Finite Group Databasebeta

← #29688987

Subgroups — #29688987

Order 15616.

1426 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic
2C2×#16839726C211yesmaximal, characteristic
3#21446287C214yesmaximal, complemented
4#21446285C214yesmaximal, complemented
5#21445897C212yesmaximal, complemented
6#21445742C212yesmaximal, complemented
7C2×C61×#407C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C22×#11758194C2212yes—
10C2×#14519756—21no—
11C2×#14519756—21no—
12C2×#14519756—21no—
13C2×#14519756—21no—
14C2×#14519756—21no—
15C2×#14519756—21no—
16C2×#14519756—21no—
17C2×#14519756—21no—
18C2×#14519756C412yes—
19C2×#14519756C412yes—
20C2×#14519756C2212yes—
21C2×#14519756C2212yes—
22C22×C4⋊Dic61C2211yescharacteristic
23C22×#11758196C412yes—
24C22×C22⋊Dic61C412yes—
25C2×#14519695C2218yescomplemented
26#16839726—21no—
27#16839726—21no—
28#16839726C2212yes—
29#16839726C2212yes—
30C2×C22⋊#92418—21no—
31C2×C22⋊#92418—21no—
32#16839530C2218yescomplemented
33C24×C4×C61C2211yescharacteristic
34C22×M16×C61C2212yes—
35C61×#407—21no—
36C61×#407—21no—
37C61×#407C2212yes—
38C61×#407C2212yes—
39C2×C22⋊C8×C61—21no—
40C2×C22⋊C8×C61—21no—
41C22×Dic122—21no—
42C22×Dic122—21no—
43C22×Dic122—21no—
44C22×Dic122—21no—
45C22×Dic122—44no—
46C22×Dic122C2×C412yes—
47C22×Dic122C2312yes—
48C23×Dic61C2×C412yes—
49C23×Dic61C2×C412yes—
50C2×#11758194—22no—
51C2×#11758194—44no—
52C2×#11758194C2314yes—
53C22×#92418—22no—
54#14519756—22no—
55#14519756—22no—
56#14519756—22no—
57#14519756—22no—
58#14519756—22no—
59#14519756—22no—
60#14519756—22no—
61#14519756—22no—
62#14519756—22no—
63#14519756—22no—
64#14519756—22no—
65#14519756—22no—
66#14519756—22no—
67#14519756—22no—
68#14519756—22no—
69#14519756—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2×C412yes—
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C2×#11758196—22no—
80C2×#11758196C2×C414yes—
81C2×#11758196C2×C414yes—
82C2×C22⋊Dic61—22no—
83C2×C22⋊Dic61—22no—
84C2×C22⋊Dic61—22no—
85C2×C22⋊Dic61—22no—
86C2×C22⋊Dic61C2×C414yes—
87C2×C22⋊Dic61C2×C414yes—
88#14519695—48no—
89C22⋊#92418—22no—
90C22⋊#92418—22no—
91C22⋊#92418—22no—
92C22⋊#92418—22no—
93C25×C61C2×C411yescharacteristic
94C23×C4×C61—44no—
95C23×C4×C61C2×C411yescharacteristic
96C23×C4×C61C2×C411yescharacteristic
97C23×C4×C61C2×C411yescharacteristic
98C23×C4×C61D411yescharacteristic
99C23×C4×C61D411yescharacteristic
100C23×C4×C61D411yescharacteristic
101C23×C4×C61D411yescharacteristic
102C23×C4×C61D411yescharacteristic
103C23×C4×C61D411yescharacteristic
104C23×C4×C61D411yescharacteristic
105C23×C4×C61D411yescharacteristic
106C23×C4×C61C2311yescharacteristic
107C23×C4×C61C2311yescharacteristic
108C23×C4×C61C2311yescharacteristic
109C2×M16×C61—22no—
110C2×M16×C61—44no—
111C2×M16×C61C2314yes—
112C22×C8×C61—22no—
113C22⋊C8×C61—22no—
114C22⋊C8×C61—22no—
115C22⋊C8×C61—22no—
116C22⋊C8×C61—22no—
117C24×C61—24no—
118C24×C61—42no—
119C24×C61—42no—
120C24×C61C22⋊C411yescharacteristic
121C24×C61C22⋊C411yescharacteristic
122C24×C61C22⋊C411yescharacteristic
123C24×C61C22⋊C411yescharacteristic
124C24×C61C22×C411yescharacteristic
125C24×C61C22×C411yescharacteristic
126C24×C61C22×C411yescharacteristic
127C22×C4×C61—24no—
128C22×C4×C61—24no—
129C22×C4×C61—24no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22⋊C411yescharacteristic
146C22×C4×C61C22⋊C411yescharacteristic
147C22×C4×C61C22⋊C411yescharacteristic
148C22×C4×C61C22⋊C411yescharacteristic
149C22×C4×C61C22⋊C411yescharacteristic
150C22×C4×C61C22⋊C411yescharacteristic
151C22×C4×C61C22⋊C411yescharacteristic
152C22×C4×C61C22⋊C411yescharacteristic
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C22×C411yescharacteristic
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2411yescharacteristic
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—44no—
174C22×Dic61—44no—
175C22×Dic61C22×C414yes—
176C22×Dic61C22×C414yes—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C2×Dic122—28no—
182C2×Dic122—44no—
183C2×Dic122—44no—
184C2×Dic122—44no—
185C2×Dic122—44no—
186C2×Dic122—44no—
187M16×C61—28no—
188M16×C61—44no—
189C2×C8×C61—24no—
190C2×C8×C61—44no—
191C22⋊Dic61—24no—
192C22⋊Dic61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199#11758196—24no—
200#11758196—24no—
201#11758196—24no—
202#11758196—24no—
203#11758194—28no—
204#11758194—44no—
205C2×#92418—24no—
206C2×#92418—44no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—416no—
228C23×C61#10114yescomplemented
229C23×C61C2×C22⋊C411yescharacteristic
230C23×C61C2×C22⋊C411yescharacteristic
231C23×C61C2×C22⋊C411yescharacteristic
232C23×C61C2×C22⋊C411yescharacteristic
233C23×C61C2×C22⋊C411yescharacteristic
234C23×C61C2×C22⋊C411yescharacteristic
235C23×C61C23×C411yescharacteristic
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
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×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C2×C22⋊C411yescharacteristic
267C2×C4×C61C2×C22⋊C411yescharacteristic
268C2×C4×C61C2×C22⋊C411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61H(5,𝔽2)14yes—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—28no—
275C2×Dic61—44no—
276C2×Dic61—44no—
277C2×Dic61—44no—
278C2×Dic61—44no—
279C2×Dic61—44no—
280Dic122—44no—
281Dic122—44no—
282Dic122—44no—
283Dic122—44no—
284C8×C61—44no—
285#92418—44no—
286#21753—611nomaximal, Sylow
287C22×C61—28no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—44no—
305C22×C61—44no—
306C22×C61—44no—
307C22×C61—44no—
308C22×C61—44no—
309C22×C61—44no—
310C22×C61C2×#10112yes—
311C22×C61C2×#10112yes—
312C22×C61C2×#10112yes—
313C22×C61#41312yes—
314C22×C61C22×C22⋊C411yescharacteristic
315C22×C61#51812yes—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61#53412yes—
323C4×C61#53412yes—
324Dic61—44no—
325Dic61—44no—
326C22×C22⋊Q8—611no—
327C2×#407—611no—
328C2×#407D6111yescharacteristic
329#1901—614no—
330#1901—614no—
331#1821—612no—
332#1784—612no—
333C2×C61—22no—
334C2×C61—22no—
335C2×C61—22no—
336C2×C61—22no—
337C2×C61—22no—
338C2×C61—41no—
339C2×C61—41no—
340C2×C61#230811yescharacteristic
341C2×C61C22×#10111yescharacteristic
342C2×C61C2×#41311yescharacteristic
343C23×Q8—611no—
344C24×C4C2×D6111yescharacteristic
345C22×M16—612no—
346C22×M16C2×D6112yes—
347C2×C22⋊Q8—612no—
348C2×C22⋊Q8—612no—
349C2×C22⋊Q8—612no—
350C2×C22⋊Q8—612no—
351C2×C22⋊Q8—1221no—
352C2×C22⋊Q8—1221no—
353C2×C22⋊Q8—1221no—
354C2×C22⋊Q8—1221no—
355C2×C22⋊Q8—1221no—
356C2×C22⋊Q8—1221no—
357C2×C22⋊Q8—1221no—
358C2×C22⋊Q8—1221no—
359C22×C4⋊C4—612no—
360C22×C4⋊C4—611no—
361C22×C22⋊C4—612no—
362C2×#101—618no—
363#407—21no—
364#407—21no—
365#407—612no—
366#407—612no—
367#407—1221no—
368#407—1221no—
369#407C2×D6112yes—
370#407C2×D6112yes—
371C2×C22⋊C8—21no—
372C2×C22⋊C8—21no—
373C2×C22⋊C8—1221no—
374C2×C22⋊C8—1221no—
375#343—618no—
376C61#2175311yescharacteristic, complemented, Sylow
377C25C4×D6111yescharacteristic
378C22×Q8—612no—
379C22×Q8—612no—
380C22×Q8—1221no—
381C22×Q8—1221no—
382C22×Q8—1221no—
383C22×Q8—1221no—
384C22×Q8—2444no—
385C23×C4—44no—
386C23×C4—612no—
387C23×C4—612no—
388C23×C4C4×D6111yescharacteristic
389C23×C4C4×D6111yescharacteristic
390C23×C4C4×D6111yescharacteristic
391C23×C4D24411yescharacteristic
392C23×C4D24411yescharacteristic
393C23×C4D24411yescharacteristic
394C23×C4D24411yescharacteristic
395C23×C4C22⋊D6111yescharacteristic
396C23×C4C22⋊D6111yescharacteristic
397C23×C4C22⋊D6111yescharacteristic
398C23×C4C22⋊D6111yescharacteristic
399C23×C4C22×D6111yescharacteristic
400C23×C4C22×D6111yescharacteristic
401C23×C4C22×D6111yescharacteristic
402C2×M16—22no—
403C2×M16—44no—
404C2×M16—614no—
405C2×M16—1222no—
406C2×M16—2444no—
407C2×M16C22×D6114yes—
408C22×C8—22no—
409C22×C8—1222no—
410C22⋊Q8—1222no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417C22⋊Q8—1222no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426C2×C4⋊C4—614no—
427C2×C4⋊C4—614no—
428C2×C4⋊C4—612no—
429C2×C4⋊C4—612no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C22⋊C4—614no—
439C2×C22⋊C4—614no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1222no—
444#101—2448no—
445C22⋊C8—22no—
446C22⋊C8—22no—
447C22⋊C8—22no—
448C22⋊C8—22no—
449C22⋊C8—1222no—
450C22⋊C8—1222no—
451C22⋊C8—1222no—
452C22⋊C8—1222no—
453C24—24no—
454C24—42no—
455C24—42no—
456C24#1175819811yescharacteristic
457C24#1175819811yescharacteristic
458C24#1175819811yescharacteristic
459C24#1175819811yescharacteristic
460C24C2×C4×D6111yescharacteristic
461C24C2×C4×D6111yescharacteristic
462C24C2×C4×D6111yescharacteristic
463C2×Q8—1222no—
464C2×Q8—1222no—
465C2×Q8—1228no—
466C2×Q8—1222no—
467C2×Q8—1222no—
468C2×Q8—2444no—
469C2×Q8—2444no—
470C2×Q8—2444no—
471C2×Q8—2444no—
472C2×Q8—2444no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—614no—
488C22×C4—614no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—2444no—
494C22×C4—2444no—
495C22×C4#1175819811yescharacteristic
496C22×C4#1175819811yescharacteristic
497C22×C4#1175819811yescharacteristic
498C22×C4#1175819811yescharacteristic
499C22×C4#1175819811yescharacteristic
500C22×C4#1175819811yescharacteristic
501C22×C4#1175819811yescharacteristic
502C22×C4#1175819811yescharacteristic
503C22×C4#1175819811yescharacteristic
504C22×C4#1175819811yescharacteristic
505C22×C4#1175819811yescharacteristic
506C22×C4#1175819811yescharacteristic
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4C2×C4×D6111yescharacteristic
509C22×C4C2×C4×D6111yescharacteristic
510C22×C4C2×D24411yescharacteristic
511C22×C4C2×D24411yescharacteristic
512C22×C4C2×D24411yescharacteristic
513C22×C4C2×D24411yescharacteristic
514C22×C4C2×D24411yescharacteristic
515C22×C4C2×D24411yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6111yescharacteristic
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C23×D6111yescharacteristic
523M16—28no—
524M16—44no—
525M16—1228no—
526M16—2444no—
527C2×C8—24no—
528C2×C8—44no—
529C2×C8—1224no—
530C2×C8—2444no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C22⋊C4—1224no—
540C22⋊C4—1224no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—416no—
564C23#1451969514yescomplemented
565C23C2×#1175819811yescharacteristic
566C23C2×#1175819811yescharacteristic
567C23C2×#1175819811yescharacteristic
568C23C2×#1175819811yescharacteristic
569C23C2×#1175819811yescharacteristic
570C23C2×#1175819811yescharacteristic
571C23C22×C4×D6111yescharacteristic
572Q8—2444no—
573Q8—2444no—
574Q8—2444no—
575Q8—2444no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1224no—
604C2×C4—1224no—
605C2×C4—1228no—
606C2×C4—2444no—
607C2×C4—2444no—
608C2×C4—2444no—
609C2×C4—2444no—
610C2×C4—2444no—
611C2×C4C2×#1175819811yescharacteristic
612C2×C4C2×#1175819811yescharacteristic
613C2×C4C2×#1175819811yescharacteristic
614C2×C4C2×#1175819811yescharacteristic
615C2×C4C2×#1175819811yescharacteristic
616C2×C4C2×#1175819811yescharacteristic
617C2×C4C22×C22⋊D6111yescharacteristic
618C2×C4C22×D24411yescharacteristic
619C2×C4#1451982214yes—
620C8—44no—
621C8—2444no—
622C22—28no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22C2×#1451969512yes—
646C22C2×#1451969512yes—
647C22C2×#1451969512yes—
648C22#1683974412yes—
649C22C22×#1175819811yescharacteristic
650C22#1684023512yes—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—2444no—
658C4—2444no—
659C4#1684032112yes—
660C4#1684032412yes—
661C2—22no—
662C2—22no—
663C2—22no—
664C2—22no—
665C2—22no—
666C2—41no—
667C2—41no—
668C2#2144839411yescharacteristic
669C2C22×#1451969511yescharacteristic
670C2C2×#1683974411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#407C2
socleC22×C61C22×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC22×C4×C61C24
hypercenterC2×#407D61
hypocenterC61#21753
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
3. #21446287C2semidirect
4. #21446285C2semidirect
5. #21445897C2semidirect
6. #21445742C2semidirect
25. C2×#14519695C22semidirect
32. #16839530C22semidirect
228. C23×C61#101semidirect
376. C61#21753semidirect
564. C23#14519695semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C2×#168397262
#214462872
#214462852
#214458972
#214457422
C2×C61×#4072
#2175361