Finite Group Databasebeta

← C2×#21457192 #30552869

Subgroups — C2×#21457192 #30552869

Order 15616.

1898 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic, complemented
2#21457192C2116yesmaximal, direct product factor
3C2×#16840066C212yesmaximal, complemented
4C2×#16840057C211yesmaximal, characteristic, complemented
5C2×#16840052C211yesmaximal, characteristic, complemented
6C2×#16840018C212yesmaximal, complemented
7C2×#16840006C212yesmaximal
8C2×#16839981C212yesmaximal, complemented
9C2×#16839972C211yesmaximal, characteristic, complemented
10C2×#16839967C211yesmaximal, characteristic, complemented
11C2×#16839786C211yesmaximal, characteristic, complemented
12C2×C61×#529C211yesmaximal, characteristic
13C22×#11758202C2212yes—
14C22×#11758200C2212yes—
15C22×#11758194C2212yes—
16#16840319—24no—
17#16840319C2218yescomplemented
18C2×#14519777C2211yescharacteristic
19C2×#14519777C2211yescharacteristic
20C2×#14519769—21no—
21C2×#14519756—21no—
22C2×#14519756C2212yes—
23C2×C4×Dic122—21no—
24C2×C4×Dic122C2212yes—
25C2×#14519736—21no—
26#16840066C22116yescomplemented
27#16840057C2218yescomplemented
28#16840052C2218yescomplemented
29#16840018C22116yescomplemented
30#16840006C22116yes—
31#16839981C22116yescomplemented
32#16839972C2218yescomplemented
33#16839967C2218yescomplemented
34C2×#14519710C2212yes—
35C2×#14519708C2212yes—
36#16839786C2218yescomplemented
37C2×C4⋊#92418C2211yescharacteristic
38C2×C4⋊#92418C2211yescharacteristic
39C2×C4⋊#92418C2212yes—
40C2×#14519701C2212yes—
41C2×#14519700C2212yes—
42C2×#14519700C2212yes—
43C2×#14519698C2212yes—
44C61×#529—24no—
45C61×#529C2218yes—
46C2×#126×C61—21no—
47C2×#125×C61C2211yescharacteristic
48C2×#124×C61C2211yescharacteristic
49C2×#123×C61—21no—
50C2×C22⋊Q8×C61C2211yescharacteristic
51C2×#121×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×C4×D4×C61—21no—
54C2×C4×D4×C61C2211yescharacteristic
55C2×#117×C61—21no—
56C22×C42×C61C2211yescharacteristic
57C22×Dic122—21no—
58C22×Dic122C2312yes—
59C2×#11758202—44no—
60C2×#11758202C2318yes—
61C2×#11758200—44no—
62C2×#11758200C2318yes—
63C2×#11758194—44no—
64C2×#11758194C2318yes—
65C22×#92418C2312yes—
66C22×#92418C2312yes—
67#14519777—22no—
68#14519777—22no—
69#14519777C2314yes—
70#14519777C2314yes—
71#14519769—28no—
72#14519756—28no—
73#14519756—24no—
74#14519756C2318yes—
75C4×Dic122—28no—
76C4×Dic122—24no—
77C4×Dic122C2318yes—
78#14519736—28no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2312yes—
83C2×#11758196—22no—
84C2×#11758196—22no—
85C2×C22⋊Dic61—22no—
86C2×C4×Dic61—22no—
87#14519710C2318yes—
88#14519708C2318yes—
89C4⋊#92418C2314yes—
90C4⋊#92418C2314yes—
91C4⋊#92418C2318yes—
92#14519701C2318yes—
93#14519700C2318yes—
94#14519700C2318yes—
95#14519698C2318yes—
96C22×Q8×C61C2311yescharacteristic
97C22×D4×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61D411yescharacteristic
101C23×C4×C61D411yescharacteristic
102C23×C4×C61C2311yescharacteristic
103#126×C61—28no—
104#125×C61—22no—
105#125×C61C2314yes—
106#124×C61—22no—
107#124×C61C2314yes—
108#123×C61—28no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61C2314yes—
111#121×C61—22no—
112#121×C61C2314yes—
113C4×Q8×C61—22no—
114C4×Q8×C61C2314yes—
115C4×D4×C61—28no—
116C4×D4×C61—22no—
117C4×D4×C61C2314yes—
118#117×C61—28no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C42×C61—21no—
131C2×C42×C61D411yescharacteristic
132C2×C42×C61D411yescharacteristic
133C2×C42×C61D414yes—
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C2×C42×C61C2314yes—
137C24×C61—21no—
138C24×C61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61C2412yes—
143C2×D4×C61—24no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—44no—
148C2×D4×C61C2412yes—
149C22×C4×C61—24no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—24no—
156C22×C4×C61—42no—
157C22×C4×C61—44no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D414yes—
164C22×C4×C61C2×D414yes—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2412yes—
167C22×C4×C61C2411yescharacteristic
168C22×Dic61—22no—
169C22×Dic61—22no—
170C2×Dic122—24no—
171C2×Dic122—24no—
172C2×Dic122—24no—
173C2×Dic122—22no—
174C2×Dic122—44no—
175C2×Dic122C2414yes—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61C2412yes—
183C4⋊C4×C61C2412yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—24no—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—24no—
189C42×C61—24no—
190C42×C61C2×D414yes—
191C42×C61C2×D412yes—
192C42×C61C2×D412yes—
193C42×C61C2×D414yes—
194C42×C61C2412yes—
195C42×C61C2412yes—
196C22⋊Dic61—28no—
197#11758202—48no—
198#11758200—48no—
199C4⋊Dic61—24no—
200C4⋊Dic61—24no—
201C4⋊Dic61—24no—
202C4⋊Dic61—22no—
203C4⋊Dic61C2414yes—
204#11758196—28no—
205#11758196—28no—
206C4×Dic61—28no—
207#11758194—48no—
208C2×#92418—24no—
209C2×#92418—24no—
210C2×#92418C2414yes—
211C2×#92418C2414yes—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61C22×D412yes—
220C23×C61C22×D411yescharacteristic
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—42no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—48no—
227D4×C61—42no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—28no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—44no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—48no—
244C2×C4×C61#13714yes—
245C2×C4×C61C22×D412yes—
246C2×C4×C61C22×D412yes—
247C2×C4×C61C22×D412yes—
248C2×C4×C61C22×D412yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D412yes—
251C2×C4×C61H(5,𝔽2)12yes—
252C2×C4×C612−1+412yes—
253C2×C4×C61C2511yescharacteristic
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258Dic122—24no—
259Dic122—24no—
260Dic122—48no—
261Dic122—44no—
262#92418—24no—
263#92418—24no—
264C2×#3252—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—44no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—84no—
273C22×C61#53712yes—
274C22×C61C2×#13712yes—
275C22×C61#57512yes—
276C22×C61C23×D411yescharacteristic
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61#53712yes—
285C4×C61C2×#13714yes—
286C4×C61C2×H(5,𝔽2)11yescharacteristic
287C4×C61C2×2−1+411yescharacteristic
288Dic61—44no—
289Dic61—44no—
290C2×#533—611no—
291C2×#529D6111yescharacteristic
292#3252—6116no—
293C2×#479—612no—
294C2×#477—611no—
295C2×#476—611no—
296C2×#470—612no—
297C2×#468—612no—
298C2×#464—612no—
299C2×#462—611no—
300C2×#461—611no—
301C2×#423—611no—
302C2×C61—42no—
303C2×C61—82no—
304C2×C61#325214yescomplemented
305C2×C61C2×#53711yescharacteristic
306C2×C61C22×#13711yescharacteristic
307C2×C61C2×#57511yescharacteristic
308C22×Q16—612no—
309C22×SD16—612no—
310C22×M16—612no—
311#533—618no—
312#533—1224no—
313C2×C4⋊Q8—611no—
314C2×C4⋊Q8—611no—
315#529—24no—
316#529C2×D6118yes—
317C2×#126—21no—
318C2×#125—1221no—
319C2×#125C2×D6111yescharacteristic
320C2×#124C2×D6111yescharacteristic
321C2×#123—21no—
322C2×C22⋊Q8—612no—
323C2×C22⋊Q8—1221no—
324C2×C22⋊Q8C2×D6111yescharacteristic
325C2×#121C2×D6111yescharacteristic
326C2×C4×Q8—612no—
327C2×C4×Q8—1221no—
328C2×C4×Q8C2×D6111yescharacteristic
329C2×C4×D4—21no—
330C2×C4×D4C2×D6111yescharacteristic
331C2×#117—21no—
332C2×#117—1221no—
333C22×C42C2×D6111yescharacteristic
334#479—6116no—
335#477—618no—
336#476—618no—
337#470—6116no—
338#468—6116no—
339#464—6116no—
340#462—618no—
341#461—618no—
342C2×#107—612no—
343C2×#106—612no—
344#423—618no—
345C2×C4⋊C8—611no—
346C2×C4⋊C8—612no—
347C2×C4⋊C8—611no—
348C2×#103—612no—
349C2×#103—612no—
350C2×#103—612no—
351C2×#102—612no—
352C61C2×#325211yescharacteristic, complemented, Sylow
353C22×Q8—612no—
354C22×Q8—1221no—
355C22×Q8C22×D6111yescharacteristic
356C22×D4—21no—
357C22×D4C22×D6111yescharacteristic
358C23×C4—21no—
359C23×C4C22⋊D6111yescharacteristic
360C23×C4C22⋊D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C2×Q16—618no—
363C2×Q16—2444no—
364C2×SD16—618no—
365C2×SD16—2444no—
366C2×M16—618no—
367C2×M16—2444no—
368C22×C8—612no—
369C22×C8—612no—
370C4⋊Q8—614no—
371C4⋊Q8—614no—
372C4⋊Q8—1222no—
373C4⋊Q8—1222no—
374#126—28no—
375#125—22no—
376#125—1228no—
377#125C22×D6114yes—
378#124—22no—
379#124C22×D6114yes—
380#123—28no—
381C22⋊Q8—22no—
382C22⋊Q8—618no—
383C22⋊Q8—1224no—
384C22⋊Q8—1228no—
385C22⋊Q8C22×D6114yes—
386#121—22no—
387#121C22×D6114yes—
388C4×Q8—22no—
389C4×Q8—618no—
390C4×Q8—1224no—
391C4×Q8—1228no—
392C4×Q8C22×D6114yes—
393C4×D4—28no—
394C4×D4—22no—
395C4×D4C22×D6114yes—
396#117—28no—
397#117—1228no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1222no—
416C2×C42—21no—
417C2×C42—1222no—
418C2×C42C22⋊D6111yescharacteristic
419C2×C42C22⋊D6111yescharacteristic
420C2×C42C22⋊D6114yes—
421C2×C42C22×D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6114yes—
424#107—618no—
425#106—618no—
426C4⋊C8—614no—
427C4⋊C8—614no—
428C4⋊C8—618no—
429#103—618no—
430#103—618no—
431#103—618no—
432#102—618no—
433C24—21no—
434C24C2×C22⋊D6111yescharacteristic
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—614no—
439C2×Q8—1224no—
440C2×Q8—1222no—
441C2×Q8—1224no—
442C2×Q8—1224no—
443C2×Q8—2444no—
444C2×Q8C23×D6112yes—
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—44no—
450C2×D4C23×D6112yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4C2×C22⋊D6112yes—
463C22×C4C2×C22⋊D6114yes—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6114yes—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C23×D6112yes—
471C22×C4C23×D6111yescharacteristic
472Q16—2448no—
473SD16—2448no—
474M16—2448no—
475C2×C8—614no—
476C2×C8—614no—
477C2×C8—1224no—
478C2×C8—1224no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—614no—
486C4⋊C4—1222no—
487C4⋊C4—1228no—
488C4⋊C4—1228no—
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—1224no—
492C4⋊C4C23×D6112yes—
493C4⋊C4C23×D6112yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—1228no—
500C42—24no—
501C42—1228no—
502C42C2×C22⋊D6114yes—
503C42C2×C22⋊D6112yes—
504C42C2×C22⋊D6112yes—
505C42C2×C22⋊D6114yes—
506C42C23×D6112yes—
507C42C23×D6112yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C22×C22⋊D6112yes—
516C23C22×C22⋊D6111yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—42no—
520Q8—1224no—
521Q8—1224no—
522Q8—2448no—
523Q8—2444no—
524D4—22no—
525D4—22no—
526D4—48no—
527D4—42no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—48no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4#1451980714yes—
549C2×C4C22×C22⋊D6112yes—
550C2×C4C22×C22⋊D6112yes—
551C2×C4C22×C22⋊D6112yes—
552C2×C4C22×C22⋊D6112yes—
553C2×C4C22×C22⋊D6112yes—
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4#1451982312yes—
556C2×C4#1451982412yes—
557C2×C4C24×D6111yescharacteristic
558C8—1224no—
559C8—1224no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—84no—
568C22#1684034012yes—
569C22C2×#1451980712yes—
570C22#1684056012yes—
571C22C23×C22⋊D6111yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—2444no—
580C4—2444no—
581C4#1684034012yes—
582C4C2×#1451980714yes—
583C4C2×#1451982311yescharacteristic
584C4C2×#1451982411yescharacteristic
585C2—42no—
586C2—82no—
587C2#2145719214yesdirect product factor
588C2C2×#1684034011yescharacteristic
589C2C22×#1451980711yescharacteristic
590C2C2×#1684056011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#529C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenterC2×#529D61
hypocenterC61C2×#3252
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#16840319C2semidirect
2. #21457192C2direct (N × H)
3. C2×#16840066C2semidirect
4. C2×#16840057C2semidirect
5. C2×#16840052C2semidirect
6. C2×#16840018C2semidirect
8. C2×#16839981C2semidirect
9. C2×#16839972C2semidirect
10. C2×#16839967C2semidirect
11. C2×#16839786C2semidirect
17. #16840319C22semidirect
26. #16840066C22semidirect
27. #16840057C22semidirect
28. #16840052C22semidirect
29. #16840018C22semidirect
31. #16839981C22semidirect
32. #16839972C22semidirect
33. #16839967C22semidirect
36. #16839786C22semidirect
304. C2×C61#3252semidirect
352. C61C2×#3252semidirect
587. C2#21457192direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403192
#214571922
C2×#168400662
C2×#168400572
C2×#168400522
C2×#168400182
C2×#168400062
C2×#168399812
C2×#168399722
C2×#168399672
C2×#168397862
C2×C61×#5292
C2×#325261