Finite Group Databasebeta

← #29681429

Subgroups — #29681429

Order 15616.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840245C211yesmaximal, characteristic
2C2×#16839758C211yesmaximal, characteristic
3C22×#14519701C211yesmaximal, characteristic, complemented
4#21448405C211yesmaximal, characteristic, complemented
5#21446162C214yesmaximal, complemented
6#21446155C214yesmaximal, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C22×#137×C61C211yesmaximal, characteristic
10C23×#92418C2211yescharacteristic
11#16840245—24no—
12#16840245—24no—
13C2×Q8×Dic61C2212yes—
14C2×D4×Dic61—22no—
15C2×D4×Dic61C2212yes—
16C2×#14519736—21no—
17C2×#14519736C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C22×C4⋊Dic61—21no—
20C22×C4⋊Dic61C2211yescharacteristic
21#16839758—24no—
22C2×#14519701—24no—
23C2×#14519701C2214yes—
24C2×#14519701C2212yes—
25C2×#14519698C2212yes—
26#16839669—21no—
27#16839669—21no—
28#16839669C2214yes—
29#16839625—22no—
30#16839525C2214yes—
31#16839516C2214yes—
32C22×#41×C61C2211yescharacteristic
33C23×Q8×C61C2211yescharacteristic
34C2×#137×C61—24no—
35C2×#137×C61—24no—
36C2×#137×C61C418yescomplemented
37C22×Q16×C61C2212yes—
38C22×SD16×C61C2212yes—
39C22×M16×C61C2211yescharacteristic
40C23×Dic61—21no—
41C23×Dic61—22no—
42C22×#92418—22no—
43C22×#92418—22no—
44C22×#92418—22no—
45C22×#92418C2312yes—
46Q8×Dic61—24no—
47Q8×Dic61—24no—
48D4×Dic61—216no—
49D4×Dic61—24no—
50D4×Dic61—24no—
51#14519736—28no—
52#14519736—22no—
53#14519736—22no—
54#14519736—22no—
55#14519736—22no—
56C2×C4⋊Dic61—28no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61D411yescharacteristic
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61D414yes—
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2312yes—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61D412yes—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2312yes—
71#14519701—24no—
72#14519701—28no—
73#14519698—24no—
74#14519684—24no—
75#14519684—24no—
76C2×#41×C61—22no—
77C2×#41×C61—22no—
78C2×#41×C61—22no—
79C2×#41×C61C2×C414yes—
80C2×#41×C61D414yes—
81C2×#41×C61D414yes—
82C22×Q8×C61—21no—
83C22×Q8×C61—22no—
84C22×Q8×C61—22no—
85C22×Q8×C61—22no—
86C22×Q8×C61—22no—
87C22×Q8×C61—22no—
88C22×Q8×C61C2×C414yescomplemented
89C22×Q8×C61C2312yes—
90C22×Q8×C61C2311yescharacteristic
91C22×D4×C61—21no—
92C22×D4×C61C2311yescharacteristic
93C23×C4×C61—21no—
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96#137×C61—232no—
97C2×Q16×C61—24no—
98C2×Q16×C61—24no—
99C2×Q16×C61C2×C418yescomplemented
100C2×SD16×C61—24no—
101C2×SD16×C61—24no—
102C2×SD16×C61C2×C418yescomplemented
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61C2×C414yescomplemented
106C22×C8×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—24no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61—28no—
119C2×Q8×C61—28no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—42no—
125C2×Q8×C61—42no—
126C2×Q8×C61—44no—
127C2×Q8×C61—44no—
128C2×Q8×C61C22×C414yes—
129C2×Q8×C61C22×C412yes—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61C2×D412yes—
132C2×D4×C61—24no—
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C22×C412yes—
139C2×D4×C61C2×D412yes—
140C2×D4×C61C2×D412yes—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61C22×C412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61#4112yes—
155C22×C4×C61#4112yes—
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—22no—
158C22×Dic61—28no—
159C22×Dic61—24no—
160C22×Dic61—22no—
161C22×Dic61—84no—
162C22×Dic61—82no—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×D412yes—
165Q16×C61—216no—
166SD16×C61—216no—
167M16×C61—28no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61C22×C414yes—
171C22⋊Dic61—28no—
172C22⋊Dic61—24no—
173C22⋊Dic61—42no—
174C4⋊Dic61—28no—
175C4⋊Dic61—22no—
176C4⋊Dic61—24no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61C2×D414yes—
182C4×Dic61—28no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185C4×Dic61—42no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—44no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61C4×D412yes—
198C23×C61#12012yes—
199C23×C61#12112yes—
200C23×C61C22×D411yescharacteristic
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61—22no—
204Q8×C61—24no—
205Q8×C61—24no—
206Q8×C61—42no—
207Q8×C61—42no—
208Q8×C61—48no—
209Q8×C61C2×C22⋊C414yes—
210D4×C61—216no—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61C2×C22⋊C414yes—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—216no—
219C2×C4×C61—24no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—48no—
223C2×C4×C61C2×C22⋊C414yes—
224C2×C4×C61C4×D414yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61#12012yes—
227C2×C4×C61#12114yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12314yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—44no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—84no—
246C2×Dic61—88no—
247C8×C61—24no—
248#92418—42no—
249#20931—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—24no—
254C22×C61—24no—
255C22×C61—44no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—44no—
259C22×C61#38614yes—
260C22×C61C2×C4×D411yescharacteristic
261C22×C61C2×#12011yescharacteristic
262C22×C61C2×#12111yescharacteristic
263C22×C61#57712yes—
264C22×C61#57812yes—
265C4×C61—24no—
266C4×C61—44no—
267C4×C61#38614yes—
268C4×C61C22×C22⋊C411yescharacteristic
269C4×C61C2×C4×D411yescharacteristic
270C4×C61C2×#12111yescharacteristic
271C4×C61C2×#12311yescharacteristic
272Dic61—42no—
273Dic61—44no—
274Dic61—42no—
275Dic61—82no—
276C22×#137D6111yescharacteristic
277C2×#520—611no—
278C2×#416—611no—
279C22×#103—611no—
280#2310—611no—
281#1878—614no—
282#1877—614no—
283#1736—611no—
284#1729—611no—
285C2×C61—24no—
286C2×C61—44no—
287C2×C61C2×#38611yescharacteristic
288C2×C61#294712yes—
289C2×C61#300911yescharacteristic
290C2×C61#301111yescharacteristic
291C2×C61#304612yes—
292C22×#41C2×D6111yescharacteristic
293C23×Q8C2×D6111yescharacteristic
294C2×#137—24no—
295C2×#137—24no—
296C2×#137Dic6118yescomplemented
297C22×Q16C2×D6112yes—
298C22×SD16C2×D6112yes—
299C22×M16C2×D6111yescharacteristic
300C23×C8—611no—
301#520—1224no—
302#520—1224no—
303C2×C4×Q8—612no—
304C2×C4×D4—612no—
305C2×C4×D4—1222no—
306C2×#117—611no—
307C2×#117—611no—
308C2×#117—1221no—
309C22×C4⋊C4—611no—
310C22×C4⋊C4—1221no—
311#416—1224no—
312C2×#103—614no—
313C2×#103—612no—
314C2×#103—1224no—
315C2×#102—612no—
316#391—614no—
317#391—1221no—
318#391—1221no—
319#381—1222no—
320#340—614no—
321#336—614no—
322C61#2093111yescharacteristic, complemented, Sylow
323C2×#41—22no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41C2×Dic6114yes—
327C2×#41C22⋊D6114yes—
328C2×#41C22⋊D6114yes—
329C22×Q8—21no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8C2×Dic6114yescomplemented
336C22×Q8C22×D6112yes—
337C22×Q8C22×D6111yescharacteristic
338C22×D4—21no—
339C22×D4C22×D6111yescharacteristic
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—1222no—
343C23×C4—1221no—
344C23×C4C22×D6111yescharacteristic
345#137—232no—
346C2×Q16—24no—
347C2×Q16—24no—
348C2×Q16C2×Dic6118yescomplemented
349C2×SD16—24no—
350C2×SD16—24no—
351C2×SD16C2×Dic6118yescomplemented
352C2×M16—22no—
353C2×M16—22no—
354C2×M16C2×Dic6114yescomplemented
355C22×C8—612no—
356C22×C8—1222no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8C22×D6112yes—
360C4×Q8—1224no—
361C4×Q8—1224no—
362C4×D4—1224no—
363C4×D4—12216no—
364C4×D4—1224no—
365#117—1228no—
366#117—1222no—
367#117—1222no—
368#117—1222no—
369#117—1222no—
370C2×C4⋊C4—614no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1228no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1222no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1221no—
382C2×C42—612no—
383C2×C42—1222no—
384C2×C42—1221no—
385#103—1228no—
386#103—1224no—
387#102—1224no—
388#95—1224no—
389#95—1224no—
390C24—21no—
391C24D4×D6111yescharacteristic
392#41—28no—
393#41—28no—
394#41—28no—
395#41C22⋊Dic61116yes—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8C22×Dic6114yes—
412C2×Q8C22×Dic6112yes—
413C2×Q8C2×C22⋊D6112yes—
414C2×Q8C2×C22⋊D6112yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C22×Dic6112yes—
422C2×D4C2×C22⋊D6112yes—
423C2×D4C2×C22⋊D6112yes—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—1228no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—4884no—
439C22×C4—4882no—
440C22×C4D4×D6112yes—
441C22×C4D4×D6111yescharacteristic
442C22×C4#1175822412yes—
443C22×C4#1175822412yes—
444C22×C4C22×Dic6112yes—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C2×C22⋊D6112yes—
447C22×C4C23×D6111yescharacteristic
448Q16—216no—
449SD16—216no—
450M16—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—2444no—
457C2×C8C22×Dic6114yes—
458C4⋊C4—614no—
459C4⋊C4—1228no—
460C4⋊C4—1222no—
461C4⋊C4—1224no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C22⋊C4—1228no—
467C22⋊C4—1224no—
468C22⋊C4—2442no—
469C42—1228no—
470C42—1222no—
471C42—1222no—
472C42—2442no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23D4×Dic6112yes—
481C23#1451974512yes—
482C23#1451974912yes—
483C23C2×D4×D6111yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—24no—
488Q8—24no—
489Q8—42no—
490Q8—42no—
491Q8—48no—
492Q8C2×C22⋊Dic6114yes—
493D4—216no—
494D4—22no—
495D4—22no—
496D4—22no—
497D4C2×C22⋊Dic6114yes—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—216no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—1222no—
507C2×C4—1224no—
508C2×C4—1222no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—4884no—
512C2×C4—4888no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2444no—
517C2×C4C2×C22⋊Dic6114yes—
518C2×C4D4×Dic6114yes—
519C2×C4D4×Dic6112yes—
520C2×C4#1451974512yes—
521C2×C4#1451974914yes—
522C2×C4#1451974912yes—
523C2×C4#1451975814yes—
524C2×C4C23×Dic6111yescharacteristic
525C2×C4C2×D4×D6111yescharacteristic
526C2×C4C22×C22⋊D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C2×#1175822411yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C8—24no—
531C8—2442no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—22no—
536C22—24no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22#1683964214yes—
542C22C2×D4×Dic6111yescharacteristic
543C22C2×#1451974511yescharacteristic
544C22C2×#1451974911yescharacteristic
545C22#1684057112yes—
546C22#1684057812yes—
547C4—24no—
548C4—44no—
549C4—4882no—
550C4—2442no—
551C4—2442no—
552C4—2444no—
553C4#1683964214yes—
554C4C22×C22⋊Dic6111yescharacteristic
555C4C2×D4×Dic6111yescharacteristic
556C4C2×#1451974911yescharacteristic
557C4C2×#1451975811yescharacteristic
558C2—24no—
559C2—44no—
560C2C2×#1683964211yescharacteristic
561C2#2145466312yes—
562C2#2145509511yescharacteristic
563C2#2145511311yescharacteristic
564C2#2145537312yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC22×#137D61
hypocenterC61#20931
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C22×#14519701C2semidirect
4. #21448405C2semidirect
5. #21446162C2semidirect
6. #21446155C2semidirect
7. #21445539C2semidirect
8. #21445517C2semidirect
36. C2×#137×C61C4semidirect
88. C22×Q8×C61C2×C4semidirect
99. C2×Q16×C61C2×C4semidirect
102. C2×SD16×C61C2×C4semidirect
105. C2×M16×C61C2×C4semidirect
296. C2×#137Dic61semidirect
322. C61#20931semidirect
335. C22×Q8C2×Dic61semidirect
348. C2×Q16C2×Dic61semidirect
351. C2×SD16C2×Dic61semidirect
354. C2×M16C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402452
C2×#168397582
C22×#145197012
#214484052
#214461622
#214461552
#214455392
#214455172
C22×#137×C612
#2093161