Finite Group Databasebeta

← #29695397

Subgroups — #29695397

Order 15616.

1410 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic
2D4×#11758194C211yesmaximal, characteristic, complemented
3C2×#16839744C212yesmaximal, complemented
4#21446663C211yesmaximal, characteristic, complemented
5#21446662C211yesmaximal, characteristic, complemented
6#21446287C212yesmaximal, complemented
7#21446285C212yesmaximal, complemented
8#21445808C211yesmaximal, characteristic, complemented
9#21445745C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic, complemented
11D4×M16×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2212yes—
13C22×#11758194C2212yes—
14#16840417—21no—
15#16840417—22no—
16#16840417—21no—
17#16840417—21no—
18#16840417—22no—
19#16840417—21no—
20C2×#14519777C2211yescharacteristic
21C2×#14519759C414yes—
22C2×#14519756C2212yes—
23C2×#14519748C412yes—
24C2×C4×D244C2211yescharacteristic
25C2×C4⋊C4×D61C412yes—
26#16839840—21no—
27#16839834—22no—
28D4×#92418—21no—
29#16839786C2211yescharacteristic
30#16839744—28no—
31C2×#14519695C2212yes—
32C2×#14519695C2212yes—
33C2×#14519695C2212yes—
34C2×#14519695C2212yes—
35C2×#14519693C2211yescharacteristic
36C2×#14519693C2212yes—
37C2×#14519693C2211yescharacteristic
38#16839726C2212yes—
39C4×#11758194C2211yescharacteristic
40#16839530C2211yescharacteristic
41#16839530C2212yes—
42#16839530C2211yescharacteristic
43C22×M16×C61C2212yes—
44C2×C4×D4×C61C2211yescharacteristic
45C61×#436—21no—
46C61×#435—22no—
47C8×D4×C61—21no—
48C61×#423C2211yescharacteristic
49C61×#407C2212yes—
50C4×M16×C61C2211yescharacteristic
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222—22no—
55C2×#11758222—24no—
56C2×#11758222—24no—
57C2×#11758222—42no—
58C2×#11758222—42no—
59C22×Dic122C2312yes—
60C22×D244C2311yescharacteristic
61C22×C22⋊D61C2×C414yes—
62C22×C4×D61C2×C414yes—
63C22×C4×D61C2×C412yes—
64C2×#11758194—22no—
65C2×#11758194—24no—
66C2×#11758194—42no—
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2312yes—
69C2×#11758194C2311yescharacteristic
70C22×#92418—22no—
71#14519777—21no—
72#14519777—21no—
73#14519777—21no—
74#14519777—21no—
75#14519759—24no—
76#14519759—24no—
77#14519759—24no—
78#14519759—24no—
79#14519756—22no—
80#14519756—22no—
81#14519756—22no—
82#14519756—22no—
83#14519748—22no—
84#14519748—22no—
85#14519748—24no—
86C4×D244—21no—
87C4×D244—21no—
88C4×D244—21no—
89C4×D244—21no—
90C2×C4⋊Dic61C2×C412yes—
91C2×C4⋊Dic61C2×C412yes—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196C2×C414yes—
94C4⋊C4×D61—24no—
95C4⋊C4×D61—22no—
96C4⋊C4×D61—22no—
97C2×C22⋊Dic61C2×C414yes—
98C2×#11758198C2×C412yes—
99C4⋊#92418—21no—
100C4⋊#92418—21no—
101#14519695—24no—
102#14519695—24no—
103#14519695—42no—
104#14519695—42no—
105#14519693—24no—
106#14519693—22no—
107#14519693D414yescomplemented
108C22⋊#92418—22no—
109C22⋊#92418—22no—
110#14519688—21no—
111C4×#92418—21no—
112C22×D4×C61C2×C411yescharacteristic
113C23×C4×C61C2312yes—
114C2×M16×C61—22no—
115C2×M16×C61—24no—
116C2×M16×C61—42no—
117C2×M16×C61C2312yes—
118C2×M16×C61C2311yescharacteristic
119C2×M16×C61C2311yescharacteristic
120C22×C8×C61—22no—
121C4×D4×C61D411yescharacteristic
122C4×D4×C61D412yes—
123C4×D4×C61D411yescharacteristic
124C4×D4×C61D411yescharacteristic
125C4×D4×C61D412yes—
126C4×D4×C61D411yescharacteristic
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C22⋊C4×C61C2×C412yes—
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61—21no—
131C4⋊C8×C61—21no—
132C22⋊C8×C61—22no—
133C22⋊C8×C61—22no—
134#97×C61—21no—
135C4×C8×C61—21no—
136C24×C61C22×C412yes—
137C23×D61C22×C412yes—
138C2×D4×C61—42no—
139C2×D4×C61C22⋊C411yescharacteristic
140C2×D4×C61C22⋊C411yescharacteristic
141C2×D4×C61C22⋊C412yes—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C412yes—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—24no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—28no—
157C2×C22⋊D61—44no—
158C22×Dic61C22×C414yes—
159C22×Dic61C22×C412yes—
160#11758222—44no—
161#11758222—44no—
162#11758222—44no—
163#11758222—44no—
164#11758222—44no—
165#11758222—44no—
166#11758222—44no—
167#11758222—44no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244C2×D412yes—
173C2×D244#4112yes—
174C2×C4×D61—28no—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—44no—
180C2×C4×D61—44no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—22no—
184C2×Dic122—22no—
185C2×Dic122—22no—
186C2×Dic122—22no—
187M16×C61—21no—
188M16×C61—22no—
189M16×C61—21no—
190M16×C61—22no—
191M16×C61—21no—
192M16×C61—44no—
193M16×C61C2×D412yes—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—22no—
197C2×C8×C61—44no—
198C4⋊C4×C61C22⋊C412yes—
199C4⋊C4×C61C22⋊C411yescharacteristic
200C4⋊C4×C61C22⋊C411yescharacteristic
201C22⋊C4×C61C22⋊C412yes—
202C22⋊C4×C61C22⋊C412yes—
203C22⋊C4×C61C22⋊C412yes—
204C22⋊C4×C61C22⋊C412yes—
205C42×C61C2×D411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C22⋊Dic61—24no—
210C22⋊Dic61—24no—
211#11758198—22no—
212#11758198—22no—
213C4⋊Dic61—22no—
214C4⋊Dic61—22no—
215C4⋊Dic61—22no—
216C4⋊Dic61—22no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219#11758196—24no—
220#11758196—24no—
221#11758194—21no—
222#11758194—22no—
223#11758194—21no—
224#11758194—21no—
225#11758194—22no—
226#11758194—44no—
227#11758194C2×D412yes—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—22no—
231C2×#92418—44no—
232C23×C61—22no—
233C23×C61—22no—
234C23×C61—42no—
235C23×C61—42no—
236C23×C61C2×C22⋊C412yes—
237C23×C61C2×C22⋊C412yes—
238C23×C61C23×C411yescharacteristic
239C22×D61—22no—
240C22×D61—22no—
241C22×D61—42no—
242C22×D61—42no—
243C22×D61C4×D414yes—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—42no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—41no—
259C2×C4×C61C2×C22⋊C412yes—
260C2×C4×C61C2×C22⋊C412yes—
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C22⋊D61—48no—
272C22⋊D61—48no—
273C2×Dic61—22no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—24no—
277C2×Dic61—42no—
278D244—22no—
279D244—22no—
280D244—22no—
281D244—22no—
282C4×D61—48no—
283C4×D61—48no—
284C4×D61—44no—
285C4×D61—44no—
286Dic122—42no—
287Dic122—42no—
288Dic122—42no—
289Dic122—42no—
290C8×C61—22no—
291C8×C61—42no—
292C8×C61—41no—
293#92418—22no—
294#92418—42no—
295#92418—41no—
296#22411—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—82no—
306C22×C61—82no—
307C22×C61C22×C22⋊C411yescharacteristic
308C22×C61C2×C4×D411yescharacteristic
309C22×C61#51811yescharacteristic
310C2×D61—24no—
311C2×D61—44no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—44no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61#41312yescomplemented
321C4×C61D4211yescharacteristic
322C4×C61D4211yescharacteristic
323C4×C61#54611yescharacteristic
324C4×C61#54611yescharacteristic
325Dic61—44no—
326Dic61—42no—
327C2×#547—611no—
328D4×M16—611no—
329D4×M16D6111yescharacteristic, complemented
330C2×#413—612no—
331#1966—611no—
332#1966—611no—
333#1901—612no—
334#1901—612no—
335#1800—611no—
336#1785—612no—
337C4×#100—611no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61—42no—
341C2×C61—42no—
342C2×C61D4×C22⋊C411yescharacteristic
343C2×C61C2×#41311yescharacteristic
344C2×C61#288911yescharacteristic
345D61—44no—
346C22×#41—612no—
347C22×M16—612no—
348C22×M16C2×D6112yes—
349#547—1221no—
350#547—1222no—
351#547—1221no—
352#547—1221no—
353#547—1222no—
354#547—1221no—
355C2×C4⋊Q8—611no—
356C2×#123—614no—
357C2×C22⋊Q8—612no—
358C2×#121—612no—
359C2×C4×D4—611no—
360C2×C4×D4C2×D6111yescharacteristic
361C22×C4⋊C4—612no—
362#436—21no—
363#436—1221no—
364#435—22no—
365#435—1222no—
366C8×D4—21no—
367C8×D4—1221no—
368#423—611no—
369#423C2×D6111yescharacteristic
370#413—1228no—
371C2×#101—612no—
372C2×#101—612no—
373C2×#101—612no—
374C2×#101—612no—
375C2×#100—611no—
376C2×#100—612no—
377C2×#100—611no—
378#407—612no—
379#407C2×D6112yes—
380C4×M16—611no—
381C4×M16C2×D6111yescharacteristic
382#343—612no—
383#343—611no—
384#343—611no—
385C61#2241111yescharacteristic, complemented, Sylow
386C2×#41—1224no—
387C2×#41—1224no—
388C2×#41—1222no—
389C2×#41—1222no—
390C2×#41—1222no—
391C2×#41—1222no—
392C2×#41—2442no—
393C2×#41—2442no—
394C22×Q8—612no—
395C22×D4—614no—
396C22×D4—611no—
397C22×D4C4×D6111yescharacteristic
398C23×C4—614no—
399C23×C4—612no—
400C23×C4C22×D6112yes—
401C2×M16—22no—
402C2×M16—24no—
403C2×M16—42no—
404C2×M16—611no—
405C2×M16—612no—
406C2×M16—611no—
407C2×M16—1222no—
408C2×M16—1224no—
409C2×M16—2442no—
410C2×M16C22×D6112yes—
411C2×M16C22×D6111yescharacteristic
412C2×M16C22×D6111yescharacteristic
413C22×C8—22no—
414C22×C8—1222no—
415C4⋊Q8—1221no—
416C4⋊Q8—1221no—
417C4⋊Q8—1221no—
418C4⋊Q8—1221no—
419#123—1224no—
420#123—1224no—
421#123—1224no—
422#123—1224no—
423C22⋊Q8—1222no—
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426C22⋊Q8—1222no—
427#121—1222no—
428#121—1222no—
429#121—1224no—
430C4×D4—1221no—
431C4×D4—1221no—
432C4×D4—1221no—
433C4×D4—1221no—
434C4×D4D24411yescharacteristic
435C4×D4D24412yes—
436C4×D4D24411yescharacteristic
437C4×D4C22⋊D6111yescharacteristic
438C4×D4C22⋊D6112yes—
439C4×D4C22⋊D6111yescharacteristic
440C2×C4⋊C4—612no—
441C2×C4⋊C4—614no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1224no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4C4×D6111yescharacteristic
448C2×C22⋊C4—614no—
449C2×C22⋊C4—612no—
450C2×C22⋊C4C4×D6112yes—
451C2×C42C22×D6111yescharacteristic
452C4⋊C8—21no—
453C4⋊C8—21no—
454C4⋊C8—1221no—
455C4⋊C8—1221no—
456#101—1224no—
457#101—1224no—
458#101—2442no—
459#101—2442no—
460#100—614no—
461#100—1222no—
462#100—1224no—
463C22⋊C8—22no—
464C22⋊C8—22no—
465C22⋊C8—1222no—
466C22⋊C8—1222no—
467#97—21no—
468#97—1221no—
469C4×C8—21no—
470C4×C8—1221no—
471C24—612no—
472C24C2×C4×D6112yes—
473#41—2444no—
474#41—2444no—
475#41—2444no—
476#41—2444no—
477#41—2444no—
478#41—2444no—
479#41—2444no—
480#41—2444no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×Q8—1222no—
484C2×Q8—1222no—
485C2×Q8—1222no—
486C2×Q8—1222no—
487C2×D4—42no—
488C2×D4—612no—
489C2×D4—612no—
490C2×D4—1228no—
491C2×D4—1224no—
492C2×D4—1224no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—2444no—
498C2×D4#1175819811yescharacteristic
499C2×D4#1175819811yescharacteristic
500C2×D4#1175819812yes—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—614no—
506C22×C4—612no—
507C22×C4—1228no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1224no—
511C22×C4—1224no—
512C22×C4—2444no—
513C22×C4—2444no—
514C22×C4C2×C4×D6111yescharacteristic
515C22×C4C2×C4×D6112yes—
516C22×C4C2×C4×D6111yescharacteristic
517C22×C4C2×D24412yes—
518C22×C4C2×D24412yes—
519C22×C4C2×C22⋊D6112yes—
520C22×C4C2×C22⋊D6112yes—
521C22×C4C23×D6111yescharacteristic
522M16—21no—
523M16—22no—
524M16—21no—
525M16—22no—
526M16—21no—
527M16—44no—
528M16—612no—
529M16—1221no—
530M16—1222no—
531M16—1221no—
532M16—1221no—
533M16—1222no—
534M16—2444no—
535M16D4×D6112yes—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—22no—
539C2×C8—44no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1222no—
543C2×C8—2444no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1222no—
549C4⋊C4—1222no—
550C4⋊C4—1222no—
551C4⋊C4—1222no—
552C4⋊C4#1175819812yes—
553C4⋊C4#1175819811yescharacteristic
554C4⋊C4#1175819811yescharacteristic
555C22⋊C4—1224no—
556C22⋊C4—1224no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4#1175819812yes—
560C22⋊C4#1175819812yes—
561C22⋊C4#1175819812yes—
562C22⋊C4#1175819812yes—
563C42C2×D24411yescharacteristic
564C42C2×D24411yescharacteristic
565C42C2×C22⋊D6111yescharacteristic
566C42C2×C22⋊D6111yescharacteristic
567C23—22no—
568C23—22no—
569C23—42no—
570C23—42no—
571C23—614no—
572C23—1222no—
573C23—1222no—
574C23—2442no—
575C23—2442no—
576C23C2×#1175819812yes—
577C23C2×#1175819812yes—
578C23C22×C4×D6111yescharacteristic
579Q8—2442no—
580Q8—2442no—
581Q8—2442no—
582Q8—2442no—
583D4—22no—
584D4—22no—
585D4—42no—
586D4—1222no—
587D4—1222no—
588D4—1222no—
589D4—1222no—
590D4—2448no—
591D4—2448no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—1222no—
605C2×C4—1224no—
606C2×C4—1224no—
607C2×C4—1224no—
608C2×C4—2448no—
609C2×C4—2448no—
610C2×C4—2444no—
611C2×C4—2444no—
612C2×C4—2442no—
613C2×C4C2×#1175819811yescharacteristic
614C2×C4C2×#1175819811yescharacteristic
615C2×C4C2×#1175819811yescharacteristic
616C2×C4C2×#1175819812yes—
617C2×C4C2×#1175819812yes—
618C2×C4C2×#1175819811yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C22×C22⋊D6111yescharacteristic
621C2×C4C22×D24411yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4#1451982211yescharacteristic
624C2×C4#1451982311yescharacteristic
625C8—22no—
626C8—42no—
627C8—41no—
628C8—1222no—
629C8—2442no—
630C8—2441no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—82no—
640C22—82no—
641C22—1224no—
642C22—2444no—
643C22—2444no—
644C22—2442no—
645C22—2442no—
646C22C22×#1175819811yescharacteristic
647C22C4×D4×D6111yescharacteristic
648C22#1684023411yescharacteristic
649C4—22no—
650C4—22no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—2444no—
655C4—2442no—
656C4#1683974412yescomplemented
657C4D4×C22⋊D6111yescharacteristic
658C4D4×D24411yescharacteristic
659C4#1684040011yescharacteristic
660C4#1684040711yescharacteristic
661C2—21no—
662C2—21no—
663C2—42no—
664C2—42no—
665C2—2444no—
666C2D4×#1175819811yescharacteristic
667C2C2×#1683974411yescharacteristic
668C2#2145430111yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingD4×M16×C61C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840417C2
cosocleC22×C4×C61C24
hypercenterD4×M16D61
hypocenterC61#22411
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
2. D4×#11758194C2semidirect
3. C2×#16839744C2semidirect
4. #21446663C2semidirect
5. #21446662C2semidirect
6. #21446287C2semidirect
7. #21446285C2semidirect
8. #21445808C2semidirect
9. #21445745C2semidirect
10. C4×#14519693C2semidirect
11. D4×M16×C61C2semidirect
107. #14519693D4semidirect
320. C4×C61#413semidirect
329. D4×M16D61semidirect
385. C61#22411semidirect
656. C4#16839744semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
D4×#117581942
C2×#168397442
#214466632
#214466622
#214462872
#214462852
#214458082
#214457452
C4×#145196932
D4×M16×C612
#2241161