Finite Group Databasebeta

← C2×#21460154 #30587694

Subgroups — C2×#21460154 #30587694

Order 15616.

7086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460154C2132yesmaximal, direct product factor
2C2×#16840512C214yesmaximal, complemented
3C2×#16840492C211yesmaximal, characteristic, complemented
4C2×#16840469C212yesmaximal, complemented
5C2×#16840451C214yesmaximal, complemented
6C2×#16840412C214yesmaximal, complemented
7C2×#16840390C214yesmaximal, complemented
8C2×#16840379C214yesmaximal, complemented
9C2×#16840332C214yesmaximal, complemented
10C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
11C2×#16840294C212yesmaximal, complemented
12C2×C61×#555C211yesmaximal, characteristic, complemented
13C22×#11758224C2214yes—
14C22×Q8×D61C2212yes—
15#16840512C22164yescomplemented
16#16840492C22116yescomplemented
17#16840469C22132yescomplemented
18#16840451C22164yescomplemented
19#16840412C22164yescomplemented
20#16840390C22164yescomplemented
21#16840379C22164yescomplemented
22#16840332C22164yescomplemented
23C2×#14519777C2214yes—
24C2×#14519777C2211yescharacteristic
25C2×#14519776C2214yes—
26C2×#14519776C2214yes—
27C2×#14519775C2212yes—
28C2×#14519775C2214yes—
29C4⋊Q8×D61C22116yescomplemented
30#16840294C22132yescomplemented
31C2×#14519770C2218yes—
32C2×#14519770C2214yes—
33C2×#14519769C2218yes—
34C2×#14519768C2212yes—
35C2×#14519768C2214yes—
36C2×#14519764C2212yes—
37C2×#14519764C2214yes—
38C2×#14519759C2218yes—
39C2×#14519759C2214yes—
40C2×#14519758C2218yes—
41C2×#14519757C2212yes—
42C2×#14519756C2214yes—
43C2×#14519754C2218yes—
44C2×#14519754C2218yes—
45C2×#14519754C2218yes—
46C2×#14519748C2212yes—
47C2×C4×Dic122C2212yes—
48C2×#14519742C2214yes—
49C2×D4×Dic61C2214yes—
50C2×C4×C22⋊D61C2214yes—
51C2×#14519736C2214yes—
52C2×#14519734C2214yes—
53C22×C4⋊Dic61C2214yes—
54C2×C4⋊C4×D61C2214yes—
55C2×C42×D61C2211yescharacteristic
56C61×#555C22116yescomplemented
57C2×C4⋊Q8×C61C2211yescharacteristic
58C2×#124×C61C2214yes—
59C2×#121×C61C2214yes—
60C2×C4×D4×C61C2212yes—
61C2×#117×C61C2214yes—
62C2×#11758224—416no—
63C2×#11758224C23132yes—
64C22×Dic122C2312yes—
65C22×Dic122C2312yes—
66C22×Dic122C2314yes—
67C22×Dic122C2314yes—
68C2×Q8×D61—48no—
69C2×Q8×D61C23116yes—
70C22×C22⋊D61C2314yes—
71C23×Dic61C2318yes—
72C22×C4×D61C2312yes—
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2314yes—
75#14519777C2318yes—
76#14519777C23132yes—
77#14519776C23132yes—
78#14519776C23132yes—
79#14519775C23132yes—
80#14519775C23116yes—
81#14519770C23164yes—
82#14519770C23132yes—
83#14519769C23164yes—
84#14519768C23132yes—
85#14519768C23116yes—
86#14519764C23116yes—
87#14519764C23132yes—
88#14519759C23164yes—
89#14519759C23132yes—
90#14519758C23164yes—
91#14519757C23116yes—
92#14519756C23132yes—
93#14519754C23164yes—
94#14519754C23164yes—
95#14519754C23164yes—
96#14519748C23116yes—
97C4×Dic122C23116yes—
98#14519742C23132yes—
99D4×Dic61C23132yes—
100C4×C22⋊D61C23132yes—
101#14519736C23132yes—
102#14519734C23132yes—
103C2×C4⋊Dic61C2314yes—
104C2×C4⋊Dic61C2312yes—
105C2×C4⋊Dic61C23132yes—
106C2×C4⋊Dic61C2318yes—
107C2×C4⋊Dic61C2314yes—
108C2×C4⋊Dic61C2314yes—
109C2×#11758196C2318yes—
110C2×#11758196C2318yes—
111C2×#11758196C2314yes—
112C2×#11758196C2318yes—
113C2×#11758196C2314yes—
114C4⋊C4×D61C23132yes—
115C2×C22⋊Dic61C2314yes—
116C2×C22⋊Dic61C2314yes—
117C2×C22⋊Dic61C2314yes—
118C2×C22⋊Dic61C2318yes—
119C2×#11758198C2314yes—
120C2×C4×Dic61C2312yes—
121C2×C4×Dic61C2311yescharacteristic
122C2×C4×Dic61C2314yes—
123C2×C4×Dic61C2314yes—
124C42×D61C2318yes—
125C22×Q8×C61C2312yes—
126C22×D4×C61C2312yes—
127C22×D4×C61C2314yes—
128C23×C4×C61C2314yes—
129C4⋊Q8×C61C2318yes—
130#124×C61C23132yes—
131#121×C61C23132yes—
132C4×D4×C61C23116yes—
133#117×C61C23132yes—
134C2×C4⋊C4×C61C2314yes—
135C2×C4⋊C4×C61C2312yes—
136C2×C22⋊C4×C61C2318yes—
137C2×C22⋊C4×C61C2314yes—
138C2×C42×C61C2311yescharacteristic
139C2×C42×C61C2314yes—
140C24×C61C2414yes—
141C23×D61C2411yescharacteristic
142C2×Q8×C61—44no—
143C2×Q8×C61C2418yes—
144C2×D4×C61—28no—
145C2×D4×C61—48no—
146C2×D4×C61C2418yes—
147C2×D4×C61C24116yes—
148C22×C4×C61—216no—
149C22×C4×C61C2411yescharacteristic
150C22×C4×C61C2414yes—
151C22×C4×C61C2414yes—
152C22×C4×C61C24116yes—
153C22×C4×C61C2412yes—
154C2×C22⋊D61—216no—
155C2×C22⋊D61C24116yes—
156C22×Dic61—416no—
157C22×Dic61C2414yes—
158C22×Dic61C2411yescharacteristic
159C22×Dic61C2412yes—
160C22×Dic61C2414yes—
161C22×Dic61C2414yes—
162C22×Dic61C24132yes—
163Q8×D61—432no—
164#11758224—464no—
165C2×C4×D61—24no—
166C2×C4×D61—216no—
167C2×C4×D61—28no—
168C2×C4×D61C2414yes—
169C2×C4×D61C24116yes—
170C2×C4×D61C2418yes—
171C2×Dic122—28no—
172C2×Dic122—48no—
173C2×Dic122—44no—
174C2×Dic122—48no—
175C2×Dic122C2418yes—
176C2×Dic122C2418yes—
177C2×Dic122C24116yes—
178C2×Dic122C24116yes—
179C4⋊C4×C61C2418yes—
180C4⋊C4×C61C24116yes—
181C22⋊C4×C61C24132yes—
182C22⋊C4×C61C24116yes—
183C42×C61C24116yes—
184C42×C61C2414yes—
185C22⋊Dic61C24116yes—
186C22⋊Dic61C24116yes—
187C22⋊Dic61C24116yes—
188C22⋊Dic61C24132yes—
189#11758198C24116yes—
190C4⋊Dic61C24116yes—
191C4⋊Dic61C2418yes—
192C4⋊Dic61C24132yes—
193C4⋊Dic61C24116yes—
194C4⋊Dic61C24116yes—
195#11758196C24132yes—
196#11758196C24132yes—
197#11758196C24116yes—
198#11758196C24132yes—
199#11758196C24116yes—
200C4×Dic61C2418yes—
201C4×Dic61C2414yes—
202C4×Dic61C24116yes—
203C4×Dic61C24116yes—
204C23×C61—28no—
205C23×C61—28no—
206C23×C61—28no—
207C23×C61C2518yes—
208C23×C61C2511yescharacteristic
209C22×D61—24no—
210C22×D61H(5,𝔽2)14yes—
211C22×D61C2512yes—
212Q8×C61—48no—
213D4×C61—216no—
214D4×C61—416no—
215C2×C4×C61—28no—
216C2×C4×C61—22no—
217C2×C4×C61—28no—
218C2×C4×C61—232no—
219C2×C4×C612−1+418yes—
220C2×C4×C61C2512yes—
221C2×C4×C61C2518yes—
222C2×C4×C61C2518yes—
223C2×C4×C61C2514yes—
224C22⋊D61—232no—
225C2×Dic61—22no—
226C2×Dic61—28no—
227C2×Dic61—28no—
228C2×Dic61—28no—
229C2×Dic61—24no—
230C2×Dic61—432no—
231C2×Dic61C2518yes—
232C2×Dic61C2514yes—
233C2×Dic61C2512yes—
234C2×Dic61C2518yes—
235C2×Dic61C2518yes—
236C4×D61—232no—
237C4×D61—28no—
238C4×D61—216no—
239Dic122—216no—
240Dic122—416no—
241Dic122—416no—
242Dic122—48no—
243C2×#3544—611nomaximal, Sylow
244C22×C61—28no—
245C22×C61—28no—
246C22×C61—28no—
247C22×C61—416no—
248C22×C61C2×H(5,𝔽2)12yes—
249C22×C61C2×2−1+414yes—
250C22×C61C2611yescharacteristic
251C2×D61—24no—
252C2×D61—28no—
253C2×D61C2×H(5,𝔽2)14yes—
254C4×C61—28no—
255C4×C61—22no—
256C4×C61—28no—
257C4×C61C2×2−1+418yes—
258Dic61—22no—
259Dic61—28no—
260Dic61—28no—
261Dic61—28no—
262Dic61—24no—
263#3544—6132no—
264C2×#563—614no—
265C2×Q82—611no—
266C2×#555D6111yescharacteristic, complemented
267C2×#554—612no—
268C2×#552—614no—
269C2×#547—614no—
270C2×#544—614no—
271C2×#542—614no—
272C2×#536—614no—
273C22×C4⋊Q8—611no—
274C2×#529—612no—
275C2×C61—48no—
276C2×C61#354414yescomplemented
277C2×C61C22×H(5,𝔽2)11yescharacteristic
278C2×C61C22×2−1+412yes—
279D61—24no—
280C22×#41—614no—
281C23×Q8—612no—
282#563—6164no—
283Q82—6116no—
284#555C2×D61116yescomplemented
285#554—6132no—
286#552—6164no—
287#547—6164no—
288#544—6164no—
289#542—6164no—
290#536—6164no—
291C2×C4⋊Q8—612no—
292C2×C4⋊Q8—614no—
293C2×C4⋊Q8—6116no—
294C2×C4⋊Q8—614no—
295C2×C4⋊Q8—614no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8—614no—
298C2×C4⋊Q8C2×D6111yescharacteristic
299#529—6132no—
300C2×#126—614no—
301C2×#126—618no—
302C2×#125—612no—
303C2×#125—614no—
304C2×#125—618no—
305C2×#124—612no—
306C2×#124—614no—
307C2×#124C2×D6114yes—
308C2×#123—618no—
309C2×#123—614no—
310C2×#123—618no—
311C2×C22⋊Q8—614no—
312C2×C22⋊Q8—612no—
313C2×C22⋊Q8—618no—
314C2×C22⋊Q8—618no—
315C2×C22⋊Q8—618no—
316C2×#121—612no—
317C2×#121C2×D6114yes—
318C2×C4×Q8—614no—
319C2×C4×Q8—612no—
320C2×C4×D4—614no—
321C2×C4×D4—614no—
322C2×C4×D4C2×D6112yes—
323C2×#117—614no—
324C2×#117—614no—
325C2×#117C2×D6114yes—
326C22×C4⋊C4—614no—
327C22×C4⋊C4—614no—
328C22×C42—611no—
329C61C2×#354411yescharacteristic, complemented, Sylow
330C2×#41—6132no—
331C2×#41—24416no—
332C22×Q8—6116no—
333C22×Q8—612no—
334C22×Q8—612no—
335C22×Q8—614no—
336C22×Q8—614no—
337C22×Q8—2448no—
338C22×Q8C22×D6112yes—
339C22×D4—614no—
340C22×D4C22×D6112yes—
341C22×D4C22×D6114yes—
342C23×C4—618no—
343C23×C4—614no—
344C23×C4—612no—
345C23×C4—611no—
346C23×C4C22×D6114yes—
347C4⋊Q8—6116no—
348C4⋊Q8—6132no—
349C4⋊Q8—6132no—
350C4⋊Q8—6132no—
351C4⋊Q8—618no—
352C4⋊Q8—6132no—
353C4⋊Q8C22×D6118yes—
354#126—6132no—
355#126—6164no—
356#125—6116no—
357#125—6132no—
358#125—6164no—
359#124—6116no—
360#124—6132no—
361#124C22×D61132yes—
362#123—6164no—
363#123—6132no—
364#123—6164no—
365C22⋊Q8—6132no—
366C22⋊Q8—6116no—
367C22⋊Q8—6164no—
368C22⋊Q8—6164no—
369C22⋊Q8—6164no—
370#121—6116no—
371#121C22×D61132yes—
372C4×Q8—6132no—
373C4×Q8—6116no—
374C4×D4—6132no—
375C4×D4—6132no—
376C4×D4C22×D61116yes—
377#117—6132no—
378#117—6132no—
379#117C22×D61132yes—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—618no—
383C2×C4⋊C4—618no—
384C2×C4⋊C4—6132no—
385C2×C4⋊C4—618no—
386C2×C4⋊C4—614no—
387C2×C4⋊C4—614no—
388C2×C4⋊C4—614no—
389C2×C4⋊C4—614no—
390C2×C4⋊C4—618no—
391C2×C4⋊C4—6132no—
392C2×C4⋊C4C22×D6114yes—
393C2×C4⋊C4C22×D6112yes—
394C2×C22⋊C4—614no—
395C2×C22⋊C4—614no—
396C2×C22⋊C4—618no—
397C2×C22⋊C4—614no—
398C2×C22⋊C4—614no—
399C2×C22⋊C4C22×D6118yes—
400C2×C22⋊C4C22×D6114yes—
401C2×C42—612no—
402C2×C42—614no—
403C2×C42—614no—
404C2×C42—611no—
405C2×C42—618no—
406C2×C42C22×D6114yes—
407C2×C42C22×D6111yescharacteristic
408C24—611no—
409C24C23×D6114yes—
410#41—24464no—
411C2×Q8—44no—
412C2×Q8—618no—
413C2×Q8—618no—
414C2×Q8—6116no—
415C2×Q8—6116no—
416C2×Q8—1228no—
417C2×Q8—2448no—
418C2×Q8—2444no—
419C2×Q8—2448no—
420C2×Q8—24432no—
421C2×Q8C23×D6118yes—
422C2×D4—28no—
423C2×D4—48no—
424C2×D4—6116no—
425C2×D4—12216no—
426C2×D4C23×D6118yes—
427C2×D4C23×D61116yes—
428C22×C4—216no—
429C22×C4—614no—
430C22×C4—612no—
431C22×C4—611no—
432C22×C4—614no—
433C22×C4—614no—
434C22×C4—6132no—
435C22×C4—618no—
436C22×C4—614no—
437C22×C4—6116no—
438C22×C4—1224no—
439C22×C4—12216no—
440C22×C4—1228no—
441C22×C4—24416no—
442C22×C4C23×D61116yes—
443C22×C4C23×D6111yescharacteristic
444C22×C4C23×D6114yes—
445C22×C4C23×D6114yes—
446C22×C4C23×D6112yes—
447C4⋊C4—6116no—
448C4⋊C4—618no—
449C4⋊C4—6132no—
450C4⋊C4—6132no—
451C4⋊C4—6132no—
452C4⋊C4—6116no—
453C4⋊C4—6116no—
454C4⋊C4—6132no—
455C4⋊C4—6116no—
456C4⋊C4—6116no—
457C4⋊C4C23×D61116yes—
458C4⋊C4C23×D6118yes—
459C22⋊C4—6116no—
460C22⋊C4—6116no—
461C22⋊C4—6132no—
462C22⋊C4—6116no—
463C22⋊C4—6116no—
464C22⋊C4C23×D61132yes—
465C22⋊C4C23×D61116yes—
466C42—618no—
467C42—614no—
468C42—6116no—
469C42—6116no—
470C42C23×D6114yes—
471C42C23×D61116yes—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—614no—
476C23—612no—
477C23—1224no—
478C23C24×D6118yes—
479C23C24×D6111yescharacteristic
480Q8—48no—
481Q8—12216no—
482Q8—24416no—
483Q8—2448no—
484Q8—24416no—
485D4—216no—
486D4—416no—
487D4—12232no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—232no—
492C2×C4—618no—
493C2×C4—614no—
494C2×C4—612no—
495C2×C4—618no—
496C2×C4—618no—
497C2×C4—1228no—
498C2×C4—12232no—
499C2×C4—1228no—
500C2×C4—1222no—
501C2×C4—1228no—
502C2×C4—1224no—
503C2×C4—1228no—
504C2×C4—12216no—
505C2×C4—24432no—
506C2×C4#1451982418yes—
507C2×C4C24×D6112yes—
508C2×C4C24×D6118yes—
509C2×C4C24×D6118yes—
510C2×C4C24×D6114yes—
511C22—28no—
512C22—28no—
513C22—28no—
514C22—416no—
515C22—614no—
516C22—1224no—
517C22—1228no—
518C22H(5,𝔽2)×D6112yes—
519C22C2×#1451982414yes—
520C22C25×D6111yescharacteristic
521C4—28no—
522C4—22no—
523C4—28no—
524C4—1228no—
525C4—1222no—
526C4—1228no—
527C4—1224no—
528C4—1228no—
529C4C2×#1451982418yes—
530C2—48no—
531C2—1224no—
532C2#2146015414yesdirect product factor
533C2C2×H(5,𝔽2)×D6111yescharacteristic
534C2C22×#1451982412yes—
subgroupG/H
centerC23C24×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC2×C61×#555C2
socleC23×C61C25
hypercenterC2×#555D61
hypocenterC61C2×#3544
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #21460154C2direct (N × H)
2. C2×#16840512C2semidirect
3. C2×#16840492C2semidirect
4. C2×#16840469C2semidirect
5. C2×#16840451C2semidirect
6. C2×#16840412C2semidirect
7. C2×#16840390C2semidirect
8. C2×#16840379C2semidirect
9. C2×#16840332C2semidirect
10. C2×C4⋊Q8×D61C2semidirect
11. C2×#16840294C2semidirect
12. C2×C61×#555C2semidirect
15. #16840512C22semidirect
16. #16840492C22semidirect
17. #16840469C22semidirect
18. #16840451C22semidirect
19. #16840412C22semidirect
20. #16840390C22semidirect
21. #16840379C22semidirect
22. #16840332C22semidirect
29. C4⋊Q8×D61C22semidirect
30. #16840294C22semidirect
56. C61×#555C22semidirect
266. C2×#555D61semidirect
276. C2×C61#3544semidirect
284. #555C2×D61semidirect
329. C61C2×#3544semidirect
532. C2#21460154direct (N × H)

Maximal subgroups

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

subgroupindex
#214601542
C2×#168405122
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