Finite Group Databasebeta

← C2×#21455630 #30547540

Subgroups — C2×#21455630 #30547540

Order 15616.

2358 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519776C211yesmaximal, characteristic, complemented
2C2×#41×Dic61C211yesmaximal, characteristic
3#21455630C2116yesmaximal, direct product factor
4C2×#16840055C212yesmaximal, complemented
5C2×#16840045C212yesmaximal, complemented
6C2×#16840050C212yesmaximal, complemented
7C2×#16840040C212yesmaximal, complemented
8C2×#16839804C211yesmaximal, characteristic, complemented
9C2×#16839802C211yesmaximal, characteristic, complemented
10C2×#16839785C211yesmaximal, characteristic, complemented
11C2×#16839763C211yesmaximal, characteristic
12C2×C61×#417C211yesmaximal, characteristic
13C23×Dic122—21no—
14C22×#11758194C2211yescharacteristic
15C2×#14519776—21no—
16C2×#14519776—21no—
17C2×#14519776—24no—
18C2×#14519776C2212yes—
19C2×#14519776C2212yes—
20C2×#14519776C2218yescomplemented
21#41×Dic61—24no—
22#41×Dic61C2218yes—
23C2×Q8×Dic61C2211yescharacteristic
24C2×Q8×Dic61C2211yescharacteristic
25C2×D4×Dic61—21no—
26C2×D4×Dic61—21no—
27C2×D4×Dic61C2211yescharacteristic
28C2×D4×Dic61C2211yescharacteristic
29C2×#14519736—21no—
30C2×#14519736C2211yescharacteristic
31C22×C4⋊Dic61C2211yescharacteristic
32C22×#11758196—21no—
33C22×C4×Dic61—21no—
34C22×C4×Dic61C2211yescharacteristic
35#16840055C22116yescomplemented
36#16840050C22116yescomplemented
37#16840045C22116yescomplemented
38#16840040C22116yescomplemented
39#16839804C2218yescomplemented
40#16839802C2218yescomplemented
41C2×#14519711C2212yes—
42C2×#14519710C2212yes—
43C2×#14519709C2212yes—
44C2×#14519708C2212yes—
45#16839785C2218yescomplemented
46C2×#14519706C2212yes—
47C2×#14519706C2212yes—
48#16839763C2218yes—
49C2×#14519701C2212yes—
50C2×#14519698C2212yes—
51C22×#41×C61C2211yescharacteristic
52C22×M16×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C61×#417C2218yes—
55C2×#103×C61C2212yes—
56C2×#102×C61C2212yes—
57C22×Dic122—22no—
58C22×Dic122—22no—
59C22×Dic122—28no—
60C22×Dic122—48no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63C23×Dic61—21no—
64C23×Dic61D411yescharacteristic
65C23×Dic61D411yescharacteristic
66C2×#11758194—24no—
67C2×#11758194C2314yes—
68C22×#92418C2312yes—
69#14519776—24no—
70#14519776—24no—
71#14519776—28no—
72#14519776—28no—
73#14519776C2318yes—
74#14519776C2318yes—
75Q8×Dic61—22no—
76Q8×Dic61—22no—
77Q8×Dic61C2314yes—
78Q8×Dic61C2314yes—
79D4×Dic61—28no—
80D4×Dic61—28no—
81D4×Dic61—22no—
82D4×Dic61—22no—
83D4×Dic61C2314yes—
84D4×Dic61C2314yes—
85#14519736—22no—
86#14519736—28no—
87#14519736C2314yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—22no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2314yes—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×#11758196—22no—
97C2×#11758196—22no—
98C2×#11758196—28no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—28no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D414yes—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2314yes—
112#14519711C2318yes—
113#14519710C2318yes—
114#14519709C2318yes—
115#14519708C2318yes—
116#14519706C2318yes—
117#14519706C2318yes—
118#14519701C2318yes—
119#14519698C2318yes—
120C2×#41×C61—22no—
121C2×#41×C61—24no—
122C2×#41×C61Q818yescomplemented
123C2×#41×C61C2314yes—
124C22×Q8×C61C2311yescharacteristic
125C22×D4×C61—21no—
126C22×D4×C61C2311yescharacteristic
127C23×C4×C61—21no—
128C23×C4×C61—21no—
129C23×C4×C61C2311yescharacteristic
130C2×M16×C61—24no—
131C2×M16×C61C2314yes—
132C22×C8×C61C2312yes—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61—22no—
135C2×C4⋊C4×C61C2312yes—
136C2×C4⋊C4×C61C2314yes—
137#103×C61C2318yes—
138#102×C61C2318yes—
139C24×C61—21no—
140C24×C61C2×D411yescharacteristic
141#41×C61—216no—
142#41×C61—28no—
143#41×C61C2×Q8116yescomplemented
144C2×Q8×C61—21no—
145C2×Q8×C61—22no—
146C2×Q8×C61C2×Q814yescomplemented
147C2×Q8×C61C2412yes—
148C2×D4×C61—28no—
149C2×D4×C61—24no—
150C2×D4×C61—21no—
151C2×D4×C61—22no—
152C2×D4×C61C2×Q814yescomplemented
153C2×D4×C61C2412yes—
154C22×C4×C61—21no—
155C22×C4×C61—28no—
156C22×C4×C61—24no—
157C22×C4×C61—24no—
158C22×C4×C61—22no—
159C22×C4×C61—44no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×Q814yes—
163C22×C4×C61#4114yes—
164C22×C4×C61C2411yescharacteristic
165C22×C4×C61C2412yes—
166C22×Dic61—24no—
167C22×Dic61—24no—
168C22×Dic61—24no—
169C22×Dic61—22no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—24no—
173C22×Dic61—24no—
174C22×Dic61—44no—
175C22×Dic61—44no—
176C22×Dic61—44no—
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D414yes—
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D414yes—
181C22×Dic61C2×D411yescharacteristic
182C22×Dic61C2×D411yescharacteristic
183C2×Dic122—28no—
184C2×Dic122—28no—
185C2×Dic122—48no—
186C2×Dic122—432no—
187C2×Dic122—48no—
188M16×C61—28no—
189C2×C8×C61—24no—
190C2×C8×C61C2414yes—
191C4⋊C4×C61—24no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61C2414yes—
194C22⋊Dic61—24no—
195C22⋊Dic61—24no—
196C22⋊Dic61—24no—
197C4⋊Dic61—24no—
198C4⋊Dic61—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—24no—
202C4⋊Dic61—24no—
203C4⋊Dic61C2414yes—
204C4⋊Dic61C2412yes—
205C4⋊Dic61C2412yes—
206#11758196—28no—
207#11758196—28no—
208C4×Dic61—24no—
209C4×Dic61—24no—
210C4×Dic61—24no—
211C4×Dic61C2×D414yes—
212C4×Dic61C2×D414yes—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2412yes—
216C4×Dic61C2412yes—
217#11758194—28no—
218C2×#92418—24no—
219C2×#92418C2414yes—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—24no—
223C23×C61—22no—
224C23×C61—42no—
225C23×C61—42no—
226C23×C61C22⋊Q814yes—
227C23×C61C22×D412yes—
228C23×C61C22×D411yescharacteristic
229Q8×C61—24no—
230Q8×C61—22no—
231Q8×C61C22×Q814yescomplemented
232D4×C61—216no—
233D4×C61—24no—
234D4×C61—22no—
235D4×C61C22×Q814yescomplemented
236C2×C4×C61—24no—
237C2×C4×C61—22no—
238C2×C4×C61—216no—
239C2×C4×C61—24no—
240C2×C4×C61—48no—
241C2×C4×C61—44no—
242C2×C4×C61C22⋊Q814yes—
243C2×C4×C61C22⋊Q818yes—
244C2×C4×C61C22×D412yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×Q812yes—
247C2×C4×C61C22×Q814yes—
248C2×C4×C61C2×#4112yes—
249C2×C4×C61C2×#4114yes—
250C2×C4×C61C2511yescharacteristic
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—28no—
254C2×Dic61—28no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—48no—
259C2×Dic61—44no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—48no—
263C2×Dic61—48no—
264C2×Dic61AGL(1,ℤ/8)14yes—
265C2×Dic61#13714yes—
266C2×Dic61C22×D412yes—
267C2×Dic61C22×D412yes—
268C2×Dic61C22×D412yes—
269C2×Dic61C22×D412yes—
270Dic122—416no—
271Dic122—416no—
272C8×C61—24no—
273#92418—24no—
274C2×#3079—611nomaximal, Sylow
275C22×C61—28no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—24no—
279C22×C61—48no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61C2×C22⋊Q814yes—
283C22×C61C2×C22⋊Q812yes—
284C22×C61C2×AGL(1,ℤ/8)12yes—
285C22×C61C2×#13712yes—
286C22×C61C23×D411yescharacteristic
287C4×C61—24no—
288C4×C61—44no—
289C4×C61C2×C22⋊Q814yes—
290C4×C61C2×C22⋊Q812yes—
291C4×C61C23×Q811yescharacteristic
292C4×C61C22×#4111yescharacteristic
293Dic61—22no—
294Dic61—22no—
295Dic61—44no—
296Dic61—42no—
297Dic61—42no—
298Dic61C2×AGL(1,ℤ/8)14yes—
299Dic61C2×#13714yes—
300C22×C4⋊Q8—611no—
301C2×C4×#41—611no—
302#3079—6116no—
303C2×#477—612no—
304C2×#475—612no—
305C2×#476—612no—
306C2×#474—612no—
307C2×#428—611no—
308C2×#428—611no—
309C2×#423—611no—
310C2×#417—611no—
311C2×#417D6111yescharacteristic
312C2×C61—24no—
313C2×C61—44no—
314C2×C61#307914yescomplemented
315C2×C61C22×C22⋊Q811yescharacteristic
316C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
317C2×C61C22×#13711yescharacteristic
318C22×#41C2×D6111yescharacteristic
319C23×Q8—1221no—
320C22×M16—611no—
321C22×M16C2×D6111yescharacteristic
322C2×C4⋊Q8—612no—
323C2×C4⋊Q8—612no—
324C2×C4⋊Q8—618no—
325C2×C4⋊Q8—1221no—
326C2×C4⋊Q8—1224no—
327C2×C4⋊Q8—1221no—
328C4×#41—618no—
329C4×#41—1224no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332C2×C4×D4—611no—
333C2×C4×D4—611no—
334C2×C4×D4—1221no—
335C2×C4×D4—1221no—
336C2×#117—611no—
337C2×#117—1221no—
338C22×C4⋊C4—611no—
339C22×C4⋊C4—1221no—
340C22×C4⋊C4C2×D6111yescharacteristic
341C22×C42—611no—
342C22×C42—1221no—
343#477—6116no—
344#476—6116no—
345#475—6116no—
346#474—6116no—
347#428—618no—
348#428—618no—
349C2×#107—612no—
350C2×#107—612no—
351C2×#106—612no—
352C2×#106—612no—
353#423—618no—
354C2×C4⋊C8—612no—
355C2×C4⋊C8—612no—
356#417—618no—
357#417C2×D6118yes—
358C2×#103—612no—
359C2×#103C2×D6112yes—
360C2×#102—612no—
361C2×#102C2×D6112yes—
362C61C2×#307911yescharacteristic, complemented, Sylow
363C2×#41—22no—
364C2×#41—24no—
365C2×#41Dic12218yescomplemented
366C2×#41C22×D6114yes—
367C22×Q8—1228no—
368C22×Q8—1222no—
369C22×Q8—1222no—
370C22×Q8—2448no—
371C22×Q8C22×D6111yescharacteristic
372C22×D4—21no—
373C22×D4C22×D6111yescharacteristic
374C23×C4—21no—
375C23×C4—21no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—1221no—
379C23×C4—1221no—
380C23×C4—1221no—
381C23×C4C22×D6111yescharacteristic
382C2×M16—24no—
383C2×M16—614no—
384C2×M16—1224no—
385C2×M16C22×D6114yes—
386C22×C8—612no—
387C22×C8C22×D6112yes—
388C4⋊Q8—618no—
389C4⋊Q8—618no—
390C4⋊Q8—1228no—
391C4⋊Q8—1228no—
392C4⋊Q8—1224no—
393C4⋊Q8—1224no—
394C4×Q8—614no—
395C4×Q8—614no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×D4—614no—
399C4×D4—614no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1228no—
403C4×D4—1228no—
404#117—614no—
405#117—1228no—
406#117—1222no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1228no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6112yes—
421C2×C4⋊C4C22×D6114yes—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426C2×C42—614no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—614no—
431C2×C42—1228no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435#107—618no—
436#107—618no—
437#106—618no—
438#106—618no—
439C4⋊C8—618no—
440C4⋊C8—618no—
441#103—618no—
442#103C22×D6118yes—
443#102—618no—
444#102C22×D6118yes—
445C24—21no—
446C24D4×D6111yescharacteristic
447#41—28no—
448#41—216no—
449#41C2×Dic122116yescomplemented
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—1228no—
453C2×Q8—1228no—
454C2×Q8—2448no—
455C2×Q8—24432no—
456C2×Q8—2448no—
457C2×Q8C2×Dic12214yescomplemented
458C2×Q8C23×D6112yes—
459C2×D4—24no—
460C2×D4—28no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4C2×Dic12214yescomplemented
464C2×D4C23×D6112yes—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1224no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—2444no—
486C22×C4—2444no—
487C22×C4—2444no—
488C22×C4C2×Dic12214yes—
489C22×C4D4×D6112yes—
490C22×C4D4×D6111yescharacteristic
491C22×C4#1175822414yes—
492C22×C4C23×D6112yes—
493C22×C4C23×D6111yescharacteristic
494M16—28no—
495M16—1228no—
496C2×C8—24no—
497C2×C8—614no—
498C2×C8—1224no—
499C2×C8C23×D6114yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—614no—
503C4⋊C4—612no—
504C4⋊C4—612no—
505C4⋊C4—1228no—
506C4⋊C4—1228no—
507C4⋊C4—1224no—
508C4⋊C4—1224no—
509C4⋊C4—1224no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4C23×D6114yes—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—1224no—
517C42—614no—
518C42—614no—
519C42—612no—
520C42—612no—
521C42—612no—
522C42—612no—
523C42—1224no—
524C42—1224no—
525C42—1224no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23#1451975414yes—
533C23C2×D4×D6112yes—
534C23C2×D4×D6111yescharacteristic
535Q8—24no—
536Q8—22no—
537Q8—24416no—
538Q8—24416no—
539Q8C22×Dic12214yescomplemented
540D4—216no—
541D4—24no—
542D4—22no—
543D4C22×Dic12214yescomplemented
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—216no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—614no—
551C2×C4—614no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1228no—
557C2×C4—1228no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1224no—
563C2×C4—2448no—
564C2×C4—2448no—
565C2×C4—2448no—
566C2×C4—2442no—
567C2×C4—2444no—
568C2×C4—2442no—
569C2×C4#1451975414yes—
570C2×C4#1451975418yes—
571C2×C4C2×D4×D6112yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C22×Dic12212yes—
574C2×C4C22×Dic12214yes—
575C2×C4C2×#1175822412yes—
576C2×C4C2×#1175822414yes—
577C2×C4C24×D6111yescharacteristic
578C8—24no—
579C8—1224no—
580C22—28no—
581C22—22no—
582C22—22no—
583C22—24no—
584C22—48no—
585C22—42no—
586C22—42no—
587C22C2×#1451975414yes—
588C22C2×#1451975412yes—
589C22AGL(1,ℤ/8)×D6112yes—
590C22#137×D6112yes—
591C22C22×D4×D6111yescharacteristic
592C4—24no—
593C4—44no—
594C4—614no—
595C4—614no—
596C4—1222no—
597C4—1222no—
598C4—2442no—
599C4—2444no—
600C4—2442no—
601C4C2×#1451975414yes—
602C4C2×#1451975412yes—
603C4C23×Dic12211yescharacteristic
604C4C22×#1175822411yescharacteristic
605C2—24no—
606C2—44no—
607C2#2145563014yesdirect product factor
608C2C22×#1451975411yescharacteristic
609C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
610C2C2×#137×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenterC2×#417D61
hypocenterC61C2×#3079
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
1. C22×#14519776C2semidirect
3. #21455630C2direct (N × H)
4. C2×#16840055C2semidirect
5. C2×#16840045C2semidirect
6. C2×#16840050C2semidirect
7. C2×#16840040C2semidirect
8. C2×#16839804C2semidirect
9. C2×#16839802C2semidirect
10. C2×#16839785C2semidirect
20. C2×#14519776C22semidirect
35. #16840055C22semidirect
36. #16840050C22semidirect
37. #16840045C22semidirect
38. #16840040C22semidirect
39. #16839804C22semidirect
40. #16839802C22semidirect
45. #16839785C22semidirect
122. C2×#41×C61Q8semidirect
143. #41×C61C2×Q8semidirect
146. C2×Q8×C61C2×Q8semidirect
152. C2×D4×C61C2×Q8semidirect
231. Q8×C61C22×Q8semidirect
235. D4×C61C22×Q8semidirect
314. C2×C61#3079semidirect
362. C61C2×#3079semidirect
365. C2×#41Dic122semidirect
449. #41C2×Dic122semidirect
457. C2×Q8C2×Dic122semidirect
463. C2×D4C2×Dic122semidirect
539. Q8C22×Dic122semidirect
543. D4C22×Dic122semidirect
607. C2#21455630direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197762
C2×#41×Dic612
#214556302
C2×#168400552
C2×#168400452
C2×#168400502
C2×#168400402
C2×#168398042
C2×#168398022
C2×#168397852
C2×#168397632
C2×C61×#4172
C2×#307961