Finite Group Databasebeta

← C2×#21457186 #30552863

Subgroups — C2×#21457186 #30552863

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×#16840296C211yesmaximal, characteristic
2#21457186C2116yesmaximal, direct product factor
3C2×#16840069C212yesmaximal, complemented
4C2×#16840054C211yesmaximal, characteristic, complemented
5C2×#16840049C211yesmaximal, characteristic, complemented
6C2×#16840018C212yesmaximal, complemented
7C2×#16840010C212yesmaximal
8C2×#16839984C212yesmaximal, complemented
9C2×#16839969C211yesmaximal, characteristic, complemented
10C2×#16839964C211yesmaximal, characteristic, complemented
11C2×#16839786C211yesmaximal, characteristic, complemented
12C2×C61×#533C211yesmaximal, characteristic, complemented
13C22×#11758202C2212yes—
14C22×#11758201C2212yes—
15C22×#11758194C2212yes—
16#16840296—24no—
17#16840296C2218yes—
18C2×#14519771—21no—
19C2×#14519769C2211yescharacteristic
20C2×#14519765C2211yescharacteristic
21C2×#14519760—21no—
22C2×#14519756C2211yescharacteristic
23C2×#14519751C2211yescharacteristic
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26C2×C4×C22⋊D61—21no—
27C2×#14519734—21no—
28#16840069C22116yescomplemented
29#16840054C2218yescomplemented
30#16840049C2218yescomplemented
31#16840018C22116yescomplemented
32#16840010C22116yes—
33#16839984C22116yescomplemented
34#16839969C2218yescomplemented
35#16839964C2218yescomplemented
36C2×#14519710C2212yes—
37C2×#14519708C2212yes—
38#16839786C2218yescomplemented
39C2×C4⋊#92418C2211yescharacteristic
40C2×C4⋊#92418C2211yescharacteristic
41C2×C4⋊#92418C2212yes—
42C2×#14519701C2212yes—
43C2×#14519701C2212yes—
44C2×#14519700C2212yes—
45C2×#14519697C2212yes—
46C61×#533—24no—
47C61×#533C2218yescomplemented
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×C4⋊Q8×C61C2211yescharacteristic
50C2×#125×C61—21no—
51C2×C22⋊Q8×C61—21no—
52C2×C22⋊Q8×C61C2212yes—
53C2×C4×Q8×C61—21no—
54C2×C4×Q8×C61C2212yes—
55C2×#117×C61—21no—
56C22×C42×C61C2211yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61C2×#11758202—44no—
62C2×#11758202C2318yes—
63C2×#11758201—44no—
64C2×#11758201C2318yes—
65C2×#11758194—44no—
66C2×#11758194C2318yes—
67C22×#92418C2312yes—
68C22×#92418C2312yes—
69#14519771—28no—
70#14519769—22no—
71#14519769C2314yes—
72#14519765—22no—
73#14519765C2314yes—
74#14519760—28no—
75#14519756—22no—
76#14519756C2314yes—
77#14519751—22no—
78#14519751C2314yes—
79C4×Dic122—22no—
80C4×Dic122C2314yes—
81C4×D244—22no—
82C4×D244C2314yes—
83C4×C22⋊D61—28no—
84#14519734—28no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97#14519710C2318yes—
98#14519708C2318yes—
99C4⋊#92418C2314yes—
100C4⋊#92418C2314yes—
101C4⋊#92418C2318yes—
102#14519701C2318yes—
103#14519701C2318yes—
104#14519700C2318yes—
105#14519697C2318yes—
106C22×Q8×C61—21no—
107C22×Q8×C61C2312yes—
108C23×C4×C61D411yescharacteristic
109C23×C4×C61D411yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C4⋊Q8×C61—22no—
112C4⋊Q8×C61—22no—
113C4⋊Q8×C61C2314yes—
114C4⋊Q8×C61C2314yes—
115#125×C61—28no—
116C22⋊Q8×C61—24no—
117C22⋊Q8×C61—28no—
118C22⋊Q8×C61C2318yes—
119C4×Q8×C61—24no—
120C4×Q8×C61—28no—
121C4×Q8×C61C2318yes—
122#117×C61—28no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61C2312yes—
129C2×C22⋊C4×C61—22no—
130C2×C42×C61—22no—
131C2×C42×C61D414yes—
132C2×C42×C61D411yescharacteristic
133C2×C42×C61D411yescharacteristic
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2314yes—
136C2×C42×C61C2311yescharacteristic
137C24×C61C2×D411yescharacteristic
138C23×D61—21no—
139C2×Q8×C61—22no—
140C2×Q8×C61—24no—
141C2×Q8×C61—24no—
142C2×Q8×C61—24no—
143C2×Q8×C61—44no—
144C2×Q8×C61C2414yes—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—42no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D414yes—
157C22×C4×C61C2×D414yes—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2412yes—
160C22×C4×C61C2411yescharacteristic
161C2×C22⋊D61—24no—
162C2×C22⋊D61—44no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—21no—
169C2×D244C2412yes—
170C2×C4×D61—24no—
171C2×C4×D61—44no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122C2412yes—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—28no—
180C4⋊C4×C61—28no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C2414yes—
183C22⋊C4×C61—28no—
184C42×C61—28no—
185C42×C61C2×D412yes—
186C42×C61C2×D414yes—
187C42×C61C2×D414yes—
188C42×C61C2×D412yes—
189C42×C61C2412yes—
190C42×C61C2412yes—
191C22⋊Dic61—24no—
192#11758202—48no—
193#11758201—48no—
194#11758198—24no—
195#11758198—24no—
196#11758198—24no—
197#11758198—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C2412yes—
201C4⋊Dic61C2412yes—
202#11758196—24no—
203#11758196—24no—
204#11758196—24no—
205#11758196—24no—
206C4×Dic61—24no—
207#11758194—48no—
208C2×#92418—24no—
209C2×#92418—24no—
210C2×#92418C2414yes—
211C2×#92418C2414yes—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61C22×D412yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—22no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221Q8×C61—24no—
222Q8×C61—24no—
223Q8×C61—48no—
224Q8×C61—44no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—28no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—44no—
236C2×C4×C61—44no—
237C2×C4×C61—44no—
238C2×C4×C61#13714yes—
239C2×C4×C61C22×D412yes—
240C2×C4×C61C22×D412yes—
241C2×C4×C61C22×D412yes—
242C2×C4×C61C22×D412yes—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D412yes—
245C2×C4×C61H(5,𝔽2)12yes—
246C2×C4×C612−1+412yes—
247C2×C4×C61C2511yescharacteristic
248C22⋊D61—48no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255D244—22no—
256D244—22no—
257D244—42no—
258C4×D61—48no—
259Dic122—22no—
260Dic122—22no—
261Dic122—42no—
262#92418—24no—
263#92418—24no—
264C2×#3252—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—44no—
269C22×C61#53712yes—
270C22×C61C2×#13712yes—
271C22×C61#57512yes—
272C22×C61C23×D411yescharacteristic
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—84no—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—44no—
282C4×C61—44no—
283C4×C61#53712yes—
284C4×C61C2×#13714yes—
285C4×C61C2×H(5,𝔽2)11yescharacteristic
286C4×C61C2×2−1+411yescharacteristic
287Dic61—42no—
288Dic61—42no—
289Dic61—42no—
290C2×#533D6111yescharacteristic, complemented
291C2×#529—611no—
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#325214yescomplemented
304C2×C61C2×#53711yescharacteristic
305C2×C61C22×#13711yescharacteristic
306C2×C61C2×#57511yescharacteristic
307D61—82no—
308C22×Q16—612no—
309C22×SD16—612no—
310C22×M16—612no—
311#533—24no—
312#533C2×D6118yescomplemented
313C2×C4⋊Q8C2×D6111yescharacteristic
314C2×C4⋊Q8C2×D6111yescharacteristic
315#529—618no—
316#529—1224no—
317C2×#126—1221no—
318C2×#125—21no—
319C2×#125—611no—
320C2×#124—611no—
321C2×#123—1221no—
322C2×C22⋊Q8—21no—
323C2×C22⋊Q8—611no—
324C2×C22⋊Q8C2×D6112yes—
325C2×#121—611no—
326C2×C4×Q8—21no—
327C2×C4×Q8—611no—
328C2×C4×Q8C2×D6112yes—
329C2×C4×D4—611no—
330C2×C4×D4—1221no—
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—21no—
354C22×Q8—611no—
355C22×Q8C22×D6112yes—
356C22×D4—611no—
357C22×D4—1221no—
358C23×C4—1221no—
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—22no—
371C4⋊Q8—22no—
372C4⋊Q8C22×D6114yes—
373C4⋊Q8C22×D6114yes—
374#126—1228no—
375#125—28no—
376#125—614no—
377#125—1222no—
378#124—614no—
379#124—1222no—
380#123—1228no—
381C22⋊Q8—28no—
382C22⋊Q8—24no—
383C22⋊Q8—614no—
384C22⋊Q8—1222no—
385C22⋊Q8C22×D6118yes—
386#121—614no—
387#121—1222no—
388C4×Q8—28no—
389C4×Q8—24no—
390C4×Q8—614no—
391C4×Q8—1222no—
392C4×Q8C22×D6118yes—
393C4×D4—614no—
394C4×D4—1228no—
395C4×D4—1222no—
396#117—28no—
397#117—1228no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22×D6112yes—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—22no—
417C2×C42—1221no—
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—1221no—
434C24C2×C22⋊D6111yescharacteristic
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—44no—
440C2×Q8—612no—
441C2×Q8—1221no—
442C2×Q8—1222no—
443C2×Q8—1222no—
444C2×Q8C23×D6114yes—
445C2×D4—612no—
446C2×D4—1224no—
447C2×D4—1221no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—2444no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1224no—
461C22×C4—2444no—
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—28no—
483C4⋊C4—28no—
484C4⋊C4—22no—
485C4⋊C4—612no—
486C4⋊C4—612no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—1224no—
492C4⋊C4—1224no—
493C4⋊C4C23×D6114yes—
494C22⋊C4—28no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C42—28no—
501C42—1224no—
502C42C2×C22⋊D6114yes—
503C42C2×C22⋊D6112yes—
504C42C2×C22⋊D6112yes—
505C42C2×C22⋊D6114yes—
506C42C23×D6112yes—
507C42C23×D6112yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—1222no—
512C23—2442no—
513C23—2442no—
514C23—2442no—
515C23C22×C22⋊D6112yes—
516C23C22×C22⋊D6111yescharacteristic
517Q8—24no—
518Q8—24no—
519Q8—48no—
520Q8—44no—
521Q8—1222no—
522Q8—1222no—
523Q8—2442no—
524D4—1222no—
525D4—1222no—
526D4—2448no—
527D4—2442no—
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—24no—
537C2×C4—24no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2448no—
548C2×C4#1451980814yes—
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—4884no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22#1684034012yes—
569C22C2×#1451980812yes—
570C22#1684056012yes—
571C22C23×C22⋊D6111yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—44no—
577C4—44no—
578C4—2442no—
579C4—2442no—
580C4—2442no—
581C4#1684034012yes—
582C4C2×#1451980814yes—
583C4C2×#1451982311yescharacteristic
584C4C2×#1451982411yescharacteristic
585C2—42no—
586C2—4882no—
587C2#2145718614yesdirect product factor
588C2C2×#1684034011yescharacteristic
589C2C22×#1451980811yescharacteristic
590C2C2×#1684056011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#533C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC2×#533D61
hypocenterC61C2×#3252
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. #21457186C2direct (N × H)
3. C2×#16840069C2semidirect
4. C2×#16840054C2semidirect
5. C2×#16840049C2semidirect
6. C2×#16840018C2semidirect
8. C2×#16839984C2semidirect
9. C2×#16839969C2semidirect
10. C2×#16839964C2semidirect
11. C2×#16839786C2semidirect
12. C2×C61×#533C2semidirect
28. #16840069C22semidirect
29. #16840054C22semidirect
30. #16840049C22semidirect
31. #16840018C22semidirect
33. #16839984C22semidirect
34. #16839969C22semidirect
35. #16839964C22semidirect
38. #16839786C22semidirect
47. C61×#533C22semidirect
290. C2×#533D61semidirect
303. C2×C61#3252semidirect
312. #533C2×D61semidirect
352. C61C2×#3252semidirect
587. C2#21457186direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402962
#214571862
C2×#168400692
C2×#168400542
C2×#168400492
C2×#168400182
C2×#168400102
C2×#168399842
C2×#168399692
C2×#168399642
C2×#168397862
C2×C61×#5332
C2×#325261