Finite Group Databasebeta

← C2×#21448720 #29839299

Subgroups — C2×#21448720 #29839299

Order 15616.

4718 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D244C211yesmaximal, characteristic
2C22×#14519739C211yesmaximal, characteristic
3C22×#14519739C211yesmaximal, characteristic
4C22×C4⋊C4×D61C212yesmaximal, complemented
5#21448720C2116yesmaximal, direct product factor
6C2×#16839661C211yesmaximal, characteristic, complemented
7C2×#16839636C212yesmaximal, complemented
8C2×#16839633C212yesmaximal, complemented
9C2×#16839631C211yesmaximal, characteristic, complemented
10C2×#16839621C212yesmaximal, complemented
11C2×C4×#11758196C211yesmaximal, characteristic, complemented
12C2×C61×#389C211yesmaximal, characteristic, complemented
13C23×D244C2211yescharacteristic
14C23×C4×D61C2212yes—
15C23×C4×D61C2212yes—
16C23×C4×D61C2212yes—
17C2×C4×D244—28no—
18C2×C4×D244C2218yes—
19C2×#14519739—28no—
20C2×#14519739C4116yescomplemented
21C2×#14519739C2218yes—
22C2×#14519739C2218yes—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×C4⋊Dic61C2212yes—
25C22×#11758196C2212yes—
26C22×#11758196C2211yescharacteristic
27C22×#11758196C2212yes—
28C22×#11758196C2211yescharacteristic
29C2×C4⋊C4×D61—216no—
30C2×C4⋊C4×D61C22116yescomplemented
31C22×#11758198C2212yes—
32C22×#11758198C2212yes—
33C22×#11758198C2212yes—
34C22×C4×Dic61C2211yescharacteristic
35C22×C4×Dic61C2211yescharacteristic
36#16839661C2218yescomplemented
37#16839636C22116yescomplemented
38#16839633C22116yescomplemented
39#16839631C2218yescomplemented
40#16839621C22116yescomplemented
41C4×#11758196C2218yescomplemented
42C2×#14519684C2211yescharacteristic
43C2×#14519684C2212yes—
44C2×#14519684C2211yescharacteristic
45C22×C4⋊C4×C61C2211yescharacteristic
46C22×C4⋊C4×C61C2212yes—
47C22×C4⋊C4×C61C2211yescharacteristic
48C22×C42×C61C2211yescharacteristic
49C61×#389C2218yescomplemented
50C2×#95×C61C2212yes—
51C24×D61C2312yes—
52C22×D244C2×C418yes—
53C22×D244D418yes—
54C22×D244Q818yes—
55C22×D244C2314yes—
56C23×Dic61C2311yescharacteristic
57C23×Dic61C2312yes—
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2311yescharacteristic
60C23×Dic61C2311yescharacteristic
61C22×C4×D61—28no—
62C22×C4×D61—28no—
63C22×C4×D61—28no—
64C22×C4×D61—28no—
65C22×C4×D61—28no—
66C22×C4×D61—28no—
67C22×C4×D61—44no—
68C22×C4×D61—44no—
69C22×C4×D61C2×C4116yescomplemented
70C22×C4×D61C2318yes—
71C22×C4×D61C2318yes—
72C22×C4×D61C2318yes—
73C4×D244—232no—
74#14519739—232no—
75#14519739C2×C4164yescomplemented
76C2×C4⋊Dic61—24no—
77C2×C4⋊Dic61—28no—
78C2×C4⋊Dic61C2318yes—
79C2×C4⋊Dic61C2314yes—
80C2×#11758196—28no—
81C2×#11758196—24no—
82C2×#11758196—44no—
83C2×#11758196D418yescomplemented
84C2×#11758196C2318yes—
85C2×#11758196C2314yes—
86C2×#11758196C2318yes—
87C2×#11758196C2314yes—
88C4⋊C4×D61—264no—
89C2×#11758198—28no—
90C2×#11758198—28no—
91C2×#11758198C2×C4116yescomplemented
92C2×#11758198C2318yes—
93C2×#11758198C2318yes—
94C2×#11758198C2318yes—
95C2×C4×Dic61—24no—
96C2×C4×Dic61C2×C418yescomplemented
97C2×C4×Dic61C2314yes—
98C2×C4×Dic61C2314yes—
99#14519684C2314yes—
100#14519684C2314yes—
101#14519684C2318yes—
102C23×C4×C61C2311yescharacteristic
103C23×C4×C61C2312yes—
104C23×C4×C61C2312yes—
105C23×C4×C61C2311yescharacteristic
106C23×C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61—28no—
108C2×C4⋊C4×C61—24no—
109C2×C4⋊C4×C61C2×C418yescomplemented
110C2×C4⋊C4×C61C2318yes—
111C2×C4⋊C4×C61C2314yes—
112C2×C4⋊C4×C61C2314yes—
113C2×C42×C61—24no—
114C2×C42×C61C2314yes—
115#95×C61C2318yes—
116C24×C61C2411yescharacteristic
117C23×D61—24no—
118C23×D61—24no—
119C23×D61—24no—
120C23×D61—24no—
121C23×D61C22×C418yes—
122C23×D61C2×D418yes—
123C23×D61C2×Q818yes—
124C23×D61C2414yes—
125C22×C4×C61—22no—
126C22×C4×C61—24no—
127C22×C4×C61—24no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61C22×C418yes—
136C22×C4×C61C22×C414yes—
137C22×C4×C61C2×D414yes—
138C22×C4×C61C2×D414yes—
139C22×C4×C61C2×Q814yes—
140C22×C4×C61C2414yes—
141C22×C4×C61C2412yes—
142C22×C4×C61C2414yes—
143C22×C4×C61C2412yes—
144C22×C4×C61C2412yes—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—24no—
150C22×Dic61—24no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—24no—
154C22×Dic61—22no—
155C22×Dic61C22×C414yes—
156C22×Dic61C22×C414yes—
157C22×Dic61C2×D414yes—
158C22×Dic61C2×D414yes—
159C22×Dic61#4114yes—
160C22×Dic61C2414yes—
161C22×Dic61C2412yes—
162C22×Dic61C2412yes—
163C22×Dic61C2412yes—
164C22×Dic61C2412yes—
165C2×D244C4⋊C4164yescomplemented
166C2×D244C22×C4116yes—
167C2×D244C2×D4116yes—
168C2×D244C2×Q8116yes—
169C2×C4×D61—264no—
170C2×C4×D61—216no—
171C2×C4×D61—216no—
172C2×C4×D61—216no—
173C2×C4×D61—216no—
174C2×C4×D61—216no—
175C2×C4×D61—216no—
176C2×C4×D61—48no—
177C2×C4×D61—48no—
178C2×C4×D61—432no—
179C2×C4×D61—432no—
180C2×C4×D61C22×C4132yescomplemented
181C4⋊C4×C61—216no—
182C4⋊C4×C61—28no—
183C4⋊C4×C61C22×C4116yescomplemented
184C42×C61—28no—
185#11758198—216no—
186#11758198—216no—
187#11758198C22×C4132yescomplemented
188C4⋊Dic61—28no—
189C4⋊Dic61—216no—
190#11758196—216no—
191#11758196—28no—
192#11758196—48no—
193#11758196C2×D4116yescomplemented
194C4×Dic61—28no—
195C4×Dic61C22×C4116yescomplemented
196C23×C61C23×C412yes—
197C23×C61C22×D412yes—
198C23×C61C22×D412yes—
199C23×C61C22×Q812yes—
200C23×C61C2×#4112yes—
201C23×C61H(5,𝔽2)12yes—
202C23×C612−1+412yes—
203C23×C61C2511yescharacteristic
204C22×D61—216no—
205C22×D61—216no—
206C22×D61—216no—
207C22×D61—216no—
208C22×D61—216no—
209C22×D61—216no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—24no—
213C22×D61—24no—
214C22×D61C2×C4⋊C4132yescomplemented
215C22×D61C23×C418yes—
216C22×D61C22×D418yes—
217C22×D61C22×Q818yes—
218C2×C4×C61—216no—
219C2×C4×C61—28no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—48no—
231C2×C4×C61—44no—
232C2×C4×C61C2×C4⋊C4116yescomplemented
233C2×C4×C61C23×C418yes—
234C2×C4×C61C23×C414yes—
235C2×C4×C61C22×D414yes—
236C2×C4×C61C22×D414yes—
237C2×C4×C61C22×Q814yes—
238C2×Dic61—28no—
239C2×Dic61—28no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—22no—
250C2×Dic61—48no—
251C2×Dic61—44no—
252C2×Dic61C4×D4116yescomplemented
253C2×Dic61C23×C414yes—
254C2×Dic61C23×C414yes—
255C2×Dic61C22×D414yes—
256C2×Dic61C22×D414yes—
257C2×Dic61C2×#4114yes—
258D244C2×C4⋊C4164yescomplemented
259C4×D61—264no—
260C4×D61—432no—
261C4×D61—432no—
262C2×D4×C4⋊C4—611nomaximal, Sylow
263C22×C61C22×C4⋊C414yes—
264C22×C61C2×C4×D414yes—
265C22×C61#52014yes—
266C22×C61D4214yes—
267C22×C61#54714yes—
268C22×C61D4×Q814yes—
269C22×C61#55414yes—
270C22×C61C24×C411yescharacteristic
271C22×C61C23×D411yescharacteristic
272C22×C61C23×D411yescharacteristic
273C22×C61C23×Q811yescharacteristic
274C22×C61C22×#4111yescharacteristic
275C22×C61C2×H(5,𝔽2)11yescharacteristic
276C22×C61C2×2−1+411yescharacteristic
277C2×D61—264no—
278C2×D61—28no—
279C2×D61—28no—
280C2×D61—28no—
281C2×D61—28no—
282C2×D61—28no—
283C2×D61—28no—
284C2×D61C22×C4⋊C4116yescomplemented
285C4×C61—28no—
286C4×C61—24no—
287C4×C61—44no—
288C4×C61—42no—
289C4×C61C22×C4⋊C418yescomplemented
290Dic61—24no—
291Dic61—24no—
292Dic61—44no—
293Dic61—42no—
294Dic61C2×C4×D418yescomplemented
295C22×C4×D4—611no—
296C22×C4×D4—611no—
297C22×C4×D4—611no—
298C23×C4⋊C4—612no—
299D4×C4⋊C4—6116no—
300C2×#389—611no—
301C2×#389D6111yescharacteristic, complemented
302C2×#385—612no—
303C2×#385—612no—
304C2×#384—611no—
305C2×#380—612no—
306C2×C4×C4⋊C4—611no—
307C2×C61D4×C4⋊C418yescomplemented
308C2×C61C23×C4⋊C411yescharacteristic
309C2×C61C22×C4×D411yescharacteristic
310C2×C61C2×#52011yescharacteristic
311C2×C61C2×D4211yescharacteristic
312C2×C61C2×#54711yescharacteristic
313C2×C61C2×D4×Q811yescharacteristic
314C2×C61C2×#55411yescharacteristic
315D61—216no—
316C23×D4—611no—
317C24×C4—612no—
318C24×C4—612no—
319C24×C4—612no—
320C2×C4×D4—6116no—
321C2×C4×D4—618no—
322C2×C4×D4—618no—
323C2×C4×D4—618no—
324C2×C4×D4—1228no—
325C2×C4×D4—1228no—
326C22×C4⋊C4—611no—
327C22×C4⋊C4—612no—
328C22×C4⋊C4—611no—
329C22×C4⋊C4—612no—
330C22×C4⋊C4—611no—
331C22×C4⋊C4—612no—
332C22×C4⋊C4—6116no—
333C22×C4⋊C4—12216no—
334C22×C4⋊C4C2×D6112yes—
335C22×C4⋊C4C2×D6111yescharacteristic
336C22×C4⋊C4C2×D6111yescharacteristic
337C22×C22⋊C4—612no—
338C22×C22⋊C4—612no—
339C22×C22⋊C4—612no—
340C22×C42—611no—
341C22×C42—611no—
342C22×C42C2×D6111yescharacteristic
343#389—618no—
344#389C2×D6118yescomplemented
345#385—6116no—
346#385—6116no—
347#384—618no—
348#380—6116no—
349C4×C4⋊C4—618no—
350C2×#95—611no—
351C2×#95—611no—
352C2×#95—612no—
353C2×#95C2×D6112yes—
354C61C2×D4×C4⋊C411yescharacteristic, complemented, Sylow
355C25—612no—
356C22×D4—618no—
357C22×D4—618no—
358C22×D4—618no—
359C22×D4—614no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—6116no—
364C23×C4—618no—
365C23×C4—611no—
366C23×C4—612no—
367C23×C4—618no—
368C23×C4—618no—
369C23×C4—1228no—
370C23×C4—1228no—
371C23×C4—1228no—
372C23×C4—1228no—
373C23×C4—1228no—
374C23×C4—1228no—
375C23×C4—2444no—
376C23×C4—2444no—
377C23×C4C22×D6111yescharacteristic
378C23×C4C22×D6112yes—
379C23×C4C22×D6112yes—
380C23×C4C22×D6111yescharacteristic
381C23×C4C22×D6111yescharacteristic
382C4×D4—6164no—
383C4×D4—12232no—
384C4×D4—12232no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—618no—
388C2×C4⋊C4—614no—
389C2×C4⋊C4—618no—
390C2×C4⋊C4—618no—
391C2×C4⋊C4—614no—
392C2×C4⋊C4—614no—
393C2×C4⋊C4—618no—
394C2×C4⋊C4—12264no—
395C2×C4⋊C4—1228no—
396C2×C4⋊C4—1228no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4—1224no—
399C2×C4⋊C4—2444no—
400C2×C4⋊C4C4×D6118yescomplemented
401C2×C4⋊C4C22×D6118yes—
402C2×C4⋊C4C22×D6114yes—
403C2×C4⋊C4C22×D6114yes—
404C2×C22⋊C4—6116no—
405C2×C22⋊C4—618no—
406C2×C22⋊C4—618no—
407C2×C22⋊C4—618no—
408C2×C22⋊C4—1228no—
409C2×C22⋊C4—1228no—
410C2×C42—24no—
411C2×C42—618no—
412C2×C42—614no—
413C2×C42—614no—
414C2×C42—1224no—
415C2×C42C22×D6114yes—
416#95—614no—
417#95—614no—
418#95—618no—
419#95C22×D6118yes—
420C24—618no—
421C24—618no—
422C24—618no—
423C24—614no—
424C24—1224no—
425C24—1224no—
426C24—1224no—
427C24—1224no—
428C24C23×D6111yescharacteristic
429C2×D4—6116no—
430C2×D4—6116no—
431C2×D4—6116no—
432C2×D4—6164no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—614no—
446C22×C4—612no—
447C22×C4—614no—
448C22×C4—614no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—6132no—
452C22×C4—612no—
453C22×C4—614no—
454C22×C4—12216no—
455C22×C4—12264no—
456C22×C4—12216no—
457C22×C4—12216no—
458C22×C4—12216no—
459C22×C4—12216no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—12216no—
471C22×C4—2448no—
472C22×C4—2448no—
473C22×C4—24432no—
474C22×C4—24432no—
475C22×C4C2×C4×D6118yes—
476C22×C4C2×C4×D6114yes—
477C22×C4D4×D6114yes—
478C22×C4D4×D6114yes—
479C22×C4Q8×D6114yes—
480C22×C4C23×D6112yes—
481C22×C4C23×D6114yes—
482C22×C4C23×D6114yes—
483C22×C4C23×D6112yes—
484C22×C4C23×D6112yes—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—6116no—
488C4⋊C4—12216no—
489C4⋊C4—12216no—
490C4⋊C4—1228no—
491C4⋊C4—1228no—
492C4⋊C4—2448no—
493C4⋊C4C2×C4×D61116yescomplemented
494C22⋊C4—6132no—
495C22⋊C4—12216no—
496C22⋊C4—12216no—
497C42—28no—
498C42—6116no—
499C42—1228no—
500C23—6132no—
501C23—618no—
502C23—618no—
503C23—618no—
504C23—12216no—
505C23—12216no—
506C23—12216no—
507C23—1224no—
508C23—1224no—
509C23—1224no—
510C23—1224no—
511C23—12216no—
512C23—12216no—
513C23—12216no—
514C23C22×C4×D6112yes—
515C23C2×D4×D6112yes—
516C23C2×D4×D6112yes—
517C23C2×Q8×D6112yes—
518C23#41×D6112yes—
519C23#1451982312yes—
520C23#1451982512yes—
521C23C24×D6111yescharacteristic
522D4—6164no—
523C2×C4—216no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—6116no—
538C2×C4—614no—
539C2×C4—614no—
540C2×C4—614no—
541C2×C4—614no—
542C2×C4—614no—
543C2×C4—12264no—
544C2×C4—1228no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1228no—
550C2×C4—1222no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1222no—
554C2×C4—1224no—
555C2×C4—1222no—
556C2×C4—24432no—
557C2×C4—24432no—
558C2×C4—2448no—
559C2×C4—2444no—
560C2×C4C4⋊C4×D61116yescomplemented
561C2×C4C22×C4×D6118yes—
562C2×C4C22×C4×D6114yes—
563C2×C4C2×D4×D6114yes—
564C2×C4C2×D4×D6114yes—
565C2×C4C2×Q8×D6114yes—
566C22—6116no—
567C22—12264no—
568C22—1228no—
569C22—1228no—
570C22—1228no—
571C22—1228no—
572C22—1228no—
573C22—1228no—
574C22C2×C4⋊C4×D6114yes—
575C22C4×D4×D6114yes—
576C22#1684024414yes—
577C22#1684039614yes—
578C22#1684041414yes—
579C22#1684042714yes—
580C22#1684046814yes—
581C22C23×C4×D6111yescharacteristic
582C22C22×D4×D6111yescharacteristic
583C22C22×D4×D6111yescharacteristic
584C22C22×Q8×D6111yescharacteristic
585C22C2×#41×D6111yescharacteristic
586C22C2×#1451982311yescharacteristic
587C22C2×#1451982511yescharacteristic
588C4—28no—
589C4—24no—
590C4—44no—
591C4—42no—
592C4—618no—
593C4—1224no—
594C4—1224no—
595C4—2444no—
596C4—2442no—
597C4C2×C4⋊C4×D6118yescomplemented
598C2—12216no—
599C2#2144872018yesdirect product factor
600C2C22×C4⋊C4×D6111yescharacteristic
601C2C2×C4×D4×D6111yescharacteristic
602C2C2×#1684024411yescharacteristic
603C2C2×#1684039611yescharacteristic
604C2C2×#1684041411yescharacteristic
605C2C2×#1684042711yescharacteristic
606C2C2×#1684046811yescharacteristic
subgroupG/H
centerC24C23×D61
commutatorC22×C61C24×C4
FrattiniC23C24×D61
FittingC2×C61×#389C2
socleC24×C61C24
prime radicalC23×D244C22
cosocleC23×C61C25
hypercenterC2×#389D61
hypocenterC61C2×D4×C4⋊C4
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊C4×D61C2semidirect
5. #21448720C2direct (N × H)
6. C2×#16839661C2semidirect
7. C2×#16839636C2semidirect
8. C2×#16839633C2semidirect
9. C2×#16839631C2semidirect
10. C2×#16839621C2semidirect
11. C2×C4×#11758196C2semidirect
12. C2×C61×#389C2semidirect
20. C2×#14519739C4semidirect
30. C2×C4⋊C4×D61C22semidirect
36. #16839661C22semidirect
37. #16839636C22semidirect
38. #16839633C22semidirect
39. #16839631C22semidirect
40. #16839621C22semidirect
41. C4×#11758196C22semidirect
49. C61×#389C22semidirect
69. C22×C4×D61C2×C4semidirect
75. #14519739C2×C4semidirect
83. C2×#11758196D4semidirect
91. C2×#11758198C2×C4semidirect
96. C2×C4×Dic61C2×C4semidirect
109. C2×C4⋊C4×C61C2×C4semidirect
165. C2×D244C4⋊C4semidirect
180. C2×C4×D61C22×C4semidirect
183. C4⋊C4×C61C22×C4semidirect
187. #11758198C22×C4semidirect
193. #11758196C2×D4semidirect
195. C4×Dic61C22×C4semidirect
214. C22×D61C2×C4⋊C4semidirect
232. C2×C4×C61C2×C4⋊C4semidirect
252. C2×Dic61C4×D4semidirect
258. D244C2×C4⋊C4semidirect
284. C2×D61C22×C4⋊C4semidirect
289. C4×C61C22×C4⋊C4semidirect
294. Dic61C2×C4×D4semidirect
301. C2×#389D61semidirect
307. C2×C61D4×C4⋊C4semidirect
344. #389C2×D61semidirect
354. C61C2×D4×C4⋊C4semidirect
400. C2×C4⋊C4C4×D61semidirect
493. C4⋊C4C2×C4×D61semidirect
560. C2×C4C4⋊C4×D61semidirect
597. C4C2×C4⋊C4×D61semidirect
599. C2#21448720direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D2442
C22×#145197392
C22×C4⋊C4×D612
#214487202
C2×#168396612
C2×#168396362
C2×#168396332
C2×#168396312
C2×#168396212
C2×C4×#117581962
C2×C61×#3892
C2×D4×C4⋊C461