Finite Group Databasebeta

← C2×#21454634 #30544083

Subgroups — C2×#21454634 #30544083

Order 15616.

2810 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic
2C2×#16840230C211yesmaximal, characteristic
3C22×Q8×Dic61C211yesmaximal, characteristic, complemented
4#21454634C2116yesmaximal, direct product factor
5C2×#16839866C212yesmaximal, complemented
6C2×#16839865C212yesmaximal, complemented
7C2×#16839854C212yesmaximal, complemented
8C2×#16839847C212yesmaximal, complemented
9C2×#16839805C211yesmaximal, characteristic, complemented
10C2×#16839764C211yesmaximal, characteristic
11C2×M16×Dic61C211yesmaximal, characteristic, complemented
12C2×C61×#419C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C2×#14519808C4116yescomplemented
15C2×#14519808C2218yes—
16C22×#11758202C2212yes—
17C22×#11758201C2212yes—
18C22×#11758194C2211yescharacteristic
19#16840230—24no—
20#16840230C2218yes—
21C2×Q8×Dic61—21no—
22C2×Q8×Dic61—21no—
23C2×Q8×Dic61—24no—
24C2×Q8×Dic61C2212yes—
25C2×Q8×Dic61C2212yes—
26C2×Q8×Dic61C2218yescomplemented
27C2×#14519742C2211yescharacteristic
28C2×#14519742C2211yescharacteristic
29C2×#14519739C2211yescharacteristic
30C2×#14519739C2211yescharacteristic
31C2×#14519737—21no—
32C2×#14519737—21no—
33C2×#14519734—21no—
34C22×C4⋊Dic61—21no—
35C22×C4⋊Dic61C2211yescharacteristic
36C22×C4×Dic61—21no—
37C22×C4×Dic61C2211yescharacteristic
38C2×C42×D61—21no—
39#16839866C22116yescomplemented
40#16839865C22116yescomplemented
41#16839854C22116yescomplemented
42#16839847C22116yescomplemented
43#16839805C2218yescomplemented
44C2×#14519710C2212yes—
45C2×#14519708C2212yes—
46#16839764C2218yes—
47C2×#14519702C2212yes—
48C2×#14519699C2212yes—
49M16×Dic61C2218yescomplemented
50C2×#14519687C2212yes—
51C2×C8×Dic61C2212yes—
52C23×Q8×C61C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C2×#117×C61C2211yescharacteristic
55C61×#419C2218yescomplemented
56C2×#103×C61C2212yes—
57C2×#103×C61C2212yes—
58C2×#11758222—22no—
59C2×#11758222—24no—
60C2×#11758222C2×C418yes—
61C2×#11758222C2314yes—
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C23×Dic61—21no—
66C23×Dic61D411yescharacteristic
67C23×Dic61D411yescharacteristic
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71#14519808C2×C4164yescomplemented
72C2×#11758202C2×C4116yescomplemented
73C2×#11758202C2318yes—
74C2×#11758201C2×C4116yescomplemented
75C2×#11758201C2318yes—
76C2×#11758194C2×C418yescomplemented
77C2×#11758194C2314yes—
78C22×#92418C2312yes—
79Q8×Dic61—24no—
80Q8×Dic61—24no—
81Q8×Dic61—28no—
82Q8×Dic61—28no—
83Q8×Dic61C2318yes—
84Q8×Dic61C2318yes—
85#14519742—22no—
86#14519742—22no—
87#14519742C2314yes—
88#14519742C2314yes—
89#14519739—22no—
90#14519739—22no—
91#14519739C2314yes—
92#14519739C2314yes—
93#14519737—28no—
94#14519737—28no—
95#14519734—28no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61—22no—
98C2×C4⋊Dic61—22no—
99C2×C4⋊Dic61—22no—
100C2×C4⋊Dic61—28no—
101C2×C4⋊Dic61C2312yes—
102C2×C4⋊Dic61C2314yes—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61—28no—
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D414yes—
115C2×C4×Dic61C2311yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117C2×C4×Dic61C2314yes—
118C42×D61—28no—
119#14519710C2318yes—
120#14519708C2318yes—
121#14519702C2318yes—
122#14519699C2318yes—
123#14519687C2318yes—
124C8×Dic61C2318yes—
125C22×Q8×C61—21no—
126C22×Q8×C61—22no—
127C22×Q8×C61—24no—
128C22×Q8×C61C2×C418yescomplemented
129C22×Q8×C61C2312yes—
130C22×Q8×C61C2314yes—
131C23×C4×C61—21no—
132C23×C4×C61C2311yescharacteristic
133C2×M16×C61—24no—
134C2×M16×C61C2314yes—
135C22×C8×C61C2312yes—
136#117×C61—22no—
137#117×C61C2314yes—
138C2×C4⋊C4×C61C2311yescharacteristic
139C2×C4⋊C4×C61C2311yescharacteristic
140C2×C22⋊C4×C61—21no—
141C2×C42×C61—21no—
142#103×C61C2318yes—
143#103×C61C2318yes—
144C24×C61C2×D411yescharacteristic
145C23×D61—21no—
146C2×Q8×C61—28no—
147C2×Q8×C61—216no—
148C2×Q8×C61—24no—
149C2×Q8×C61—22no—
150C2×Q8×C61—24no—
151C2×Q8×C61—28no—
152C2×Q8×C61C22×C418yes—
153C2×Q8×C61C22×C4116yescomplemented
154C2×Q8×C61C2414yes—
155C22×C4×C61—28no—
156C22×C4×C61—22no—
157C22×C4×C61—24no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61C22×C414yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D412yes—
163C22×C4×C61#4114yes—
164C22×C4×C61C2411yescharacteristic
165C22×C4×C61C2412yes—
166C2×C22⋊D61—28no—
167C2×C22⋊D61—24no—
168C22×Dic61—21no—
169C22×Dic61—24no—
170C22×Dic61—24no—
171C22×Dic61—22no—
172C22×Dic61—42no—
173C22×Dic61—44no—
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D411yescharacteristic
176C22×Dic61C2×D414yes—
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D414yes—
179C22×Dic61C2×D411yescharacteristic
180#11758222—216no—
181#11758222—28no—
182#11758222C22×C4116yes—
183C2×D244—21no—
184C2×D244—22no—
185C2×D244C22×C414yes—
186C2×D244C2412yes—
187C2×C4×D61—28no—
188C2×C4×D61—24no—
189C2×C4×D61—24no—
190C2×C4×D61—24no—
191C2×C4×D61—44no—
192C2×C4×D61—44no—
193C2×Dic122—21no—
194C2×Dic122—22no—
195C2×Dic122C22×C414yes—
196C2×Dic122C2412yes—
197M16×C61—28no—
198C2×C8×C61—24no—
199C2×C8×C61C2414yes—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61C2412yes—
203C4⋊C4×C61C2412yes—
204C22⋊C4×C61—24no—
205C42×C61—24no—
206#11758202C22×C4132yescomplemented
207#11758201C22×C4132yescomplemented
208#11758198—24no—
209#11758198—24no—
210C4⋊Dic61—24no—
211C4⋊Dic61—22no—
212C4⋊Dic61—28no—
213C4⋊Dic61—28no—
214C4⋊Dic61C2414yes—
215#11758196—24no—
216#11758196—24no—
217C4×Dic61—24no—
218C4×Dic61—24no—
219C4×Dic61—28no—
220C4×Dic61—28no—
221C4×Dic61C2×D414yes—
222C4×Dic61C2×D414yes—
223C4×Dic61C2×D412yes—
224C4×Dic61C2×D412yes—
225C4×Dic61C2412yes—
226C4×Dic61C2412yes—
227#11758194C22×C4116yescomplemented
228C2×#92418C22×C418yes—
229C2×#92418C2414yes—
230C23×C61—22no—
231C23×C61—22no—
232C23×C61C4×D414yes—
233C23×C61C22×D412yes—
234C23×C61C22×D411yescharacteristic
235C22×D61—24no—
236C22×D61—22no—
237C22×D61—42no—
238C22×D61—42no—
239Q8×C61—28no—
240Q8×C61—216no—
241Q8×C61—24no—
242Q8×C61C23×C418yes—
243C2×C4×C61—28no—
244C2×C4×C61—216no—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C4×D414yes—
251C2×C4×C61C4×D418yes—
252C2×C4×C61C23×C412yes—
253C2×C4×C61C23×C414yes—
254C2×C4×C61C22×D412yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4112yes—
257C2×C4×C61C2×#4114yes—
258C2×C4×C61C2511yescharacteristic
259C22⋊D61—216no—
260C2×Dic61—24no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—28no—
266C2×Dic61—24no—
267C2×Dic61—44no—
268C2×Dic61—48no—
269C2×Dic61—44no—
270C2×Dic61#13714yescomplemented
271C2×Dic61C22×D412yes—
272C2×Dic61C22×D412yes—
273C2×Dic61C22×D412yes—
274C2×Dic61C22×D412yes—
275D244—24no—
276D244—22no—
277D244C23×C414yes—
278C4×D61—216no—
279C4×D61—48no—
280C4×D61—48no—
281Dic122—24no—
282Dic122—22no—
283Dic122C23×C414yes—
284C8×C61—24no—
285#92418C23×C418yes—
286C2×C4×#137—611nomaximal, Sylow
287C22×C61—28no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61C2×C4×D414yes—
291C22×C61C2×C4×D412yes—
292C22×C61C2×#13712yes—
293C22×C61#57512yes—
294C22×C61C23×D411yescharacteristic
295C2×D61—24no—
296C2×D61—48no—
297C2×D61—42no—
298C2×D61—42no—
299C4×C61—28no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61C2×C4×D412yes—
303C4×C61C2×C4×D414yes—
304C4×C61C24×C411yescharacteristic
305C4×C61C22×#4111yescharacteristic
306Dic61—24no—
307Dic61—22no—
308Dic61—22no—
309Dic61—22no—
310Dic61—44no—
311Dic61C2×#13714yescomplemented
312C22×#137—611no—
313C2×C4×#41—611no—
314C22×C4×Q8—611no—
315C4×#137—6116no—
316C2×#441—612no—
317C2×#440—612no—
318C2×C4×Q16—612no—
319C2×C4×SD16—612no—
320C2×#428—611no—
321C2×#417—611no—
322C2×#419D6111yescharacteristic, complemented
323C2×C4×M16—611no—
324C2×C61—24no—
325C2×C61C4×#13714yescomplemented
326C2×C61C22×C4×D411yescharacteristic
327C2×C61C22×#13711yescharacteristic
328C2×C61C2×#57511yescharacteristic
329D61—44no—
330C22×#41—611no—
331C23×Q8C2×D6111yescharacteristic
332C2×#137—6116no—
333C2×#137—618no—
334C22×Q16—612no—
335C22×SD16—612no—
336C22×M16—611no—
337C22×M16C2×D6111yescharacteristic
338C4×#41—618no—
339C4×#41—1224no—
340C2×C4×Q8—612no—
341C2×C4×Q8—612no—
342C2×C4×Q8—618no—
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—1221no—
346C2×C4×Q8—1224no—
347C2×C4×Q8—1221no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C2×C4×D4—1221no—
351C2×C4×D4—1221no—
352C2×#117—1221no—
353C2×#117C2×D6111yescharacteristic
354C22×C4⋊C4—611no—
355C22×C4⋊C4—1221no—
356C22×C42—611no—
357C22×C42—1221no—
358C22×C42—1221no—
359#441—6116no—
360#440—6116no—
361C4×Q16—6116no—
362C4×SD16—6116no—
363#428—618no—
364C2×#107—612no—
365C2×#106—612no—
366#419C2×D6118yescomplemented
367#417—618no—
368C2×#103—612no—
369C2×#103C2×D6112yes—
370C2×#103C2×D6112yes—
371C2×#102—612no—
372C4×M16—618no—
373C2×#97—612no—
374C2×C4×C8—612no—
375C61C2×C4×#13711yescharacteristic, complemented, Sylow
376C2×#41—618no—
377C2×#41—614no—
378C2×#41—1222no—
379C2×#41—1224no—
380C22×Q8—22no—
381C22×Q8—21no—
382C22×Q8—24no—
383C22×Q8—611no—
384C22×Q8C4×D6118yescomplemented
385C22×Q8C22×D6112yes—
386C22×Q8C22×D6114yes—
387C22×D4—611no—
388C22×D4—1221no—
389C23×C4—21no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—1221no—
393C23×C4—1221no—
394C23×C4—1221no—
395C23×C4—1221no—
396C23×C4C22×D6111yescharacteristic
397#137—6164no—
398C2×Q16—6116no—
399C2×Q16—618no—
400C2×SD16—6116no—
401C2×SD16—618no—
402C2×M16—24no—
403C2×M16—618no—
404C2×M16—614no—
405C2×M16C22×D6114yes—
406C22×C8—612no—
407C22×C8C22×D6112yes—
408C4×Q8—618no—
409C4×Q8—618no—
410C4×Q8—614no—
411C4×Q8—614no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1228no—
415C4×Q8—1228no—
416C4×Q8—1224no—
417C4×Q8—1224no—
418C4×D4—614no—
419C4×D4—614no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1228no—
423C4×D4—1228no—
424#117—22no—
425#117—1228no—
426#117C22×D6114yes—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—614no—
429C2×C4⋊C4—1228no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—21no—
442C2×C42—611no—
443C2×C42—614no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—614no—
448C2×C42—1228no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1228no—
452C2×C42—1222no—
453C2×C42—1222no—
454#107—618no—
455#106—618no—
456#103—618no—
457#103C22×D6118yes—
458#103C22×D6118yes—
459#102—618no—
460#97—618no—
461C4×C8—618no—
462C24—1221no—
463C24D4×D6111yescharacteristic
464#41—6116no—
465#41—1228no—
466#41—12216no—
467C2×Q8—24no—
468C2×Q8—22no—
469C2×Q8—24no—
470C2×Q8—28no—
471C2×Q8—28no—
472C2×Q8—216no—
473C2×Q8—614no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1222no—
477C2×Q8C2×C4×D6118yes—
478C2×Q8C2×C4×D61116yescomplemented
479C2×Q8C23×D6114yes—
480C2×D4—614no—
481C2×D4—612no—
482C2×D4—1224no—
483C2×D4—1228no—
484C2×D4—1221no—
485C2×D4—1222no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—28no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—614no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—614no—
497C22×C4—1224no—
498C22×C4—1224no—
499C22×C4—1224no—
500C22×C4—1221no—
501C22×C4—1224no—
502C22×C4—1228no—
503C22×C4—1224no—
504C22×C4—1222no—
505C22×C4—2444no—
506C22×C4—2444no—
507C22×C4—2442no—
508C22×C4—2444no—
509C22×C4C2×C4×D6114yes—
510C22×C4D4×D6112yes—
511C22×C4D4×D6111yescharacteristic
512C22×C4#1175822414yes—
513C22×C4C23×D6112yes—
514C22×C4C23×D6111yescharacteristic
515Q16—6132no—
516SD16—6132no—
517M16—28no—
518M16—6116no—
519C2×C8—24no—
520C2×C8—618no—
521C2×C8—614no—
522C2×C8C23×D6114yes—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—614no—
526C4⋊C4—1228no—
527C4⋊C4—1228no—
528C4⋊C4—1224no—
529C4⋊C4—1224no—
530C4⋊C4—1224no—
531C4⋊C4—1222no—
532C4⋊C4C23×D6112yes—
533C4⋊C4C23×D6112yes—
534C22⋊C4—24no—
535C22⋊C4—1224no—
536C22⋊C4—1224no—
537C42—24no—
538C42—614no—
539C42—614no—
540C42—612no—
541C42—612no—
542C42—612no—
543C42—612no—
544C42—1224no—
545C42—1224no—
546C42—1228no—
547C42—1228no—
548C23—22no—
549C23—22no—
550C23—1224no—
551C23—1222no—
552C23—2442no—
553C23—2442no—
554C23#1451973714yes—
555C23C2×D4×D6112yes—
556C23C2×D4×D6111yescharacteristic
557Q8—28no—
558Q8—216no—
559Q8—24no—
560Q8—614no—
561Q8—1224no—
562Q8—1222no—
563Q8C22×C4×D6118yes—
564D4—614no—
565D4—12216no—
566D4—1224no—
567D4—1222no—
568C2×C4—28no—
569C2×C4—216no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—614no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—1228no—
581C2×C4—1224no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—12216no—
587C2×C4—1224no—
588C2×C4—2448no—
589C2×C4—2448no—
590C2×C4—2444no—
591C2×C4—2448no—
592C2×C4—2444no—
593C2×C4#1451973714yes—
594C2×C4#1451973718yes—
595C2×C4C22×C4×D6112yes—
596C2×C4C22×C4×D6114yes—
597C2×C4C2×D4×D6112yes—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×#1175822412yes—
600C2×C4C2×#1175822414yes—
601C2×C4C24×D6111yescharacteristic
602C8—24no—
603C8—618no—
604C22—28no—
605C22—22no—
606C22—22no—
607C22—1224no—
608C22—2448no—
609C22—2442no—
610C22—2442no—
611C22C2×#1451973714yes—
612C22C2×#1451973712yes—
613C22#137×D6112yes—
614C22#1684056112yes—
615C22C22×D4×D6111yescharacteristic
616C4—28no—
617C4—42no—
618C4—42no—
619C4—614no—
620C4—1224no—
621C4—1222no—
622C4—1222no—
623C4—1222no—
624C4—2444no—
625C4C2×#1451973714yes—
626C4C2×#1451973712yes—
627C4C23×C4×D6111yescharacteristic
628C4C22×#1175822411yescharacteristic
629C2—24no—
630C2—2444no—
631C2#2145463414yesdirect product factor
632C2C22×#1451973711yescharacteristic
633C2C2×#137×D6111yescharacteristic
634C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC2×C4×C61C25
hypercenterC2×#419D61
hypocenterC61C2×C4×#137
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C22×Q8×Dic61C2semidirect
4. #21454634C2direct (N × H)
5. C2×#16839866C2semidirect
6. C2×#16839865C2semidirect
7. C2×#16839854C2semidirect
8. C2×#16839847C2semidirect
9. C2×#16839805C2semidirect
11. C2×M16×Dic61C2semidirect
12. C2×C61×#419C2semidirect
14. C2×#14519808C4semidirect
26. C2×Q8×Dic61C22semidirect
39. #16839866C22semidirect
40. #16839865C22semidirect
41. #16839854C22semidirect
42. #16839847C22semidirect
43. #16839805C22semidirect
49. M16×Dic61C22semidirect
55. C61×#419C22semidirect
71. #14519808C2×C4semidirect
72. C2×#11758202C2×C4semidirect
74. C2×#11758201C2×C4semidirect
76. C2×#11758194C2×C4semidirect
128. C22×Q8×C61C2×C4semidirect
153. C2×Q8×C61C22×C4semidirect
206. #11758202C22×C4semidirect
207. #11758201C22×C4semidirect
227. #11758194C22×C4semidirect
270. C2×Dic61#137semidirect
311. Dic61C2×#137semidirect
322. C2×#419D61semidirect
325. C2×C61C4×#137semidirect
366. #419C2×D61semidirect
375. C61C2×C4×#137semidirect
384. C22×Q8C4×D61semidirect
478. C2×Q8C2×C4×D61semidirect
631. C2#21454634direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168402302
C22×Q8×Dic612
#214546342
C2×#168398662
C2×#168398652
C2×#168398542
C2×#168398472
C2×#168398052
C2×#168397642
C2×M16×Dic612
C2×C61×#4192
C2×C4×#13761