Finite Group Databasebeta

← C2×#21461729 #30615116

Subgroups — C2×#21461729 #30615116

Order 15616.

1354 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840748C211yesmaximal, characteristic
2C2×#16840741C211yesmaximal, characteristic
3#21461729C218yesmaximal, direct product factor
4C2×#16840633C212yesmaximal, complemented
5C2×#16840629C212yesmaximal, complemented
6C2×#16840230C211yesmaximal, characteristic
7C2×D61⋊M16C2212yes—
8C2×#14519856C2212yes—
9#16840748—22no—
10#16840748C2214yes—
11C2×#14519838—21no—
12C2×#14519837—21no—
13#16840741—22no—
14#16840741C2214yes—
15C2×#14519836—21no—
16C2×#14519835—21no—
17#16840633C2218yescomplemented
18#16840629C2218yescomplemented
19C22×#11758222C2211yescharacteristic
20#16840230—22no—
21#16840230—22no—
22#16840230C414yescomplemented
23#16840230C2214yes—
24C2×#14519742—21no—
25C2×#14519739—21no—
26C2×#14519737—21no—
27C2×#14519737—21no—
28C2×#14519734C411yescharacteristic, complemented
29C2×#14519734C2211yescharacteristic
30C22×C4×Dic61C411yescharacteristic, complemented
31C2×C42×D61C411yescharacteristic, complemented
32C2×C42×D61C2211yescharacteristic
33C2×#117×C61C411yescharacteristic, complemented
34C2×#11758213—22no—
35D61⋊M16—22no—
36D61⋊M16—22no—
37D61⋊M16—22no—
38D61⋊M16C2314yes—
39C2×#11758217—22no—
40C2×D61⋊C8—22no—
41#14519856—22no—
42#14519856—22no—
43#14519856—22no—
44#14519856C2314yes—
45C2×C4⋊#6923—22no—
46C2×C22⋊#6923—22no—
47C2×C4×#6923—22no—
48#14519838—24no—
49#14519837—24no—
50#14519836—24no—
51#14519835—24no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222—22no—
55C2×#11758222—22no—
56C2×#11758222C2×C412yes—
57C2×#11758222D412yes—
58C2×#11758222D412yes—
59C2×#11758222C2312yes—
60C22×Dic122D411yescharacteristic
61C22×D244D411yescharacteristic
62C22×C22⋊D61D411yescharacteristic
63C22×C22⋊D61D411yescharacteristic
64C23×Dic61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61C2×C411yescharacteristic
68C22×C4×D61C2311yescharacteristic
69#14519742—24no—
70#14519742—24no—
71#14519739—24no—
72#14519739—24no—
73#14519737—24no—
74#14519737—24no—
75#14519737—24no—
76#14519737—24no—
77#14519734—22no—
78#14519734—21no—
79#14519734—21no—
80#14519734C2×C412yes—
81#14519734C2×C414yescomplemented
82#14519734C2312yes—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—22no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61C2×C414yescomplemented
94C42×D61—22no—
95C42×D61—21no—
96C42×D61—21no—
97C42×D61C2×C412yes—
98C42×D61C2×C414yescomplemented
99C42×D61C2312yes—
100C23×C4×C61C2×C411yescharacteristic
101#117×C61—22no—
102#117×C61C2×C414yescomplemented
103C2×C4⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2×C411yescharacteristic
106C2×C42×C61C2×C411yescharacteristic
107C24×C61C22⋊C411yescharacteristic
108C23×D61C22⋊C411yescharacteristic
109C23×D61C2×D411yescharacteristic
110C22×#6923—42no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61C22⋊C411yescharacteristic
118C22×C4×C61C22⋊C412yes—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61C22⋊C412yes—
126C2×C22⋊D61C22⋊C412yes—
127C2×C22⋊D61C2×D412yes—
128C2×C22⋊D61C2×D412yes—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—24no—
132C22×Dic61—44no—
133C22×Dic61C22⋊C411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135#11758222—24no—
136#11758222—24no—
137#11758222—24no—
138#11758222—24no—
139#11758222—24no—
140#11758222—24no—
141#11758222—24no—
142#11758222—24no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244C22⋊C412yes—
146C2×D244C2×D412yes—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—22no—
152C2×C4×D61—24no—
153C2×C4×D61—22no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61C22⋊C412yes—
157C2×C4×D61C22×C412yes—
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61#4112yes—
162C2×C4×D61#4112yes—
163C2×C4×D61#4112yes—
164C2×C4×D61#4112yes—
165C2×C4×D61C2411yescharacteristic
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122C22⋊C412yes—
169C2×Dic122C2×D412yes—
170C22⋊#6923—24no—
171C22⋊#6923—42no—
172#11758217—24no—
173#11758217—42no—
174C2×#92420—42no—
175C4⋊#6923—24no—
176C4⋊#6923—42no—
177C4×#6923—24no—
178C4×#6923—42no—
179#11758213—24no—
180#11758213—42no—
181D61⋊C8—24no—
182D61⋊C8—42no—
183C4⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61C22×C412yes—
188C42×C61C22×C412yes—
189#11758198—22no—
190#11758198—24no—
191#11758198—22no—
192#11758196—22no—
193#11758196—24no—
194#11758196—22no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—24no—
198C4×Dic61—22no—
199C4×Dic61—24no—
200C4×Dic61—22no—
201C4×Dic61C22×C412yes—
202C4×Dic61C22×C412yes—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—42no—
206C23×C61C2×C22⋊C411yescharacteristic
207C23×C61C2×C22⋊C412yes—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61C2×C22⋊C412yes—
213C22×D61C2×C22⋊C411yescharacteristic
214C22×D61#12312yes—
215C22×D61#12312yes—
216C22×D61C22×D411yescharacteristic
217C2×#6923—44no—
218C2×#6923—84no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C2×C22⋊C412yes—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61#11711yescharacteristic
232C2×C4×C61#11711yescharacteristic
233C2×C4×C61#11711yescharacteristic
234C2×C4×C61#11711yescharacteristic
235C2×C4×C61C23×C411yescharacteristic
236C22⋊D61—24no—
237C22⋊D61—24no—
238C22⋊D61—24no—
239C22⋊D61—24no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—48no—
246C2×Dic61—42no—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61#12312yes—
250C2×Dic61#12312yes—
251C2×Dic61C22×D411yescharacteristic
252D244—24no—
253D244—24no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—44no—
259C4×D61—44no—
260C4×D61—48no—
261C4×D61#11712yes—
262C4×D61#11712yes—
263C4×D61#12312yes—
264C4×D61#12312yes—
265C4×D61#12312yes—
266C4×D61#12312yes—
267C4×D61C2×#4112yes—
268C4×D61C2×#4112yes—
269Dic122—24no—
270Dic122—24no—
271#92420—44no—
272C2×#1827—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61#38111yescharacteristic
279C22×C61#38111yescharacteristic
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—24no—
282C2×D61—22no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61#38112yes—
290C2×D61C2×#12312yes—
291#6923—84no—
292C4×C61—22no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61#38111yescharacteristic
296C4×C61#38111yescharacteristic
297C4×C61C2×#11711yescharacteristic
298C4×C61C2×#11711yescharacteristic
299Dic61—24no—
300Dic61—22no—
301Dic61—42no—
302Dic61#38112yes—
303Dic61C2×#12312yes—
304C2×C4×#41—611no—
305C2×#413—611no—
306C2×#410—611no—
307#1827—618no—
308C2×#339—612no—
309C2×H(3,ℤ/4)—612no—
310C2×C61—21no—
311C2×C61—21no—
312C2×C61—42no—
313C2×C61#182712yescomplemented
314C2×C61C2×#38111yescharacteristic
315D61—22no—
316D61—22no—
317D61—42no—
318C22×#41—611no—
319C22×M16—612no—
320C4×#41—614no—
321C4×#41—614no—
322C4×#41—1222no—
323C4×#41—1222no—
324C2×C4×Q8—1221no—
325C2×C4×D4—1221no—
326C2×C4×D4—1221no—
327C2×C4×D4—1221no—
328C2×#117—611no—
329C2×#117—611no—
330C2×#117—612no—
331C2×#117#692311yescharacteristic, complemented
332C22×C42—611no—
333C22×C42—611no—
334C22×C42—611no—
335#413—614no—
336#413—1222no—
337C2×#101—1221no—
338C2×#100—1221no—
339#410—614no—
340#410—1222no—
341C2×#99—1221no—
342C2×#99—1221no—
343#339—618no—
344H(3,ℤ/4)—618no—
345C61C2×#182711yescharacteristic, complemented, Sylow
346C2×#41—612no—
347C2×#41—612no—
348C2×#41—612no—
349C2×#41—612no—
350C2×#41—1222no—
351C2×#41—1222no—
352C2×#41—1222no—
353C2×#41—1222no—
354C22×Q8—611no—
355C22×D4—611no—
356C22×D4—611no—
357C22×D4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—1221no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4C2×#692311yescharacteristic
364C2×M16—614no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—1222no—
368C2×M16—1222no—
369C2×M16—1222no—
370C22×C8—1222no—
371C4×Q8—1224no—
372C4×Q8—1224no—
373C4×D4—1224no—
374C4×D4—1224no—
375C4×D4—1224no—
376C4×D4—1224no—
377C4×D4—1224no—
378C4×D4—1224no—
379#117—22no—
380#117—612no—
381#117—612no—
382#117—614no—
383#117—614no—
384#117—1222no—
385#117—1221no—
386#117—1221no—
387#117—1222no—
388#117—1222no—
389#117—1222no—
390#117C2×#692314yescomplemented
391C2×C4⋊C4—21no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1222no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1222no—
399C2×C42—611no—
400C2×C42—614no—
401C2×C42—611no—
402C2×C42—612no—
403C2×C42—612no—
404C2×C42—614no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42—1222no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1222no—
412C2×C42—1222no—
413C2×C42C2×#692311yescharacteristic
414C2×C42C2×#692311yescharacteristic
415#101—1224no—
416#100—1224no—
417#99—1224no—
418#99—1224no—
419C24—611no—
420C24—611no—
421C24C22⋊#692311yescharacteristic
422#41—1224no—
423#41—1224no—
424#41—1224no—
425#41—1224no—
426#41—1224no—
427#41—1224no—
428#41—1224no—
429#41—1224no—
430C2×Q8—612no—
431C2×Q8—612no—
432C2×Q8—1222no—
433C2×Q8—1222no—
434C2×D4—612no—
435C2×D4—612no—
436C2×D4—612no—
437C2×D4—612no—
438C2×D4—612no—
439C2×D4—612no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—612no—
463C22×C4—612no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1224no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1224no—
474C22×C4—2444no—
475C22×C4—2444no—
476C22×C4—2444no—
477C22×C4—2442no—
478C22×C4C22⋊#692311yescharacteristic
479C22×C4C22⋊#692312yes—
480C22×C4C22×#692312yes—
481C22×C4C22×#692311yescharacteristic
482M16—1224no—
483M16—1224no—
484M16—2442no—
485M16—2442no—
486C2×C8—1224no—
487C2×C8—2442no—
488C2×C8—2442no—
489C4⋊C4—24no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1224no—
493C4⋊C4—1224no—
494C4⋊C4—2442no—
495C22⋊C4—24no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C22⋊C4—2442no—
501C42—21no—
502C42—21no—
503C42—612no—
504C42—612no—
505C42—1222no—
506C42—1222no—
507C42—1224no—
508C42—1224no—
509C42—1221no—
510C42—1221no—
511C42—1224no—
512C42—2442no—
513C42C22×#692312yes—
514C42C22×#692312yes—
515C23—21no—
516C23—21no—
517C23—42no—
518C23—612no—
519C23—612no—
520C23—611no—
521C23—611no—
522C23—612no—
523C23—1221no—
524C23—1221no—
525C23—2442no—
526C23—2442no—
527C23C2×C22⋊#692311yescharacteristic
528C23C2×C22⋊#692312yes—
529Q8—1224no—
530Q8—1224no—
531D4—1224no—
532D4—1224no—
533D4—1224no—
534D4—1224no—
535D4—1224no—
536D4—1224no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—1224no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—4884no—
570C2×C4—2444no—
571C2×C4—2444no—
572C2×C4—2448no—
573C2×C4—2448no—
574C2×C4—2444no—
575C2×C4—2442no—
576C2×C4C2×C22⋊#692312yes—
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4#1451985611yescharacteristic
579C2×C4#1451985611yescharacteristic
580C2×C4#1451985611yescharacteristic
581C2×C4#1451985611yescharacteristic
582C2×C4C23×#692311yescharacteristic
583C8—2444no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—612no—
590C22—612no—
591C22—1224no—
592C22—1222no—
593C22—2441no—
594C22—2441no—
595C22—2442no—
596C22—2442no—
597C22—2442no—
598C22—2442no—
599C22#1684070511yescharacteristic
600C22#1684070511yescharacteristic
601C22C22×C22⋊#692311yescharacteristic
602C4—22no—
603C4—42no—
604C4—42no—
605C4—612no—
606C4—612no—
607C4—1224no—
608C4—1222no—
609C4—4884no—
610C4—2442no—
611C4#1684070511yescharacteristic
612C4#1684070511yescharacteristic
613C4C2×#1451985611yescharacteristic
614C4C2×#1451985611yescharacteristic
615C2—21no—
616C2—21no—
617C2—42no—
618C2—1222no—
619C2—1222no—
620C2—2442no—
621C2#2146172912yesdirect product factor
622C2C2×#1684070511yescharacteristic
subgroupG/H
centerC2×C4#14519856
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#117×C61C4
socleC22×C61#381
prime radicalC22×#11758222C22
cosocleC2×C4×D61C24
hypercenterC2×#117#6923
hypocenterC61C2×#1827
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21461729C2direct (N × H)
4. C2×#16840633C2semidirect
5. C2×#16840629C2semidirect
17. #16840633C22semidirect
18. #16840629C22semidirect
22. #16840230C4semidirect
28. C2×#14519734C4semidirect
30. C22×C4×Dic61C4semidirect
31. C2×C42×D61C4semidirect
33. C2×#117×C61C4semidirect
81. #14519734C2×C4semidirect
93. C2×C4×Dic61C2×C4semidirect
98. C42×D61C2×C4semidirect
102. #117×C61C2×C4semidirect
313. C2×C61#1827semidirect
331. C2×#117#6923semidirect
345. C61C2×#1827semidirect
390. #117C2×#6923semidirect
621. C2#21461729direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407482
C2×#168407412
#214617292
C2×#168406332
C2×#168406292
C2×#168402302
C2×#182761