Finite Group Databasebeta

← #29714944

Subgroups — #29714944

Order 15616.

1380 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840434C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic, complemented
3C2×#16839860C211yesmaximal, characteristic, complemented
4#21447053C212yesmaximal, complemented
5#21446906C212yesmaximal, complemented
6#21446760C211yesmaximal, characteristic, complemented
7#21446642C211yesmaximal, characteristic, complemented
8#21446156C212yesmaximal, complemented
9#21446065C212yesmaximal, complemented
10#21445826C211yesmaximal, characteristic, complemented
11C61×#2581C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13C22×#11758201C2211yescharacteristic
14C22×#11758201C2212yes—
15#16840434—24no—
16#16840434—24no—
17C2×#14519774C2212yes—
18C2×#14519774C2211yescharacteristic
19C2×#14519752—22no—
20C2×#14519752—21no—
21C2×C4×D244C2212yes—
22C2×C4×D244C2211yescharacteristic
23#16839962—24no—
24#16839860—24no—
25C2×#14519710C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#400—21no—
40C61×#395—21no—
41C61×#391C2212yes—
42C61×#389C2212yes—
43C61×#387—22no—
44C61×#382C2211yescharacteristic
45C2×#11758226—28no—
46C2×#11758226—48no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244—22no—
50C22×D244—21no—
51C22×D244C2312yes—
52C22×D244C2311yescharacteristic
53C22×C4×D61—22no—
54C22×C4×D61—21no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C2×#11758201—24no—
58C2×#11758201—24no—
59C2×#11758201—22no—
60C2×#11758201—42no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2312yes—
63C22×#92418C2311yescharacteristic
64#14519774—24no—
65#14519774—24no—
66#14519774—22no—
67#14519774D414yes—
68#14519752—28no—
69#14519752—216no—
70C4×D244—24no—
71C4×D244—22no—
72C4×D244—24no—
73C4×D244D414yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758198—22no—
76C2×#11758198—21no—
77#14519710—22no—
78C4⋊#92418—22no—
79#14519701—22no—
80#14519697—24no—
81#14519697—22no—
82#14519697—24no—
83#14519697—42no—
84#14519697—42no—
85#14519688—22no—
86C22×Q8×C61D411yescharacteristic
87C22×Q8×C61C2311yescharacteristic
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61D412yes—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—22no—
104#95×C61—22no—
105#95×C61—22no—
106C23×D61—22no—
107C23×D61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61#4112yescomplemented
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—21no—
125#11758226—432no—
126C2×D244—22no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—22no—
130C2×D244—21no—
131C2×D244—22no—
132C2×D244—21no—
133C2×D244—22no—
134C2×D244—28no—
135C2×D244—24no—
136C2×D244—22no—
137C2×D244—21no—
138C2×D244—28no—
139C2×D244—24no—
140C2×D244—44no—
141C2×D244—44no—
142C2×D244—44no—
143C2×D244—82no—
144C2×D244—82no—
145C2×D244—82no—
146C2×D244C2×D414yes—
147C2×D244C2×D412yes—
148C2×C4×D61—28no—
149C2×C4×D61—24no—
150C2×C4×D61—44no—
151C2×C4×D61—44no—
152C2×C4×D61—44no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—44no—
161C42×C61—22no—
162C42×C61—22no—
163C42×C61—42no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166#11758201—28no—
167#11758201—48no—
168#11758198—28no—
169#11758198—24no—
170C4⋊Dic61—21no—
171C4⋊Dic61C2×D412yes—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—21no—
180C2×#92418#4112yescomplemented
181C23×C61C22×D411yescharacteristic
182C22×D61—24no—
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—41no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194Q8×C61—28no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61—41no—
198Q8×C61—41no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
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×C61AGL(1,ℤ/8)14yes—
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—22no—
224C2×Dic61—42no—
225D244—22no—
226D244—22no—
227D244—24no—
228D244—24no—
229D244—48no—
230D244—416no—
231D244—42no—
232D244—42no—
233D244—41no—
234D244—41no—
235D244—84no—
236D244—88no—
237C4×D61—48no—
238C4×D61—416no—
239#92418—22no—
240#92418—41no—
241#92418—42no—
242#24200—611nomaximal, Sylow
243C22×C61C2×#12011yescharacteristic
244C22×C61#54811yescharacteristic
245C22×C61#55011yescharacteristic
246C22×C61C2×AGL(1,ℤ/8)12yes—
247C22×C61#57712yes—
248C2×D61—42no—
249C2×D61—44no—
250C2×D61—42no—
251C2×D61—44no—
252C2×D61—44no—
253C2×D61—42no—
254C2×D61—82no—
255C2×D61—82no—
256C2×D61—82no—
257C2×D61—82no—
258C2×D61—82no—
259C2×D61—82no—
260C4×C61—22no—
261C4×C61—24no—
262C4×C61—22no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—82no—
266C4×C61—81no—
267C4×C61#52511yescharacteristic
268C4×C61#53711yescharacteristic
269C4×C61#54611yescharacteristic
270C4×C61#54711yescharacteristic
271Dic61—42no—
272C2×#550—611no—
273C2×#460—611no—
274C2×#440—611no—
275#2581D6111yescharacteristic, complemented
276#2041—612no—
277#2014—612no—
278#1985—611no—
279#1961—611no—
280#1877—612no—
281#1862—612no—
282#1804—611no—
283C2×C61#240711yescharacteristic
284C2×C61#300311yescharacteristic
285C2×C61#300911yescharacteristic
286C2×C61#336812yes—
287C2×C61#339712yes—
288D61—82no—
289D61—84no—
290C22×#41—1221no—
291C22×SD16—611no—
292C22×SD16—612no—
293#550—1224no—
294#550—1224no—
295C2×#127—611no—
296C2×#127—612no—
297C2×#121—1222no—
298C2×#121—1221no—
299C2×C4×Q8C2×D6111yescharacteristic
300C2×C4×Q8C2×D6111yescharacteristic
301C2×C4×D4—611no—
302C2×C4×D4—612no—
303#460—1224no—
304#440—1224no—
305C2×#107—611no—
306C2×C4⋊C8—611no—
307C2×#103—611no—
308C2×#102—612no—
309C2×#102—612no—
310C2×#102—611no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#97—611no—
314#400—21no—
315#395—21no—
316#391C2×D6112yes—
317#389C2×D6112yes—
318#387—22no—
319#382C2×D6111yescharacteristic
320#340—612no—
321#336—612no—
322#336—612no—
323C61#2420011yescharacteristic, complemented, Sylow
324C2×#41—1228no—
325C2×#41—2448no—
326C22×Q8C22⋊D6111yescharacteristic
327C22×Q8C22×D6111yescharacteristic
328C22×D4—611no—
329C22×D4—612no—
330C22×D4—1222no—
331C22×D4—1221no—
332C22×D4—1221no—
333C22×D4—1222no—
334C23×C4—1222no—
335C23×C4—1221no—
336C2×SD16—1222no—
337C2×SD16—1222no—
338C2×SD16—1222no—
339C2×SD16—1224no—
340C2×SD16—1224no—
341C2×SD16—2442no—
342C22×C8—611no—
343C22×C8—612no—
344C22×C8—611no—
345#127—614no—
346#127—1224no—
347#127—1224no—
348#127—1222no—
349#121—1228no—
350#121—12216no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×D4—614no—
356C4×D4—1224no—
357C4×D4—1224no—
358C4×D4—1222no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4C22⋊D6112yes—
363C2×C4⋊C4C22⋊D6111yescharacteristic
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C4⋊C4C22×D6112yes—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1221no—
368C2×C42—21no—
369C2×C42C22×D6112yes—
370C2×C42C22×D6111yescharacteristic
371#107—1222no—
372C4⋊C8—1222no—
373#103—1222no—
374#102—1224no—
375#102—1222no—
376#102—1224no—
377#102—2442no—
378#102—2442no—
379#97—1222no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385C24—1222no—
386C24—1221no—
387#41—24432no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8#1175822612yescomplemented
396C2×D4—612no—
397C2×D4—614no—
398C2×D4—1228no—
399C2×D4—1228no—
400C2×D4—1222no—
401C2×D4—1221no—
402C2×D4—1222no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—1222no—
406C2×D4—1221no—
407C2×D4—1222no—
408C2×D4—1221no—
409C2×D4—1222no—
410C2×D4—1224no—
411C2×D4—1224no—
412C2×D4—4882no—
413C2×D4—4882no—
414C2×D4—4882no—
415C2×D4—2444no—
416C2×D4—2444no—
417C2×D4—2444no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—1228no—
422C22×C4—1221no—
423C22×C4—1224no—
424C22×C4—2444no—
425C22×C4—2444no—
426C22×C4—2444no—
427C22×C4C2×C22⋊D6112yes—
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6112yes—
431C22×C4C23×D6111yescharacteristic
432SD16—1228no—
433SD16—2448no—
434C2×C8—612no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—1221no—
441C2×C8—1222no—
442C2×C8—1221no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—612no—
452C4⋊C4—1221no—
453C22⋊C4—1228no—
454C22⋊C4—1224no—
455C42—22no—
456C42—22no—
457C42—42no—
458C42D4×D6112yes—
459C42D4×D6112yes—
460C23—1224no—
461C23—1222no—
462C23—2442no—
463C23—2442no—
464C23—2442no—
465C23—2442no—
466C23—2442no—
467C23—2442no—
468C23—2441no—
469C23—2442no—
470C23—2442no—
471C23—2441no—
472C23C22×C22⋊D6111yescharacteristic
473Q8—28no—
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478D4—1222no—
479D4—1222no—
480D4—1224no—
481D4—1224no—
482D4—4884no—
483D4—4888no—
484D4—2448no—
485D4—24416no—
486D4—2442no—
487D4—2442no—
488D4—2441no—
489D4—2441no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—1222no—
506C2×C4—2448no—
507C2×C4—24416no—
508C2×C4—2442no—
509C2×C4#1451974512yes—
510C2×C4#1451974512yes—
511C2×C4#1451980214yes—
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4#41×D6111yescharacteristic
515C2×C4C2×#1175822611yescharacteristic
516C2×C4#1451982311yescharacteristic
517C2×C4#1451982511yescharacteristic
518C8—1222no—
519C8—2441no—
520C8—2442no—
521C22—4882no—
522C22—4882no—
523C22—4882no—
524C22—4882no—
525C22—4882no—
526C22—4882no—
527C22—2442no—
528C22—2444no—
529C22—2442no—
530C22—2444no—
531C22—2444no—
532C22—2442no—
533C22C2×#1451974511yescharacteristic
534C22#1684042311yescharacteristic
535C22#1684043111yescharacteristic
536C22C2×#1451980212yes—
537C22#1684057012yes—
538C4—22no—
539C4—24no—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—81no—
545C4—2442no—
546C4#1684027511yescharacteristic
547C4#1684034211yescharacteristic
548C4#1684040311yescharacteristic
549C4#1684041411yescharacteristic
550C2—4882no—
551C2—4884no—
552C2#2144916611yescharacteristic
553C2#2145504511yescharacteristic
554C2#2145509411yescharacteristic
555C2#2145854812yes—
556C2#2145883912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2581C2
socleC23×C61C22×D4
prime radicalC2×#16840434C2
hypercenter#2581D61
hypocenterC61#24200
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C2×#16839962C2semidirect
3. C2×#16839860C2semidirect
4. #21447053C2semidirect
5. #21446906C2semidirect
6. #21446760C2semidirect
7. #21446642C2semidirect
8. #21446156C2semidirect
9. #21446065C2semidirect
10. #21445826C2semidirect
11. C61×#2581C2semidirect
115. C2×Q8×C61#41semidirect
180. C2×#92418#41semidirect
275. #2581D61semidirect
323. C61#24200semidirect
395. C2×Q8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404342
C2×#168399622
C2×#168398602
#214470532
#214469062
#214467602
#214466422
#214461562
#214460652
#214458262
C61×#25812
#2420061