Finite Group Databasebeta

← #29714094

Subgroups — #29714094

Order 15616.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840416C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3C2×#16839860C211yesmaximal, characteristic, complemented
4#21447001C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446791C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446127C212yesmaximal, complemented
9#21446006C212yesmaximal, complemented
10#21445826C211yesmaximal, characteristic, complemented
11C61×#2536C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758201C2211yescharacteristic
14#16840416—24no—
15#16840416—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519761—22no—
18C2×#14519757—21no—
19C2×#14519751C2212yes—
20C2×C4×D244C2211yescharacteristic
21C22×C4⋊Dic61C2212yes—
22#16840057—24no—
23#16839860—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2212yes—
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×C22⋊Q8×C61C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#398—21no—
41C61×#394—21no—
42C61×#388—22no—
43C61×#382C2211yescharacteristic
44C61×#380C2212yes—
45C2×#11758224—28no—
46C2×#11758224—48no—
47C22×Dic122—21no—
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—22no—
50C23×Dic61—22no—
51C22×C4×D61—21no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C22×#92418C2311yescharacteristic
56C22×#92418C2312yes—
57C22×#92418C2311yescharacteristic
58#14519777—22no—
59#14519777D414yes—
60#14519761—216no—
61#14519757—28no—
62#14519751—24no—
63#14519751—24no—
64C4×D244—22no—
65C4×D244D414yes—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×#11758198—21no—
73C2×#11758198—22no—
74#14519710—24no—
75#14519710—22no—
76C4⋊#92418—22no—
77#14519701—24no—
78#14519701—22no—
79#14519697—22no—
80#14519697—42no—
81C22⋊#92418—24no—
82C22⋊#92418—42no—
83#14519688—22no—
84C22×Q8×C61C2311yescharacteristic
85C22×D4×C61D411yescharacteristic
86C23×C4×C61C2312yes—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—42no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D412yes—
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#95×C61—22no—
104#95×C61—21no—
105#95×C61—21no—
106C24×C61C2×D412yes—
107C23×D61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61#4112yescomplemented
114C2×D4×C61—24no—
115C2×D4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—28no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C22×Dic61—28no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—44no—
136C22×Dic61—44no—
137#11758224—432no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C2×D412yes—
144C2×C4×D61—24no—
145C2×C4×D61—44no—
146C2×Dic122—24no—
147C2×Dic122—44no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—44no—
154C4⋊C4×C61—42no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61—44no—
158C42×C61—42no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161#11758201—28no—
162#11758198—28no—
163#11758198—24no—
164C4⋊Dic61—28no—
165C4⋊Dic61—24no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D412yes—
169C4⋊Dic61C2×D414yes—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418—22no—
177C2×#92418—21no—
178C2×#92418#4112yescomplemented
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61AGL(1,ℤ/8)14yes—
186C23×C61C22×D411yescharacteristic
187C22×D61—22no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—42no—
191C22×D61—42no—
192Q8×C61—22no—
193Q8×C61—22no—
194Q8×C61—41no—
195Q8×C61—41no—
196D4×C61—28no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C22⋊D61—416no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—416no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229D244—22no—
230D244—22no—
231D244—41no—
232D244—41no—
233C4×D61—48no—
234Dic122—48no—
235#92418—22no—
236#92418—41no—
237#92418—42no—
238#24132—611nomaximal, Sylow
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—24no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61C2×#12011yescharacteristic
247C22×C61#54611yescharacteristic
248C22×C61#54711yescharacteristic
249C22×C61C2×AGL(1,ℤ/8)12yes—
250C22×C61#57812yes—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—82no—
255C2×D61—82no—
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—42no—
260C4×C61—81no—
261C4×C61—82no—
262C4×C61#52511yescharacteristic
263C4×C61#53711yescharacteristic
264C4×C61#54811yescharacteristic
265C4×C61#55011yescharacteristic
266Dic61—44no—
267Dic61—42no—
268C2×#547—611no—
269C2×#477—611no—
270C2×#440—611no—
271#2536D6111yescharacteristic, complemented
272#2033—612no—
273#2000—612no—
274#1993—611no—
275#1955—611no—
276#1872—612no—
277#1852—612no—
278#1804—611no—
279C2×C61—44no—
280C2×C61#240711yescharacteristic
281C2×C61#300311yescharacteristic
282C2×C61#301111yescharacteristic
283C2×C61#330812yes—
284C2×C61#334312yes—
285D61—82no—
286C22×#41—1221no—
287C22×SD16—611no—
288#547—1224no—
289#547—1224no—
290C2×C4⋊Q8—611no—
291C2×#123—1222no—
292C2×C22⋊Q8—1221no—
293C2×C22⋊Q8C2×D6112yes—
294C2×#121—612no—
295C2×C4×Q8C2×D6111yescharacteristic
296C2×C4×D4—611no—
297C2×C4×D4C2×D6111yescharacteristic
298C22×C4⋊C4—612no—
299#477—1224no—
300#440—1224no—
301C2×#107—612no—
302C2×#107—611no—
303C2×C4⋊C8—611no—
304C2×#103—612no—
305C2×#103—611no—
306C2×#102—611no—
307C2×#102—612no—
308C2×C22⋊C8—612no—
309C2×C22⋊C8—612no—
310C2×#97—611no—
311#398—21no—
312#394—21no—
313#388—22no—
314#382C2×D6111yescharacteristic
315#380C2×D6112yes—
316#340—612no—
317#340—612no—
318#340—612no—
319C61#2413211yescharacteristic, complemented, Sylow
320C2×#41—1228no—
321C2×#41—2448no—
322C22×Q8—1221no—
323C22×Q8C22×D6111yescharacteristic
324C22×D4—611no—
325C22×D4—1222no—
326C22×D4C22⋊D6111yescharacteristic
327C23×C4—1222no—
328C23×C4—1221no—
329C23×C4C22×D6112yes—
330C2×SD16—1222no—
331C2×SD16—1222no—
332C2×SD16—1222no—
333C22×C8—611no—
334C22×C8—611no—
335C22×C8—612no—
336C4⋊Q8—614no—
337C4⋊Q8—1222no—
338#123—12216no—
339C22⋊Q8—24no—
340C22⋊Q8—42no—
341C22⋊Q8—1228no—
342#121—1224no—
343#121—1224no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—614no—
349C4×D4—1222no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4C22⋊D6111yescharacteristic
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C4⋊C4C22×D6112yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4C22⋊D6112yes—
366C2×C42—21no—
367C2×C42C22×D6111yescharacteristic
368#107—1222no—
369#107—1224no—
370C4⋊C8—1222no—
371#103—1222no—
372#103—1224no—
373#102—1222no—
374#102—2442no—
375C22⋊C8—1224no—
376C22⋊C8—2442no—
377#97—1222no—
378#95—22no—
379#95—21no—
380#95—21no—
381C24—1221no—
382C24C2×C22⋊D6112yes—
383#41—24432no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—1224no—
390C2×Q8—2444no—
391C2×Q8#1175822412yescomplemented
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—612no—
395C2×D4—1228no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—2444no—
402C2×D4—2444no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—1228no—
413C22×C4—1221no—
414C22×C4—1222no—
415C22×C4—1224no—
416C22×C4—2444no—
417C22×C4—2444no—
418C22×C4—2444no—
419C22×C4C2×C22⋊D6112yes—
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C23×D6111yescharacteristic
423SD16—1228no—
424C2×C8—612no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—1221no—
431C2×C8—1222no—
432C2×C8—1221no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—612no—
441C4⋊C4—614no—
442C4⋊C4—1228no—
443C4⋊C4—1221no—
444C4⋊C4—1222no—
445C4⋊C4—1224no—
446C22⋊C4—24no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—1228no—
450C22⋊C4—1224no—
451C42—42no—
452C42D4×D6112yes—
453C42D4×D6112yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—1222no—
461C23—2442no—
462C23—2442no—
463C23—2441no—
464C23—2441no—
465C23#1451980214yes—
466C23C22×C22⋊D6111yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—41no—
470Q8—41no—
471Q8—2448no—
472D4—28no—
473D4—1222no—
474D4—1222no—
475D4—24416no—
476D4—2441no—
477D4—2441no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—1222no—
496C2×C4—1224no—
497C2×C4—2448no—
498C2×C4—24416no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974512yes—
503C2×C4#1451974512yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4C2×#1175822411yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982411yescharacteristic
510C8—1222no—
511C8—2441no—
512C8—2442no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—4882no—
521C22—4882no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22C2×#1451974511yescharacteristic
526C22#1684040611yescharacteristic
527C22#1684041111yescharacteristic
528C22C2×#1451980212yes—
529C22#1684057712yes—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—82no—
535C4—81no—
536C4—2442no—
537C4—2444no—
538C4#1684027211yescharacteristic
539C4#1684034111yescharacteristic
540C4#1684041811yescharacteristic
541C4#1684043011yescharacteristic
542C2—44no—
543C2—4882no—
544C2#2144916011yescharacteristic
545C2#2145504511yescharacteristic
546C2#2145511211yescharacteristic
547C2#2145781012yes—
548C2#2145825912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2536C2
socleC23×C61C22×D4
prime radicalC2×#16840416C2
hypercenter#2536D61
hypocenterC61#24132
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
3. C2×#16839860C2semidirect
4. #21447001C2semidirect
5. #21446820C2semidirect
6. #21446791C2semidirect
7. #21446618C2semidirect
8. #21446127C2semidirect
9. #21446006C2semidirect
10. #21445826C2semidirect
11. C61×#2536C2semidirect
113. C2×Q8×C61#41semidirect
178. C2×#92418#41semidirect
271. #2536D61semidirect
319. C61#24132semidirect
391. C2×Q8#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
C2×#168400572
C2×#168398602
#214470012
#214468202
#214467912
#214466182
#214461272
#214460062
#214458262
C61×#25362
#2413261