Finite Group Databasebeta

← C2×#21446827 #29706809

Subgroups — C2×#21446827 #29706809

Order 15616.

2182 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758202C211yesmaximal, characteristic, complemented
2C22×#14519756C211yesmaximal, characteristic
3C22×#14519701C211yesmaximal, characteristic, complemented
4C22×#14519700C211yesmaximal, characteristic, complemented
5C22×C22⋊#92418C211yesmaximal, characteristic
6#21446827C218yesmaximal, direct product factor
7C2×#16839524C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C61C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C22×#11758202—24no—
11C22×#11758202—24no—
12C22×#11758202—24no—
13C22×#11758202C2214yescomplemented
14C23×#92418C2211yescharacteristic
15C23×#92418C2211yescharacteristic
16C2×#14519756—24no—
17C2×#14519756—22no—
18C2×#14519756—24no—
19C2×#14519756C2214yes—
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758196—21no—
22C22×C22⋊Dic61—21no—
23C2×#14519701—24no—
24C2×#14519701C2214yescomplemented
25C2×#14519700—24no—
26C2×#14519700C2214yescomplemented
27C2×C22⋊#92418—24no—
28C2×C22⋊#92418C2214yes—
29#16839524C2214yescomplemented
30C23×Q8×C61C2211yescharacteristic
31C24×C4×C61C2211yescharacteristic
32C2×C22⋊Q8×C61—24no—
33C2×C22⋊Q8×C61—22no—
34C2×C22⋊Q8×C61—24no—
35C2×C22⋊Q8×C61C2214yes—
36C22×C4⋊C4×C61—21no—
37C22×C4⋊C4×C61C2211yescharacteristic
38C22×C22⋊C4×C61—21no—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—21no—
44C22×Dic122—22no—
45C22×Dic122D414yes—
46C22×Dic122C2312yes—
47C23×Dic61—21no—
48C23×Dic61—21no—
49C2×#11758202—232no—
50C2×#11758202—28no—
51C2×#11758202—28no—
52C2×#11758202—28no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—22no—
56C22×#92418—22no—
57C22×#92418—22no—
58C22×#92418D414yes—
59C22×#92418C2312yes—
60C22×#92418C2312yes—
61#14519756—216no—
62#14519756—28no—
63#14519756—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D414yes—
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—28no—
69C2×#11758196—24no—
70C2×C22⋊Dic61—28no—
71C2×C22⋊Dic61—24no—
72#14519701—28no—
73#14519700—28no—
74C22⋊#92418—28no—
75C25×C61D411yescharacteristic
76C22×Q8×C61—22no—
77C22×Q8×C61—21no—
78C22×Q8×C61—22no—
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61—22no—
82C22×Q8×C61D414yes—
83C22×Q8×C61C2312yes—
84C23×C4×C61—21no—
85C23×C4×C61—21no—
86C23×C4×C61—22no—
87C23×C4×C61—22no—
88C23×C4×C61—22no—
89C23×C4×C61—22no—
90C23×C4×C61—22no—
91C23×C4×C61—22no—
92C23×C4×C61D411yescharacteristic
93C23×C4×C61D412yes—
94C23×C4×C61C2312yes—
95C23×C4×C61C2311yescharacteristic
96C22⋊Q8×C61—216no—
97C22⋊Q8×C61—28no—
98C22⋊Q8×C61—28no—
99C2×C4⋊C4×C61—28no—
100C2×C4⋊C4×C61—24no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61D414yes—
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—28no—
106C2×C22⋊C4×C61—24no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—22no—
112C24×C61—22no—
113C24×C61Q1614yescomplemented
114C24×C61C2×D412yes—
115C24×C61C2×D411yescharacteristic
116C2×Q8×C61—24no—
117C2×Q8×C61—22no—
118C2×Q8×C61—28no—
119C2×Q8×C61—28no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—42no—
125C2×Q8×C61—42no—
126C2×Q8×C61—44no—
127C2×Q8×C61—44no—
128C2×Q8×C61C2×D414yes—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—24no—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—44no—
150C22×C4×C61—44no—
151C22×C4×C61—44no—
152C22×C4×C61—44no—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D412yes—
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61#4112yes—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61—42no—
169C22×Dic61—42no—
170C2×Dic122—28no—
171C2×Dic122—28no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—22no—
176C2×Dic122—24no—
177C2×Dic122—22no—
178C2×Dic122—42no—
179C2×Dic122—44no—
180C2×Dic122—42no—
181C2×Dic122—44no—
182C2×Dic122C2×D414yes—
183C4⋊C4×C61—216no—
184C4⋊C4×C61—24no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C2×D414yes—
187C22⋊C4×C61—216no—
188C22⋊Dic61—216no—
189#11758202—232no—
190C4⋊Dic61—24no—
191C4⋊Dic61—22no—
192C4⋊Dic61C2×D414yes—
193#11758196—216no—
194C2×#92418—28no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—44no—
201C2×#92418C2×D414yes—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—28no—
209C23×C61—28no—
210C23×C61—28no—
211C23×C61—44no—
212C23×C61—44no—
213C23×C61—44no—
214C23×C61—44no—
215C23×C61#12012yes—
216C23×C61#12012yes—
217C23×C61#12112yes—
218C23×C61C2×Q1612yes—
219C23×C61C2×Q1614yescomplemented
220C23×C61#13512yes—
221C23×C61#13712yes—
222C23×C61#13712yes—
223C23×C61C22×D411yescharacteristic
224Q8×C61—24no—
225Q8×C61—24no—
226Q8×C61—42no—
227Q8×C61—42no—
228Q8×C61—48no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—44no—
239C2×C4×C61—48no—
240C2×C4×C61—48no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—48no—
246C2×C4×C61#12014yes—
247C2×C4×C61#12012yes—
248C2×C4×C61#12012yes—
249C2×C4×C61#12114yes—
250C2×C4×C61#12114yes—
251C2×C4×C61#12112yes—
252C2×C4×C61#12714yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—48no—
266C2×Dic61—48no—
267Dic122—24no—
268Dic122—24no—
269Dic122—42no—
270Dic122—42no—
271Dic122—48no—
272#92418—24no—
273#92418—42no—
274C2×#2001—611nomaximal, Sylow
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—24no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—416no—
282C22×C61—42no—
283C22×C61#39214yes—
284C22×C61#45114yes—
285C22×C61#45114yes—
286C22×C61#45214yes—
287C22×C61#45214yes—
288C22×C61#46714yes—
289C22×C61#47014yes—
290C22×C61C2×#12011yescharacteristic
291C22×C61C2×#12011yescharacteristic
292C22×C61C2×#12111yescharacteristic
293C22×C61C22×Q1611yescharacteristic
294C22×C61C2×#13511yescharacteristic
295C22×C61C2×#13711yescharacteristic
296C22×C61C2×#13711yescharacteristic
297C4×C61—42no—
298C4×C61—44no—
299C4×C61—44no—
300C4×C61#39214yes—
301C4×C61C2×#12011yescharacteristic
302C4×C61C2×#12111yescharacteristic
303C4×C61C2×#12111yescharacteristic
304C4×C61C2×#12711yescharacteristic
305Dic61—44no—
306Dic61—44no—
307C23×Q16—611no—
308C22×C22⋊Q8—611no—
309C22×C22⋊Q8D6111yescharacteristic
310C22×#103—611no—
311C22×#103—611no—
312C22×C22⋊C8—611no—
313#2001—618no—
314C2×#340—611no—
315C2×C61—44no—
316C2×C61#200118yescomplemented
317C2×C61C2×#39211yescharacteristic
318C2×C61C2×#45111yescharacteristic
319C2×C61C2×#45111yescharacteristic
320C2×C61C2×#45211yescharacteristic
321C2×C61C2×#45211yescharacteristic
322C2×C61C2×#46711yescharacteristic
323C2×C61C2×#47011yescharacteristic
324C23×Q8—611no—
325C23×Q8C2×D6111yescharacteristic
326C24×C4C2×D6111yescharacteristic
327C22×Q16—614no—
328C22×Q16—1224no—
329C22×Q16—1224no—
330C22×Q16—1224no—
331C23×C8—611no—
332C23×C8—611no—
333C2×C22⋊Q8—24no—
334C2×C22⋊Q8—22no—
335C2×C22⋊Q8—24no—
336C2×C22⋊Q8—614no—
337C2×C22⋊Q8—1224no—
338C2×C22⋊Q8—1224no—
339C2×C22⋊Q8—1222no—
340C2×C22⋊Q8C2×D6114yes—
341C22×C4⋊C4—21no—
342C22×C4⋊C4—611no—
343C22×C4⋊C4—1221no—
344C22×C4⋊C4C2×D6111yescharacteristic
345C22×C22⋊C4—21no—
346C22×C22⋊C4—1221no—
347C2×#103—614no—
348C2×#103—614no—
349C2×#103—1224no—
350C2×#103—1224no—
351C2×C22⋊C8—614no—
352C2×C22⋊C8—1224no—
353#340—614no—
354C61C2×#200111yescharacteristic, complemented, Sylow
355C25C22⋊D6111yescharacteristic
356C22×Q8—21no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—614no—
363C22×Q8—612no—
364C22×Q8—1222no—
365C22×Q8—1221no—
366C22×Q8—1222no—
367C22×Q8—1222no—
368C22×Q8—1222no—
369C22×Q8—1222no—
370C22×Q8C22⋊D6114yes—
371C22×Q8C22×D6112yes—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—1221no—
381C23×C4—1221no—
382C23×C4C22⋊D6112yes—
383C23×C4C22⋊D6111yescharacteristic
384C23×C4C22×D6112yes—
385C23×C4C22×D6111yescharacteristic
386C2×Q16—1228no—
387C2×Q16—1228no—
388C2×Q16—12232no—
389C2×Q16—1228no—
390C22×C8—614no—
391C22×C8—612no—
392C22×C8—612no—
393C22×C8—1222no—
394C22×C8—1222no—
395C22×C8—1222no—
396C22×C8—1222no—
397C22×C8—1222no—
398C22⋊Q8—216no—
399C22⋊Q8—28no—
400C22⋊Q8—28no—
401C22⋊Q8—1228no—
402C22⋊Q8—1228no—
403C22⋊Q8—12216no—
404C2×C4⋊C4—28no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—614no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—1228no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1224no—
414C2×C4⋊C4C22⋊D6114yes—
415C2×C4⋊C4C22×D6112yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—1228no—
419C2×C22⋊C4—1224no—
420#103—1228no—
421#103—1228no—
422C22⋊C8—1228no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24#1175820214yescomplemented
430C24C2×C22⋊D6112yes—
431C24C2×C22⋊D6111yescharacteristic
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—28no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—614no—
445C2×Q8—1222no—
446C2×Q8—1224no—
447C2×Q8—1222no—
448C2×Q8—1222no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×Q8—1228no—
452C2×Q8—1228no—
453C2×Q8—2442no—
454C2×Q8—2442no—
455C2×Q8—2444no—
456C2×Q8—2444no—
457C2×Q8C2×C22⋊D6114yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4D4×D6112yes—
491C22×C4D4×D6112yes—
492C22×C4D4×D6112yes—
493C22×C4#1175822412yes—
494C22×C4C2×C22⋊D6112yes—
495C22×C4C2×C22⋊D6112yes—
496C22×C4C2×C22⋊D6112yes—
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499Q16—12232no—
500C2×C8—614no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1228no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—2444no—
508C4⋊C4—216no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—614no—
512C4⋊C4—12216no—
513C4⋊C4—1222no—
514C4⋊C4—1224no—
515C4⋊C4C2×C22⋊D6114yes—
516C22⋊C4—216no—
517C22⋊C4—12216no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—28no—
526C23—28no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23#1451974512yes—
532C23#1451974612yes—
533C23#1451974912yes—
534C23C2×#1175820212yes—
535C23C2×#1175820214yescomplemented
536C23#1451979712yes—
537C23#1451980712yes—
538C23#1451980812yes—
539C23C22×C22⋊D6111yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—42no—
544Q8—48no—
545Q8—1224no—
546Q8—1224no—
547Q8—2442no—
548Q8—2442no—
549Q8—2448no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2448no—
574C2×C4—2448no—
575C2×C4#1451974514yes—
576C2×C4#1451974512yes—
577C2×C4#1451974612yes—
578C2×C4#1451974814yes—
579C2×C4#1451974914yes—
580C2×C4#1451974912yes—
581C2×C4#1451977314yes—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C22×C22⋊D6111yescharacteristic
586C2×C4C22×C22⋊D6111yescharacteristic
587C2×C4C2×#1175822411yescharacteristic
588C8—1224no—
589C8—2442no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—416no—
596C22—42no—
597C22—42no—
598C22#1683967014yes—
599C22#1683991714yes—
600C22#1683991814yes—
601C22#1683992214yes—
602C22#1683992414yes—
603C22#1684000114yes—
604C22#1684001814yes—
605C22C2×#1451974511yescharacteristic
606C22C2×#1451974611yescharacteristic
607C22C2×#1451974911yescharacteristic
608C22C22×#1175820211yescharacteristic
609C22C2×#1451979711yescharacteristic
610C22C2×#1451980711yescharacteristic
611C22C2×#1451980811yescharacteristic
612C4—42no—
613C4—44no—
614C4—44no—
615C4—2444no—
616C4—2444no—
617C4#1683967014yes—
618C4C2×#1451974511yescharacteristic
619C4C2×#1451974811yescharacteristic
620C4C2×#1451974911yescharacteristic
621C4C2×#1451977311yescharacteristic
622C2—44no—
623C2#2144682718yesdirect product factor
624C2C2×#1683967011yescharacteristic
625C2C2×#1683991711yescharacteristic
626C2C2×#1683991811yescharacteristic
627C2C2×#1683992211yescharacteristic
628C2C2×#1683992411yescharacteristic
629C2C2×#1684000111yescharacteristic
630C2C2×#1684001811yescharacteristic
subgroupG/H
centerC24C2×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×C22⋊Q8×C61C2
socleC24×C61C2×D4
prime radicalC25×C61D4
hypercenterC22×C22⋊Q8D61
hypocenterC61C2×#2001
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11758202C2semidirect
3. C22×#14519701C2semidirect
4. C22×#14519700C2semidirect
6. #21446827C2direct (N × H)
7. C2×#16839524C2semidirect
13. C22×#11758202C22semidirect
24. C2×#14519701C22semidirect
26. C2×#14519700C22semidirect
29. #16839524C22semidirect
113. C24×C61Q16semidirect
219. C23×C61C2×Q16semidirect
316. C2×C61#2001semidirect
354. C61C2×#2001semidirect
429. C24#11758202semidirect
535. C23C2×#11758202semidirect
623. C2#21446827direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117582022
C22×#145197562
C22×#145197012
C22×#145197002
C22×C22⋊#924182
#214468272
C2×#168395242
C22×C22⋊Q8×C612
C2×#200161