes—
167C42×C61C2×D412yes—
168C42×C61C2414yes—
169C42×C61C2412yes—
170#11758200C2×Q8116yescomplemented
171C4⋊Dic61—28no—
172C4⋊Dic61—24no—
173C4⋊Dic61—22no—
174C4⋊Dic61—21no—
175C4⋊Dic61C2412yes—
176C4⋊Dic61C2414yes—
177#11758196—24no—
178#11758196—28no—
179C4×Dic61—28no—
180C4×Dic61—24no—
181C2×#92418—22no—
182C2×#92418—24no—
183C2×#92418C2×Q814yes—
184C2×#92418C2412yes—
185C2×#92418C2414yes—
186C2×#92418C2412yes—
187C23×C61—24no—
188C23×C61—22no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61C22×D411yescharacteristic
192Q8×C61—28no—
193Q8×C61—84no—
194D4×C61—24no—
195D4×C61—22no—
196D4×C61C22×Q814yes—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—48no—
205C2×C4×C61—88no—
206C2×C4×C61#13714yes—
207C2×C4×C61C22×D414yes—
208C2×C4×C61C22×D412yes—
209C2×C4×C61C22×D414yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D412yes—
212C2×C4×C61C22×Q812yes—
213C2×C4×C612−1+412yes—
214C2×C4×C61C2511yescharacteristic
215C2×Dic61—24no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—22no—
219C2×Dic61—42no—
220C2×Dic61—44no—
221Dic122—24no—
222Dic122—22no—
223Dic122—88no—
224Dic122—84no—
225Dic122C22×Q814yes—
226#92418—22no—
227#92418—24no—
228#92418C22×Q814yes—
229C2×#3385—611nomaximal, Sylow
230C22×C61—24no—
231C22×C61—48no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61D4×Q812yes—
235C22×C61C2×#13712yes—
236C22×C61#57612yes—
237C22×C61C23×D411yescharacteristic
238C4×C61—24no—
239C4×C61—24no—
240C4×C61—22no—
241C4×C61—42no—
242C4×C61—44no—
243C4×C61D4×Q812yes—
244C4×C61C2×#13714yes—
245C4×C61C23×Q811yescharacteristic
246C4×C61C2×2−1+411yescharacteristic
247Dic61—24no—
248Dic61—42no—
249Dic61—44no—
250C2×Q82—611no—
251C2×#554D6111yescharacteristic
252#3385—6116no—
253C2×#501—612no—
254C2×#500—611no—
255C2×#479—612no—
256C2×#477—611no—
257C2×#476—611no—
258C2×#474—612no—
259C2×#446—611no—
260C2×#440—612no—
261C2×C4×SD16—611no—
262C2×C61—44no—
263C2×C61#338514yescomplemented
264C2×C61C2×D4×Q811yescharacteristic
265C2×C61C22×#13711yescharacteristic
266C2×C61C2×#57611yescharacteristic
267C22×SD16—611no—
268Q82—618no—
269Q82—1224no—
270#554—24no—
271#554C2×D6118yes—
272C2×C4⋊Q8—611no—
273C2×C4⋊Q8—612no—
274C2×C4⋊Q8—1221no—
275C2×C4⋊Q8—1222no—
276C2×C4⋊Q8C2×D6111yescharacteristic
277C2×#125C2×D6112yes—
278C2×C22⋊Q8—21no—
279C2×C22⋊Q8—22no—
280C2×C4×Q8—611no—
281C2×C4×Q8—612no—
282C2×C4×Q8—1221no—
283C2×C4×Q8C2×D6111yescharacteristic
284C2×C4×D4C2×D6111yescharacteristic
285C2×C4×D4C2×D6112yes—
286C22×C4⋊C4—21no—
287#501—6116no—
288#500—618no—
289#479—6116no—
290#477—618no—
291#476—618no—
292#474—6116no—
293#446—618no—
294#440—6116no—
295C4×SD16—618no—
296C2×#107—612no—
297C2×#107—612no—
298C2×#107—612no—
299C2×#106—612no—
300C2×#106—611no—
301C2×C4⋊C8—611no—
302C2×C4⋊C8—612no—
303C2×#103—612no—
304C2×#103—611no—
305C2×#102—611no—
306C2×#102—612no—
307C2×#97—612no—
308C2×C4×C8—611no—
309C61C2×#338511yescharacteristic, complemented, Sylow
310C22×Q8—21no—
311C22×Q8—611no—
312C22×Q8—1221no—
313C22×Q8—1222no—
314C22×Q8C22⋊D6111yescharacteristic
315C22×D4C22×D6111yescharacteristic
316C23×C4—21no—
317C23×C4—22no—
318C2×SD16—618no—
319C2×SD16—614no—
320C22×C8—611no—
321C22×C8—611no—
322C22×C8—612no—
323C4⋊Q8—22no—
324C4⋊Q8—618no—
325C4⋊Q8—614no—
326C4⋊Q8—1228no—
327C4⋊Q8—12216no—
328C4⋊Q8—1224no—
329C4⋊Q8—1222no—
330C4⋊Q8C22×D6114yes—
331#125—24no—
332#125C22×D6118yes—
333C22⋊Q8—28no—
334C22⋊Q8—216no—
335C4×Q8—614no—
336C4×Q8—618no—
337C4×Q8—1228no—
338C4×Q8—1224no—
339C4×Q8—1222no—
340C4×Q8C22⋊D6114yes—
341C4×Q8C22×D6114yes—
342C4×D4—24no—
343C4×D4—22no—
344C4×D4C22×D6114yes—
345C4×D4C22×D6118yes—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4C22⋊D6111yescharacteristic
357C2×C4⋊C4C22⋊D6112yes—
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C4⋊C4C22×D6112yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C42—1222no—
363C2×C42—1221no—
364C2×C42C22×D6112yes—
365C2×C42C22×D6111yescharacteristic
366#107—618no—
367#107—618no—
368#107—618no—
369#106—614no—
370#106—618no—
371C4⋊C8—618no—
372C4⋊C8—614no—
373#103—614no—
374#103—618no—
375#102—614no—
376#102—618no—
377#97—618no—
378C4×C8—614no—
379C24—21no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—82no—
383C2×Q8—614no—
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1224no—
387C2×Q8—1228no—
388C2×Q8—1222no—
389C2×Q8—4884no—
390C2×Q8—4882no—
391C2×Q8C2×C22⋊D6114yes—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4Q8×D6114yes—
395C2×D4C23×D6112yes—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—44no—
401C22×C4—84no—
402C22×C4—1221no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4C2×C22⋊D6112yes—
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6112yes—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410SD16—6116no—
411C2×C8—614no—
412C2×C8—612no—
413C2×C8—612no—
414C2×C8—614no—
415C2×C8—1222no—
416C2×C8—1224no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—612no—
423C4⋊C4—614no—
424C4⋊C4—1221no—
425C4⋊C4—1222no—
426C4⋊C4—1224no—
427C4⋊C4—1228no—
428C4⋊C4—1228no—
429C4⋊C4—1224no—
430C4⋊C4C2×C22⋊D6114yes—
431C4⋊C4C2×C22⋊D6118yes—
432C4⋊C4C23×D6112yes—
433C4⋊C4C23×D6114yes—
434C22⋊C4—24no—
435C22⋊C4—28no—
436C42—1228no—
437C42—1224no—
438C42C2×C22⋊D6114yes—
439C42C2×C22⋊D6112yes—
440C42C23×D6112yes—
441C42C23×D6114yes—
442C23—22no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23C22×C22⋊D6111yescharacteristic
447Q8—28no—
448Q8—84no—
449Q8—614no—
450Q8—1222no—
451Q8—1224no—
452Q8—4888no—
453Q8—4884no—
454D4—22no—
455D4—24no—
456D4C2×Q8×D6114yes—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—48no—
465C2×C4—88no—
466C2×C4—1222no—
467C2×C4—1224no—
468C2×C4—1224no—
469C2×C4—1222no—
470C2×C4—2442no—
471C2×C4—2444no—
472C2×C4#1451980714yes—
473C2×C4C22×C22⋊D6114yes—
474C2×C4C22×C22⋊D6112yes—
475C2×C4C22×C22⋊D6114yes—
476C2×C4C22×C22⋊D6112yes—
477C2×C4C22×C22⋊D6111yescharacteristic
478C2×C4C2×Q8×D6112yes—
479C2×C4#1451982512yes—
480C2×C4C24×D6111yescharacteristic
481C8—614no—
482C8—1222no—
483C8—1224no—
484C22—24no—
485C22—48no—
486C22—42no—
487C22—42no—
488C22Q8×C22⋊D6112yes—
489C22C2×#1451980712yes—
490C22#1684056512yes—
491C22C23×C22⋊D6111yescharacteristic
492C4—24no—
493C4—24no—
494C4—22no—
495C4—42no—
496C4—44no—
497C4—1224no—
498C4—2442no—
499C4—2444no—
500C4Q8×C22⋊D6112yes—
501C4C2×#1451980714yes—
502C4C22×Q8×D6111yescharacteristic
503C4C2×#1451982511yescharacteristic
504C2—44no—
505C2#2145872914yesdirect product factor
506C2C2×Q8×C22⋊D6111yescharacteristic
507C2C22×#1451980711yescharacteristic
508C2C2×#1684056511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#554D61
hypocenterC61C2×#3385
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic122C2semidirect
2. #21458729C2direct (N × H)
3. C2×#16840172C2semidirect
4. C2×#16840165C2semidirect
5. C2×#16840066C2semidirect
6. C2×#16840054C2semidirect
9. C2×#16839893C2semidirect
15. Q8×Dic122C22semidirect
23. #16840172C22semidirect
24. #16840165C22semidirect
25. #16840066C22semidirect
26. #16840054C22semidirect
29. #16839893C22semidirect
58. C2×#11758200Q8semidirect
170. #11758200C2×Q8semidirect
263. C2×C61#3385semidirect
309. C61C2×#3385semidirect
505. C2#21458729direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1222
#214587292
C2×#168401722
C2×#168401652
C2×#168400662
C2×#168400542
C2×#168400522
C2×#168400422
C2×#168398932
C2×#168398622
C2×C4×#117582002
C2×C61×#5542
C2×#338561