Finite Group Databasebeta

← #29681391

Subgroups — #29681391

Order 15616.

1830 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519807C211yesmaximal, characteristic
2C2×#16839763C211yesmaximal, characteristic
3C2×#16839770C211yesmaximal, characteristic, complemented
4C2×#16839668C211yesmaximal, characteristic, complemented
5#21446160C214yesmaximal, complemented
6#21446153C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445513C211yesmaximal, characteristic, complemented
9C2×C4×#41×C61C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519807—24no—
12C2×#14519807—24no—
13C2×#14519807C418yescomplemented
14C22×#11758202C2212yes—
15C22×#11758200C2212yes—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19#16839770—24no—
20#16839763—24no—
21C2×#14519701C2212yes—
22C2×#14519700C2214yes—
23C2×#14519698C2212yes—
24#16839668—21no—
25#16839668—21no—
26#16839668C2214yes—
27C2×#14519684—22no—
28#16839524C2214yes—
29#16839517C2214yes—
30C22×#41×C61C2211yescharacteristic
31C4×#41×C61—24no—
32C4×#41×C61—24no—
33C2×C4×Q8×C61C2212yes—
34C2×C4×D4×C61—22no—
35C2×C4×D4×C61C2212yes—
36C2×#117×C61—21no—
37C2×#117×C61C2211yescharacteristic
38C22×C42×C61—21no—
39C22×C42×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—
49C23×Dic61—21no—
50#14519807—232no—
51C2×#11758202—24no—
52C2×#11758202—24no—
53C2×#11758202C2×C418yescomplemented
54C2×#11758200—24no—
55C2×#11758200—24no—
56C2×#11758200C2×C418yescomplemented
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194C2×C414yescomplemented
63C22×#92418C2312yes—
64C22×#92418C2312yes—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2312yes—
69#14519701—24no—
70#14519700—28no—
71#14519698—24no—
72#14519684—24no—
73#14519684—24no—
74C2×#41×C61—22no—
75C2×#41×C61—22no—
76C2×#41×C61—22no—
77C2×#41×C61C2×C414yes—
78C2×#41×C61D414yes—
79C2×#41×C61D414yes—
80C22×Q8×C61C2311yescharacteristic
81C22×D4×C61—21no—
82C22×D4×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C23×C4×C61—22no—
85C23×C4×C61D412yes—
86C23×C4×C61C2311yescharacteristic
87C4×Q8×C61—24no—
88C4×Q8×C61—24no—
89C4×D4×C61—216no—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92#117×C61—28no—
93#117×C61—22no—
94#117×C61D414yes—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61C2312yes—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—22no—
99C2×C42×C61—28no—
100C2×C42×C61—21no—
101C2×C42×C61—22no—
102C2×C42×C61—22no—
103C2×C42×C61—22no—
104C2×C42×C61D411yescharacteristic
105C2×C42×C61D411yescharacteristic
106C2×C42×C61C2312yes—
107C24×C61—21no—
108C24×C61C2×D411yescharacteristic
109#41×C61—28no—
110#41×C61—28no—
111#41×C61—28no—
112#41×C61C22⋊C4116yes—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C412yes—
117C2×Q8×C61C2×D412yes—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61C22×C412yes—
126C2×D4×C61C2×D412yes—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—21no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—28no—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—42no—
137C22×C4×C61—44no—
138C22×C4×C61—44no—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61#4112yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—24no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61—44no—
155C22×Dic61—42no—
156C2×Dic122—28no—
157C2×Dic122—28no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—24no—
163C2×Dic122—22no—
164C2×Dic122—42no—
165C2×Dic122—44no—
166C2×Dic122—42no—
167C2×Dic122—44no—
168C2×Dic122C22×C414yes—
169C4⋊C4×C61—28no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61C2×D414yes—
172C22⋊C4×C61—28no—
173C22⋊C4×C61—24no—
174C42×C61—28no—
175C42×C61—22no—
176C42×C61—24no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180#11758202—216no—
181#11758200—216no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—42no—
185#11758194—28no—
186#11758194—44no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418C22×C414yes—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61C4×D412yes—
201C23×C61#12012yes—
202C23×C61#12112yes—
203C23×C61C22×D411yescharacteristic
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61C2×C22⋊C414yes—
208D4×C61—216no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61C2×C22⋊C414yes—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—216no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—48no—
224C2×C4×C61—416no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61C2×C22⋊C414yes—
228C2×C4×C61C4×D414yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61#12012yes—
231C2×C4×C61#12114yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12314yes—
234C2×C4×C61#13714yescomplemented
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×Dic61—24no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—48no—
245C2×Dic61—42no—
246C2×Dic61—44no—
247C2×Dic61—84no—
248Dic122—24no—
249Dic122—24no—
250Dic122—42no—
251Dic122—42no—
252Dic122—48no—
253#92418—24no—
254#92418—42no—
255#20927—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—24no—
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61#38614yes—
266C22×C61C2×C4×D411yescharacteristic
267C22×C61C2×#12011yescharacteristic
268C22×C61C2×#12111yescharacteristic
269C22×C61C2×#13712yes—
270C22×C61#57512yes—
271C4×C61—24no—
272C4×C61—24no—
273C4×C61—42no—
274C4×C61—44no—
275C4×C61#38614yes—
276C4×C61C22×C22⋊C411yescharacteristic
277C4×C61C2×C4×D411yescharacteristic
278C4×C61C2×#12111yescharacteristic
279C4×C61C2×#12311yescharacteristic
280Dic61—44no—
281Dic61—82no—
282C22×#137—611no—
283C2×C4×#41D6111yescharacteristic
284C2×#417—611no—
285C2×#419—611no—
286C2×#391—611no—
287#1878—614no—
288#1877—614no—
289#1736—611no—
290#1728—611no—
291C2×C61—24no—
292C2×C61—44no—
293C2×C61C2×#38611yescharacteristic
294C2×C61C4×#13712yes—
295C2×C61#300712yes—
296C2×C61#303911yescharacteristic
297C2×C61#304011yescharacteristic
298C22×#41C2×D6111yescharacteristic
299C23×Q8—611no—
300C2×#137—618no—
301C2×#137—1224no—
302C2×#137—1224no—
303C22×Q16—612no—
304C22×SD16—612no—
305C22×M16—611no—
306C22×M16—611no—
307C4×#41—24no—
308C4×#41—24no—
309C2×C4×Q8C2×D6112yes—
310C2×C4×D4—22no—
311C2×C4×D4C2×D6112yes—
312C2×#117—21no—
313C2×#117C2×D6111yescharacteristic
314C22×C4⋊C4—611no—
315C22×C42—21no—
316C22×C42C2×D6111yescharacteristic
317#419—1224no—
318#417—1224no—
319C2×#103—612no—
320C2×#103—614no—
321C2×#102—612no—
322#391—614no—
323#391—1221no—
324#391—1221no—
325C2×#95—1222no—
326#340—614no—
327#336—614no—
328C61#2092711yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41C4×D6114yes—
333C2×#41D24414yes—
334C2×#41C22⋊D6114yes—
335C22×Q8—614no—
336C22×Q8—612no—
337C22×Q8—1222no—
338C22×Q8—1221no—
339C22×Q8—1222no—
340C22×Q8—1222no—
341C22×Q8—1222no—
342C22×Q8—1222no—
343C22×Q8C22×D6111yescharacteristic
344C22×D4—21no—
345C22×D4C22×D6111yescharacteristic
346C23×C4—21no—
347C23×C4—22no—
348C23×C4—1221no—
349C23×C4—1221no—
350C23×C4C22⋊D6112yes—
351C23×C4C22×D6111yescharacteristic
352#137—12232no—
353C2×Q16—618no—
354C2×Q16—1224no—
355C2×Q16—1224no—
356C2×SD16—618no—
357C2×SD16—1224no—
358C2×SD16—1224no—
359C2×M16—614no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16—1222no—
364C2×M16—1222no—
365C22×C8—612no—
366C22×C8—612no—
367C4×Q8—24no—
368C4×Q8—24no—
369C4×D4—216no—
370C4×D4—24no—
371C4×D4—24no—
372#117—28no—
373#117—22no—
374#117C22⋊D6114yes—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6112yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C42—28no—
384C2×C42—21no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42C22⋊D6111yescharacteristic
389C2×C42C22⋊D6111yescharacteristic
390C2×C42C22×D6112yes—
391#103—1224no—
392#103—1228no—
393#102—1224no—
394#95—1224no—
395#95—1224no—
396C24—21no—
397C24C2×C22⋊D6111yescharacteristic
398#41—28no—
399#41—28no—
400#41—28no—
401#41#11758198116yes—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—614no—
406C2×Q8—1222no—
407C2×Q8—1224no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—1228no—
413C2×Q8—1228no—
414C2×Q8—2442no—
415C2×Q8—2442no—
416C2×Q8—2444no—
417C2×Q8—2444no—
418C2×Q8C2×C4×D6112yes—
419C2×Q8C2×D24412yes—
420C2×Q8C2×C22⋊D6112yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4C2×C4×D6112yes—
428C2×D4C2×D24412yes—
429C2×D4C2×C22⋊D6112yes—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—1222no—
444C22×C4—2444no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4C2×C4×D6112yes—
449C22×C4C2×D24412yes—
450C22×C4#1175822212yes—
451C22×C4#1175822212yes—
452C22×C4C2×C22⋊D6112yes—
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6112yes—
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458Q16—12216no—
459SD16—12216no—
460M16—1228no—
461M16—2444no—
462C2×C8—614no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C4⋊C4—28no—
469C4⋊C4—22no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—2442no—
473C4⋊C4C2×C22⋊D6114yes—
474C22⋊C4—28no—
475C22⋊C4—24no—
476C42—28no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23C4×C22⋊D6112yes—
490C23#1451974612yes—
491C23#1451975112yes—
492C23C22×C22⋊D6111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—22no—
496Q8—1224no—
497Q8—1224no—
498Q8—2442no—
499Q8—2442no—
500Q8—2448no—
501Q8C2×#1175819814yes—
502D4—216no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4C2×#1175819814yes—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—216no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—48no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—1224no—
522C2×C4—4884no—
523C2×C4—2442no—
524C2×C4—2444no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2448no—
528C2×C4C2×#1175819814yes—
529C2×C4C4×C22⋊D6114yes—
530C2×C4C4×C22⋊D6112yes—
531C2×C4#1451974612yes—
532C2×C4#1451975114yes—
533C2×C4#1451975112yes—
534C2×C4#1451976014yes—
535C2×C4#1451980714yescomplemented
536C2×C4C22×C4×D6111yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C2×C4C22×C22⋊D6111yescharacteristic
539C2×C4C22×D24411yescharacteristic
540C2×C4C2×#1175822211yescharacteristic
541C2×C4C2×#1175822211yescharacteristic
542C8—1224no—
543C8—2442no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—24no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22#1683964114yes—
554C22C2×C4×C22⋊D6111yescharacteristic
555C22C2×#1451974611yescharacteristic
556C22C2×#1451975111yescharacteristic
557C22C2×#1451980712yes—
558C22#1684056012yes—
559C4—24no—
560C4—24no—
561C4—42no—
562C4—44no—
563C4—4882no—
564C4—2444no—
565C4#1683964114yes—
566C4C22×#1175819811yescharacteristic
567C4C2×C4×C22⋊D6111yescharacteristic
568C4C2×#1451975111yescharacteristic
569C4C2×#1451976011yescharacteristic
570C2—24no—
571C2—44no—
572C2C2×#1683964111yescharacteristic
573C2C4×#1451980712yes—
574C2#2145507912yes—
575C2#2145530611yescharacteristic
576C2#2145531311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C4×#41×C61C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×C4×#41D61
hypocenterC61#20927
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
3. C2×#16839770C2semidirect
4. C2×#16839668C2semidirect
5. #21446160C2semidirect
6. #21446153C2semidirect
7. #21445540C2semidirect
8. #21445513C2semidirect
13. C2×#14519807C4semidirect
46. C22×Dic122C2×C4semidirect
53. C2×#11758202C2×C4semidirect
56. C2×#11758200C2×C4semidirect
62. C2×#11758194C2×C4semidirect
234. C2×C4×C61#137semidirect
328. C61#20927semidirect
535. C2×C4#14519807semidirect

Maximal subgroups

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

subgroupindex
C22×#145198072
C2×#168397632
C2×#168397702
C2×#168396682
#214461602
#214461532
#214455402
#214455132
C2×C4×#41×C612
#2092761