Finite Group Databasebeta

← #29681265

Subgroups — #29681265

Order 15616.

1778 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840336C211yesmaximal, characteristic
2C2×#16840243C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839728C211yesmaximal, characteristic
5#21446128C214yesmaximal, complemented
6#21446123C214yesmaximal, complemented
7#21445538C211yesmaximal, characteristic, complemented
8#21445536C211yesmaximal, characteristic, complemented
9C22×#103×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840336—24no—
13#16840336C418yescomplemented
14C2×#14519775—21no—
15C2×#14519766—21no—
16C2×#14519760—22no—
17C2×#14519753C2214yes—
18#16840243—24no—
19#16840243—24no—
20C2×#14519742—21no—
21C2×#14519739—21no—
22C2×C4×C22⋊D61—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25C2×#14519735C2212yes—
26C2×C4⋊C4×D61C2212yes—
27#16839771—24no—
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30#16839728—24no—
31C2×#14519689C2214yes—
32#16839525C2212yes—
33#16839525C2212yes—
34#16839524C2214yes—
35C23×Q8×C61C2211yescharacteristic
36C23×C8×C61C2211yescharacteristic
37C22×C4⋊C4×C61C2211yescharacteristic
38C2×#103×C61—24no—
39C2×#103×C61C2214yes—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222C2×C414yes—
45C2×#11758222D414yes—
46C22×Dic122D411yescharacteristic
47C22×D244D411yescharacteristic
48C22×C22⋊D61D412yes—
49C22×C4×D61—22no—
50C22×C4×D61C2312yes—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C22×#92418C2312yes—
55#14519775—28no—
56#14519766—28no—
57#14519760—216no—
58#14519753—28no—
59#14519753C2×C4116yescomplemented
60#14519742—28no—
61#14519739—28no—
62C4×C22⋊D61—216no—
63#14519736—22no—
64#14519736—22no—
65#14519736C2×C414yescomplemented
66#14519736D414yes—
67#14519735—24no—
68#14519735—24no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—22no—
73C2×#11758196—24no—
74C4⋊C4×D61—24no—
75C4⋊C4×D61—24no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—22no—
79C2×#11758198—24no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83#14519701—24no—
84#14519701—24no—
85#14519689—28no—
86C22×Q8×C61—21no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61C2×C414yescomplemented
93C22×Q8×C61C2312yes—
94C23×C4×C61—21no—
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61—22no—
98C22×C8×C61—22no—
99C22×C8×C61—22no—
100C22×C8×C61C2312yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61D414yes—
104C2×C4⋊C4×C61C2312yes—
105#103×C61—28no—
106C24×C61C2×D411yescharacteristic
107C23×D61C2×D412yes—
108C2×Q8×C61—24no—
109C2×Q8×C61—22no—
110C2×Q8×C61—28no—
111C2×Q8×C61—28no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—42no—
117C2×Q8×C61—44no—
118C2×Q8×C61—44no—
119C2×Q8×C61—42no—
120C2×Q8×C61C22×C414yes—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—24no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—82no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—24no—
137C2×C22⋊D61—24no—
138C2×C22⋊D61C22⋊C418yes—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61C2×D412yes—
143#11758222—28no—
144#11758222—28no—
145#11758222—28no—
146#11758222—28no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244C22⋊C414yes—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—28no—
155C2×C4×D61—84no—
156C2×C4×D61C22×C414yes—
157C2×C4×D61C2×D414yes—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122C22⋊C414yes—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—44no—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61C2×D414yes—
168C22⋊Dic61—24no—
169C22⋊Dic61—24no—
170#11758198—28no—
171#11758198—216no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61C22×C414yes—
176C4⋊Dic61C2×D412yes—
177C4⋊Dic61C2×D412yes—
178#11758196—28no—
179#11758196—216no—
180C4×Dic61—28no—
181C4×Dic61—24no—
182C4×Dic61—24no—
183#11758194—44no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61C2×C22⋊C412yes—
190C23×C61#12012yes—
191C23×C61#12312yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—24no—
196C22×D61—24no—
197C22×D61C2×C22⋊C414yes—
198Q8×C61—24no—
199Q8×C61—24no—
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—48no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—44no—
208C2×C4×C61—48no—
209C2×C4×C61—84no—
210C2×C4×C61C2×C22⋊C412yes—
211C2×C4×C61C4×D414yes—
212C2×C4×C61C4×D414yes—
213C2×C4×C61#12012yes—
214C2×C4×C61#12114yes—
215C2×C4×C61#12114yes—
216C2×C4×C61#12312yes—
217C2×C4×C61C23×C411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C22⋊D61—216no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61C2×C22⋊C414yes—
234D244—28no—
235C4×D61—24no—
236C4×D61—24no—
237C4×D61—24no—
238C4×D61—24no—
239C4×D61—88no—
240Dic122—28no—
241C8×C61—42no—
242#92418—42no—
243#20913—611nomaximal, Sylow
244C22×C61—24no—
245C22×C61—24no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61#38614yes—
249C22×C61C22×C22⋊C411yescharacteristic
250C22×C61C2×#12011yescharacteristic
251C22×C61C2×#12311yescharacteristic
252C22×C61#57712yes—
253C22×C61#57812yes—
254C2×D61—28no—
255C2×D61—24no—
256C2×D61—44no—
257C2×D61—42no—
258C2×D61—42no—
259C4×C61—44no—
260C4×C61—44no—
261C4×C61#38614yes—
262C4×C61C2×C4×D411yescharacteristic
263C4×C61C2×C4×D411yescharacteristic
264C4×C61C2×#12111yescharacteristic
265C4×C61C2×#12111yescharacteristic
266Dic61—24no—
267Dic61—44no—
268Dic61—42no—
269Dic61—42no—
270C2×#536—611no—
271C2×#520—611no—
272C2×#419—611no—
273C22×#103D6111yescharacteristic, complemented
274C2×#408—611no—
275#1873—614no—
276#1872—614no—
277#1736—611no—
278#1735—611no—
279C2×C61—24no—
280C2×C61C2×#38611yescharacteristic
281C2×C61#289812yes—
282C2×C61#300911yescharacteristic
283C2×C61#301111yescharacteristic
284C2×C61#309712yes—
285D61—44no—
286C22×#41—611no—
287C23×Q8C2×D6111yescharacteristic
288C22×M16—611no—
289C23×C8C2×D6111yescharacteristic
290#536—618no—
291#536—1224no—
292C2×C4⋊Q8—1221no—
293C2×#124—1221no—
294C2×#123—1222no—
295C2×C22⋊Q8—614no—
296#520—1224no—
297#520—1224no—
298C2×C4×Q8—1221no—
299C2×C4×D4—1221no—
300C2×C4×D4—1222no—
301C2×#117—611no—
302C2×#117—611no—
303C2×#117—612no—
304C22×C4⋊C4—612no—
305C22×C4⋊C4C2×D6111yescharacteristic
306#419—1224no—
307C2×#103—24no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#103C2×D6114yes—
311#408—1224no—
312C2×C22⋊C8—614no—
313#340—612no—
314#340—612no—
315#340—614no—
316C61#2091311yescharacteristic, complemented, Sylow
317C2×#41—614no—
318C2×#41—614no—
319C2×#41—1222no—
320C2×#41—1222no—
321C2×#41—1222no—
322C2×#41—1222no—
323C22×Q8—21no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—611no—
330C22×Q8C4×D6114yescomplemented
331C22×Q8C22×D6112yes—
332C22×D4—612no—
333C22×D4—611no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—612no—
337C23×C4—1222no—
338C23×C4C22×D6111yescharacteristic
339C2×M16—1222no—
340C2×M16—1222no—
341C2×M16—1222no—
342C22×C8—22no—
343C22×C8—22no—
344C22×C8—22no—
345C22×C8—612no—
346C22×C8C22×D6112yes—
347C4⋊Q8—1228no—
348#124—1228no—
349#123—12216no—
350C22⋊Q8—6116no—
351C22⋊Q8—1228no—
352C4×Q8—1228no—
353C4×D4—1228no—
354C4×D4—12216no—
355#117—614no—
356#117—614no—
357#117—1224no—
358#117—1222no—
359#117—1222no—
360#117—1224no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1224no—
370C2×C4⋊C4C22⋊D6114yes—
371C2×C4⋊C4C22×D6112yes—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1224no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C42—1222no—
377C2×C42—1221no—
378C2×C42—1221no—
379#103—28no—
380#103—1224no—
381#103—1224no—
382C22⋊C8—1228no—
383C24—612no—
384C24D4×D6111yescharacteristic
385#41—1228no—
386#41—1228no—
387#41—1228no—
388#41—1228no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—614no—
402C2×Q8—1222no—
403C2×Q8—1222no—
404C2×Q8C2×C4×D6114yes—
405C2×D4—618no—
406C2×D4—614no—
407C2×D4—1224no—
408C2×D4—1224no—
409C2×D4—1222no—
410C2×D4—1222no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4—612no—
419C22×C4—614no—
420C22×C4—614no—
421C22×C4—1228no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—1221no—
429C22×C4—4884no—
430C22×C4C2×C4×D6112yes—
431C22×C4#1175822212yes—
432C22×C4D4×D6112yes—
433C22×C4D4×D6112yes—
434C22×C4D4×D6111yescharacteristic
435C22×C4#1175822412yes—
436C22×C4C2×C22⋊D6112yes—
437C22×C4C23×D6111yescharacteristic
438M16—2444no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—614no—
449C4⋊C4—612no—
450C4⋊C4—612no—
451C4⋊C4—12216no—
452C4⋊C4—1228no—
453C4⋊C4—1222no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4C2×C22⋊D6114yes—
457C22⋊C4—12216no—
458C22⋊C4—1228no—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C42—1228no—
462C42—1224no—
463C42—1224no—
464C23—22no—
465C23—22no—
466C23—614no—
467C23—1222no—
468C23—1222no—
469C23—1224no—
470C23—1224no—
471C23C22⋊C4×D6112yes—
472C23#1451974512yes—
473C23#1451975912yes—
474C23C2×D4×D6111yescharacteristic
475Q8—24no—
476Q8—24no—
477Q8—42no—
478Q8—42no—
479Q8—48no—
480Q8—1228no—
481D4—12216no—
482D4—1228no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—84no—
490C2×C4—614no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1224no—
494C2×C4—1224no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—1224no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—4888no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4C22⋊C4×D6112yes—
506C2×C4#1451973714yes—
507C2×C4C4×C22⋊D6114yes—
508C2×C4#1451974512yes—
509C2×C4#1451974814yes—
510C2×C4#1451975014yes—
511C2×C4#1451975912yes—
512C2×C4C22×C4×D6111yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4C2×#1175822411yescharacteristic
517C2×C4C2×#1175822211yescharacteristic
518C8—42no—
519C8—2442no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—22no—
524C22—1228no—
525C22—1224no—
526C22—2442no—
527C22—2444no—
528C22—2442no—
529C22#1683963814yes—
530C22C2×C22⋊C4×D6111yescharacteristic
531C22C2×#1451974511yescharacteristic
532C22C2×#1451975911yescharacteristic
533C22#1684056812yes—
534C22#1684057512yes—
535C4—44no—
536C4—44no—
537C4—1224no—
538C4—2442no—
539C4—2444no—
540C4—2442no—
541C4#1683963814yes—
542C4C2×#1451973711yescharacteristic
543C4C2×C4×C22⋊D6111yescharacteristic
544C4C2×#1451974811yescharacteristic
545C4C2×#1451975011yescharacteristic
546C2—24no—
547C2—2444no—
548C2C2×#1683963811yescharacteristic
549C2#2145434712yes—
550C2#2145509211yescharacteristic
551C2#2145511011yescharacteristic
552C2#2145578412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC22×#103D61
hypocenterC61#20913
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
5. #21446128C2semidirect
6. #21446123C2semidirect
7. #21445538C2semidirect
8. #21445536C2semidirect
9. C22×#103×C61C2semidirect
13. #16840336C4semidirect
59. #14519753C2×C4semidirect
65. #14519736C2×C4semidirect
92. C22×Q8×C61C2×C4semidirect
273. C22×#103D61semidirect
316. C61#20913semidirect
330. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403362
C2×#168402432
C2×#168397712
C2×#168397282
#214461282
#214461232
#214455382
#214455362
C22×#103×C612
#2091361