Finite Group Databasebeta

← C2×#21460215 #30587805

Subgroups — C2×#21460215 #30587805

Order 15616.

7810 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460215C2132yesmaximal, direct product factor
2C2×#16840510C211yesmaximal, characteristic, complemented
3C2×#16840506C212yesmaximal, complemented
4C2×#16840466C214yesmaximal, complemented
5C2×#16840430C214yesmaximal, complemented
6C2×#16840410C214yesmaximal, complemented
7C2×#16840364C214yesmaximal, complemented
8C2×#16840342C214yesmaximal, complemented
9C2×#16840339C214yesmaximal, complemented
10C2×#16840318C211yesmaximal, characteristic, complemented
11C2×#16840311C212yesmaximal, complemented
12C2×C61×#562C211yesmaximal, characteristic, complemented
13C22×#11758226C2212yes—
14C22×#11758222C2214yes—
15C22×#11758224C2214yes—
16C22×#11758224C2212yes—
17#16840510C22116yescomplemented
18#16840506C22132yescomplemented
19#16840466C22164yescomplemented
20#16840430C22164yescomplemented
21#16840410C22164yescomplemented
22#16840364C22164yescomplemented
23#16840342C22164yescomplemented
24#16840339C22164yescomplemented
25#16840318C22116yescomplemented
26#16840311C22132yescomplemented
27C2×#14519776C2212yes—
28C2×#14519775C2212yes—
29C2×#14519773C2212yes—
30C2×#14519773C2214yes—
31C2×#14519769C2211yescharacteristic
32C2×#14519767C2214yes—
33C2×#14519765C2212yes—
34C2×#14519762C2214yes—
35C2×#14519761C2214yes—
36C2×#14519759C2214yes—
37C2×#14519755C2214yes—
38C2×#14519755C2214yes—
39C2×#14519755C2214yes—
40C2×#14519754C2214yes—
41C2×#14519754C2214yes—
42C2×#14519753C2214yes—
43C2×#14519752C2214yes—
44C2×#14519752C2214yes—
45C2×#14519751C2214yes—
46C2×#14519750C2214yes—
47C2×#14519750C2214yes—
48C2×#14519750C2214yes—
49C2×#14519750C2214yes—
50C2×#14519750C2214yes—
51C2×#14519749C2214yes—
52C2×#14519749C2214yes—
53C2×#14519749C2214yes—
54C2×#14519742C2214yes—
55C2×C4×C22⋊D61C2214yes—
56C2×#14519737C2214yes—
57C2×#14519737C2214yes—
58C2×#14519737C2214yes—
59C2×#14519735C2212yes—
60C22×#11758196C2214yes—
61C2×C4⋊C4×D61C2214yes—
62C2×C42×D61C2211yescharacteristic
63C61×#562C22116yescomplemented
64C2×#125×C61C2211yescharacteristic
65C2×#124×C61C2212yes—
66C2×#123×C61C2214yes—
67C2×C22⋊Q8×C61C2214yes—
68C2×#121×C61C2214yes—
69C2×#11758226—48no—
70C2×#11758226C23116yes—
71C2×#11758222—416no—
72C2×#11758222C23132yes—
73C2×#11758224—416no—
74C2×#11758224—48no—
75C2×#11758224C23116yes—
76C2×#11758224C23132yes—
77C22×Dic122C2314yes—
78C22×Dic122C2312yes—
79C22×D244C2314yes—
80C22×D244C2312yes—
81C22×D244C2314yes—
82C22×C22⋊D61C2314yes—
83C22×C22⋊D61C2314yes—
84C22×C22⋊D61C2314yes—
85C22×C22⋊D61C2314yes—
86C22×C22⋊D61C2314yes—
87C23×Dic61C2314yes—
88C23×Dic61C2314yes—
89C22×C4×D61C2311yescharacteristic
90C22×C4×D61C2312yes—
91C22×C4×D61C2314yes—
92C22×C4×D61C2314yes—
93C22×C4×D61C2314yes—
94#14519776C23116yes—
95#14519775C23116yes—
96#14519773C23116yes—
97#14519773C23132yes—
98#14519769C2318yes—
99#14519767C23132yes—
100#14519765C23116yes—
101#14519762C23132yes—
102#14519761C23132yes—
103#14519759C23132yes—
104#14519755C23132yes—
105#14519755C23132yes—
106#14519755C23132yes—
107#14519754C23132yes—
108#14519754C23132yes—
109#14519753C23132yes—
110#14519752C23132yes—
111#14519752C23132yes—
112#14519751C23132yes—
113#14519750C23132yes—
114#14519750C23132yes—
115#14519750C23132yes—
116#14519750C23132yes—
117#14519750C23132yes—
118#14519749C23132yes—
119#14519749C23132yes—
120#14519749C23132yes—
121#14519742C23132yes—
122C4×C22⋊D61C23132yes—
123#14519737C23132yes—
124#14519737C23132yes—
125#14519737C23132yes—
126#14519735C23116yes—
127C2×C4⋊Dic61C2314yes—
128C2×C4⋊Dic61C2312yes—
129C2×#11758196C23132yes—
130C2×#11758196C2314yes—
131C2×#11758196C2314yes—
132C2×#11758196C2314yes—
133C2×#11758196C2314yes—
134C2×#11758196C2314yes—
135C2×#11758196C2314yes—
136C4⋊C4×D61C23132yes—
137C2×C22⋊Dic61C2314yes—
138C2×#11758198C2314yes—
139C2×#11758198C2314yes—
140C2×#11758198C2314yes—
141C2×#11758198C2314yes—
142C2×#11758198C2314yes—
143C2×C4×Dic61C2311yescharacteristic
144C2×C4×Dic61C2312yes—
145C2×C4×Dic61C2314yes—
146C42×D61C2318yes—
147C22×Q8×C61C2312yes—
148C22×D4×C61C2314yes—
149C22×D4×C61C2312yes—
150C23×C4×C61C2314yes—
151#125×C61C2318yes—
152#124×C61C23116yes—
153#123×C61C23132yes—
154C22⋊Q8×C61C23132yes—
155#121×C61C23132yes—
156C2×C4⋊C4×C61C2314yes—
157C2×C4⋊C4×C61C2312yes—
158C2×C4⋊C4×C61C2314yes—
159C2×C22⋊C4×C61C2314yes—
160C2×C22⋊C4×C61C2314yes—
161C2×C22⋊C4×C61C2314yes—
162C2×C42×C61C2311yescharacteristic
163C24×C61C2414yes—
164C23×D61C2414yes—
165C23×D61C2411yescharacteristic
166C2×Q8×C61—44no—
167C2×Q8×C61C2418yes—
168C2×D4×C61—48no—
169C2×D4×C61—44no—
170C2×D4×C61C24116yes—
171C2×D4×C61C2418yes—
172C22×C4×C61—48no—
173C22×C4×C61C2414yes—
174C22×C4×C61C2412yes—
175C22×C4×C61C2414yes—
176C22×C4×C61C2411yescharacteristic
177C22×C4×C61C24116yes—
178C2×C22⋊D61—216no—
179C2×C22⋊D61—216no—
180C2×C22⋊D61—48no—
181C2×C22⋊D61—48no—
182C2×C22⋊D61—48no—
183C2×C22⋊D61C24116yes—
184C2×C22⋊D61C24116yes—
185C2×C22⋊D61C24116yes—
186C2×C22⋊D61C24116yes—
187C2×C22⋊D61C24116yes—
188C22×Dic61—216no—
189C22×Dic61—48no—
190C22×Dic61C2414yes—
191C22×Dic61C2414yes—
192C22×Dic61C24116yes—
193C22×Dic61C24116yes—
194C22×Dic61C2412yes—
195C22×Dic61C2411yescharacteristic
196#11758226—432no—
197#11758224—464no—
198#11758224—432no—
199#11758222—464no—
200C2×D244—48no—
201C2×D244—44no—
202C2×D244—48no—
203C2×D244C24116yes—
204C2×D244C24116yes—
205C2×D244C2418yes—
206C2×C4×D61—28no—
207C2×C4×D61—216no—
208C2×C4×D61—24no—
209C2×C4×D61—48no—
210C2×C4×D61—48no—
211C2×C4×D61C2418yes—
212C2×C4×D61C24116yes—
213C2×C4×D61C2414yes—
214C2×C4×D61C24116yes—
215C2×C4×D61C24116yes—
216C2×Dic122—216no—
217C2×Dic122—44no—
218C2×Dic122C24116yes—
219C2×Dic122C2418yes—
220C4⋊C4×C61C24116yes—
221C4⋊C4×C61C24116yes—
222C4⋊C4×C61C2418yes—
223C22⋊C4×C61C24116yes—
224C22⋊C4×C61C24116yes—
225C22⋊C4×C61C24116yes—
226C42×C61C2414yes—
227C22⋊Dic61C24116yes—
228#11758198C24116yes—
229#11758198C24116yes—
230#11758198C24116yes—
231#11758198C24116yes—
232#11758198C24116yes—
233C4⋊Dic61C24116yes—
234C4⋊Dic61C2418yes—
235#11758196C24116yes—
236#11758196C24116yes—
237#11758196C24116yes—
238#11758196C24116yes—
239#11758196C24116yes—
240#11758196C24116yes—
241C4×Dic61C2414yes—
242C4×Dic61C2418yes—
243C4×Dic61C24116yes—
244C23×C61—28no—
245C23×C61—28no—
246C23×C61—28no—
247C23×C61C2518yes—
248C23×C61C2511yescharacteristic
249C22×D61—28no—
250C22×D61—24no—
251C22×D61—28no—
252C22×D61—22no—
253C22×D61—28no—
254C22×D61C2518yes—
255C22×D61C2512yes—
256Q8×C61—48no—
257D4×C61—416no—
258D4×C61—48no—
259C2×C4×C61—28no—
260C2×C4×C61—24no—
261C2×C4×C61—28no—
262C2×C4×C61—22no—
263C2×C4×C61—416no—
264C2×C4×C61C2514yes—
265C2×C4×C61C2518yes—
266C2×C4×C61C2512yes—
267C2×C4×C61C2518yes—
268C22⋊D61—232no—
269C22⋊D61—232no—
270C22⋊D61—416no—
271C22⋊D61—416no—
272C22⋊D61—416no—
273C2×Dic61—28no—
274C2×Dic61—28no—
275C2×Dic61—232no—
276C2×Dic61—416no—
277C2×Dic61H(5,𝔽2)18yes—
278C2×Dic612−1+414yes—
279C2×Dic61C2518yes—
280C2×Dic61C2518yes—
281C2×Dic61C2514yes—
282C2×Dic61C2512yes—
283D244—416no—
284D244—48no—
285D244—416no—
286C4×D61—232no—
287C4×D61—216no—
288C4×D61—28no—
289C4×D61—416no—
290C4×D61—416no—
291Dic122—232no—
292Dic122—48no—
293C2×#3548—611nomaximal, Sylow
294C22×C61—28no—
295C22×C61—28no—
296C22×C61—28no—
297C22×C61—416no—
298C22×C61C2×H(5,𝔽2)14yes—
299C22×C61C2×2−1+412yes—
300C22×C61C2611yescharacteristic
301C2×D61—28no—
302C2×D61—24no—
303C2×D61—28no—
304C2×D61—22no—
305C2×D61—28no—
306C2×D61—416no—
307C2×D61—44no—
308C4×C61—28no—
309C4×C61—24no—
310C4×C61—28no—
311C4×C61—22no—
312Dic61—28no—
313Dic61—28no—
314Dic61C2×H(5,𝔽2)18yes—
315Dic61C2×2−1+414yes—
316#3548—6132no—
317C2×#562—611no—
318C2×#562—612no—
319C2×#562D6111yescharacteristic, complemented
320C2×#554—614no—
321C2×#550—614no—
322C2×#547—614no—
323C2×#540—614no—
324C2×#537—614no—
325C2×#537—614no—
326C2×#533—611no—
327C2×#532—612no—
328C2×C61—48no—
329C2×C61#354814yescomplemented
330C2×C61C22×H(5,𝔽2)12yes—
331C2×C61C22×2−1+411yescharacteristic
332D61—48no—
333D61—42no—
334C22×#41—612no—
335C22×#41—612no—
336C22×#41—614no—
337C22×#41—614no—
338#562—6116no—
339#562—6132no—
340#562C2×D61116yescomplemented
341#554—6164no—
342#550—6164no—
343#547—6164no—
344#540—6164no—
345#537—6164no—
346#537—6164no—
347#533—6116no—
348#532—6132no—
349C2×C4⋊Q8—612no—
350C2×C4⋊Q8—612no—
351C2×#127—612no—
352C2×#127—614no—
353C2×#125—614no—
354C2×#125—611no—
355C2×#125C2×D6111yescharacteristic
356C2×#124—612no—
357C2×#124C2×D6112yes—
358C2×#123—614no—
359C2×#123—614no—
360C2×#123—614no—
361C2×#123C2×D6114yes—
362C2×C22⋊Q8—614no—
363C2×C22⋊Q8—614no—
364C2×C22⋊Q8—614no—
365C2×C22⋊Q8—614no—
366C2×C22⋊Q8—614no—
367C2×C22⋊Q8—614no—
368C2×C22⋊Q8C2×D6114yes—
369C2×#121—614no—
370C2×#121—614no—
371C2×#121—614no—
372C2×#121—614no—
373C2×#121—614no—
374C2×#121—614no—
375C2×#121—614no—
376C2×#121—614no—
377C2×#121—614no—
378C2×#121—614no—
379C2×#121—614no—
380C2×#121C2×D6114yes—
381C2×C4×Q8—614no—
382C2×C4×D4—614no—
383C2×C4×D4—614no—
384C2×C4×D4—614no—
385C2×C4×D4—614no—
386C2×#117—612no—
387C22×C4⋊C4—614no—
388C22×C4⋊C4—614no—
389C22×C42—611no—
390C61C2×#354811yescharacteristic, complemented, Sylow
391C2×#41—6116no—
392C2×#41—6116no—
393C2×#41—6132no—
394C2×#41—6132no—
395C2×#41—24416no—
396C2×#41—2448no—
397C2×#41—2448no—
398C2×#41—24416no—
399C22×Q8—614no—
400C22×Q8—612no—
401C22×Q8C22×D6112yes—
402C22×D4—614no—
403C22×D4—614no—
404C22×D4—614no—
405C22×D4—614no—
406C22×D4—614no—
407C22×D4—614no—
408C22×D4—614no—
409C22×D4—612no—
410C22×D4C22×D6114yes—
411C22×D4C22×D6112yes—
412C23×C4—614no—
413C23×C4—614no—
414C23×C4—614no—
415C23×C4—611no—
416C23×C4—612no—
417C23×C4—614no—
418C23×C4—614no—
419C23×C4C22×D6114yes—
420C4⋊Q8—6116no—
421C4⋊Q8—6116no—
422#127—6116no—
423#127—6132no—
424#125—6132no—
425#125—618no—
426#125C22×D6118yes—
427#124—6116no—
428#124C22×D61116yes—
429#123—6132no—
430#123—6132no—
431#123—6132no—
432#123C22×D61132yes—
433C22⋊Q8—6132no—
434C22⋊Q8—6132no—
435C22⋊Q8—6132no—
436C22⋊Q8—6132no—
437C22⋊Q8—6132no—
438C22⋊Q8—6132no—
439C22⋊Q8C22×D61132yes—
440#121—6132no—
441#121—6132no—
442#121—6132no—
443#121—6132no—
444#121—6132no—
445#121—6132no—
446#121—6132no—
447#121—6132no—
448#121—6132no—
449#121—6132no—
450#121—6132no—
451#121C22×D61132yes—
452C4×Q8—6132no—
453C4×D4—6132no—
454C4×D4—6132no—
455C4×D4—6132no—
456C4×D4—6132no—
457#117—6116no—
458C2×C4⋊C4—614no—
459C2×C4⋊C4—614no—
460C2×C4⋊C4—614no—
461C2×C4⋊C4—614no—
462C2×C4⋊C4—614no—
463C2×C4⋊C4—6132no—
464C2×C4⋊C4—612no—
465C2×C4⋊C4—614no—
466C2×C4⋊C4—614no—
467C2×C4⋊C4—6132no—
468C2×C4⋊C4C22×D6114yes—
469C2×C4⋊C4C22×D6112yes—
470C2×C4⋊C4C22×D6114yes—
471C2×C22⋊C4—614no—
472C2×C22⋊C4—614no—
473C2×C22⋊C4—614no—
474C2×C22⋊C4—614no—
475C2×C22⋊C4—614no—
476C2×C22⋊C4—614no—
477C2×C22⋊C4C22×D6114yes—
478C2×C22⋊C4C22×D6114yes—
479C2×C22⋊C4C22×D6114yes—
480C2×C42—614no—
481C2×C42—611no—
482C2×C42—612no—
483C2×C42—618no—
484C2×C42C22×D6111yescharacteristic
485C24—611no—
486C24—614no—
487C24C23×D6114yes—
488#41—24464no—
489#41—24432no—
490#41—24432no—
491#41—24464no—
492C2×Q8—44no—
493C2×Q8—6116no—
494C2×Q8—618no—
495C2×Q8—12216no—
496C2×Q8—2444no—
497C2×Q8C23×D6118yes—
498C2×D4—48no—
499C2×D4—44no—
500C2×D4—6116no—
501C2×D4—6116no—
502C2×D4—6116no—
503C2×D4—6116no—
504C2×D4—6116no—
505C2×D4—6116no—
506C2×D4—6116no—
507C2×D4—618no—
508C2×D4—12216no—
509C2×D4—12216no—
510C2×D4—2448no—
511C2×D4—2448no—
512C2×D4—2444no—
513C2×D4—2448no—
514C2×D4—2448no—
515C2×D4—2448no—
516C2×D4C23×D61116yes—
517C2×D4C23×D6118yes—
518C22×C4—48no—
519C22×C4—611no—
520C22×C4—612no—
521C22×C4—6116no—
522C22×C4—614no—
523C22×C4—614no—
524C22×C4—6116no—
525C22×C4—614no—
526C22×C4—618no—
527C22×C4—6116no—
528C22×C4—6116no—
529C22×C4—6116no—
530C22×C4—1224no—
531C22×C4—12216no—
532C22×C4—1228no—
533C22×C4—12216no—
534C22×C4—2448no—
535C22×C4—2448no—
536C22×C4—2448no—
537C22×C4C23×D6111yescharacteristic
538C22×C4C23×D6112yes—
539C22×C4C23×D6114yes—
540C22×C4C23×D6114yes—
541C22×C4C23×D61116yes—
542C4⋊C4—6116no—
543C4⋊C4—6116no—
544C4⋊C4—6116no—
545C4⋊C4—6116no—
546C4⋊C4—6116no—
547C4⋊C4—6116no—
548C4⋊C4—6116no—
549C4⋊C4—618no—
550C4⋊C4C23×D61116yes—
551C4⋊C4C23×D6118yes—
552C4⋊C4C23×D61116yes—
553C22⋊C4—6116no—
554C22⋊C4—6116no—
555C22⋊C4—6116no—
556C22⋊C4—6116no—
557C22⋊C4—6116no—
558C22⋊C4—6116no—
559C22⋊C4C23×D61116yes—
560C22⋊C4C23×D61116yes—
561C22⋊C4C23×D61116yes—
562C42—614no—
563C42—618no—
564C42—6116no—
565C42C23×D6114yes—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—612no—
570C23—618no—
571C23—1228no—
572C23—1224no—
573C23—1228no—
574C23—1222no—
575C23—1228no—
576C23C24×D6118yes—
577C23C24×D6111yescharacteristic
578Q8—48no—
579Q8—12232no—
580Q8—2448no—
581D4—416no—
582D4—48no—
583D4—12232no—
584D4—12232no—
585D4—24416no—
586D4—24416no—
587D4—2448no—
588D4—24416no—
589D4—24416no—
590D4—24416no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—618no—
597C2×C4—614no—
598C2×C4—612no—
599C2×C4—614no—
600C2×C4—618no—
601C2×C4—618no—
602C2×C4—1228no—
603C2×C4—12232no—
604C2×C4—12216no—
605C2×C4—12232no—
606C2×C4—1228no—
607C2×C4—1228no—
608C2×C4—24416no—
609C2×C4—24416no—
610C2×C4—24416no—
611C2×C4C24×D6112yes—
612C2×C4C24×D6114yes—
613C2×C4C24×D6118yes—
614C2×C4C24×D6118yes—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—416no—
619C22—1228no—
620C22—1224no—
621C22—1228no—
622C22—1222no—
623C22—1228no—
624C22—2444no—
625C22—24416no—
626C22H(5,𝔽2)×D6114yes—
627C222−1+4×D6112yes—
628C22C25×D6111yescharacteristic
629C4—28no—
630C4—28no—
631C4—24no—
632C4—22no—
633C4—618no—
634C4—614no—
635C4—1228no—
636C4—1228no—
637C2—48no—
638C2—2442no—
639C2—2448no—
640C2#2146021514yesdirect product factor
641C2C2×H(5,𝔽2)×D6112yes—
642C2C2×2−1+4×D6111yescharacteristic
subgroupG/H
centerC23C24×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC2×C61×#562C2
socleC23×C61C25
hypercenterC2×#562D61
hypocenterC61C2×#3548
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #21460215C2direct (N × H)
2. C2×#16840510C2semidirect
3. C2×#16840506C2semidirect
4. C2×#16840466C2semidirect
5. C2×#16840430C2semidirect
6. C2×#16840410C2semidirect
7. C2×#16840364C2semidirect
8. C2×#16840342C2semidirect
9. C2×#16840339C2semidirect
10. C2×#16840318C2semidirect
11. C2×#16840311C2semidirect
12. C2×C61×#562C2semidirect
17. #16840510C22semidirect
18. #16840506C22semidirect
19. #16840466C22semidirect
20. #16840430C22semidirect
21. #16840410C22semidirect
22. #16840364C22semidirect
23. #16840342C22semidirect
24. #16840339C22semidirect
25. #16840318C22semidirect
26. #16840311C22semidirect
63. C61×#562C22semidirect
319. C2×#562D61semidirect
329. C2×C61#3548semidirect
340. #562C2×D61semidirect
390. C61C2×#3548semidirect
640. C2#21460215direct (N × H)

Maximal subgroups

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

subgroupindex
#214602152
C2×#168405102
C2×#168405062
C2×#168404662
C2×#168404302
C2×#168404102
C2×#168403642
C2×#168403422
C2×#168403392
C2×#168403182
C2×#168403112
C2×C61×#5622
C2×#354861