Finite Group Databasebeta

← #29714371

Subgroups — #29714371

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3C4×#14519753C211yesmaximal, characteristic
4#21447020C212yesmaximal
5#21446826C212yesmaximal
6#21446766C211yesmaximal, characteristic, complemented
7#21446743C211yesmaximal, characteristic, complemented
8#21446162C212yesmaximal, complemented
9#21446071C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic
11C2×C61×#462C211yesmaximal, characteristic, complemented
12#16840317—24no—
13#16840317—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768—21no—
16C2×#14519757—21no—
17C2×#14519753C2212yes—
18C2×#14519753C2211yescharacteristic
19C2×Q8×Dic61C2212yes—
20C2×#14519742—21no—
21C2×#14519735—21no—
22C2×C42×D61C2211yescharacteristic
23#16840057—24no—
24C2×#14519710C2212yes—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519701C2212yes—
29C2×#14519689C2212yes—
30#16839669C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839643—21no—
33#16839633—21no—
34#16839619C2211yescharacteristic
35C4×#11758196—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37C4×#11758198—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40#16839516C2212yes—
41C22×Q16×C61C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4×Q8×C61C2211yescharacteristic
44C61×#462—24no—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#103×C61C2212yes—
47C22×Dic122—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777D414yes—
54#14519768—28no—
55#14519757—28no—
56#14519753—24no—
57#14519753—22no—
58#14519753—24no—
59#14519753—22no—
60Q8×Dic61—24no—
61Q8×Dic61—24no—
62#14519742—28no—
63#14519735—28no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—22no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C42×D61—22no—
81C42×D61—22no—
82#14519710—24no—
83C4⋊#92418—22no—
84#14519701—24no—
85#14519701—24no—
86#14519701—42no—
87#14519689—42no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93C22×Q8×C61C2312yes—
94C22×Q8×C61C2311yescharacteristic
95C2×Q16×C61—24no—
96C2×Q16×C61—24no—
97C2×Q16×C61—42no—
98C22×C8×C61C2312yes—
99C4⋊Q8×C61—22no—
100C4⋊Q8×C61D414yes—
101C4×Q8×C61—22no—
102C4×Q8×C61—22no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C4⋊C8×C61—22no—
110#103×C61—24no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—21no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—21no—
122C2×Q8×C61C2×D414yes—
123C2×Q8×C61#4112yes—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61Q1614yescomplemented
148C2×Dic122—24no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151Q16×C61—48no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C42×C61—42no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163#11758198—28no—
164#11758198—24no—
165#11758198—42no—
166C4⋊Dic61—28no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61C2×D414yes—
173C4⋊Dic61#4112yes—
174#11758196—28no—
175#11758196—24no—
176#11758196—42no—
177C4×Dic61—28no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—42no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61C22×D411yescharacteristic
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61C2×Q1612yes—
194Q8×C61—22no—
195Q8×C61—24no—
196Q8×C61—24no—
197Q8×C61—22no—
198Q8×C61—41no—
199Q8×C61—42no—
200Q8×C61—42no—
201Q8×C61—41no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61#12012yes—
214C2×C4×C61#12012yes—
215C2×C4×C61C2×Q1612yes—
216C2×C4×C61#13712yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—24no—
228C2×Dic61—21no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61C2×Q1612yes—
241C4×D61—28no—
242C4×D61—44no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—41no—
246C4×D61—41no—
247Dic122—48no—
248C8×C61—42no—
249#92418—42no—
250#24155—611nomaximal, Sylow
251C22×C61C2×#12011yescharacteristic
252C22×C61#52511yescharacteristic
253C22×C61#53711yescharacteristic
254C22×C61C22×Q1611yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61#57511yescharacteristic
257C22×C61#57511yescharacteristic
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C2×D61—41no—
265C4×C61—22no—
266C4×C61—44no—
267C4×C61—42no—
268C4×C61—81no—
269C4×C61#45114yes—
270C4×C61#54611yescharacteristic
271C4×C61#54711yescharacteristic
272C4×C61#54811yescharacteristic
273C4×C61#55011yescharacteristic
274Dic61—22no—
275Dic61—42no—
276Dic61—41no—
277Dic61—44no—
278Dic61—41no—
279Dic61—81no—
280C2×#533—611no—
281C2×#477—611no—
282C2×#462D6111yescharacteristic, complemented
283C4×C22⋊Q8—611no—
284#2036—612no—
285#2001—612no—
286#1987—611no—
287#1981—611no—
288#1878—612no—
289#1863—612no—
290#1797—611no—
291C2×C61#240711yescharacteristic
292C2×C61C2×#45111yescharacteristic
293C2×C61#300511yescharacteristic
294C2×C61#310411yescharacteristic
295C2×C61#311611yescharacteristic
296C2×C61#324911yescharacteristic
297C2×C61#325211yescharacteristic
298D61—42no—
299C22×Q16C2×D6112yes—
300#533—1224no—
301#533—1224no—
302C2×C4⋊Q8—611no—
303C2×C4⋊Q8C2×D6111yescharacteristic
304C2×#125—1221no—
305C2×C22⋊Q8—612no—
306C2×C22⋊Q8—611no—
307C2×C22⋊Q8—1221no—
308C2×C4×Q8—612no—
309C2×C4×Q8—1221no—
310C2×C4×Q8C2×D6111yescharacteristic
311C2×#117—1221no—
312C22×C42—611no—
313#477—1224no—
314#462—24no—
315C2×#107—612no—
316C2×C4⋊C8—611no—
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#103—612no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#103C2×D6112yes—
322C2×C22⋊C8—612no—
323#391—611no—
324#389—611no—
325#387—1221no—
326#385—1221no—
327#380—611no—
328C4×C4⋊C4—611no—
329C4×C4⋊C4—1221no—
330C4×C22⋊C4—1221no—
331#340—612no—
332#340—612no—
333#336—612no—
334C61#2415511yescharacteristic, complemented, Sylow
335C22×Q8—1221no—
336C22×Q8C22×D6111yescharacteristic
337C22×Q8C22×D6112yes—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C2×Q16—24no—
342C2×Q16—24no—
343C2×Q16—42no—
344C22×C8—612no—
345C22×C8C22×D6112yes—
346C4⋊Q8—22no—
347C4⋊Q8—614no—
348C4⋊Q8—1222no—
349C4⋊Q8C22⋊D6114yes—
350#125—1228no—
351C22⋊Q8—1228no—
352C22⋊Q8—1224no—
353C22⋊Q8—1222no—
354C22⋊Q8—1222no—
355C22⋊Q8—1224no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—1228no—
359C4×Q8—1224no—
360C4×Q8—1224no—
361#117—1228no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C42—21no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—1222no—
382C2×C42—1222no—
383C2×C42—1222no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386#107—1224no—
387C4⋊C8—22no—
388C4⋊C8—1222no—
389#103—24no—
390#103—1224no—
391#103—1224no—
392#103—2442no—
393C22⋊C8—2442no—
394#95—1221no—
395#95—1221no—
396#95—1221no—
397#95—1221no—
398#95—1221no—
399C24—611no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—1224no—
411C2×Q8—2442no—
412C2×Q8—2442no—
413C2×Q8#1175822412yes—
414C2×Q8C2×C22⋊D6114yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—611no—
419C22×C4—614no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—614no—
452C4⋊C4—612no—
453C4⋊C4—1228no—
454C4⋊C4—1224no—
455C4⋊C4—1228no—
456C4⋊C4—1222no—
457C4⋊C4—1221no—
458C4⋊C4—1224no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—2442no—
462C22⋊C4—1228no—
463C22⋊C4—1224no—
464C22⋊C4—2442no—
465C42—42no—
466C42—1228no—
467C42—1221no—
468C42—1221no—
469C42—1222no—
470C42—1222no—
471C42—2442no—
472C42D4×D6112yes—
473C42C2×C22⋊D6112yes—
474C23—612no—
475C23—1221no—
476C23—1221no—
477C23—1221no—
478C23—1221no—
479C23—1221no—
480C23—1221no—
481C23C2×D4×D6111yescharacteristic
482Q8—22no—
483Q8—24no—
484Q8—24no—
485Q8—22no—
486Q8—41no—
487Q8—42no—
488Q8—42no—
489Q8—41no—
490Q8—2448no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—612no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1228no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1224no—
509C2×C4—1222no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—2444no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4#1451974512yes—
527C2×C4#1451974512yes—
528C2×C4Q16×D6112yes—
529C2×C4#1451980912yes—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4C2×#1175822411yescharacteristic
534C2×C4#1451982211yescharacteristic
535C2×C4#1451982411yescharacteristic
536C8—42no—
537C8—2442no—
538C22—1222no—
539C22—1222no—
540C22—1222no—
541C22—2441no—
542C22—2441no—
543C22—2441no—
544C22—2441no—
545C22C2×#1451974511yescharacteristic
546C22#1684027211yescharacteristic
547C22#1684034111yescharacteristic
548C22C2×Q16×D6111yescharacteristic
549C22C2×#1451980911yescharacteristic
550C22#1684056111yescharacteristic
551C22#1684056111yescharacteristic
552C4—22no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4—1222no—
557C4—4881no—
558C4—2442no—
559C4—2441no—
560C4—2441no—
561C4—2444no—
562C4#1683991614yes—
563C4#1684040611yescharacteristic
564C4#1684041111yescharacteristic
565C4#1684041811yescharacteristic
566C4#1684043011yescharacteristic
567C2—2442no—
568C2#2144916011yescharacteristic
569C2C2×#1683991611yescharacteristic
570C2#2145506011yescharacteristic
571C2#2145585711yescharacteristic
572C2#2145598011yescharacteristic
573C2#2145716211yescharacteristic
574C2#2145719011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#462C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#462D61
hypocenterC61#24155
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
6. #21446766C2semidirect
7. #21446743C2semidirect
8. #21446162C2semidirect
9. #21446071C2semidirect
11. C2×C61×#462C2semidirect
147. C2×C4×D61Q16semidirect
282. C2×#462D61semidirect
334. C61#24155semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400572
C4×#145197532
#214470202
#214468262
#214467662
#214467432
#214461622
#214460712
#214457962
C2×C61×#4622
#2415561