Finite Group Databasebeta

← #29681269

Subgroups — #29681269

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×#16840333C211yesmaximal, characteristic
2C2×#16839770C211yesmaximal, characteristic, complemented
3C22×#14519700C211yesmaximal, characteristic, complemented
4C2×#16839729C211yesmaximal, characteristic
5#21446131C214yesmaximal, complemented
6#21446126C214yesmaximal, complemented
7#21445538C211yesmaximal, characteristic, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C2×C61×#520C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13#16840333—24no—
14#16840333C418yescomplemented
15C2×#14519775—21no—
16C2×#14519764—21no—
17C2×#14519758—22no—
18C2×#14519756C2214yes—
19C2×#14519736C2211yescharacteristic
20#16839770—24no—
21C2×#14519700—24no—
22C2×#14519700C2214yes—
23C2×#14519700C2212yes—
24C2×#14519700C2212yes—
25#16839729—24no—
26C2×C22⋊#92418C2214yes—
27#16839524C2212yes—
28#16839524C2212yes—
29#16839524C2214yes—
30C22×#41×C61C2211yescharacteristic
31C61×#520—24no—
32C61×#520—24no—
33C2×C4×Q8×C61—21no—
34C2×C4×D4×C61—22no—
35C2×C4×D4×C61—21no—
36C2×#117×C61C2212yes—
37C2×#117×C61C2211yescharacteristic
38C22×C4⋊C4×C61C2212yes—
39C22×C4⋊C4×C61C2211yescharacteristic
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122C2×C414yescomplemented
47C22×Dic122C2312yes—
48C23×Dic61—21no—
49C2×#11758194—22no—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C22×#92418—22no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519775—28no—
58#14519764—28no—
59#14519758—216no—
60#14519756—28no—
61#14519756C2×C4116yescomplemented
62#14519736—22no—
63#14519736C2×C414yescomplemented
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—24no—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61—24no—
68C2×C4×Dic61—21no—
69#14519700—28no—
70#14519700—24no—
71#14519700—24no—
72C22⋊#92418—28no—
73C2×#41×C61—22no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61—22no—
77C2×#41×C61C2×C414yes—
78C2×#41×C61D414yes—
79C22×Q8×C61D411yescharacteristic
80C22×D4×C61D412yes—
81C22×D4×C61D411yescharacteristic
82C23×C4×C61—22no—
83C23×C4×C61—21no—
84C23×C4×C61C2312yes—
85C23×C4×C61C2311yescharacteristic
86C4×Q8×C61—28no—
87C4×D4×C61—216no—
88C4×D4×C61—28no—
89#117×C61—22no—
90#117×C61—24no—
91#117×C61—24no—
92#117×C61D414yes—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—24no—
97C2×C4⋊C4×C61—24no—
98C2×C4⋊C4×C61D414yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—21no—
104C2×C42×C61—22no—
105C2×C42×C61—21no—
106C24×C61C2×D412yes—
107C24×C61C2×D411yescharacteristic
108#41×C61—28no—
109#41×C61—28no—
110#41×C61—28no—
111#41×C61—28no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22⋊C414yes—
115C2×D4×C61—24no—
116C2×D4×C61—24no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61C22⋊C418yes—
120C2×D4×C61C22⋊C414yes—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—28no—
129C22×C4×C61—24no—
130C22×C4×C61—82no—
131C22×C4×C61—84no—
132C22×C4×C61C22×C414yes—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D414yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—24no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C2×Dic122—28no—
149C2×Dic122—28no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—24no—
155C2×Dic122—22no—
156C2×Dic122—42no—
157C2×Dic122—44no—
158C2×Dic122—44no—
159C2×Dic122—42no—
160C2×Dic122C22×C414yes—
161C4⋊C4×C61—28no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D414yes—
168C4⋊C4×C61C2×D412yes—
169C22⋊C4×C61—28no—
170C22⋊C4×C61—24no—
171C42×C61—28no—
172C42×C61—24no—
173C22⋊Dic61—24no—
174C22⋊Dic61—216no—
175C4⋊Dic61—22no—
176C4⋊Dic61C22×C414yes—
177#11758196—216no—
178C4×Dic61—24no—
179#11758194—44no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—44no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—24no—
192C23×C61—24no—
193C23×C61C2×C22⋊C414yes—
194C23×C61C2×C22⋊C412yes—
195C23×C61#12012yes—
196C23×C61#12312yes—
197C23×C61C22×D411yescharacteristic
198Q8×C61—28no—
199D4×C61—216no—
200D4×C61—28no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—44no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—84no—
214C2×C4×C61—88no—
215C2×C4×C61C2×C22⋊C414yes—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C4×D414yes—
218C2×C4×C61C4×D414yes—
219C2×C4×C61#12012yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12114yes—
222C2×C4×C61#12312yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—48no—
236Dic122—24no—
237Dic122—24no—
238Dic122—42no—
239Dic122—42no—
240Dic122—48no—
241#92418—42no—
242#92418—42no—
243#20913—611nomaximal, Sylow
244C22×C61—24no—
245C22×C61—24no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—28no—
249C22×C61—24no—
250C22×C61—44no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61#38614yes—
254C22×C61C22×C22⋊C411yescharacteristic
255C22×C61C2×#12011yescharacteristic
256C22×C61C2×#12311yescharacteristic
257C22×C61#57712yes—
258C22×C61#57812yes—
259C4×C61—24no—
260C4×C61—44no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61#38614yes—
264C4×C61C2×C4×D411yescharacteristic
265C4×C61C2×C4×D411yescharacteristic
266C4×C61C2×#12111yescharacteristic
267C4×C61C2×#12111yescharacteristic
268Dic61—44no—
269Dic61—44no—
270C2×#536—611no—
271C2×#520D6111yescharacteristic
272C2×#419—611no—
273C22×#103—611no—
274C2×#408—611no—
275#1873—614no—
276#1872—614no—
277#1736—611no—
278#1735—611no—
279C2×C61—24no—
280C2×C61—44no—
281C2×C61C2×#38611yescharacteristic
282C2×C61#289812yes—
283C2×C61#300911yescharacteristic
284C2×C61#301111yescharacteristic
285C2×C61#309712yes—
286C22×#41C2×D6111yescharacteristic
287C23×Q8—611no—
288C22×M16—611no—
289C23×C8—611no—
290#536—618no—
291#536—1224no—
292C2×C4⋊Q8—1221no—
293C2×#124—1221no—
294C2×#123—1222no—
295C2×C22⋊Q8—614no—
296#520—24no—
297#520—24no—
298C2×C4×Q8—21no—
299C2×C4×D4—22no—
300C2×C4×D4—21no—
301C2×#117—611no—
302C2×#117C2×D6112yes—
303C2×#117C2×D6111yescharacteristic
304C22×C4⋊C4C2×D6112yes—
305C22×C4⋊C4C2×D6111yescharacteristic
306#419—1224no—
307C2×#103—614no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#103—1224no—
311#408—1224no—
312C2×C22⋊C8—614no—
313#340—612no—
314#340—614no—
315#340—612no—
316C61#2091311yescharacteristic, complemented, Sylow
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41C4×D6114yes—
322C2×#41D24414yes—
323C22×Q8—614no—
324C22×Q8—612no—
325C22×Q8—1222no—
326C22×Q8—1221no—
327C22×Q8—1222no—
328C22×Q8—1222no—
329C22×Q8—1222no—
330C22×Q8—1222no—
331C22×Q8C22⋊D6111yescharacteristic
332C22×D4C22⋊D6112yes—
333C22×D4C22⋊D6111yescharacteristic
334C23×C4—22no—
335C23×C4—21no—
336C23×C4—1221no—
337C23×C4C22×D6112yes—
338C23×C4C22×D6111yescharacteristic
339C2×M16—1222no—
340C2×M16—1222no—
341C2×M16—1222no—
342C22×C8—612no—
343C22×C8—612no—
344C22×C8—1222no—
345C22×C8—1222no—
346C22×C8—1222no—
347C4⋊Q8—1228no—
348#124—1228no—
349#123—12216no—
350C22⋊Q8—6116no—
351C22⋊Q8—1228no—
352C4×Q8—28no—
353C4×D4—216no—
354C4×D4—28no—
355#117—22no—
356#117—24no—
357#117—24no—
358#117—614no—
359#117—1222no—
360#117C22⋊D6114yes—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1224no—
368C2×C4⋊C4C22⋊D6114yes—
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C4⋊C4C22×D6112yes—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1224no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—1221no—
379#103—1228no—
380#103—1224no—
381#103—1224no—
382C22⋊C8—1228no—
383C24C2×C22⋊D6112yes—
384C24C2×C22⋊D6111yescharacteristic
385#41—28no—
386#41—28no—
387#41—28no—
388#41—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—614no—
392C2×Q8—1222no—
393C2×Q8—1224no—
394C2×Q8—1222no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—1228no—
399C2×Q8—1228no—
400C2×Q8—2442no—
401C2×Q8—2444no—
402C2×Q8—2444no—
403C2×Q8—2442no—
404C2×Q8#1175819814yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4#1175819818yes—
410C2×D4#1175819814yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—82no—
421C22×C4—84no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1224no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4C2×C4×D6114yes—
428C22×C4C2×C4×D6112yes—
429C22×C4C2×D24414yes—
430C22×C4C2×D24412yes—
431C22×C4#1175822212yes—
432C22×C4#1175822212yes—
433C22×C4C2×C22⋊D6112yes—
434C22×C4C2×C22⋊D6112yes—
435C22×C4C2×C22⋊D6112yes—
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C23×D6111yescharacteristic
438M16—2444no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—2444no—
446C4⋊C4—28no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—24no—
451C4⋊C4—614no—
452C4⋊C4—1222no—
453C4⋊C4—12216no—
454C4⋊C4C2×C22⋊D6112yes—
455C4⋊C4C2×C22⋊D6114yes—
456C4⋊C4C2×C22⋊D6112yes—
457C22⋊C4—28no—
458C22⋊C4—24no—
459C22⋊C4—1224no—
460C22⋊C4—12216no—
461C42—28no—
462C42—24no—
463C42—1224no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23C2×#1175819814yes—
471C23C2×#1175819812yes—
472C23#1451974612yes—
473C23#1451976012yes—
474C23C22×C22⋊D6111yescharacteristic
475Q8—28no—
476Q8—1224no—
477Q8—1224no—
478Q8—2442no—
479Q8—2442no—
480Q8—2448no—
481D4—216no—
482D4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—88no—
497C2×C4—1224no—
498C2×C4—1224no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2448no—
504C2×C4C2×#1175819814yes—
505C2×C4C2×#1175819812yes—
506C2×C4C4×C22⋊D6114yes—
507C2×C4C4×C22⋊D6114yes—
508C2×C4#1451974612yes—
509C2×C4#1451975114yes—
510C2×C4#1451975114yes—
511C2×C4#1451976012yes—
512C2×C4C22×C4×D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C22×D24411yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C2×C4C2×#1175822211yescharacteristic
518C8—2442no—
519C8—2442no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—22no—
524C22—28no—
525C22—24no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22#1683964114yes—
530C22C22×#1175819811yescharacteristic
531C22C2×#1451974611yescharacteristic
532C22C2×#1451976011yescharacteristic
533C22#1684057312yes—
534C22#1684057712yes—
535C4—24no—
536C4—42no—
537C4—44no—
538C4—42no—
539C4—2444no—
540C4—2444no—
541C4#1683964114yes—
542C4C2×C4×C22⋊D6111yescharacteristic
543C4C2×C4×C22⋊D6111yescharacteristic
544C4C2×#1451975111yescharacteristic
545C4C2×#1451975111yescharacteristic
546C2—24no—
547C2—44no—
548C2C2×#1683964111yescharacteristic
549C2#2145434912yes—
550C2#2145509611yescharacteristic
551C2#2145511411yescharacteristic
552C2#2145578812yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#520C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#520D61
hypocenterC61#20913
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. C2×#16839770C2semidirect
3. C22×#14519700C2semidirect
5. #21446131C2semidirect
6. #21446126C2semidirect
7. #21445538C2semidirect
8. #21445535C2semidirect
14. #16840333C4semidirect
46. C22×Dic122C2×C4semidirect
61. #14519756C2×C4semidirect
63. #14519736C2×C4semidirect
316. C61#20913semidirect

Maximal subgroups

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

subgroupindex
C2×#168403332
C2×#168397702
C22×#145197002
C2×#168397292
#214461312
#214461262
#214455382
#214455352
C2×C61×#5202
#2091361