Finite Group Databasebeta

← #29711963

Subgroups — #29711963

Order 15616.

1334 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840367C211yesmaximal, characteristic, complemented
2C2×#16839907C212yesmaximal, complemented
3#21446970C211yesmaximal, characteristic, complemented
4#21446965C211yesmaximal, characteristic, complemented
5#21446906C211yesmaximal, characteristic, complemented
6#21446889C211yesmaximal, characteristic, complemented
7#21446170C212yesmaximal, complemented
8#21446156C212yesmaximal, complemented
9#21446136C212yesmaximal, complemented
10#21445890C211yesmaximal, characteristic, complemented
11C61×#2311C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×#11758199C2212yes—
14#16840367—28no—
15C2×#14519774C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519762—22no—
18C2×#14519751C2212yes—
19C2×#14519750—22no—
20C2×C4×D244C2212yes—
21#16839907—28no—
22C2×#14519701C2212yes—
23C2×#14519698C2212yes—
24C2×#14519697C2212yes—
25C2×#14519697C2212yes—
26C2×C22⋊#92418C2212yes—
27#16839524C2212yes—
28#16839524C2212yes—
29#16839517C2212yes—
30#16839517C2211yescharacteristic
31#16839517C2211yescharacteristic
32C22×#41×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#391C2211yescharacteristic
36C61×#391C2211yescharacteristic
37C61×#390C2211yescharacteristic
38C61×#390C2211yescharacteristic
39C61×#386—22no—
40C61×#380—22no—
41C22×D244—22no—
42C22×D244C2312yes—
43C22×C22⋊D61—22no—
44C22×C4×D61—22no—
45C2×#11758201—24no—
46C2×#11758201—24no—
47C2×#11758201—42no—
48C2×#11758199—24no—
49C2×#11758199—24no—
50C2×#11758199—42no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519774—24no—
54#14519769—24no—
55#14519762—216no—
56#14519751—24no—
57#14519751D418yes—
58#14519750—216no—
59C4×D244—24no—
60C4×D244—24no—
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65#14519701—24no—
66#14519698—24no—
67#14519697—42no—
68#14519697—42no—
69C22⋊#92418—24no—
70C2×#41×C61—22no—
71C2×#41×C61—22no—
72C2×#41×C61—24no—
73C2×#41×C61—42no—
74C22×Q8×C61C2311yescharacteristic
75C22×D4×C61—21no—
76C22×D4×C61C2311yescharacteristic
77C23×C4×C61—21no—
78C23×C4×C61—21no—
79C23×C4×C61C2311yescharacteristic
80#117×C61—24no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—42no—
83C2×C4⋊C4×C61D412yes—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61D412yes—
89C2×C42×C61C2311yescharacteristic
90C2×C42×C61C2311yescharacteristic
91#95×C61—22no—
92#95×C61—22no—
93C24×C61—21no—
94C24×C61C2×D411yescharacteristic
95C23×D61—22no—
96#41×C61—48no—
97#41×C61—48no—
98C2×Q8×C61—21no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61#4112yes—
102C2×D4×C61—24no—
103C2×D4×C61—22no—
104C2×D4×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—42no—
107C2×D4×C61—42no—
108C2×D4×C61#4112yes—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—24no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—82no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D414yes—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—28no—
126C2×C22⋊D61—82no—
127C2×C22⋊D61—82no—
128C22×Dic61—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—28no—
135C2×D244—82no—
136C2×D244—82no—
137C2×D244C2×D414yes—
138C2×C4×D61—28no—
139C2×C4×D61—82no—
140C2×C4×D61—82no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C22⋊C4×C61—24no—
146C22⋊C4×C61—82no—
147C22⋊C4×C61—82no—
148C42×C61—22no—
149C42×C61—22no—
150#11758201—48no—
151#11758199—48no—
152#11758198—28no—
153#11758198—28no—
154C4⋊Dic61—22no—
155C4⋊Dic61C2×D414yes—
156#11758196—28no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61#12012yes—
173C23×C61C22×D411yescharacteristic
174C22×D61—24no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181Q8×C61—24no—
182Q8×C61—42no—
183D4×C61—24no—
184D4×C61—48no—
185D4×C61—42no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—48no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—84no—
205C2×C4×C61#12012yes—
206C2×C4×C61C22×D412yes—
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61H(5,𝔽2)12yes—
210C22⋊D61—88no—
211C2×Dic61—24no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214D244—24no—
215D244—24no—
216D244—42no—
217D244—42no—
218D244—88no—
219C4×D61—88no—
220#92418—42no—
221#92418—42no—
222#23941—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—24no—
225C22×C61—42no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—44no—
229C22×C61—42no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—82no—
233C22×C61C2×#12011yescharacteristic
234C22×C61#52511yescharacteristic
235C22×C61#53511yescharacteristic
236C22×C61#57612yes—
237C22×C61#57712yes—
238C2×D61—44no—
239C2×D61—44no—
240C2×D61—44no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—81no—
250C4×C61—81no—
251C4×C61#54612yes—
252C4×C61#54612yes—
253Dic61—44no—
254C2×#540—611no—
255C2×#449—612no—
256#2311D6111yescharacteristic, complemented
257#2026—611no—
258#2025—611no—
259#2014—611no—
260#2011—611no—
261#1879—612no—
262#1877—612no—
263#1874—612no—
264#1818—611no—
265C2×C61—42no—
266C2×C61—82no—
267C2×C61#240411yescharacteristic
268C2×C61#301212yes—
269C2×C61#311912yes—
270C2×C61#322011yescharacteristic
271C2×C61#322111yescharacteristic
272D61—84no—
273C22×#41C2×D6111yescharacteristic
274C22×SD16—612no—
275C22×D8—612no—
276#540—1228no—
277C2×#127—611no—
278C2×#125—611no—
279C2×#123—1222no—
280C2×#121—612no—
281C2×#121—1222no—
282C2×C4×D4—612no—
283C2×#117C2×D6111yescharacteristic
284C22×C4⋊C4C2×D6111yescharacteristic
285#449—1228no—
286C2×#103—612no—
287C2×#102—612no—
288C2×#102—612no—
289C2×#102—612no—
290C2×C22⋊C8—612no—
291#391C2×D6111yescharacteristic
292#391C2×D6111yescharacteristic
293#390C2×D6111yescharacteristic
294#390C2×D6111yescharacteristic
295#386—22no—
296#380—22no—
297#340—612no—
298#340—612no—
299#336—612no—
300#336—611no—
301#336—611no—
302C61#2394111yescharacteristic, complemented, Sylow
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—24no—
306C2×#41—42no—
307C22×Q8C22×D6111yescharacteristic
308C22×D4—21no—
309C22×D4—612no—
310C22×D4—1222no—
311C22×D4—1222no—
312C22×D4C22×D6111yescharacteristic
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—1222no—
316C23×C4C22×D6111yescharacteristic
317C2×SD16—1224no—
318C2×SD16—1224no—
319C2×SD16—2442no—
320C2×D8—1224no—
321C2×D8—1224no—
322C2×D8—2442no—
323C22×C8—612no—
324C22×C8—612no—
325#127—1224no—
326#125—1224no—
327#123—12216no—
328#121—618no—
329#121—12216no—
330#121—1224no—
331C4×D4—1224no—
332C4×D4—1224no—
333#117—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4C22⋊D6112yes—
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C4⋊C4C22×D6111yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4C22⋊D6112yes—
346C2×C42C22×D6111yescharacteristic
347C2×C42C22×D6111yescharacteristic
348#103—1224no—
349#102—1224no—
350#102—2442no—
351#102—2442no—
352C22⋊C8—1224no—
353#95—22no—
354#95—22no—
355C24—21no—
356C24—1222no—
357C24C2×C22⋊D6111yescharacteristic
358#41—48no—
359#41—48no—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8#1175822212yes—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—614no—
371C2×D4—1228no—
372C2×D4—1228no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4—1222no—
378C2×D4—4882no—
379C2×D4—4882no—
380C2×D4—4882no—
381C2×D4—4882no—
382C2×D4#1175822212yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—82no—
394C22×C4—1228no—
395C22×C4—1222no—
396C22×C4—4882no—
397C22×C4—4882no—
398C22×C4D4×D6114yes—
399C22×C4C2×C22⋊D6112yes—
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403SD16—2448no—
404D8—2448no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—614no—
416C4⋊C4—1228no—
417C4⋊C4—1222no—
418C22⋊C4—24no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4—1228no—
422C22⋊C4—1228no—
423C42—22no—
424C42—22no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—21no—
429C23—22no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—1224no—
435C23—2442no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23—2442no—
440C23—2442no—
441C23#1451974512yes—
442C23C22×C22⋊D6111yescharacteristic
443Q8—24no—
444Q8—42no—
445D4—24no—
446D4—48no—
447D4—42no—
448D4—1224no—
449D4—1224no—
450D4—4888no—
451D4—4888no—
452D4—2442no—
453D4—2442no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—48no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—84no—
473C2×C4—1224no—
474C2×C4—4888no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4#1451974512yes—
478C2×C4C2×D4×D6112yes—
479C2×C4C2×#1175822211yescharacteristic
480C2×C4C2×#1175822211yescharacteristic
481C2×C4#1451982212yes—
482C8—2442no—
483C8—2442no—
484C22—24no—
485C22—22no—
486C22—42no—
487C22—41no—
488C22—41no—
489C22—44no—
490C22—42no—
491C22—82no—
492C22—81no—
493C22—81no—
494C22—4882no—
495C22—4882no—
496C22—4882no—
497C22—4882no—
498C22—2444no—
499C22—2444no—
500C22—2444no—
501C22C2×#1451974511yescharacteristic
502C22#1684027311yescharacteristic
503C22#1684032911yescharacteristic
504C22#1684056512yes—
505C22#1684057012yes—
506C4—22no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—81no—
511C4—81no—
512C4—2444no—
513C4#1684040212yes—
514C4#1684040212yes—
515C2—42no—
516C2—82no—
517C2—4884no—
518C2#2144913811yescharacteristic
519C2#2145512012yes—
520C2#2145601812yes—
521C2#2145686911yescharacteristic
522C2#2145688011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC23×C61C22×D4
hypercenter#2311D61
hypocenterC61#23941
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. C2×#16840367C2semidirect
2. C2×#16839907C2semidirect
3. #21446970C2semidirect
4. #21446965C2semidirect
5. #21446906C2semidirect
6. #21446889C2semidirect
7. #21446170C2semidirect
8. #21446156C2semidirect
9. #21446136C2semidirect
10. #21445890C2semidirect
11. C61×#2311C2semidirect
256. #2311D61semidirect
302. C61#23941semidirect

Maximal subgroups

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

subgroupindex
C2×#168403672
C2×#168399072
#214469702
#214469652
#214469062
#214468892
#214461702
#214461562
#214461362
#214458902
C61×#23112
#2394161