Finite Group Databasebeta

← C2×#21455670 #30547594

Subgroups — C2×#21455670 #30547594

Order 15616.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic, complemented
2#21455670C2116yesmaximal, direct product factor
3C2×#16840069C212yesmaximal, complemented
4C2×#16840057C212yesmaximal, complemented
5C2×#16840062C212yesmaximal, complemented
6C2×#16840042C212yesmaximal, complemented
7C2×#16839802C211yesmaximal, characteristic, complemented
8C22×#14519710C211yesmaximal, characteristic, complemented
9C2×#16839789C211yesmaximal, characteristic, complemented
10C2×#16839763C211yesmaximal, characteristic
11C2×#16839758C211yesmaximal, characteristic
12C2×C61×#520C211yesmaximal, characteristic
13C22×#11758194C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840390—24no—
16#16840390C2218yescomplemented
17C2×#14519777C2212yes—
18C2×#14519776—21no—
19C2×#14519769C2212yes—
20C2×#14519768—21no—
21C2×#14519754—22no—
22C2×#14519736C2211yescharacteristic
23C22×C4⋊Dic61C2211yescharacteristic
24#16840069C22116yescomplemented
25#16840062C22116yescomplemented
26#16840057C22116yescomplemented
27#16840042C22116yescomplemented
28#16839802C2218yescomplemented
29C2×#14519710C2212yes—
30C2×#14519710C2212yes—
31C2×#14519710C2212yes—
32C2×#14519710C2218yescomplemented
33C2×#14519708C2212yes—
34#16839789C2218yescomplemented
35C2×C4⋊#92418C2212yes—
36C2×C4⋊#92418C2212yes—
37#16839763C2218yes—
38#16839758C2218yes—
39C2×#14519701C2211yescharacteristic
40C2×#14519701C2212yes—
41C2×#14519701C2211yescharacteristic
42C2×#14519698C2211yescharacteristic
43C2×#14519698C2212yes—
44C2×#14519698C2211yescharacteristic
45C22×#41×C61C2211yescharacteristic
46C61×#520—24no—
47C61×#520C2218yes—
48C2×C4×Q8×C61C2212yes—
49C2×C4×D4×C61—22no—
50C2×C4×D4×C61C2212yes—
51C2×#117×C61—21no—
52C2×#117×C61C2211yescharacteristic
53C22×C4⋊C4×C61—21no—
54C22×C4⋊C4×C61C2211yescharacteristic
55C22×Dic122—22no—
56C23×Dic61—21no—
57C2×#11758194—24no—
58C2×#11758194C2314yes—
59C22×#92418—24no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2312yes—
63C22×#92418C2314yes—
64#14519777—24no—
65#14519777C2318yes—
66#14519776—28no—
67#14519769—24no—
68#14519769C2318yes—
69#14519768—28no—
70#14519754—216no—
71#14519736—22no—
72#14519736C2314yes—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2314yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—22no—
81C2×#11758196—22no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84#14519710C2318yes—
85#14519710C2318yes—
86#14519710C2318yes—
87#14519708C2318yes—
88C4⋊#92418C2318yes—
89C4⋊#92418C2318yes—
90#14519701C2314yes—
91#14519701C2314yes—
92#14519701C2318yes—
93#14519698C2314yes—
94#14519698C2314yes—
95#14519698C2318yes—
96C2×#41×C61—22no—
97C2×#41×C61—24no—
98C2×#41×C61Q818yescomplemented
99C2×#41×C61C2314yes—
100C22×Q8×C61C2311yescharacteristic
101C22×D4×C61—21no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61—21no—
105C23×C4×C61—22no—
106C23×C4×C61C2311yescharacteristic
107C4×Q8×C61—24no—
108C4×Q8×C61C2318yes—
109C4×D4×C61—216no—
110C4×D4×C61—24no—
111C4×D4×C61C2318yes—
112#117×C61—28no—
113#117×C61D414yes—
114#117×C61C2314yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61—22no—
118C2×C4⋊C4×C61—28no—
119C2×C4⋊C4×C61D411yescharacteristic
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61C2312yes—
122C2×C4⋊C4×C61C2314yes—
123C2×C22⋊C4×C61—22no—
124C2×C22⋊C4×C61D412yes—
125C2×C42×C61—22no—
126C2×C42×C61C2312yes—
127C24×C61—21no—
128C24×C61C2×D411yescharacteristic
129#41×C61—216no—
130#41×C61—28no—
131#41×C61C2×Q8116yescomplemented
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61C2×Q814yes—
135C2×Q8×C61C2412yes—
136C2×D4×C61—28no—
137C2×D4×C61—24no—
138C2×D4×C61—21no—
139C2×D4×C61—22no—
140C2×D4×C61C2×Q814yes—
141C2×D4×C61C2412yes—
142C22×C4×C61—21no—
143C22×C4×C61—28no—
144C22×C4×C61—24no—
145C22×C4×C61—22no—
146C22×C4×C61—24no—
147C22×C4×C61—28no—
148C22×C4×C61—84no—
149C22×C4×C61—82no—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×Q814yes—
155C22×C4×C61#4114yes—
156C22×C4×C61C2411yescharacteristic
157C22×C4×C61C2412yes—
158C22×Dic61—22no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—24no—
162C22×Dic61—82no—
163C2×Dic122—28no—
164C2×Dic122—84no—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—28no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C2×D414yes—
169C4⋊C4×C61C2×D414yes—
170C4⋊C4×C61C2414yes—
171C22⋊C4×C61—28no—
172C22⋊C4×C61C2×D418yes—
173C42×C61—28no—
174C42×C61C2×D414yes—
175C42×C61C2414yes—
176C22⋊Dic61—24no—
177C4⋊Dic61—24no—
178C4⋊Dic61—22no—
179C4⋊Dic61—28no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C2412yes—
183C4⋊Dic61C2414yes—
184C4⋊Dic61C2412yes—
185#11758196—28no—
186#11758196—28no—
187C4×Dic61—24no—
188#11758194—28no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—28no—
192C2×#92418—24no—
193C2×#92418C2414yes—
194C2×#92418C2412yes—
195C2×#92418C2412yes—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—24no—
199C23×C61—22no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61C22⋊Q814yes—
203C23×C61C22×D412yes—
204C23×C61C22×D411yescharacteristic
205Q8×C61—24no—
206Q8×C61—22no—
207Q8×C61C22×Q814yes—
208D4×C61—216no—
209D4×C61—24no—
210D4×C61—22no—
211D4×C61C22×Q814yes—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—216no—
217C2×C4×C61—24no—
218C2×C4×C61—44no—
219C2×C4×C61—88no—
220C2×C4×C61—84no—
221C2×C4×C61C22⋊Q814yes—
222C2×C4×C61C22⋊Q818yes—
223C2×C4×C61C22×D414yes—
224C2×C4×C61C22×D414yes—
225C2×C4×C61C22×D412yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×Q812yes—
228C2×C4×C61C22×Q814yes—
229C2×C4×C61C2×#4112yes—
230C2×C4×C61C2×#4114yes—
231C2×C4×C61C2511yescharacteristic
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—42no—
236C2×Dic61—44no—
237C2×Dic61—42no—
238C2×Dic61—84no—
239Dic122—88no—
240#92418—22no—
241#92418—22no—
242#92418—24no—
243C2×#3083—611nomaximal, Sylow
244C22×C61—28no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—24no—
248C22×C61—48no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61C2×C22⋊Q814yes—
252C22×C61C2×C22⋊Q812yes—
253C22×C61#57612yes—
254C22×C61#57812yes—
255C22×C61C23×D411yescharacteristic
256C4×C61—24no—
257C4×C61—24no—
258C4×C61—24no—
259C4×C61—44no—
260C4×C61C2×C22⋊Q814yes—
261C4×C61C2×C22⋊Q812yes—
262C4×C61C23×Q811yescharacteristic
263C4×C61C22×#4111yescharacteristic
264Dic61—42no—
265Dic61—44no—
266Dic61—42no—
267C2×#544—611no—
268C2×#520D6111yescharacteristic
269#3083—6116no—
270C2×#479—612no—
271C2×#477—612no—
272C2×#478—612no—
273C2×#474—612no—
274C2×#428—611no—
275C22×#107—611no—
276C2×#424—611no—
277C2×#417—611no—
278C2×#416—611no—
279C2×C61—24no—
280C2×C61—44no—
281C2×C61#308314yescomplemented
282C2×C61C22×C22⋊Q811yescharacteristic
283C2×C61C2×#57611yescharacteristic
284C2×C61C2×#57811yescharacteristic
285C22×#41C2×D6111yescharacteristic
286C22×M16—611no—
287C23×C8—611no—
288#544—618no—
289#544—1224no—
290C2×C4⋊Q8—612no—
291C2×C4⋊Q8—1221no—
292C2×#125—612no—
293C2×#125—1221no—
294C2×C22⋊Q8—1222no—
295#520—24no—
296#520C2×D6118yes—
297C2×C4×Q8C2×D6112yes—
298C2×C4×D4—22no—
299C2×C4×D4C2×D6112yes—
300C2×#117—21no—
301C2×#117—611no—
302C2×#117C2×D6111yescharacteristic
303C22×C4⋊C4—21no—
304C22×C4⋊C4—611no—
305C22×C4⋊C4C2×D6111yescharacteristic
306#479—6116no—
307#478—6116no—
308#477—6116no—
309#474—6116no—
310#428—618no—
311C2×#107—612no—
312C2×#107—612no—
313C2×#107—612no—
314C2×#107—618no—
315C2×#106—612no—
316#424—618no—
317C2×C4⋊C8—612no—
318C2×C4⋊C8—612no—
319#417—618no—
320#416—618no—
321C2×#103—611no—
322C2×#103—612no—
323C2×#103—611no—
324C2×#102—611no—
325C2×#102—612no—
326C2×#102—611no—
327C61C2×#308311yescharacteristic, complemented, Sylow
328C2×#41—24no—
329C2×#41—22no—
330C2×#41Dic12218yescomplemented
331C2×#41C22×D6114yes—
332C22×Q8—1222no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4—21no—
335C22×D4C22×D6111yescharacteristic
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—22no—
339C23×C4—1221no—
340C23×C4C22×D6111yescharacteristic
341C2×M16—614no—
342C2×M16—1224no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8—614no—
346C22×C8—612no—
347C22×C8—1224no—
348C4⋊Q8—618no—
349C4⋊Q8—1228no—
350C4⋊Q8—1224no—
351#125—618no—
352#125—1228no—
353#125—1224no—
354C22⋊Q8—12216no—
355C4×Q8—24no—
356C4×Q8C22×D6118yes—
357C4×D4—216no—
358C4×D4—24no—
359C4×D4C22×D6118yes—
360#117—28no—
361#117—614no—
362#117—1222no—
363#117C22⋊D6114yes—
364#117C22×D6114yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—28no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—614no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4C22⋊D6111yescharacteristic
379C2×C4⋊C4C22⋊D6111yescharacteristic
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6114yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4C22⋊D6112yes—
385C2×C42—22no—
386C2×C42—1221no—
387C2×C42C22×D6112yes—
388#107—618no—
389#107—618no—
390#107—618no—
391#106—618no—
392C4⋊C8—618no—
393C4⋊C8—618no—
394#103—614no—
395#103—614no—
396#103—618no—
397#102—614no—
398#102—614no—
399#102—618no—
400C24—21no—
401C24C2×C22⋊D6111yescharacteristic
402#41—28no—
403#41—216no—
404#41C2×Dic122116yescomplemented
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—1228no—
408C2×Q8—4884no—
409C2×Q8C2×Dic12214yes—
410C2×Q8C23×D6112yes—
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4C2×Dic12214yes—
416C2×D4C23×D6112yes—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—84no—
424C22×C4—82no—
425C22×C4—1224no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4—4882no—
430C22×C4C2×Dic12214yes—
431C22×C4#1175822214yes—
432C22×C4C2×C22⋊D6112yes—
433C22×C4C2×C22⋊D6112yes—
434C22×C4C2×C22⋊D6112yes—
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C23×D6112yes—
437C22×C4C23×D6111yescharacteristic
438M16—1228no—
439C2×C8—614no—
440C2×C8—612no—
441C2×C8—612no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1228no—
445C2×C8—1224no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—22no—
449C4⋊C4—612no—
450C4⋊C4—614no—
451C4⋊C4—612no—
452C4⋊C4—1224no—
453C4⋊C4—1222no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1228no—
457C4⋊C4—1228no—
458C4⋊C4—1228no—
459C4⋊C4C2×C22⋊D6114yes—
460C4⋊C4C2×C22⋊D6114yes—
461C4⋊C4C23×D6114yes—
462C22⋊C4—28no—
463C22⋊C4—1224no—
464C22⋊C4C2×C22⋊D6118yes—
465C42—28no—
466C42—1224no—
467C42C2×C22⋊D6114yes—
468C42C23×D6114yes—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—42no—
474C23—42no—
475C23#1451975614yes—
476C23C22×C22⋊D6112yes—
477C23C22×C22⋊D6111yescharacteristic
478Q8—24no—
479Q8—22no—
480Q8—4888no—
481Q8C22×Dic12214yes—
482D4—216no—
483D4—24no—
484D4—22no—
485D4C22×Dic12214yes—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—216no—
491C2×C4—24no—
492C2×C4—44no—
493C2×C4—88no—
494C2×C4—84no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1224no—
498C2×C4—4884no—
499C2×C4—2444no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451975614yes—
503C2×C4#1451975618yes—
504C2×C4C22×C22⋊D6114yes—
505C2×C4C22×C22⋊D6114yes—
506C2×C4C22×C22⋊D6112yes—
507C2×C4C22×C22⋊D6111yescharacteristic
508C2×C4C22×Dic12212yes—
509C2×C4C22×Dic12214yes—
510C2×C4C2×#1175822212yes—
511C2×C4C2×#1175822214yes—
512C2×C4C24×D6111yescharacteristic
513C8—1222no—
514C8—1222no—
515C8—1224no—
516C22—28no—
517C22—22no—
518C22—22no—
519C22—24no—
520C22—48no—
521C22—42no—
522C22—42no—
523C22C2×#1451975614yes—
524C22C2×#1451975612yes—
525C22#1684056512yes—
526C22#1684057712yes—
527C22C23×C22⋊D6111yescharacteristic
528C4—24no—
529C4—24no—
530C4—24no—
531C4—44no—
532C4—2444no—
533C4—2442no—
534C4—2442no—
535C4C2×#1451975614yes—
536C4C2×#1451975612yes—
537C4C23×Dic12211yescharacteristic
538C4C22×#1175822211yescharacteristic
539C2—24no—
540C2—44no—
541C2#2145567014yesdirect product factor
542C2C22×#1451975611yescharacteristic
543C2C2×#1684056511yescharacteristic
544C2C2×#1684057711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#520C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenterC2×#520D61
hypocenterC61C2×#3083
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16840390C2semidirect
2. #21455670C2direct (N × H)
3. C2×#16840069C2semidirect
4. C2×#16840057C2semidirect
5. C2×#16840062C2semidirect
6. C2×#16840042C2semidirect
7. C2×#16839802C2semidirect
8. C22×#14519710C2semidirect
9. C2×#16839789C2semidirect
16. #16840390C22semidirect
24. #16840069C22semidirect
25. #16840062C22semidirect
26. #16840057C22semidirect
27. #16840042C22semidirect
28. #16839802C22semidirect
32. C2×#14519710C22semidirect
34. #16839789C22semidirect
98. C2×#41×C61Q8semidirect
131. #41×C61C2×Q8semidirect
281. C2×C61#3083semidirect
327. C61C2×#3083semidirect
330. C2×#41Dic122semidirect
404. #41C2×Dic122semidirect
541. C2#21455670direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403902
#214556702
C2×#168400692
C2×#168400572
C2×#168400622
C2×#168400422
C2×#168398022
C22×#145197102
C2×#168397892
C2×#168397632
C2×#168397582
C2×C61×#5202
C2×#308361