Finite Group Databasebeta

← #29696833

Subgroups — #29696833

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×#16840484C211yesmaximal, characteristic
2C2×#16839891C211yesmaximal, characteristic, complemented
3#21448731C211yesmaximal, characteristic, complemented
4#21446749C211yesmaximal, characteristic
5#21446748C211yesmaximal, characteristic
6#21446740C211yesmaximal, characteristic, complemented
7#21446633C211yesmaximal, characteristic
8#21446632C211yesmaximal, characteristic
9#21446619C211yesmaximal, characteristic, complemented
10#21445823C211yesmaximal, characteristic, complemented
11#21445818C211yesmaximal, characteristic
12#21445649C211yesmaximal, characteristic, complemented
13#21445648C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic
15C61×#1803C211yesmaximal, characteristic, complemented
16#16840484—24no—
17#16840484C418yescomplemented
18C2×#14519775C2211yescharacteristic
19C2×#14519772—21no—
20C2×#14519771—21no—
21C2×#14519766C2211yescharacteristic
22C2×#14519765C2211yescharacteristic
23C2×#14519755—21no—
24C2×C4×Dic122C2211yescharacteristic
25C2×#14519742—21no—
26C2×#14519742C2211yescharacteristic
27C2×#14519739C2211yescharacteristic
28C2×#14519734—21no—
29#16839891—24no—
30C2×#14519706C2211yescharacteristic
31C2×#14519706C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519699C2211yescharacteristic
39C2×#14519699C2211yescharacteristic
40C2×#14519697C2211yescharacteristic
41C2×#14519697C2211yescharacteristic
42C2×#14519688C2211yescharacteristic
43C2×#14519687C2211yescharacteristic
44C2×C8×Dic61C2211yescharacteristic
45#16839658—21no—
46#16839657—21no—
47#16839657—21no—
48#16839631C2211yescharacteristic
49#16839631C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51C4⋊C4×Dic61C2211yescharacteristic
52C4×C4⋊Dic61C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#384C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122D411yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519775—22no—
69#14519775C2×C414yes—
70#14519772—28no—
71#14519771—28no—
72#14519766—22no—
73#14519766C2×C414yes—
74#14519765—22no—
75#14519765C2×C414yes—
76#14519755—28no—
77C4×Dic122C2×C414yes—
78C4×Dic122D414yes—
79#14519742—22no—
80#14519742—44no—
81#14519742C2×C414yes—
82#14519739—22no—
83#14519739C2×C414yes—
84#14519734—28no—
85C2×C4⋊Dic61D411yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196D411yescharacteristic
96C2×#11758196D411yescharacteristic
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108#14519706—22no—
109#14519706—22no—
110#14519702—22no—
111#14519702—41no—
112#14519701—22no—
113#14519701—22no—
114#14519700—22no—
115#14519700—41no—
116#14519699—22no—
117#14519699—41no—
118#14519697—22no—
119#14519697—41no—
120#14519688—22no—
121#14519687—22no—
122C8×Dic61—22no—
123#14519684—21no—
124#14519684—21no—
125#14519684—21no—
126C22×Q8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C4×Q8×C61—22no—
130C4×Q8×C61C2×C414yes—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C42×C61—21no—
138C2×C42×C61C2311yescharacteristic
139C4⋊C8×C61—22no—
140#103×C61—41no—
141#103×C61—41no—
142C23×D61—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—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244C22×C412yes—
169C2×C4×D61—24no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
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—24no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—42no—
194C4⋊C4×C61—81no—
195C4⋊C4×C61C22×C412yes—
196C42×C61—24no—
197C42×C61C22×C412yes—
198C42×C61C2×D412yes—
199#11758198—24no—
200#11758198—24no—
201#11758198—24no—
202#11758198—24no—
203#11758198—24no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C22⋊C414yes—
208C4⋊Dic61#4112yes—
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—42no—
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×C61C22×D411yescharacteristic
233C22×D61—22no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—41no—
243Q8×C61—41no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
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×C61C2×C22⋊C412yes—
261C2×C4×C61C2×C22⋊C412yes—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61C2×C22⋊C412yes—
284C2×Dic61C2×C22⋊C412yes—
285C2×Dic61C2×C22⋊C412yes—
286C2×Dic61C2×C22⋊C412yes—
287C2×Dic61AGL(1,ℤ/8)12yes—
288C2×Dic61#13712yes—
289D244—22no—
290D244—22no—
291D244—41no—
292D244—41no—
293C4×D61—48no—
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×C61C22×C22⋊C411yescharacteristic
305C22×C61#54811yescharacteristic
306C22×C61#55011yescharacteristic
307C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C61C2×#13711yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C4×C61—22no—
319C4×C61—22no—
320C4×C61—42no—
321C4×C61—42no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C4×#4111yescharacteristic
325C4×C61#52011yescharacteristic
326C4×C61#54811yescharacteristic
327C4×C61#55011yescharacteristic
328Dic61—22no—
329Dic61—22no—
330Dic61—22no—
331Dic61—42no—
332Dic61—42no—
333Dic61—81no—
334Dic61#41714yes—
335C2×#556—611no—
336C2×#446—611no—
337Q8×C4⋊C4—611no—
338#1983—611no—
339#1983—611no—
340#1981—611no—
341#1959—611no—
342#1959—611no—
343#1956—611no—
344#1803—611no—
345#1803D6111yescharacteristic, complemented
346#1802—611no—
347#1763—611no—
348#1763—611no—
349C4×#102—611no—
350C2×C61#234111yescharacteristic
351C2×C61C2×#41711yescharacteristic
352C2×C61#289811yescharacteristic
353C2×C61#336711yescharacteristic
354C2×C61#336911yescharacteristic
355C2×C61#339711yescharacteristic
356C2×C61#339811yescharacteristic
357D61—82no—
358#556—618no—
359#556—1224no—
360C2×C4⋊Q8—611no—
361C2×#126—1221no—
362C2×#126—1221no—
363C2×#124—611no—
364C2×#124—611no—
365C2×C22⋊Q8—1221no—
366C2×C4×Q8—611no—
367C2×C4×Q8—611no—
368C2×C4×Q8—1221no—
369C2×C4×Q8C2×D6111yescharacteristic
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×#102—611no—
387C2×#102—611no—
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#384C2×D6111yescharacteristic
397C4×C4⋊C4—611no—
398C4×C4⋊C4—611no—
399C4×C4⋊C4—611no—
400#340—611no—
401#340—611no—
402#340C2×D6111yescharacteristic
403#340C2×D6111yescharacteristic
404C61#2254411yescharacteristic, complemented, Sylow
405C22×Q8—611no—
406C22×Q8—611no—
407C22×Q8C22×D6111yescharacteristic
408C22×D4—611no—
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—614no—
419#124—614no—
420#124—1222no—
421#124—1222no—
422C22⋊Q8—1228no—
423C4×Q8—22no—
424C4×Q8—614no—
425C4×Q8—614no—
426C4×Q8—614no—
427C4×Q8—1222no—
428C4×Q8—2444no—
429C4×Q8C4×D6114yes—
430C4×D4—614no—
431C4×D4—1222no—
432#117—1228no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
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—41no—
468#103—41no—
469#103—1222no—
470#103—1222no—
471#103—1222no—
472#103—1222no—
473#103—2441no—
474#103—2441no—
475#102—1222no—
476#102—1222no—
477#102—2441no—
478#102—2441no—
479#97—1222no—
480#97—1222no—
481C4×C8—1222no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485C24—1221no—
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—612no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—1224no—
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—24no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—42no—
543C4⋊C4—81no—
544C4⋊C4—614no—
545C4⋊C4—614no—
546C4⋊C4—614no—
547C4⋊C4—612no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—4881no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4C2×C4×D6112yes—
559C22⋊C4—1224no—
560C22⋊C4—1224no—
561C22⋊C4—1224no—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C42—24no—
565C42—612no—
566C42—612no—
567C42—612no—
568C42—612no—
569C42—1224no—
570C42—2442no—
571C42—2442no—
572C42—2442no—
573C42C2×C4×D6112yes—
574C42D4×D6112yes—
575C23—1222no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23C2×D4×D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—1228no—
588Q8—1222no—
589Q8—1222no—
590Q8—2441no—
591Q8—2441no—
592D4—1222no—
593D4—1222no—
594D4—2441no—
595D4—2441no—
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—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—612no—
616C2×C4—612no—
617C2×C4—612no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—2448no—
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×C4#41×D6111yescharacteristic
640C2×C4#1451982211yescharacteristic
641C2×C4#1451982511yescharacteristic
642C8—41no—
643C8—41no—
644C8—1222no—
645C8—2441no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22C2×C22⋊C4×D6111yescharacteristic
654C22#1684042211yescharacteristic
655C22#1684043011yescharacteristic
656C22AGL(1,ℤ/8)×D6111yescharacteristic
657C22#137×D6111yescharacteristic
658C22#1684056811yescharacteristic
659C22#1684057511yescharacteristic
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—614no—
667C4—1222no—
668C4—1222no—
669C4—1222no—
670C4—4881no—
671C4—2442no—
672C4—2442no—
673C4#1684023011yescharacteristic
674C4#1684024311yescharacteristic
675C4#1684042211yescharacteristic
676C4#1684043011yescharacteristic
677C2—4882no—
678C2#2144865611yescharacteristic
679C2#417×D6111yescharacteristic
680C2#2145434711yescharacteristic
681C2#2145854011yescharacteristic
682C2#2145856211yescharacteristic
683C2#2145883411yescharacteristic
684C2#2145884511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#22544
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16839891C2semidirect
3. #21448731C2semidirect
6. #21446740C2semidirect
9. #21446619C2semidirect
10. #21445823C2semidirect
12. #21445649C2semidirect
13. #21445648C2semidirect
15. C61×#1803C2semidirect
17. #16840484C4semidirect
345. #1803D61semidirect
404. C61#22544semidirect

Maximal subgroups

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

subgroupindex
C2×#168404842
C2×#168398912
#214487312
#214467492
#214467482
#214467402
#214466332
#214466322
#214466192
#214458232
#214458182
#214456492
#214456482
#214456332
C61×#18032
#2254461