Finite Group Databasebeta

← #29696841

Subgroups — #29696841

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840481C211yesmaximal, characteristic
2C2×#16839891C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
4#21446750C211yesmaximal, characteristic
5#21446740C211yesmaximal, characteristic, complemented
6#21446636C211yesmaximal, characteristic
7#21446634C211yesmaximal, characteristic
8#21446619C211yesmaximal, characteristic, complemented
9#21445825C211yesmaximal, characteristic, complemented
10#21445822C211yesmaximal, characteristic, complemented
11#21445817C211yesmaximal, characteristic
12#21445652C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#1983C211yesmaximal, characteristic
16#16840481—24no—
17#16840481C418yescomplemented
18C2×#14519775C2211yescharacteristic
19C2×#14519770—21no—
20C2×#14519770—21no—
21C2×#14519764C2211yescharacteristic
22C2×#14519754—21no—
23C2×Q8×Dic61C2211yescharacteristic
24C2×C4×Dic122—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×Dic122C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C2×#14519733—21no—
29#16839891—24no—
30C2×#14519706C2211yescharacteristic
31C2×#14519706C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519698C2211yescharacteristic
40C2×#14519688C2211yescharacteristic
41C2×#14519687C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43#16839660—21no—
44#16839659—21no—
45#16839659—21no—
46#16839632C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839631C2211yescharacteristic
49C4⋊C4×Dic61C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#124×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C4×C4⋊C4×C61C2211yescharacteristic
62C22×Dic122D411yescharacteristic
63C22×Dic122C2311yescharacteristic
64C23×Dic61—21no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519775—22no—
68#14519775C2×C414yes—
69#14519770—28no—
70#14519770—28no—
71#14519764—22no—
72#14519764C2×C414yes—
73#14519754—28no—
74Q8×Dic61—22no—
75Q8×Dic61C2×C414yes—
76C4×Dic122—22no—
77C4×Dic122—44no—
78C4×Dic122C2×C414yes—
79C4×Dic122C2×C414yes—
80C4×Dic122D414yes—
81D4×Dic61—22no—
82D4×Dic61C2×C414yes—
83#14519733—28no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61D411yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196D411yescharacteristic
99C2×#11758196D411yescharacteristic
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—21no—
102C2×C22⋊Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109#14519706—22no—
110#14519706—22no—
111#14519702—41no—
112#14519702—41no—
113#14519701—22no—
114#14519701—41no—
115#14519700—22no—
116#14519700—22no—
117#14519698—22no—
118#14519698—41no—
119#14519688—22no—
120#14519687—22no—
121C8×Dic61—22no—
122#14519684—21no—
123#14519684—21no—
124C22×Q8×C61C2311yescharacteristic
125C22×D4×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128#124×C61—22no—
129#124×C61C2×C414yes—
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—22no—
137#103×C61—22no—
138#103×C61—41no—
139#102×C61—22no—
140#102×C61—41no—
141#95×C61—21no—
142C24×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C22×C412yes—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61C22×C412yes—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—24no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C22⋊C414yes—
180C2×Dic122C22×C412yes—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61#4112yes—
191C22⋊C4×C61—24no—
192C22⋊C4×C61—24no—
193C42×C61—42no—
194C42×C61C22×C412yes—
195C42×C61C2×D412yes—
196C22⋊Dic61—24no—
197C22⋊Dic61—24no—
198C22⋊Dic61—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—24no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206C4⋊Dic61—81no—
207C4⋊Dic61C22⋊C414yes—
208C4⋊Dic61C22×C412yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—42no—
213#11758196—42no—
214#11758196—81no—
215#11758196—81no—
216#11758196C22⋊C414yes—
217#11758196C22⋊C414yes—
218C4×Dic61—24no—
219C4×Dic61—24no—
220C4×Dic61—42no—
221C4×Dic61—42no—
222C4×Dic61C22×C412yes—
223C4×Dic61C22×C412yes—
224C4×Dic61C2×D412yes—
225C4×Dic61C2×D412yes—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418#4112yes—
232C23×C61—22no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61C22×D411yescharacteristic
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—41no—
243Q8×C61—41no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61C2×C22⋊C412yes—
262C2×C4×C61C2×C22⋊C412yes—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—48no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61C2×C22⋊C412yes—
289C2×Dic61C2×C22⋊C412yes—
290C2×Dic61C2×C22⋊C412yes—
291C2×Dic61C2×C22⋊C412yes—
292C2×Dic61AGL(1,ℤ/8)12yes—
293C2×Dic61#13712yes—
294Dic122—28no—
295Dic122—22no—
296Dic122—22no—
297Dic122—41no—
298Dic122—41no—
299C8×C61—41no—
300C8×C61—41no—
301#92418—22no—
302#92418—41no—
303#22544—611nomaximal, Sylow
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61C22×C22⋊C411yescharacteristic
312C22×C61#54811yescharacteristic
313C22×C61#55011yescharacteristic
314C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C61C2×#13711yescharacteristic
316C22×C61#57711yescharacteristic
317C22×C61#57811yescharacteristic
318C4×C61—22no—
319C4×C61—22no—
320C4×C61—42no—
321C4×C61—42no—
322C4×C61—81no—
323C4×C61C4×#4111yescharacteristic
324C4×C61#52011yescharacteristic
325C4×C61#54811yescharacteristic
326C4×C61#55011yescharacteristic
327Dic61—22no—
328Dic61—22no—
329Dic61—22no—
330Dic61—42no—
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334Dic61#41714yes—
335C2×#556—611no—
336C2×#446—611no—
337Q8×C4⋊C4—611no—
338#1983—611no—
339#1983D6111yescharacteristic
340#1981—611no—
341#1959—611no—
342#1959—611no—
343#1956—611no—
344#1803—611no—
345#1803—611no—
346#1802—611no—
347#1763—611no—
348#1763—611no—
349C4×#102—611no—
350C2×C61—82no—
351C2×C61#234111yescharacteristic
352C2×C61C2×#41711yescharacteristic
353C2×C61#289811yescharacteristic
354C2×C61#336711yescharacteristic
355C2×C61#336911yescharacteristic
356C2×C61#339711yescharacteristic
357C2×C61#339811yescharacteristic
358#556—618no—
359#556—1224no—
360C2×C4⋊Q8—611no—
361C2×#126—1221no—
362C2×#126—1221no—
363C2×#124—611no—
364C2×#124C2×D6111yescharacteristic
365C2×C22⋊Q8—1221no—
366C2×C4×Q8—611no—
367C2×C4×Q8—611no—
368C2×C4×Q8—611no—
369C2×C4×Q8—1221no—
370C2×C4×D4—611no—
371C2×#117—1221no—
372#446—1224no—
373C2×C4⋊C8—611no—
374C2×C4⋊C8—611no—
375C2×C4⋊C8C2×D6111yescharacteristic
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103C2×D6111yescharacteristic
383C2×#103C2×D6111yescharacteristic
384C2×#102—611no—
385C2×#102—611no—
386C2×#102C2×D6111yescharacteristic
387C2×#102C2×D6111yescharacteristic
388C2×#97—611no—
389C2×#97—611no—
390C2×C4×C8—611no—
391#389—1221no—
392#389—1221no—
393#389—1221no—
394#384—611no—
395#384—611no—
396#384—611no—
397C4×C4⋊C4—611no—
398C4×C4⋊C4—611no—
399C4×C4⋊C4C2×D6111yescharacteristic
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404C61#2254411yescharacteristic, complemented, Sylow
405C22×Q8—611no—
406C22×Q8—611no—
407C22×Q8C22×D6111yescharacteristic
408C22×D4C22×D6111yescharacteristic
409C23×C4—1221no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C4⋊Q8—614no—
415C4⋊Q8—1222no—
416#126—1228no—
417#126—1228no—
418#124—22no—
419#124—614no—
420#124—1222no—
421#124C4×D6114yes—
422C22⋊Q8—1228no—
423C4×Q8—614no—
424C4×Q8—614no—
425C4×Q8—614no—
426C4×Q8—614no—
427C4×Q8—1222no—
428C4×Q8—1222no—
429C4×Q8—2444no—
430C4×D4—614no—
431C4×D4—1222no—
432#117—1228no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C42—21no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—1221no—
460C2×C42—1221no—
461C2×C42—1221no—
462C2×C42—1221no—
463C2×C42C22×D6111yescharacteristic
464C4⋊C8—22no—
465C4⋊C8—1222no—
466C4⋊C8—1222no—
467#103—22no—
468#103—41no—
469#103—1222no—
470#103—1222no—
471#103—1222no—
472#103—2441no—
473#103—2441no—
474#103—2441no—
475#102—22no—
476#102—41no—
477#102—1222no—
478#102—2441no—
479#97—1222no—
480#97—1222no—
481C4×C8—1222no—
482#95—21no—
483#95—1221no—
484#95—1221no—
485C24—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—614no—
492C2×Q8—612no—
493C2×Q8—1222no—
494C2×Q8—1222no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8C2×C4×D6112yes—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4C2×C4×D6112yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—1224no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4D4×D6111yescharacteristic
522C22×C4D4×D6111yescharacteristic
523C22×C4C23×D6111yescharacteristic
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—612no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—614no—
540C4⋊C4—614no—
541C4⋊C4—614no—
542C4⋊C4—612no—
543C4⋊C4—1224no—
544C4⋊C4—1224no—
545C4⋊C4—1224no—
546C4⋊C4—1221no—
547C4⋊C4—1224no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—4881no—
553C4⋊C4—4881no—
554C4⋊C4—4881no—
555C4⋊C4—2442no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4#1175822412yes—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—1224no—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C42—42no—
565C42—612no—
566C42—612no—
567C42—612no—
568C42—612no—
569C42—1224no—
570C42—1224no—
571C42—2442no—
572C42—2442no—
573C42C2×C4×D6112yes—
574C42D4×D6112yes—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C2×D4×D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—1228no—
588Q8—1222no—
589Q8—1222no—
590Q8—2441no—
591Q8—2441no—
592D4—22no—
593D4—22no—
594D4—41no—
595D4—41no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—2448no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4C22⋊C4×D6112yes—
635C2×C4C22⋊C4×D6112yes—
636C2×C4C22×C4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4C2×#1175822411yescharacteristic
640C2×C4#1451982211yescharacteristic
641C2×C4#1451982411yescharacteristic
642C8—41no—
643C8—41no—
644C8—1222no—
645C8—2441no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C2×C22⋊C4×D6111yescharacteristic
654C22#1684041811yescharacteristic
655C22#1684043011yescharacteristic
656C22AGL(1,ℤ/8)×D6111yescharacteristic
657C22#137×D6111yescharacteristic
658C22#1684056911yescharacteristic
659C22#1684057611yescharacteristic
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—614no—
666C4—1222no—
667C4—1222no—
668C4—1222no—
669C4—4881no—
670C4—4881no—
671C4—2442no—
672C4—2442no—
673C4C4×#1175822411yescharacteristic
674C4#1684024111yescharacteristic
675C4#1684042011yescharacteristic
676C4#1684043211yescharacteristic
677C2—82no—
678C2#2144865811yescharacteristic
679C2#417×D6111yescharacteristic
680C2#2145434811yescharacteristic
681C2#2145853211yescharacteristic
682C2#2145855411yescharacteristic
683C2#2145884211yescharacteristic
684C2#2145885311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#22544
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#16839891C2semidirect
3. C4⋊C4×Dic122C2semidirect
5. #21446740C2semidirect
8. #21446619C2semidirect
9. #21445825C2semidirect
10. #21445822C2semidirect
12. #21445652C2semidirect
13. #21445651C2semidirect
17. #16840481C4semidirect
404. C61#22544semidirect

Maximal subgroups

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

subgroupindex
C2×#168404812
C2×#168398912
C4⋊C4×Dic1222
#214467502
#214467402
#214466362
#214466342
#214466192
#214458252
#214458222
#214458172
#214456522
#214456512
#102×Dic612
C61×#19832
#2254461