Finite Group Databasebeta

← #29696886

Subgroups — #29696886

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2#21448776C211yesmaximal, characteristic, complemented
3#21446749C211yesmaximal, characteristic
4#21446749C211yesmaximal, characteristic
5#21446741C211yesmaximal, characteristic, complemented
6#21446639C211yesmaximal, characteristic, complemented
7#21446635C211yesmaximal, characteristic
8#21446630C211yesmaximal, characteristic
9#21445835C211yesmaximal, characteristic, complemented
10#21445830C211yesmaximal, characteristic
11#21445825C211yesmaximal, characteristic, complemented
12#21445650C211yesmaximal, characteristic, complemented
13#21445645C211yesmaximal, characteristic
14C4×#14519702C211yesmaximal, characteristic, complemented
15C2×C61×#446C211yesmaximal, characteristic, complemented
16#16840485—24no—
17#16840485C418yescomplemented
18C2×#14519777C2211yescharacteristic
19C2×#14519772—21no—
20C2×#14519772—21no—
21C2×#14519765C2211yescharacteristic
22C2×#14519765C2211yescharacteristic
23C2×#14519753—21no—
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×D244C2211yescharacteristic
27C2×#14519735—21no—
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519699C2211yescharacteristic
39C2×#14519699C2211yescharacteristic
40#16839669—21no—
41#16839648—21no—
42#16839648—21no—
43#16839632C2211yescharacteristic
44#16839628C2211yescharacteristic
45#16839628C2211yescharacteristic
46C4⋊C4×Dic61—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49C4×C4⋊Dic61C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#446—24no—
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C2×C4×C8×C61C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66#14519777—22no—
67#14519777C2×C414yes—
68#14519772—28no—
69#14519772—28no—
70#14519765—22no—
71#14519765—22no—
72#14519765C2×C414yes—
73#14519765C2×C414yes—
74#14519753—28no—
75C4×Dic122—22no—
76C4×Dic122—22no—
77C4×Dic122C2×C414yes—
78C4×Dic122C2×C414yes—
79C4×D244—22no—
80C4×D244C2×C414yes—
81#14519735—28no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519702—22no—
107#14519702—41no—
108#14519702—41no—
109#14519702—41no—
110#14519702—41no—
111#14519699—22no—
112#14519699—22no—
113#14519699—41no—
114#14519699—41no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122C22×Q8×C61D411yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C4×Q8×C61C2×C414yes—
128C4×Q8×C61D414yes—
129C2×C4⋊C4×C61D411yescharacteristic
130C2×C4⋊C4×C61D411yescharacteristic
131C2×C4⋊C4×C61D411yescharacteristic
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—22no—
136C4⋊C8×C61—22no—
137C4⋊C8×C61—22no—
138#97×C61—22no—
139#97×C61—22no—
140C4×C8×C61—22no—
141C23×D61—21no—
142C2×Q8×C61—41no—
143C2×Q8×C61—41no—
144C2×Q8×C61C22⋊C414yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244C22×C412yes—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C22×C412yes—
179C2×Dic122C22×C412yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61#4112yes—
192C4⋊C4×C61C22⋊C414yes—
193C4⋊C4×C61C22⋊C414yes—
194C4⋊C4×C61C22⋊C414yes—
195C42×C61C22×C412yes—
196C42×C61C22×C412yes—
197C42×C61C22×C412yes—
198C42×C61C2×D412yes—
199C42×C61C2×D412yes—
200C42×C61C2×D412yes—
201#11758198—24no—
202#11758198—24no—
203#11758198—24no—
204#11758198—24no—
205#11758198—24no—
206C4⋊Dic61—24no—
207C4⋊Dic61—24no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—42no—
217C4⋊Dic61—81no—
218C4⋊Dic61C22×C412yes—
219C4⋊Dic61#4112yes—
220#11758196—24no—
221#11758196—24no—
222C4×Dic61—24no—
223C4×Dic61—24no—
224C4×Dic61—42no—
225C4×Dic61—81no—
226C4×Dic61—81no—
227C23×C61C22×D411yescharacteristic
228C22×D61—22no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235Q8×C61—44no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281D244—22no—
282D244—22no—
283D244—41no—
284D244—41no—
285C4×D61—48no—
286Dic122—22no—
287Dic122—22no—
288Dic122—22no—
289Dic122—22no—
290Dic122—41no—
291Dic122—41no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—22no—
295C8×C61—41no—
296C8×C61—41no—
297C8×C61—41no—
298#22548—611nomaximal, Sylow
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61#54811yescharacteristic
301C22×C61#55011yescharacteristic
302C22×C61#57511yescharacteristic
303C22×C61#57511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57811yescharacteristic
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61—22no—
319C4×C61C4×#4111yescharacteristic
320C4×C61#52011yescharacteristic
321C4×C61#54811yescharacteristic
322C4×C61#55011yescharacteristic
323Dic61—42no—
324Dic61—42no—
325Dic61—42no—
326Dic61—42no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330C2×#556—611no—
331C2×#446D6111yescharacteristic, complemented
332#2355—611no—
333#1983—611no—
334#1983—611no—
335#1981—611no—
336#1960—611no—
337#1959—611no—
338#1958—611no—
339#1805—611no—
340#1804—611no—
341#1803—611no—
342#1763—611no—
343#1762—611no—
344C4×#103—611no—
345C2×C61#234111yescharacteristic
346C2×C61#289611yescharacteristic
347C2×C61#289811yescharacteristic
348C2×C61#337611yescharacteristic
349C2×C61#337711yescharacteristic
350C2×C61#340111yescharacteristic
351C2×C61#340211yescharacteristic
352D61—82no—
353#556—618no—
354#556—1224no—
355C2×C4⋊Q8—611no—
356C2×#126—1221no—
357C2×#126—1221no—
358C2×#124—611no—
359C2×#124—611no—
360C2×C22⋊Q8—1221no—
361C2×C4×Q8—611no—
362C2×C4×Q8—611no—
363C2×C4×Q8C2×D6111yescharacteristic
364C2×C4×D4—611no—
365C2×#117—1221no—
366#446—24no—
367C2×C4⋊C8C2×D6111yescharacteristic
368C2×C4⋊C8C2×D6111yescharacteristic
369C2×C4⋊C8C2×D6111yescharacteristic
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#97C2×D6111yescharacteristic
383C2×#97C2×D6111yescharacteristic
384C2×C4×C8C2×D6111yescharacteristic
385#391—1221no—
386#387—1221no—
387#387—1221no—
388#384—611no—
389#382—611no—
390#382—611no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—611no—
393C4×C4⋊C4—611no—
394C4×C4⋊C4—1221no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399C61#2254811yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C22×Q8—611no—
402C22×Q8D24411yescharacteristic
403C22×D4—611no—
404C23×C4—1221no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C4⋊Q8—614no—
410C4⋊Q8—1222no—
411#126—1228no—
412#126—1228no—
413#124—614no—
414#124—614no—
415#124—1222no—
416#124—1222no—
417C22⋊Q8—1228no—
418C4×Q8—614no—
419C4×Q8—614no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422C4×Q8C4×D6114yes—
423C4×Q8C22⋊D6114yes—
424C4×D4—614no—
425C4×D4—1222no—
426#117—1228no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4D24411yescharacteristic
439C2×C4⋊C4D24411yescharacteristic
440C2×C4⋊C4D24411yescharacteristic
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42C22×D6111yescharacteristic
452C2×C42C22×D6111yescharacteristic
453C2×C42C22×D6111yescharacteristic
454C4⋊C8—22no—
455C4⋊C8—22no—
456C4⋊C8—22no—
457#103—1222no—
458#103—1222no—
459#103—1222no—
460#103—1222no—
461#103—2441no—
462#103—2441no—
463#103—2441no—
464#103—2441no—
465#102—1222no—
466#102—1222no—
467#102—2441no—
468#102—2441no—
469#97—22no—
470#97—22no—
471C4×C8—22no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—1221no—
480C2×Q8—41no—
481C2×Q8—41no—
482C2×Q8—612no—
483C2×Q8—612no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8#1175819814yes—
495C2×D4—612no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1224no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4C2×D24411yescharacteristic
512C22×C4C2×D24411yescharacteristic
513C22×C4C2×D24411yescharacteristic
514C22×C4C2×D24411yescharacteristic
515C22×C4C2×D24411yescharacteristic
516C22×C4C2×D24411yescharacteristic
517C22×C4C23×D6111yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8#1175822612yes—
530C4⋊C4—612no—
531C4⋊C4—612no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—4881no—
545C4⋊C4—2442no—
546C4⋊C4#1175819814yes—
547C4⋊C4#1175819814yes—
548C4⋊C4#1175819814yes—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1224no—
553C22⋊C4—1224no—
554C42—1224no—
555C42—1224no—
556C42—4881no—
557C42—4881no—
558C42—2442no—
559C42C2×C4×D6112yes—
560C42C2×C4×D6112yes—
561C42C2×C4×D6112yes—
562C42C2×C22⋊D6112yes—
563C42C2×C22⋊D6112yes—
564C42C2×C22⋊D6112yes—
565C23—1222no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23C22×D24411yescharacteristic
573Q8—44no—
574Q8—1222no—
575Q8—1222no—
576Q8—1222no—
577Q8—1222no—
578Q8—2441no—
579Q8—2441no—
580Q8—2441no—
581Q8—2441no—
582D4—1222no—
583D4—1222no—
584D4—2441no—
585D4—2441no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—2448no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4C2×#1175819812yes—
621C2×C4C2×#1175819812yes—
622C2×C4C2×#1175819812yes—
623C2×C4C2×#1175819812yes—
624C2×C4C2×#1175819812yes—
625C2×C4C2×#1175819812yes—
626C2×C4C22×C4×D6111yescharacteristic
627C2×C4C22×C22⋊D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4C2×#1175822611yescharacteristic
630C2×C4#1451982311yescharacteristic
631C2×C4#1451982511yescharacteristic
632C8—22no—
633C8—41no—
634C8—41no—
635C8—41no—
636C22—4881no—
637C22—4881no—
638C22—4881no—
639C22—4881no—
640C22—2442no—
641C22—2442no—
642C22—2442no—
643C22C22×#1175819811yescharacteristic
644C22#1684042411yescharacteristic
645C22#1684043411yescharacteristic
646C22#1684056211yescharacteristic
647C22#1684056211yescharacteristic
648C22#1684057411yescharacteristic
649C22#1684057911yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—4881no—
657C4—4881no—
658C4—4881no—
659C4—2442no—
660C4—2442no—
661C4—2442no—
662C4—2442no—
663C4C4×#1175822611yescharacteristic
664C4#1684024411yescharacteristic
665C4#1684042311yescharacteristic
666C4#1684043111yescharacteristic
667C2—4882no—
668C2#2144865711yescharacteristic
669C2#2145433911yescharacteristic
670C2#2145435311yescharacteristic
671C2#2145864811yescharacteristic
672C2#2145866311yescharacteristic
673C2#2145888711yescharacteristic
674C2#2145890011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#446C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterC2×#446D61
hypocenterC61#22548
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #21448776C2semidirect
5. #21446741C2semidirect
6. #21446639C2semidirect
9. #21445835C2semidirect
11. #21445825C2semidirect
12. #21445650C2semidirect
14. C4×#14519702C2semidirect
15. C2×C61×#446C2semidirect
17. #16840485C4semidirect
331. C2×#446D61semidirect
399. C61#22548semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
#214487762
#214467492
#214467412
#214466392
#214466352
#214466302
#214458352
#214458302
#214458252
#214456502
#214456452
C4×#145197022
C2×C61×#4462
#2254861