Finite Group Databasebeta

← #29665401

Subgroups — #29665401

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455628C211yesmaximal, characteristic, complemented
2#21455268C211yesmaximal, characteristic
3#21454927C211yesmaximal, characteristic, complemented
4#21445246C212yesmaximal, complemented
5#21445239C212yesmaximal, complemented
6#21445164C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444590C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10C2×#16839508C211yesmaximal, characteristic, complemented
11C61×#3206C211yesmaximal, characteristic, complemented
12C2×#14519797—21no—
13C4×#11758222C2211yescharacteristic
14#16840054—21no—
15#16840049—21no—
16#16840044C2212yes—
17#16840039C2212yes—
18#16839981—21no—
19#16839974—21no—
20#16839969C2211yescharacteristic
21#16839964C2211yescharacteristic
22#16839959C2211yescharacteristic
23#16839954C2211yescharacteristic
24C8×Dic122C2212yes—
25C8×D244C2212yes—
26#16839825—22no—
27#16839814—21no—
28#16839802—21no—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C4⋊C8×D61—21no—
33#16839761—21no—
34#16839758—21no—
35#16839513C2212yes—
36#16839513C2212yes—
37#16839508C2212yes—
38#16839508C2212yes—
39#16839508C2212yes—
40#16839506C2212yes—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#492C2212yes—
44C61×#487C2212yes—
45C61×#461—21no—
46C61×#461—21no—
47C61×#448—22no—
48C61×#431C2211yescharacteristic
49C2×#11758222D411yescharacteristic
50#14519797—44no—
51C2×#11758202—21no—
52C2×#11758200—21no—
53C2×#11758201—21no—
54C2×#11758199—21no—
55C2×#11758194—21no—
56C22×#92418—21no—
57C22×#92418—21no—
58C2×C8×D61—22no—
59C4×Dic122D411yescharacteristic
60C4×Dic122C2311yescharacteristic
61C4×D244D411yescharacteristic
62C4×D244C2311yescharacteristic
63C4×C22⋊D61—21no—
64C4×C22⋊D61—21no—
65#14519736D411yescharacteristic
66#14519734—21no—
67C42×D61—21no—
68#14519710—22no—
69#14519708—22no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2312yes—
76C4⋊#92418C2312yes—
77#14519706—22no—
78#14519701—21no—
79#14519701—21no—
80#14519700—22no—
81#14519698—21no—
82#14519698—21no—
83#14519697—22no—
84#14519689—22no—
85C8×Dic61—22no—
86C2×#41×C61—21no—
87C22×Q8×C61—21no—
88C2×SD16×C61—22no—
89C2×SD16×C61—22no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4⋊Q8×C61C2311yescharacteristic
95#127×C61C2311yescharacteristic
96#124×C61—21no—
97#121×C61—21no—
98#121×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61D411yescharacteristic
106#103×C61—22no—
107#103×C61—22no—
108C22⋊C8×C61D412yes—
109C4×C8×C61C2312yes—
110#41×C61—42no—
111#41×C61—42no—
112C2×Q8×C61—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—41no—
115C2×Q8×C61—81no—
116C2×Q8×C61—81no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C2×C22⋊D61—21no—
127#11758222—22no—
128#11758222SD1614yescomplemented
129C2×D244C2×D411yescharacteristic
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×Dic122C2×D411yescharacteristic
134SD16×C61—82no—
135SD16×C61—82no—
136C2×C8×C61C2×D412yes—
137C2×C8×C61C2×D412yes—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61#4111yescharacteristic
149C42×C61C2411yescharacteristic
150C22⋊Dic61—21no—
151#11758202—42no—
152#11758201—42no—
153#11758200—42no—
154#11758199—42no—
155#11758198—21no—
156#11758198—21no—
157C4⋊Dic61C2×D411yescharacteristic
158C4⋊Dic61C2×D411yescharacteristic
159#11758196—21no—
160#11758196—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—42no—
170C2×#92418—81no—
171C8×D61—82no—
172C23×C61—41no—
173C23×C61#12111yescharacteristic
174C22×D61—21no—
175Q8×C61—41no—
176Q8×C61—41no—
177Q8×C61—42no—
178Q8×C61—42no—
179Q8×C61—82no—
180D4×C61—41no—
181D4×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—81no—
185D4×C61—82no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—82no—
192C2×C4×C61—82no—
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61C2×SD1612yes—
197C2×C4×C61#13512yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C22⋊D61—24no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205D244—21no—
206D244C2×SD1612yes—
207C4×D61—24no—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—81no—
211Dic122—21no—
212Dic122C2×SD1612yes—
213C8×C61—22no—
214C8×C61—22no—
215#92418—41no—
216#92418—41no—
217#92418—42no—
218#92418—42no—
219#19408—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61#46512yes—
226C22×C61C2×#12111yescharacteristic
227C2×D61—22no—
228C2×D61—41no—
229C2×D61—41no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—82no—
234C4×C61—81no—
235C4×C61#46512yes—
236C4×C61C22×SD1611yescharacteristic
237C4×C61C2×#13511yescharacteristic
238C4×C61#57611yescharacteristic
239C4×C61#57711yescharacteristic
240Dic61—22no—
241Dic61—41no—
242Dic61—41no—
243#3206D6111yescharacteristic, complemented
244#3079—611no—
245#3035—611no—
246#2988—611no—
247#1680—612no—
248#1679—612no—
249#1666—612no—
250#1665—612no—
251#1570—611no—
252#1547—611no—
253C2×#332—611no—
254C2×C61—22no—
255C2×C61—161no—
256C2×C61#304511yescharacteristic
257C2×C61C2×#46511yescharacteristic
258C2×C61#306911yescharacteristic
259D61—42no—
260C2×#135—1221no—
261#532C2×D6111yescharacteristic
262C2×C4⋊Q8C2×D6111yescharacteristic
263C4×#41—611no—
264#492C2×D6112yes—
265#487C2×D6112yes—
266#477—1221no—
267#476—1221no—
268#475—612no—
269#474—612no—
270#464—1221no—
271#463—1221no—
272#462—611no—
273#461—21no—
274#461—21no—
275#461—611no—
276#460—611no—
277#459—611no—
278#448—22no—
279C8×Q8—612no—
280C8×D4—612no—
281C8×D4—1222no—
282#431—1221no—
283#431C2×D6111yescharacteristic
284#428—1221no—
285#423—611no—
286C2×C4⋊C8—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—1221no—
289#417—1221no—
290#416—1221no—
291#334—612no—
292#334—612no—
293#332—612no—
294#332—612no—
295#332—612no—
296#331—612no—
297C61#1940811yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41—611no—
300C22×Q8—21no—
301#135—2444no—
302C2×Q16—1221no—
303C2×SD16—22no—
304C2×SD16—22no—
305C2×SD16—1221no—
306C2×SD16—1221no—
307C2×D8—1221no—
308C2×M16—1221no—
309C22×C8—1221no—
310C22×C8—1221no—
311C22×C8—1222no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8C22×D6112yes—
316C4⋊Q8C22×D6111yescharacteristic
317#127C22×D6111yescharacteristic
318#124—21no—
319#121—21no—
320#121—21no—
321C4×Q8—611no—
322C4×Q8—611no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—611no—
326C4×D4—611no—
327C4×D4—1221no—
328C4×D4—1221no—
329#117—611no—
330#117—1221no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C42—1221no—
334C2×C42C22×D6111yescharacteristic
335#107—1222no—
336#106—1222no—
337C4⋊C8—612no—
338C4⋊C8—612no—
339C4⋊C8—611no—
340C4⋊C8—611no—
341C4⋊C8—1222no—
342C4⋊C8—1221no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8C22⋊D6111yescharacteristic
346C4⋊C8C22⋊D6111yescharacteristic
347#103—22no—
348#103—22no—
349#103—1222no—
350#103—1221no—
351#103—1221no—
352#102—1222no—
353#102—1221no—
354#102—1221no—
355C22⋊C8—1222no—
356C22⋊C8C22⋊D6112yes—
357C4×C8—1222no—
358C4×C8C22×D6112yes—
359#41—42no—
360#41—42no—
361#41—614no—
362#41—1222no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8—81no—
367C2×Q8—81no—
368C2×Q8—611no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—611no—
374C2×D4—1221no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4D4×D6111yescharacteristic
381C22×C4#1175822411yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383Q16—2442no—
384SD16—82no—
385SD16—82no—
386SD16—2442no—
387SD16—2442no—
388D8—2442no—
389M16—2442no—
390C2×C8—1222no—
391C2×C8—1221no—
392C2×C8—1221no—
393C2×C8—1222no—
394C2×C8—4882no—
395C2×C8—4881no—
396C2×C8—2442no—
397C2×C8C2×C22⋊D6112yes—
398C2×C8C2×C22⋊D6112yes—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—611no—
406C4⋊C4—611no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—1221no—
412C22⋊C4—1221no—
413C22⋊C4—1221no—
414C42—1221no—
415C42—1221no—
416C42—1221no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—41no—
422C23—1221no—
423C23#1451974811yescharacteristic
424Q8—41no—
425Q8—41no—
426Q8—42no—
427Q8—42no—
428Q8—82no—
429Q8—612no—
430Q8—1221no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—81no—
436D4—82no—
437D4—612no—
438D4—1224no—
439D4—1221no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—1221no—
448C2×C4—1224no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—4881no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4#1451974811yescharacteristic
455C2×C4#1451974811yescharacteristic
456C2×C4#1451974811yescharacteristic
457C2×C4SD16×D6112yes—
458C2×C4#1451979812yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C2×#1175822411yescharacteristic
462C8—22no—
463C8—22no—
464C8—2441no—
465C8—2441no—
466C8—2442no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—1222no—
474C22—2441no—
475C22—2441no—
476C22#1683998812yes—
477C22C2×#1451974811yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—81no—
482C4—82no—
483C4—1222no—
484C4—2441no—
485C4—2441no—
486C4#1683998812yes—
487C4C2×SD16×D6111yescharacteristic
488C4C2×#1451979811yescharacteristic
489C4#1684056511yescharacteristic
490C4#1684057311yescharacteristic
491C2—22no—
492C2—161no—
493C2—2442no—
494C2#2145535411yescharacteristic
495C2C2×#1683998811yescharacteristic
496C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3206C2
socleC22×C61C2×#121
prime radical#21455268C2
hypercenter#3206D61
hypocenterC61#19408
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
1. #21455628C2semidirect
3. #21454927C2semidirect
4. #21445246C2semidirect
5. #21445239C2semidirect
6. #21445164C2semidirect
7. #21445157C2semidirect
8. #21444590C2semidirect
9. #21444496C2semidirect
10. C2×#16839508C2semidirect
11. C61×#3206C2semidirect
128. #11758222SD16semidirect
243. #3206D61semidirect
297. C61#19408semidirect

Maximal subgroups

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

subgroupindex
#214556282
#214552682
#214549272
#214452462
#214452392
#214451642
#214451572
#214445902
#214444962
C2×#168395082
C61×#32062
#1940861