Finite Group Databasebeta

← C2×#21456217 #30549308

Subgroups — C2×#21456217 #30549308

Order 15616.

2182 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840308C211yesmaximal, characteristic
2C22×#14519768C211yesmaximal, characteristic, complemented
3#21456217C2116yesmaximal, direct product factor
4C2×#16840129C212yesmaximal, complemented
5C2×#16840125C212yesmaximal, complemented
6C2×#16840118C212yesmaximal, complemented
7C2×#16840115C212yesmaximal, complemented
8C2×#16839764C211yesmaximal, characteristic, complemented
9C2×#16839763C211yesmaximal, characteristic, complemented
10C2×#16839761C211yesmaximal, characteristic, complemented
11C2×M16×Dic61C211yesmaximal, characteristic, complemented
12C2×C61×#417C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15#16840308—24no—
16#16840308C2218yes—
17C2×#14519775C2211yescharacteristic
18C2×#14519773C2211yescharacteristic
19C2×#14519768—21no—
20C2×#14519768—21no—
21C2×#14519768—24no—
22C2×#14519768C2212yes—
23C2×#14519768C2212yes—
24C2×#14519768C2218yescomplemented
25C2×#14519766C2211yescharacteristic
26C2×#14519764C2211yescharacteristic
27C2×#14519749—21no—
28C2×#14519749—21no—
29C2×C4×C22⋊D61—21no—
30C2×C4×C22⋊D61—21no—
31C22×C4⋊Dic61—21no—
32C22×C4⋊Dic61C2211yescharacteristic
33C22×#11758196—21no—
34C22×C4×Dic61C2211yescharacteristic
35#16840129C22116yescomplemented
36#16840125C22116yescomplemented
37#16840118C22116yescomplemented
38#16840115C22116yescomplemented
39#16839764C2218yescomplemented
40#16839763C2218yescomplemented
41#16839761C2218yescomplemented
42C2×#14519702C2212yes—
43C2×#14519701C2212yes—
44C2×#14519700C2212yes—
45C2×#14519699C2212yes—
46C2×#14519698C2212yes—
47C2×#14519697C2212yes—
48M16×Dic61C2218yescomplemented
49C2×#14519687C2212yes—
50C2×C8×Dic61C2212yes—
51C22×#41×C61C2211yescharacteristic
52C22×M16×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C61×#417C2218yescomplemented
55C2×#103×C61C2212yes—
56C2×#102×C61C2212yes—
57C2×#11758222—22no—
58C2×#11758222—24no—
59C2×#11758222—24no—
60C2×#11758222C2314yes—
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C22⋊D61—21no—
64C22×C22⋊D61—21no—
65C22×C22⋊D61—21no—
66C23×Dic61—21no—
67C23×Dic61D411yescharacteristic
68C23×Dic61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758194—24no—
71C2×#11758194C2314yes—
72C22×#92418C2312yes—
73#14519775—22no—
74#14519775C2314yes—
75#14519773—22no—
76#14519773C2314yes—
77#14519768—24no—
78#14519768—24no—
79#14519768—28no—
80#14519768—28no—
81#14519768C2318yes—
82#14519768C2318yes—
83#14519766—22no—
84#14519766C2314yes—
85#14519764—22no—
86#14519764C2314yes—
87#14519749—28no—
88#14519749—28no—
89C4×C22⋊D61—28no—
90C4×C22⋊D61—28no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—22no—
93C2×C4⋊Dic61—22no—
94C2×C4⋊Dic61—22no—
95C2×C4⋊Dic61—28no—
96C2×C4⋊Dic61C2312yes—
97C2×C4⋊Dic61C2314yes—
98C2×#11758196—22no—
99C2×#11758196—22no—
100C2×#11758196—28no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×C22⋊Dic61—21no—
104C2×C22⋊Dic61—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D414yes—
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2314yes—
113#14519702C2318yes—
114#14519701C2318yes—
115#14519700C2318yes—
116#14519699C2318yes—
117#14519698C2318yes—
118#14519697C2318yes—
119#14519687C2318yes—
120C8×Dic61C2318yes—
121C2×#41×C61—22no—
122C2×#41×C61—24no—
123C2×#41×C61—24no—
124C2×#41×C61C2314yes—
125C22×Q8×C61C2311yescharacteristic
126C22×D4×C61—21no—
127C22×D4×C61C2311yescharacteristic
128C23×C4×C61—21no—
129C23×C4×C61—21no—
130C23×C4×C61C2311yescharacteristic
131C2×M16×C61—24no—
132C2×M16×C61C2314yes—
133C22×C8×C61C2312yes—
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61—22no—
136C2×C4⋊C4×C61C2312yes—
137C2×C4⋊C4×C61C2314yes—
138#103×C61C2318yes—
139#102×C61C2318yes—
140C24×C61—21no—
141C24×C61C2×D411yescharacteristic
142C23×D61—21no—
143#41×C61—28no—
144#41×C61—28no—
145#41×C61—48no—
146C2×Q8×C61—21no—
147C2×Q8×C61—22no—
148C2×Q8×C61—22no—
149C2×Q8×C61C2412yes—
150C2×D4×C61—24no—
151C2×D4×C61—21no—
152C2×D4×C61—22no—
153C2×D4×C61—22no—
154C2×D4×C61—44no—
155C2×D4×C61C2412yes—
156C22×C4×C61—24no—
157C22×C4×C61—22no—
158C22×C4×C61—21no—
159C22×C4×C61—24no—
160C22×C4×C61—44no—
161C22×C4×C61—44no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D412yes—
164C22×C4×C61#4114yes—
165C22×C4×C61#4114yes—
166C22×C4×C61C2411yescharacteristic
167C22×C4×C61C2412yes—
168C2×C22⋊D61—24no—
169C2×C22⋊D61—24no—
170C2×C22⋊D61—24no—
171C2×C22⋊D61—44no—
172C2×C22⋊D61—44no—
173C2×C22⋊D61—44no—
174C22×Dic61—21no—
175C22×Dic61—24no—
176C22×Dic61—22no—
177C22×Dic61—42no—
178C22×Dic61—42no—
179C22×Dic61—44no—
180C22×Dic61C2×D411yescharacteristic
181C22×Dic61C2×D411yescharacteristic
182C22×Dic61C2×D411yescharacteristic
183C22×Dic61C2×D411yescharacteristic
184C22×Dic61C2×D414yes—
185C22×Dic61C2×D414yes—
186#11758222—28no—
187#11758222—28no—
188#11758222—48no—
189C2×D244—22no—
190C2×D244—22no—
191C2×D244—21no—
192C2×D244C2412yes—
193C2×C4×D61—24no—
194C2×C4×D61—44no—
195C2×Dic122—22no—
196C2×Dic122—22no—
197C2×Dic122—21no—
198C2×Dic122C2412yes—
199M16×C61—28no—
200C2×C8×C61—24no—
201C2×C8×C61C2414yes—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—24no—
204C4⋊C4×C61C2414yes—
205C22⋊Dic61—24no—
206C22⋊Dic61—24no—
207#11758198—24no—
208#11758198—24no—
209C4⋊Dic61—22no—
210C4⋊Dic61—24no—
211C4⋊Dic61—28no—
212C4⋊Dic61—28no—
213C4⋊Dic61C2414yes—
214#11758196—28no—
215#11758196—28no—
216#11758196—24no—
217#11758196—24no—
218C4×Dic61C2×D414yes—
219C4×Dic61C2×D414yes—
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D412yes—
222C4×Dic61C2412yes—
223C4×Dic61C2412yes—
224#11758194—28no—
225C2×#92418—24no—
226C2×#92418C2414yes—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—42no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61#12414yes—
234C23×C61C22×D412yes—
235C23×C61C22×D411yescharacteristic
236C22×D61—22no—
237C22×D61—42no—
238C22×D61—42no—
239C22×D61—42no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—42no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—48no—
246D4×C61—42no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—48no—
250C2×C4×C61—42no—
251C2×C4×C61—44no—
252C2×C4×C61—48no—
253C2×C4×C61#12414yes—
254C2×C4×C61#12418yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D412yes—
257C2×C4×C61C2×#4112yes—
258C2×C4×C61C2×#4114yes—
259C2×C4×C61C2×#4112yes—
260C2×C4×C61C2×#4114yes—
261C2×C4×C61C2511yescharacteristic
262C22⋊D61—48no—
263C22⋊D61—48no—
264C22⋊D61—48no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—24no—
270C2×Dic61—22no—
271C2×Dic61—44no—
272C2×Dic61—44no—
273C2×Dic61—48no—
274C2×Dic61—42no—
275C2×Dic61—44no—
276C2×Dic61C22×D412yes—
277C2×Dic61C22×D412yes—
278C2×Dic61C22×D412yes—
279C2×Dic61C22×D412yes—
280D244—22no—
281D244—22no—
282D244—42no—
283C4×D61—48no—
284Dic122—22no—
285Dic122—22no—
286Dic122—42no—
287C8×C61—24no—
288#92418—24no—
289C2×#3142—611nomaximal, Sylow
290C22×C61—28no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—84no—
297C22×C61C2×#12414yes—
298C22×C61C2×#12412yes—
299C22×C61#57512yes—
300C22×C61#57512yes—
301C22×C61C23×D411yescharacteristic
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—84no—
306C4×C61—42no—
307C4×C61—44no—
308C4×C61C2×#12412yes—
309C4×C61C2×#12414yes—
310C4×C61C22×#4111yescharacteristic
311C4×C61C22×#4111yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61—44no—
318C2×#532—611no—
319C22×#125—611no—
320#3142—6116no—
321C2×#491—612no—
322C2×#490—612no—
323C2×#488—612no—
324C2×#488—612no—
325C2×#417—611no—
326C2×#417—611no—
327C2×#417—611no—
328C2×#417D6111yescharacteristic, complemented
329C2×C4×M16—611no—
330C2×C61—24no—
331C2×C61—82no—
332C2×C61#314214yescomplemented
333C2×C61C22×#12411yescharacteristic
334C2×C61C2×#57511yescharacteristic
335C2×C61C2×#57511yescharacteristic
336D61—82no—
337C22×#41—611no—
338C22×#41C2×D6111yescharacteristic
339C22×M16—611no—
340C22×M16C2×D6111yescharacteristic
341#532—618no—
342#532—1224no—
343C2×C4⋊Q8—611no—
344C2×#127—611no—
345C2×#125—612no—
346C2×#125—612no—
347C2×#125—618no—
348C2×#125—1221no—
349C2×#125—1221no—
350C2×#125—1224no—
351C2×#124—611no—
352C2×#124—611no—
353C2×#121—1221no—
354C2×#121—1221no—
355C2×C4×D4—1221no—
356C2×C4×D4—1221no—
357C22×C4⋊C4—611no—
358C22×C4⋊C4—1221no—
359C22×C4⋊C4—1221no—
360C22×C4⋊C4C2×D6111yescharacteristic
361C22×C42—611no—
362#491—6116no—
363#490—6116no—
364#488—6116no—
365#488—6116no—
366#417—618no—
367#417—618no—
368#417—618no—
369#417C2×D6118yescomplemented
370C2×#103—612no—
371C2×#103—612no—
372C2×#103—612no—
373C2×#103C2×D6112yes—
374C2×#102—612no—
375C2×#102—612no—
376C2×#102—612no—
377C2×#102C2×D6112yes—
378C4×M16—618no—
379C2×#97—612no—
380C2×C4×C8—612no—
381C61C2×#314211yescharacteristic, complemented, Sylow
382C2×#41—22no—
383C2×#41—24no—
384C2×#41—24no—
385C2×#41—614no—
386C2×#41—1222no—
387C2×#41—1224no—
388C2×#41—1224no—
389C2×#41C22×D6114yes—
390C22×Q8—611no—
391C22×Q8C22×D6111yescharacteristic
392C22×D4—21no—
393C22×D4—611no—
394C22×D4—1221no—
395C22×D4—1221no—
396C22×D4—1221no—
397C22×D4C22×D6111yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4—611no—
401C23×C4—611no—
402C23×C4—1221no—
403C23×C4—1221no—
404C23×C4C22×D6111yescharacteristic
405C2×M16—24no—
406C2×M16—614no—
407C2×M16—1224no—
408C2×M16C22×D6114yes—
409C22×C8—612no—
410C22×C8C22×D6112yes—
411C4⋊Q8—614no—
412C4⋊Q8—1222no—
413#127—614no—
414#127—1222no—
415#125—618no—
416#125—618no—
417#125—1228no—
418#125—1228no—
419#125—1224no—
420#125—1224no—
421#124—614no—
422#124—614no—
423#124—1222no—
424#124—1222no—
425#121—1228no—
426#121—1228no—
427C4×D4—1228no—
428C4×D4—1228no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—614no—
433C2×C4⋊C4—1228no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1228no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4—1222no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1222no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4C22×D6112yes—
444C2×C4⋊C4C22×D6114yes—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—614no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—614no—
455#103—618no—
456#103—618no—
457#103—618no—
458#103C22×D6118yes—
459#102—618no—
460#102—618no—
461#102—618no—
462#102C22×D6118yes—
463#97—618no—
464C4×C8—618no—
465C24—21no—
466C24—1221no—
467C24D4×D6111yescharacteristic
468#41—28no—
469#41—28no—
470#41—48no—
471#41—1228no—
472#41—1228no—
473#41—2448no—
474C2×Q8—21no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—612no—
478C2×Q8—1221no—
479C2×Q8—1222no—
480C2×Q8—1222no—
481C2×Q8C23×D6112yes—
482C2×D4—24no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—44no—
487C2×D4—612no—
488C2×D4—1224no—
489C2×D4—1224no—
490C2×D4—1224no—
491C2×D4—1221no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—2444no—
495C2×D4—2444no—
496C2×D4—2444no—
497C2×D4C23×D6112yes—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—614no—
509C22×C4—614no—
510C22×C4—1224no—
511C22×C4—1221no—
512C22×C4—1222no—
513C22×C4—1224no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2444no—
517C22×C4—2444no—
518C22×C4#1175822214yes—
519C22×C4D4×D6112yes—
520C22×C4D4×D6111yescharacteristic
521C22×C4#1175822414yes—
522C22×C4C23×D6112yes—
523C22×C4C23×D6111yescharacteristic
524M16—28no—
525M16—1228no—
526C2×C8—24no—
527C2×C8—614no—
528C2×C8—1224no—
529C2×C8C23×D6114yes—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—614no—
533C4⋊C4—1228no—
534C4⋊C4—1228no—
535C4⋊C4—1228no—
536C4⋊C4—1228no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1222no—
541C4⋊C4C23×D6114yes—
542C22⋊C4—1224no—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C42—614no—
547C42—614no—
548C42—612no—
549C42—612no—
550C42—612no—
551C42—612no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—1222no—
559C23—2442no—
560C23—2442no—
561C23—2442no—
562C23#1451976314yes—
563C23C2×D4×D6112yes—
564C23C2×D4×D6111yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—42no—
568Q8—1222no—
569Q8—1222no—
570Q8—2442no—
571D4—22no—
572D4—22no—
573D4—48no—
574D4—42no—
575D4—1222no—
576D4—1222no—
577D4—2448no—
578D4—2448no—
579D4—2448no—
580D4—2442no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1224no—
597C2×C4—2444no—
598C2×C4—2444no—
599C2×C4—2448no—
600C2×C4—2442no—
601C2×C4—2444no—
602C2×C4—2448no—
603C2×C4#1451976314yes—
604C2×C4#1451976318yes—
605C2×C4C2×D4×D6112yes—
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×#1175822412yes—
608C2×C4C2×#1175822414yes—
609C2×C4C2×#1175822212yes—
610C2×C4C2×#1175822214yes—
611C2×C4C24×D6111yescharacteristic
612C8—24no—
613C8—1224no—
614C22—22no—
615C22—22no—
616C22—28no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—84no—
621C22—4884no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22C2×#1451976314yes—
626C22C2×#1451976312yes—
627C22#1684055812yes—
628C22#1684056112yes—
629C22C22×D4×D6111yescharacteristic
630C4—44no—
631C4—42no—
632C4—1222no—
633C4—1222no—
634C4—1222no—
635C4—1222no—
636C4—2444no—
637C4—2442no—
638C4C2×#1451976314yes—
639C4C2×#1451976312yes—
640C4C22×#1175822411yescharacteristic
641C4C22×#1175822211yescharacteristic
642C2—24no—
643C2—82no—
644C2—4882no—
645C2#2145621714yesdirect product factor
646C2C22×#1451976311yescharacteristic
647C2C2×#1684055811yescharacteristic
648C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC2×#16840308C2
hypercenterC2×#417D61
hypocenterC61C2×#3142
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#14519768C2semidirect
3. #21456217C2direct (N × H)
4. C2×#16840129C2semidirect
5. C2×#16840125C2semidirect
6. C2×#16840118C2semidirect
7. C2×#16840115C2semidirect
8. C2×#16839764C2semidirect
9. C2×#16839763C2semidirect
10. C2×#16839761C2semidirect
11. C2×M16×Dic61C2semidirect
12. C2×C61×#417C2semidirect
24. C2×#14519768C22semidirect
35. #16840129C22semidirect
36. #16840125C22semidirect
37. #16840118C22semidirect
38. #16840115C22semidirect
39. #16839764C22semidirect
40. #16839763C22semidirect
41. #16839761C22semidirect
48. M16×Dic61C22semidirect
54. C61×#417C22semidirect
328. C2×#417D61semidirect
332. C2×C61#3142semidirect
369. #417C2×D61semidirect
381. C61C2×#3142semidirect
645. C2#21456217direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403082
C22×#145197682
#214562172
C2×#168401292
C2×#168401252
C2×#168401182
C2×#168401152
C2×#168397642
C2×#168397632
C2×#168397612
C2×M16×Dic612
C2×C61×#4172
C2×#314261