Finite Group Databasebeta

← #29714801

Subgroups — #29714801

Order 15616.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic
3#21451189C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal
5#21446910C212yesmaximal
6#21446754C211yesmaximal, characteristic, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446162C212yesmaximal, complemented
9#21446085C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic
11C2×C61×#462C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840486—24no—
14#16840486—24no—
15C2×#14519772—22no—
16C2×#14519766C2212yes—
17C2×#14519757—21no—
18C2×Q8×Dic61C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519735—21no—
22C4×#11758201—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839695—22no—
31#16839669C2212yes—
32#16839668—21no—
33#16839660—21no—
34#16839658C2212yes—
35#16839632C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×Q16×C61C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×Q8×C61C2211yescharacteristic
42C61×#462—24no—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519772—216no—
54#14519766—24no—
55#14519766—24no—
56#14519757—28no—
57Q8×Dic61—24no—
58Q8×Dic61—24no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×D244—22no—
62C4×D244D414yes—
63#14519735—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C2×#11758196D412yes—
72C2×#11758198—21no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61C2312yes—
77#14519708—22no—
78#14519701—22no—
79#14519701—24no—
80#14519701—42no—
81#14519699—42no—
82#14519697—22no—
83C4×#92418—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684—22no—
87C22×Q8×C61C2312yes—
88C22×Q8×C61C2311yescharacteristic
89C2×Q16×C61—24no—
90C2×Q16×C61—24no—
91C2×Q16×C61—42no—
92C22×C8×C61C2312yes—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yes—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#103×C61—24no—
105C23×D61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D414yes—
117C2×Q8×C61#4112yes—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C2×D412yes—
127C22×Dic61C2×D412yes—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244C2×D412yes—
134C2×C4×D61—24no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140Q16×C61—48no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—42no—
149C42×C61—42no—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152#11758201—28no—
153#11758198—28no—
154#11758198—28no—
155#11758198—24no—
156C4⋊Dic61—28no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—42no—
162C4⋊Dic61—81no—
163C4⋊Dic61C2×D412yes—
164#11758196—24no—
165#11758196—82no—
166C4×Dic61—28no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418#4112yes—
175C23×C61C22×D411yescharacteristic
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183Q8×C61—22no—
184Q8×C61—22no—
185Q8×C61—24no—
186Q8×C61—24no—
187Q8×C61—42no—
188Q8×C61—42no—
189Q8×C61—41no—
190Q8×C61—41no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—24no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61#12012yes—
203C2×C4×C61#12012yes—
204C2×C4×C61#13512yes—
205C2×C4×C61#13712yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224D244—22no—
225D244—22no—
226D244—41no—
227D244—41no—
228C4×D61—48no—
229Dic122—44no—
230C8×C61—42no—
231#92418—22no—
232#92418—41no—
233#24189—611nomaximal, Sylow
234C22×C61C2×#12011yescharacteristic
235C22×C61#54811yescharacteristic
236C22×C61#55011yescharacteristic
237C22×C61C2×#13511yescharacteristic
238C22×C61C2×#13711yescharacteristic
239C22×C61#57711yescharacteristic
240C22×C61#57811yescharacteristic
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C4×C61—22no—
249C4×C61—44no—
250C4×C61—42no—
251C4×C61—81no—
252C4×C61#45214yes—
253C4×C61#52511yescharacteristic
254C4×C61#53711yescharacteristic
255C4×C61#54611yescharacteristic
256C4×C61#54711yescharacteristic
257Dic61—42no—
258Dic61—42no—
259Dic61—42no—
260Dic61—82no—
261C2×#556—611no—
262C2×#462D6111yescharacteristic, complemented
263C2×C4×SD16—611no—
264#2582—611no—
265#2042—612no—
266#2015—612no—
267#1984—611no—
268#1956—611no—
269#1878—612no—
270#1865—612no—
271#1802—611no—
272C2×C61#240711yescharacteristic
273C2×C61C2×#45211yescharacteristic
274C2×C61#301311yescharacteristic
275C2×C61#336411yescharacteristic
276C2×C61#336911yescharacteristic
277C2×C61#339411yescharacteristic
278C2×C61#339811yescharacteristic
279D61—82no—
280C22×Q16C2×D6112yes—
281C22×SD16—611no—
282#556—1224no—
283#556—1224no—
284C2×C4⋊Q8C2×D6111yescharacteristic
285C2×#126—1222no—
286C2×#124—612no—
287C2×C22⋊Q8—1221no—
288C2×C4×Q8—611no—
289C2×C4×Q8—612no—
290C2×C4×Q8C2×D6111yescharacteristic
291C2×C4×D4—611no—
292C2×#117—1221no—
293#462—24no—
294C4×SD16—1224no—
295C2×#106—611no—
296C2×C4⋊C8C2×D6111yescharacteristic
297C2×#103—612no—
298C2×#103—611no—
299C2×#103—612no—
300C2×#103C2×D6112yes—
301C2×#102—611no—
302C2×#102—612no—
303C2×C4×C8—611no—
304#398—1222no—
305#391—612no—
306#391—1221no—
307#389—612no—
308#389—1221no—
309#384—611no—
310#340—612no—
311#336—612no—
312#336—612no—
313C61#2418911yescharacteristic, complemented, Sylow
314C22×Q8—611no—
315C22×Q8C22×D6111yescharacteristic
316C22×Q8C22×D6112yes—
317C22×D4—611no—
318C23×C4—1221no—
319C2×Q16—24no—
320C2×Q16—24no—
321C2×Q16—42no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×SD16—1222no—
325C22×C8—611no—
326C22×C8—611no—
327C22×C8C22×D6112yes—
328C4⋊Q8—22no—
329C4⋊Q8C22⋊D6114yes—
330#126—12216no—
331#124—1224no—
332#124—1224no—
333C22⋊Q8—1228no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—1224no—
337C4×Q8—1224no—
338C4×Q8—1222no—
339C4×Q8—1222no—
340C4×D4—614no—
341C4×D4—1222no—
342#117—1228no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4—1221no—
358C2×C42—21no—
359C2×C42—612no—
360C2×C42—1222no—
361C2×C42C22×D6111yescharacteristic
362#106—1222no—
363C4⋊C8—22no—
364#103—24no—
365#103—1222no—
366#103—1224no—
367#103—2442no—
368#102—1222no—
369#102—2442no—
370C4×C8—1222no—
371#95—1222no—
372#95—1222no—
373#95—1222no—
374C24—1221no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8#1175822412yes—
389C2×Q8C2×C22⋊D6114yes—
390C2×D4—612no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—1221no—
402C22×C4—1224no—
403C22×C4—1222no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4D4×D6111yescharacteristic
407C22×C4D4×D6111yescharacteristic
408C22×C4C23×D6111yescharacteristic
409Q16—48no—
410SD16—1228no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—612no—
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8—1221no—
418C2×C8—1221no—
419C2×C8—1221no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—42no—
425C4⋊C4—612no—
426C4⋊C4—1228no—
427C4⋊C4—1221no—
428C4⋊C4—1224no—
429C4⋊C4—1222no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—4881no—
433C4⋊C4—4882no—
434C4⋊C4—2442no—
435C22⋊C4—1228no—
436C22⋊C4—1228no—
437C22⋊C4—1224no—
438C42—42no—
439C42—1228no—
440C42—1222no—
441C42—1222no—
442C42D4×D6112yes—
443C42C2×C22⋊D6112yes—
444C23—1222no—
445C23—2441no—
446C23—2441no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23C2×D4×D6111yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—24no—
455Q8—24no—
456Q8—42no—
457Q8—42no—
458Q8—41no—
459Q8—41no—
460Q8—2444no—
461D4—1222no—
462D4—1222no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2448no—
489C2×C4#1451974512yes—
490C2×C4#1451974512yes—
491C2×C4#1451980012yes—
492C2×C4#1451980912yes—
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4C22×C22⋊D6111yescharacteristic
495C2×C4#41×D6111yescharacteristic
496C2×C4C2×#1175822411yescharacteristic
497C2×C4#1451982211yescharacteristic
498C2×C4#1451982411yescharacteristic
499C8—42no—
500C8—1222no—
501C8—2441no—
502C22—4881no—
503C22—4881no—
504C22—4881no—
505C22—4881no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22C2×#1451974511yescharacteristic
510C22#1684041811yescharacteristic
511C22#1684043011yescharacteristic
512C22C2×#1451980011yescharacteristic
513C22C2×#1451980911yescharacteristic
514C22#1684057111yescharacteristic
515C22#1684057811yescharacteristic
516C4—22no—
517C4—44no—
518C4—42no—
519C4—81no—
520C4—4882no—
521C4—2442no—
522C4—2442no—
523C4—2442no—
524C4#1683992314yes—
525C4#1684027211yescharacteristic
526C4#1684034111yescharacteristic
527C4#1684040611yescharacteristic
528C4#1684041111yescharacteristic
529C2—4882no—
530C2#2144916011yescharacteristic
531C2C2×#1683992311yescharacteristic
532C2#2145513311yescharacteristic
533C2#2145850811yescharacteristic
534C2#2145856311yescharacteristic
535C2#2145881411yescharacteristic
536C2#2145884911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#462C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#462D61
hypocenterC61#24189
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
3. #21451189C2semidirect
6. #21446754C2semidirect
7. #21446622C2semidirect
8. #21446162C2semidirect
9. #21446085C2semidirect
11. C2×C61×#462C2semidirect
262. C2×#462D61semidirect
313. C61#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×C4×#117582012
#214511892
#214470562
#214469102
#214467542
#214466222
#214461622
#214460852
#214458202
C2×C61×#4622
#2418961