Finite Group Databasebeta

← C2×#21446332 #29681942

Subgroups — C2×#21446332 #29681942

Order 15616.

1698 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519807C211yesmaximal, characteristic
2C22×#14519784C211yesmaximal, characteristic
3C22×#14519700C211yesmaximal, characteristic, complemented
4C22×#14519695C211yesmaximal, characteristic, complemented
5#21446332C218yesmaximal, direct product factor
6C2×#16839531C211yesmaximal, characteristic, complemented
7C2×#16839526C211yesmaximal, characteristic, complemented
8C2×C61×#417C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C2×#14519807—24no—
11C2×#14519807—22no—
12C2×#14519807C418yescomplemented
13C2×#14519807C2214yes—
14C22×#11758202—21no—
15C22×#11758200—21no—
16C2×#14519784—24no—
17C2×#14519784—22no—
18C2×#14519784—24no—
19C2×#14519784C2214yes—
20C22×#11758194—21no—
21C22×#11758194C2211yescharacteristic
22C22×#11758194C2211yescharacteristic
23C23×#92418—21no—
24C23×#92418C2211yescharacteristic
25C2×#14519700—21no—
26C2×#14519700—21no—
27C2×#14519700—22no—
28C2×#14519700C2214yescomplemented
29C2×#14519695—24no—
30C2×#14519695C2214yescomplemented
31#16839531C2214yescomplemented
32#16839526C2214yescomplemented
33C22×#41×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C22×C4⋊C4×C61C2211yescharacteristic
36C61×#417—22no—
37C61×#417C2214yes—
38C2×#103×C61—21no—
39C2×#102×C61—21no—
40C22×Dic122—21no—
41C22×Dic122—21no—
42C22×Dic122—21no—
43C22×Dic122—22no—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122—22no—
47C22×Dic122—21no—
48C22×Dic122C2×C414yes—
49C22×Dic122C2312yes—
50C23×Dic61—21no—
51#14519807—216no—
52#14519807—28no—
53#14519807C2×C4116yescomplemented
54C2×#11758202—28no—
55C2×#11758202—24no—
56C2×#11758200—28no—
57C2×#11758200—24no—
58#14519784—216no—
59#14519784—28no—
60#14519784—28no—
61C2×#11758194—28no—
62C2×#11758194—22no—
63C2×#11758194—21no—
64C2×#11758194—22no—
65C2×#11758194—21no—
66C2×#11758194—24no—
67C2×#11758194C2×C414yes—
68C2×#11758194D414yes—
69C2×#11758194C2312yes—
70C2×#11758194C2312yes—
71C22×#92418—28no—
72C22×#92418—21no—
73C22×#92418—22no—
74C22×#92418—22no—
75C22×#92418—21no—
76C22×#92418—24no—
77C22×#92418D411yescharacteristic
78C22×#92418D411yescharacteristic
79C22×#92418D412yes—
80C22×#92418C2312yes—
81#14519700—24no—
82#14519700—24no—
83#14519695—28no—
84C2×#41×C61—22no—
85C2×#41×C61—21no—
86C2×#41×C61—22no—
87C2×#41×C61—21no—
88C2×#41×C61C2×C414yes—
89C2×#41×C61D414yes—
90C2×#41×C61D412yes—
91C2×#41×C61C2312yes—
92C22×Q8×C61D411yescharacteristic
93C22×D4×C61—21no—
94C22×D4×C61D411yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—21no—
99C2×M16×C61—22no—
100C2×M16×C61—22no—
101C2×M16×C61C2312yes—
102C22×C8×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2312yes—
109#103×C61—24no—
110#102×C61—24no—
111C24×C61—21no—
112C24×C61#4111yescharacteristic
113#41×C61—24no—
114#41×C61—24no—
115#41×C61—24no—
116#41×C61—24no—
117#41×C61—24no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61C22⋊C418yes—
121#41×C61C22×C414yes—
122#41×C61C2×D414yes—
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61C22⋊C414yes—
126C2×Q8×C61C2×D412yes—
127C2×D4×C61—24no—
128C2×D4×C61—22no—
129C2×D4×C61—24no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61C22⋊C414yes—
134C2×D4×C61C2×D412yes—
135C22×C4×C61—21no—
136C22×C4×C61—24no—
137C22×C4×C61—24no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61—82no—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61#4111yescharacteristic
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—24no—
155C22×Dic61—22no—
156C22×Dic61—41no—
157C22×Dic61—41no—
158C22×Dic61—42no—
159C2×Dic122—24no—
160C2×Dic122—24no—
161C2×Dic122—24no—
162C2×Dic122—24no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—24no—
168C2×Dic122—22no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—41no—
174C2×Dic122—41no—
175C2×Dic122—42no—
176C2×Dic122C22×C414yes—
177M16×C61—22no—
178M16×C61—22no—
179M16×C61—42no—
180C2×C8×C61—22no—
181C2×C8×C61—42no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186#11758202—216no—
187#11758200—216no—
188#11758194—216no—
189#11758194—24no—
190#11758194—24no—
191#11758194—22no—
192#11758194—22no—
193#11758194C22×C414yes—
194#11758194C2×D414yes—
195C2×#92418—216no—
196C2×#92418—24no—
197C2×#92418—22no—
198C2×#92418—24no—
199C2×#92418—22no—
200C2×#92418—24no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418C2×D412yes—
206C2×#92418C2×D412yes—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—24no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61C4×D412yes—
214C23×C61#12111yescharacteristic
215C23×C61#12111yescharacteristic
216C23×C61#12312yes—
217C23×C61C2×#4111yescharacteristic
218Q8×C61—22no—
219Q8×C61—24no—
220Q8×C61C2×C22⋊C414yes—
221D4×C61—24no—
222D4×C61—24no—
223D4×C61—28no—
224D4×C61—22no—
225D4×C61—24no—
226D4×C61C2×C22⋊C414yes—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—28no—
232C2×C4×C61—42no—
233C2×C4×C61—82no—
234C2×C4×C61—82no—
235C2×C4×C61—82no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C4×D412yes—
241C2×C4×C61#12012yes—
242C2×C4×C61#12012yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61#12312yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×Dic61—24no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—44no—
258C2×Dic61—44no—
259Dic122—24no—
260Dic122—24no—
261Dic122—42no—
262Dic122—42no—
263Dic122—44no—
264Dic122—44no—
265C8×C61—42no—
266#92418—24no—
267#92418—22no—
268#92418—24no—
269#92418—22no—
270C2×#1908—611nomaximal, Sylow
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—24no—
278C22×C61—48no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61#38612yes—
282C22×C61#38612yes—
283C22×C61#47112yes—
284C22×C61#47312yes—
285C22×C61C2×C4×D411yescharacteristic
286C22×C61C2×#12111yescharacteristic
287C22×C61C2×#12311yescharacteristic
288C4×C61—24no—
289C4×C61—82no—
290C4×C61#38612yes—
291C4×C61#38612yes—
292C4×C61C22×C22⋊C411yescharacteristic
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×#12011yescharacteristic
295C4×C61C2×#12111yescharacteristic
296Dic61—44no—
297C22×#137—611no—
298C22×#131—611no—
299C2×#417D6111yescharacteristic
300C22×#103—611no—
301C22×#101—611no—
302#1908—618no—
303C2×#343—611no—
304C2×#341—611no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—44no—
309C2×C61#190814yescomplemented
310C2×C61C2×#38611yescharacteristic
311C2×C61C2×#47111yescharacteristic
312C2×C61C2×#47311yescharacteristic
313C22×#41C2×D6111yescharacteristic
314C23×Q8—611no—
315C2×#137—618no—
316C2×#137—614no—
317C2×#137—1224no—
318C2×#137—1222no—
319C22×Q16—1221no—
320C22×SD16—1221no—
321C2×#131—614no—
322C2×#131—1222no—
323C2×#131—1224no—
324C2×#131—1224no—
325C22×M16—611no—
326C22×M16—611no—
327C22×M16—1221no—
328C22×M16C2×D6111yescharacteristic
329C23×C8—611no—
330C23×C8—1221no—
331C22×C4⋊C4C2×D6111yescharacteristic
332#417—22no—
333#417C2×D6114yes—
334C2×#103—21no—
335C2×#103—614no—
336C2×#103—1221no—
337C2×#103—1221no—
338C2×#103—1222no—
339C2×#102—21no—
340C2×#101—614no—
341C2×#101—1224no—
342#343—614no—
343#341—614no—
344C61C2×#190811yescharacteristic, complemented, Sylow
345C2×#41—21no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—21no—
349C2×#41C4×D6114yes—
350C2×#41D24414yes—
351C2×#41C22⋊D6112yes—
352C2×#41C22×D6112yes—
353C22×Q8—614no—
354C22×Q8—612no—
355C22×Q8—1222no—
356C22×Q8—1221no—
357C22×Q8—1222no—
358C22×Q8—1221no—
359C22×Q8—1221no—
360C22×Q8—1222no—
361C22×Q8—1221no—
362C22×Q8—1221no—
363C22×Q8C22⋊D6111yescharacteristic
364C22×D4—21no—
365C22×D4C22⋊D6111yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—1221no—
369C23×C4C22×D6111yescharacteristic
370#137—6116no—
371#137—12216no—
372#137—1228no—
373C2×Q16—1228no—
374C2×Q16—1224no—
375C2×SD16—1228no—
376C2×SD16—1224no—
377#131—1228no—
378#131—12216no—
379#131—1228no—
380C2×M16—21no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—614no—
384C2×M16—614no—
385C2×M16—612no—
386C2×M16—612no—
387C2×M16—1224no—
388C2×M16—1228no—
389C2×M16—1222no—
390C2×M16—1222no—
391C2×M16—1221no—
392C2×M16—1221no—
393C2×M16C22×D6112yes—
394C22×C8—21no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—612no—
398C22×C8—612no—
399C22×C8—1224no—
400C22×C8—1228no—
401C22×C8—1222no—
402C22×C8—1222no—
403C22×C8—1221no—
404C22×C8—1221no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4C22×D6112yes—
411#103—24no—
412#103—1224no—
413#103—1224no—
414#102—24no—
415#101—1228no—
416C24—21no—
417C24#1175822411yescharacteristic
418#41—22no—
419#41—22no—
420#41—24no—
421#41—24no—
422#41—24no—
423#41—24no—
424#41—24no—
425#41#1175819818yes—
426#41C2×C4×D6114yes—
427#41C2×D24414yes—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—614no—
431C2×Q8—1222no—
432C2×Q8—1224no—
433C2×Q8—1222no—
434C2×Q8—1222no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×Q8—1224no—
438C2×Q8—1224no—
439C2×Q8—1224no—
440C2×Q8—1224no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C2×Q8—2441no—
444C2×Q8—2441no—
445C2×Q8—2442no—
446C2×Q8—2442no—
447C2×Q8—2442no—
448C2×Q8#1175819814yes—
449C2×Q8C2×C22⋊D6112yes—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4#1175819814yes—
457C2×D4C2×C22⋊D6112yes—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—82no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4—2441no—
470C22×C4—2441no—
471C22×C4—2442no—
472C22×C4C2×C4×D6112yes—
473C22×C4C2×D24412yes—
474C22×C4D4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6112yes—
477C22×C4#1175822412yes—
478C22×C4#1175822411yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C23×D6111yescharacteristic
482Q16—12216no—
483SD16—12216no—
484M16—22no—
485M16—22no—
486M16—42no—
487M16—614no—
488M16—614no—
489M16—12216no—
490M16—1222no—
491M16—1222no—
492M16—1224no—
493M16—1224no—
494C2×C8—22no—
495C2×C8—42no—
496C2×C8—612no—
497C2×C8—612no—
498C2×C8—12216no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1224no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1224no—
507C2×C8—1224no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23#1451973712yes—
519C23#1451974911yescharacteristic
520C23#1451974911yescharacteristic
521C23#1451976112yes—
522C23C2×#1175822411yescharacteristic
523Q8—22no—
524Q8—24no—
525Q8—1224no—
526Q8—1224no—
527Q8—2442no—
528Q8—2442no—
529Q8—2444no—
530Q8—2444no—
531Q8C2×#1175819814yes—
532D4—24no—
533D4—24no—
534D4—28no—
535D4—22no—
536D4—24no—
537D4C2×#1175819814yes—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—1224no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4C2×#1175819812yes—
553C2×C4C2×#1175819812yes—
554C2×C4#1451973712yes—
555C2×C4#1451973712yes—
556C2×C4#1451973712yes—
557C2×C4#1451974712yes—
558C2×C4#1451974712yes—
559C2×C4#1451974912yes—
560C2×C4#1451974912yes—
561C2×C4#1451974911yescharacteristic
562C2×C4#1451974911yescharacteristic
563C2×C4#1451976112yes—
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×D24411yescharacteristic
569C2×C4C2×#1175822411yescharacteristic
570C8—42no—
571C8—1224no—
572C8—1222no—
573C8—1222no—
574C8—1224no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—48no—
585C22#1683963912yes—
586C22#1683963912yes—
587C22#1684002312yes—
588C22#1684003512yes—
589C22C2×#1451973711yescharacteristic
590C22C2×#1451974911yescharacteristic
591C22C2×#1451976111yescharacteristic
592C4—24no—
593C4—82no—
594C4—2444no—
595C4#1683963912yes—
596C4#1683963912yes—
597C4C22×#1175819811yescharacteristic
598C4C2×#1451973711yescharacteristic
599C4C2×#1451974711yescharacteristic
600C4C2×#1451974911yescharacteristic
601C2—21no—
602C2—21no—
603C2—22no—
604C2—44no—
605C2#2144633214yesdirect product factor
606C2C2×#1683963911yescharacteristic
607C2C2×#1684002311yescharacteristic
608C2C2×#1684003511yescharacteristic
subgroupG/H
centerC23#14519749
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61#121
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#417D61
hypocenterC61C2×#1908
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
3. C22×#14519700C2semidirect
4. C22×#14519695C2semidirect
5. #21446332C2direct (N × H)
6. C2×#16839531C2semidirect
7. C2×#16839526C2semidirect
12. C2×#14519807C4semidirect
28. C2×#14519700C22semidirect
30. C2×#14519695C22semidirect
31. #16839531C22semidirect
32. #16839526C22semidirect
53. #14519807C2×C4semidirect
309. C2×C61#1908semidirect
344. C61C2×#1908semidirect
605. C2#21446332direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198072
C22×#145197842
C22×#145197002
C22×#145196952
#214463322
C2×#168395312
C2×#168395262
C2×C61×#4172
C2×#190861