Finite Group Databasebeta

← #29698368

Subgroups — #29698368

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839864C211yesmaximal, characteristic
2C2×#16839864C211yesmaximal, characteristic
3C2×C4×#11758190C211yesmaximal, characteristic
4#21448739C211yesmaximal, characteristic, complemented
5C4⋊C4×D244C211yesmaximal, characteristic
6#21446557C211yesmaximal, characteristic, complemented
7#21446552C211yesmaximal, characteristic, complemented
8#21446090C211yesmaximal, characteristic, complemented
9#21446083C211yesmaximal, characteristic
10#21446076C211yesmaximal, characteristic, complemented
11#21446069C211yesmaximal, characteristic
12#21445987C211yesmaximal, characteristic, complemented
13#21445830C211yesmaximal, characteristic
14#21445825C211yesmaximal, characteristic, complemented
15C61×#1771C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4⋊C4×D61—21no—
24#16839864—24no—
25#16839864C418yescomplemented
26C4×#11758190—24no—
27C2×#14519711C2211yescharacteristic
28C2×#14519711C2211yescharacteristic
29C2×#14519711C2211yescharacteristic
30C2×#14519709C2211yescharacteristic
31C2×#14519709C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839657—21no—
43#16839657—21no—
44#16839636—21no—
45#16839636—21no—
46#16839632C2211yescharacteristic
47#16839628C2211yescharacteristic
48#16839626—21no—
49#16839619—21no—
50#16839613—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54#16839525C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190C2×C414yes—
68C2×#11758190D414yes—
69C2×#11758190Q814yes—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73C4×Dic122—41no—
74C4×Dic122—41no—
75C4×Dic122C2×C414yescomplemented
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244—41no—
80C4×D244—41no—
81C4×D244C2×C414yes—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C4⋊C4×D61—44no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104#14519711—22no—
105#14519711—41no—
106#14519711C2×C414yescomplemented
107#14519709—22no—
108#14519709—41no—
109#14519702—22no—
110#14519702—22no—
111#14519702C2×C414yescomplemented
112#14519699—22no—
113#14519699—22no—
114#14519699C2×C414yes—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61—22no—
131#97×C61—22no—
132#97×C61C2×C414yescomplemented
133C4×C8×C61—22no—
134#95×C61D411yescharacteristic
135#95×C61D411yescharacteristic
136C23×D61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C22×C412yes—
155C2×D244C2×D412yes—
156C2×D244C2×Q812yes—
157C2×C4×D61—24no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—81no—
165C2×C4×D61—81no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170C2×Dic122C2×D412yes—
171C2×Dic122C2×Q812yes—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C22×C412yes—
181C2×C8×C61C22×C412yes—
182C2×C8×C61C2×D412yes—
183C2×C8×C61C2×Q812yes—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61C22×C412yes—
191C42×C61#4112yes—
192#11758198—24no—
193#11758198—42no—
194#11758198—42no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206C4⋊Dic61—81no—
207C4⋊Dic61C22×C412yes—
208C4⋊Dic61C22×C412yes—
209#11758196—24no—
210#11758196—42no—
211#11758196—42no—
212#11758196—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196—42no—
216C4×Dic61—24no—
217C4×Dic61—42no—
218C4×Dic61—42no—
219#11758190C4⋊C4116yescomplemented
220C23×C61C22×D411yescharacteristic
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
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×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61AGL(1,ℤ/8)12yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×Q811yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
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—
273D244—22no—
274D244—22no—
275D244—22no—
276D244C2×C4⋊C414yes—
277C4×D61—48no—
278C4×D61—84no—
279C4×D61—84no—
280Dic122—22no—
281Dic122—22no—
282Dic122—22no—
283Dic122C2×C4⋊C414yescomplemented
284C8×C61—22no—
285C8×C61—41no—
286C8×C61—41no—
287C8×C61C2×C4⋊C414yescomplemented
288#22677—611nomaximal, Sylow
289C22×C61C2×C4×D411yescharacteristic
290C22×C61D4211yescharacteristic
291C22×C61D4×Q811yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61#57511yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57811yescharacteristic
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—22no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61C22×C4⋊C411yescharacteristic
310C4×C61#52011yescharacteristic
311C4×C61#54711yescharacteristic
312C4×C61#55411yescharacteristic
313Dic61—24no—
314Dic61—42no—
315Dic61—42no—
316Dic61—81no—
317Dic61—81no—
318Dic61—81no—
319Dic61—81no—
320C2×#440—611no—
321C2×#440—611no—
322C2×C4×SD16—611no—
323#2351—611no—
324D4×C4⋊C4—611no—
325#1943—611no—
326#1942—611no—
327#1865—611no—
328#1864—611no—
329#1863—611no—
330#1862—611no—
331#1847—611no—
332#1804—611no—
333#1803—611no—
334#1771D6111yescharacteristic, complemented
335C2×C61D4×C4⋊C411yescharacteristic
336C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
337C2×C61#294711yescharacteristic
338C2×C61#326511yescharacteristic
339C2×C61#326911yescharacteristic
340C2×C61#338611yescharacteristic
341C2×C61#338811yescharacteristic
342D61—44no—
343C22×SD16—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C22×C4⋊C4—1221no—
351#440—618no—
352#440—1224no—
353C4×SD16—1224no—
354C2×#107—611no—
355C2×#107—611no—
356C2×#107—611no—
357C2×#106—611no—
358C2×#106—611no—
359C2×C4⋊C8C2×D6111yescharacteristic
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#97C2×D6111yescharacteristic
367C2×#97C2×D6111yescharacteristic
368C2×C4×C8C2×D6111yescharacteristic
369#389—611no—
370#389—611no—
371#389—611no—
372#389—611no—
373#389—1221no—
374#389—1221no—
375#385—1221no—
376#385—1221no—
377#384—611no—
378#382—611no—
379#382—1221no—
380#380—1221no—
381C4×C4⋊C4—1221no—
382C4×C4⋊C4C2×D6111yescharacteristic
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#336C2×D6111yescharacteristic
388#336C2×D6111yescharacteristic
389C61#2267711yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×D4—611no—
392C23×C4—1221no—
393C23×C4—1221no—
394C23×C4—1221no—
395C2×SD16—614no—
396C2×SD16—614no—
397C2×SD16—614no—
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402C4×Q8—614no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—2441no—
407C4×Q8—2441no—
408C4×D4—614no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—2441no—
413C4×D4—2441no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—2444no—
430C2×C4⋊C4D24411yescharacteristic
431C2×C4⋊C4D24411yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439C2×C42C22×D6111yescharacteristic
440C2×C42C22×D6111yescharacteristic
441#107—614no—
442#107—1222no—
443#107—2441no—
444#106—1222no—
445#106—2441no—
446C4⋊C8—22no—
447#103—614no—
448#103—1222no—
449#103—1222no—
450#102—614no—
451#102—1222no—
452#102—1222no—
453#97—22no—
454#97C4×D6114yescomplemented
455C4×C8—22no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95D24411yescharacteristic
463#95D24411yescharacteristic
464C24—1221no—
465C2×Q8—612no—
466C2×Q8—612no—
467C2×Q8—612no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×D4—612no—
472C2×D4—612no—
473C2×D4—612no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1224no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—4881no—
486C22×C4—4881no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4C2×D24411yescharacteristic
494C22×C4C2×D24411yescharacteristic
495C22×C4C2×D24411yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C2×D24411yescharacteristic
498C22×C4C2×D24411yescharacteristic
499C22×C4C23×D6111yescharacteristic
500SD16—6116no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8C2×C4×D6112yes—
510C2×C8C2×C4×D6112yes—
511C2×C8D4×D6112yes—
512C2×C8Q8×D6112yes—
513C4⋊C4—22no—
514C4⋊C4—41no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1224no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—4881no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C22⋊C4—1224no—
537C22⋊C4—2442no—
538C22⋊C4—2442no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—1224no—
544C42—2442no—
545C42—2442no—
546C42C2×C4×D6112yes—
547C42#1175822212yes—
548C23—1222no—
549C23—1222no—
550C23—1222no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23C22×D24411yescharacteristic
556Q8—614no—
557Q8—1222no—
558Q8—1222no—
559Q8—1222no—
560D4—614no—
561D4—1222no—
562D4—1222no—
563D4—1222no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—4884no—
585C2×C4—4884no—
586C2×C4—2448no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4C4×D24412yes—
604C2×C4C4×D24412yes—
605C2×C4#1451980512yes—
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4C2×Q8×D6111yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4#1451982311yescharacteristic
611C2×C4#1451982411yescharacteristic
612C8—22no—
613C8—41no—
614C8—41no—
615C8C4⋊C4×D6114yescomplemented
616C22—1224no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22—2442no—
622C22—2442no—
623C22C2×C4×D24411yescharacteristic
624C22D4×D24411yescharacteristic
625C22Q8×D24411yescharacteristic
626C22C2×#1451980511yescharacteristic
627C22#1684056211yescharacteristic
628C22#1684057411yescharacteristic
629C22#1684057911yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—1224no—
637C4—4881no—
638C4—4881no—
639C4—4881no—
640C4—4881no—
641C4—2442no—
642C4—2442no—
643C4C2×C4⋊C4×D6111yescharacteristic
644C4#1684024211yescharacteristic
645C4#1684041711yescharacteristic
646C4#1684047211yescharacteristic
647C2—2444no—
648C2C4⋊C4×D24411yescharacteristic
649C2C4×#1451980511yescharacteristic
650C2#2145466411yescharacteristic
651C2#2145731911yescharacteristic
652C2#2145737911yescharacteristic
653C2#2145874211yescharacteristic
654C2#2145876611yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1771C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1771D61
hypocenterC61#22677
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
4. #21448739C2semidirect
6. #21446557C2semidirect
7. #21446552C2semidirect
8. #21446090C2semidirect
10. #21446076C2semidirect
12. #21445987C2semidirect
14. #21445825C2semidirect
15. C61×#1771C2semidirect
25. #16839864C4semidirect
75. C4×Dic122C2×C4semidirect
106. #14519711C2×C4semidirect
111. #14519702C2×C4semidirect
132. #97×C61C2×C4semidirect
219. #11758190C4⋊C4semidirect
283. Dic122C2×C4⋊C4semidirect
287. C8×C61C2×C4⋊C4semidirect
334. #1771D61semidirect
389. C61#22677semidirect
454. #97C4×D61semidirect
615. C8C4⋊C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
C2×C4×#117581902
#214487392
C4⋊C4×D2442
#214465572
#214465522
#214460902
#214460832
#214460762
#214460692
#214459872
#214458302
#214458252
C61×#17712
#2267761