Finite Group Databasebeta

← #29714903

Subgroups — #29714903

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839964C211yesmaximal, characteristic, complemented
2C2×#16839872C211yesmaximal, characteristic
3#21451177C211yesmaximal, characteristic, complemented
4#21447055C212yesmaximal, complemented
5#21446896C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446640C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal, complemented
9#21446087C212yesmaximal, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×#11758199C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16839964—24no—
16#16839872—24no—
17C2×#14519708C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×#14519688C2211yescharacteristic
26#16839697—21no—
27#16839673—21no—
28#16839664C2212yes—
29#16839660C2212yes—
30#16839633—22no—
31#16839628C2211yescharacteristic
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#556—24no—
36C61×#556—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#126×C61—22no—
39C2×#124×C61C2212yes—
40C2×C22⋊Q8×C61—21no—
41C2×C4×Q8×C61C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61—21no—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758201—24no—
48C2×#11758201—24no—
49C2×#11758201—42no—
50C2×#11758199—22no—
51C2×#11758199—22no—
52C2×#11758199—22no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55C22×#92418C2311yescharacteristic
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64#14519708—22no—
65C4⋊#92418—22no—
66#14519701—24no—
67#14519701—24no—
68#14519701—42no—
69#14519698—22no—
70#14519698—42no—
71#14519697—22no—
72#14519688—22no—
73#14519684—21no—
74#14519684—21no—
75#14519684—22no—
76C22×Q8×C61D411yescharacteristic
77C22×Q8×C61C2312yes—
78C22×D4×C61C2311yescharacteristic
79C23×C4×C61—21no—
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61D414yes—
82#126×C61—216no—
83#124×C61—24no—
84#124×C61—24no—
85C22⋊Q8×C61—28no—
86C4×Q8×C61—24no—
87C4×Q8×C61—24no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61D414yes—
92#117×C61—28no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61D412yes—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—22no—
102C2×C42×C61—22no—
103C2×C42×C61C2312yes—
104C2×C42×C61C2311yescharacteristic
105C24×C61—21no—
106C23×D61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—42no—
114C2×Q8×C61C2×D414yes—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61C2×D412yes—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61—44no—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—22no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244#4112yes—
138C2×C4×D61—42no—
139C2×C4×D61—82no—
140C4⋊C4×C61—28no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—28no—
148C22⋊C4×C61—28no—
149C22⋊C4×C61—24no—
150C42×C61—28no—
151C42×C61—22no—
152C42×C61—22no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155#11758201—48no—
156#11758199—28no—
157#11758198—44no—
158#11758198—42no—
159#11758198—82no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164#11758196—82no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418#4112yes—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184C22×D61—42no—
185Q8×C61—24no—
186Q8×C61—24no—
187Q8×C61—44no—
188Q8×C61—42no—
189Q8×C61—42no—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—41no—
193D4×C61—41no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—48no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×C4×C612−1+411yescharacteristic
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221D244—22no—
222D244—22no—
223D244—41no—
224D244—41no—
225C4×D61—84no—
226#92418—22no—
227#92418—41no—
228#92418—42no—
229#24197—611nomaximal, Sylow
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61C2×#12011yescharacteristic
236C22×C61#54811yescharacteristic
237C22×C61#55011yescharacteristic
238C22×C61#57512yes—
239C22×C61#57712yes—
240C2×D61—44no—
241C2×D61—42no—
242C2×D61—82no—
243C2×D61—81no—
244C2×D61—81no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—44no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61#52511yescharacteristic
252C4×C61#53711yescharacteristic
253C4×C61#54611yescharacteristic
254C4×C61#54711yescharacteristic
255Dic61—82no—
256Dic61—81no—
257C2×#556D6111yescharacteristic, complemented
258C2×#461—611no—
259C2×#442—611no—
260#2580—611no—
261#2042—612no—
262#2012—612no—
263#1985—611no—
264#1960—611no—
265#1880—612no—
266#1865—612no—
267#1804—611no—
268C2×C61—82no—
269C2×C61#240711yescharacteristic
270C2×C61#300511yescharacteristic
271C2×C61#301011yescharacteristic
272C2×C61#337612yes—
273C2×C61#340112yes—
274D61—82no—
275C22×SD16—612no—
276C22×D8—611no—
277#556—24no—
278#556—24no—
279C2×C4⋊Q8C2×D6111yescharacteristic
280C2×#126—22no—
281C2×#124C2×D6112yes—
282C2×C22⋊Q8—21no—
283C2×C4×Q8C2×D6112yes—
284C2×C4×Q8C2×D6111yescharacteristic
285C2×C4×D4—611no—
286C2×C4×D4C2×D6111yescharacteristic
287C2×#117—21no—
288#461—1224no—
289#442—1224no—
290C2×#106—611no—
291C2×C4⋊C8—611no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103—612no—
295C2×#102—611no—
296C2×#102—611no—
297C2×#102—612no—
298C2×#97—611no—
299#399—1221no—
300#393—1221no—
301#390—612no—
302#389—612no—
303#385—1222no—
304#382—611no—
305#340—612no—
306#336—612no—
307#336—612no—
308C61#2419711yescharacteristic, complemented, Sylow
309C22×Q8C22⋊D6111yescharacteristic
310C22×Q8C22×D6112yes—
311C22×D4—611no—
312C22×D4C22×D6111yescharacteristic
313C23×C4—21no—
314C23×C4—1221no—
315C2×SD16—1224no—
316C2×SD16—1224no—
317C2×SD16—2442no—
318C2×D8—1222no—
319C2×D8—1222no—
320C2×D8—1222no—
321C22×C8—611no—
322C22×C8—612no—
323C22×C8—611no—
324C4⋊Q8—22no—
325C4⋊Q8C22⋊D6114yes—
326#126—216no—
327#124—24no—
328#124—24no—
329C22⋊Q8—28no—
330C4×Q8—24no—
331C4×Q8—24no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×D4—22no—
335C4×D4—1222no—
336C4×D4—1222no—
337C4×D4C22⋊D6114yes—
338#117—28no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4C22⋊D6111yescharacteristic
346C2×C4⋊C4C22⋊D6112yes—
347C2×C4⋊C4C22×D6111yescharacteristic
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1222no—
354C2×C42—22no—
355C2×C42C22×D6112yes—
356C2×C42C22×D6111yescharacteristic
357#106—1222no—
358C4⋊C8—1222no—
359#103—1224no—
360#103—1224no—
361#103—2442no—
362#102—1222no—
363#102—1222no—
364#102—2442no—
365#97—1222no—
366#95—1221no—
367#95—1221no—
368#95—1222no—
369C24—21no—
370C24—1221no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—42no—
378C2×Q8C2×C22⋊D6114yes—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—612no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4C2×C22⋊D6112yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—44no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—4882no—
398C22×C4—2442no—
399C22×C4C2×C22⋊D6112yes—
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404SD16—2448no—
405D8—1228no—
406C2×C8—612no—
407C2×C8—1221no—
408C2×C8—1221no—
409C2×C8—1221no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1221no—
414C2×C8—1221no—
415C4⋊C4—28no—
416C4⋊C4—21no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—4882no—
426C4⋊C4C2×C22⋊D6112yes—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—24no—
430C22⋊C4—4882no—
431C22⋊C4—2444no—
432C22⋊C4—2442no—
433C42—28no—
434C42—22no—
435C42—22no—
436C42C2×C22⋊D6112yes—
437C42C2×C22⋊D6112yes—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—1222no—
444C23—2441no—
445C23—2442no—
446C23—2442no—
447C23—2441no—
448C23C22×C22⋊D6111yescharacteristic
449Q8—24no—
450Q8—24no—
451Q8—44no—
452Q8—42no—
453Q8—42no—
454D4—22no—
455D4—22no—
456D4—41no—
457D4—41no—
458D4—1222no—
459D4—1222no—
460D4—2441no—
461D4—2441no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—4884no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4#1451974612yes—
483C2×C4#1451974612yes—
484C2×C4C22×C22⋊D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4#41×D6111yescharacteristic
487C2×C4#41×D6111yescharacteristic
488C2×C4#1451982311yescharacteristic
489C2×C4#1451982411yescharacteristic
490C8—1222no—
491C8—2441no—
492C8—2442no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—82no—
497C22—82no—
498C22—4881no—
499C22—4882no—
500C22—4881no—
501C22—2442no—
502C22—2444no—
503C22C2×#1451974611yescharacteristic
504C22#1684041911yescharacteristic
505C22#1684043111yescharacteristic
506C22#1684056012yes—
507C22#1684057012yes—
508C4—22no—
509C4—22no—
510C4—42no—
511C4—44no—
512C4—42no—
513C4—42no—
514C4—4881no—
515C4—4882no—
516C4#1684026911yescharacteristic
517C4#1684034011yescharacteristic
518C4#1684040011yescharacteristic
519C4#1684041111yescharacteristic
520C2—82no—
521C2—4882no—
522C2#2144915811yescharacteristic
523C2#2145506111yescharacteristic
524C2#2145510511yescharacteristic
525C2#2145864012yes—
526C2#2145888512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#556D61
hypocenterC61#24197
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16839964C2semidirect
3. #21451177C2semidirect
4. #21447055C2semidirect
6. #21446760C2semidirect
7. #21446640C2semidirect
8. #21446174C2semidirect
9. #21446087C2semidirect
10. #21445829C2semidirect
11. C2×C61×#556C2semidirect
257. C2×#556D61semidirect
308. C61#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
C2×#168398722
#214511772
#214470552
#214468962
#214467602
#214466402
#214461742
#214460872
#214458292
C2×C61×#5562
#2419761