Finite Group Databasebeta

← #29665409

Subgroups — #29665409

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#21456744C211yesmaximal, characteristic, complemented
2#21455274C211yesmaximal, characteristic
3#21454930C211yesmaximal, characteristic, complemented
4#21445247C212yesmaximal, complemented
5#21445240C212yesmaximal, complemented
6#21445169C212yesmaximal, complemented
7#21445162C212yesmaximal, complemented
8#21444591C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic, complemented
10C2×#16839508C211yesmaximal, characteristic, complemented
11C61×#3079C211yesmaximal, characteristic, complemented
12C2×#14519797—21no—
13#16840312C2211yescharacteristic
14#16840136C2212yes—
15#16840112C2212yes—
16#16839984—21no—
17#16839977—21no—
18#16839972C2211yescharacteristic
19#16839967C2211yescharacteristic
20#16839964—21no—
21#16839964—21no—
22#16839962C2211yescharacteristic
23#16839957C2211yescharacteristic
24#16839901—22no—
25Q8×#92418C2212yes—
26D4×#92418—22no—
27D4×#92418C2212yes—
28#16839815—21no—
29#16839815C2211yescharacteristic
30C2×C4⋊#92418—21no—
31C2×C4⋊#92418C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839756—21no—
34#16839512C2212yes—
35#16839512C2212yes—
36#16839509C2212yes—
37#16839508C2212yes—
38#16839508C2212yes—
39#16839505C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C4×#41×C61C2211yescharacteristic
42C61×#477—21no—
43C61×#476—21no—
44C61×#475C2212yes—
45C61×#474C2212yes—
46C61×#428—21no—
47C61×#423C2211yescharacteristic
48C61×#417—21no—
49C2×#11758222—21no—
50#14519797—44no—
51C2×#11758202—21no—
52C2×#11758200—21no—
53C2×#11758201—21no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758199—21no—
57C22×#92418—21no—
58C22×#92418—21no—
59C22×#92418—22no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67C4⋊#92418—21no—
68C4⋊#92418—21no—
69C4⋊#92418—22no—
70C4⋊#92418D411yescharacteristic
71C4⋊#92418D411yescharacteristic
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75#14519701—22no—
76#14519701—22no—
77#14519700—21no—
78#14519700—21no—
79#14519697—21no—
80#14519697—21no—
81C22⋊#92418—22no—
82C22⋊#92418D412yes—
83C4×#92418—22no—
84C4×#92418C2312yes—
85C2×#41×C61D411yescharacteristic
86C22×Q8×C61—21no—
87C2×M16×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61C2312yes—
92C4×Q8×C61D411yescharacteristic
93C4×Q8×C61C2311yescharacteristic
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D411yescharacteristic
97C4×D4×C61C2311yescharacteristic
98#117×C61—21no—
99#117×C61D411yescharacteristic
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105#106×C61—22no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#103×C61—22no—
109#102×C61—22no—
110#41×C61—22no—
111#41×C61SD1614yescomplemented
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×Q8×C61—81no—
115C2×Q8×C61—81no—
116C2×Q8×C61C2×D411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61C2×D411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C2×C22⋊D61—41no—
127#11758222—42no—
128#11758222—42no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×C4×D61—41no—
133C2×Dic122—21no—
134M16×C61—42no—
135C2×C8×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61C2×D411yescharacteristic
143C4⋊C4×C61C2×D411yescharacteristic
144C22⋊C4×C61—21no—
145C22⋊C4×C61—21no—
146C22⋊C4×C61—21no—
147C42×C61—21no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758202—42no—
155#11758201—42no—
156#11758201—82no—
157#11758201—82no—
158#11758200—42no—
159#11758199—42no—
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C2×#92418—22no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—42no—
168C2×#92418—82no—
169C2×#92418—81no—
170C2×#92418C2×D412yes—
171C2×#92418C2×D412yes—
172C23×C61—21no—
173C23×C61#12111yescharacteristic
174C22×D61—41no—
175Q8×C61—21no—
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—82no—
179Q8×C61C2×SD1612yes—
180D4×C61—24no—
181D4×C61—21no—
182D4×C61C2×SD1612yes—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—24no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—81no—
194C2×C4×C61—82no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61C2×SD1612yes—
199C2×C4×C61#13512yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C22⋊D61—82no—
204C2×Dic61—41no—
205D244—41no—
206D244—41no—
207D244—41no—
208D244—41no—
209D244—81no—
210C4×D61—82no—
211Dic122—41no—
212Dic122—41no—
213C8×C61—42no—
214#92418—22no—
215#92418—22no—
216#92418—41no—
217#92418—41no—
218#92418—42no—
219#19408—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—22no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61#46512yes—
226C22×C61C2×#12111yescharacteristic
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—21no—
231C4×C61—22no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—41no—
235C4×C61—41no—
236C4×C61—82no—
237C4×C61#46512yes—
238C4×C61C22×SD1611yescharacteristic
239C4×C61C2×#13511yescharacteristic
240C4×C61#57611yescharacteristic
241C4×C61#57711yescharacteristic
242Dic61—81no—
243#3206—611no—
244#3079D6111yescharacteristic, complemented
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—42no—
256C2×C61#304511yescharacteristic
257C2×C61C2×#46511yescharacteristic
258C2×C61#306911yescharacteristic
259D61—161no—
260C2×#135—1221no—
261#532—611no—
262C2×C4⋊Q8C2×D6111yescharacteristic
263C4×#41C2×D6111yescharacteristic
264#492—612no—
265#487—612no—
266#477—21no—
267#476—21no—
268#475C2×D6112yes—
269#474C2×D6112yes—
270#464—1221no—
271#463—1221no—
272#462—611no—
273#461—611no—
274#461—1221no—
275#461—1221no—
276#460—611no—
277#459—611no—
278#448—1222no—
279C8×Q8—612no—
280C8×D4—612no—
281C8×D4—1222no—
282#431—611no—
283#431—1221no—
284#428—21no—
285#423C2×D6111yescharacteristic
286C2×C4⋊C8—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—1221no—
289#417—21no—
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—1221no—
299C2×#41C22⋊D6111yescharacteristic
300C22×Q8—21no—
301#135—2444no—
302C2×Q16—1221no—
303C2×SD16—1222no—
304C2×SD16—1222no—
305C2×SD16—1221no—
306C2×SD16—1221no—
307C2×D8—1221no—
308C2×M16—21no—
309C22×C8—1222no—
310C22×C8—1221no—
311C22×C8—1221no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—611no—
316C4⋊Q8C22×D6112yes—
317#127—611no—
318#124—1221no—
319#121—1221no—
320#121—1221no—
321C4×Q8C22⋊D6111yescharacteristic
322C4×Q8C22×D6111yescharacteristic
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—1221no—
326C4×D4—1221no—
327C4×D4C22⋊D6111yescharacteristic
328C4×D4C22×D6111yescharacteristic
329#117—21no—
330#117C22⋊D6111yescharacteristic
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C42—21no—
334C2×C42C22×D6111yescharacteristic
335#107—22no—
336#106—22no—
337C4⋊C8—21no—
338C4⋊C8—611no—
339C4⋊C8—611no—
340C4⋊C8—612no—
341C4⋊C8—611no—
342C4⋊C8—611no—
343C4⋊C8—1222no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8C22×D6112yes—
347#103—22no—
348#103—1222no—
349#103—1222no—
350#103—1221no—
351#103—1221no—
352#102—22no—
353#102—1221no—
354#102—1221no—
355C22⋊C8—612no—
356C22⋊C8—1222no—
357C4×C8—612no—
358C4×C8—1222no—
359#41—22no—
360#41—2442no—
361#41—2442no—
362#41#1175820114yescomplemented
363C2×Q8—22no—
364C2×Q8—41no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×Q8—1221no—
368C2×Q8C2×C22⋊D6111yescharacteristic
369C2×D4—21no—
370C2×D4—1221no—
371C2×D4—1221no—
372C2×D4—1221no—
373C2×D4—2441no—
374C2×D4C2×C22⋊D6111yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—2441no—
380C22×C4D4×D6111yescharacteristic
381C22×C4#1175822411yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383Q16—2442no—
384SD16—4882no—
385SD16—4882no—
386SD16—2442no—
387SD16—2442no—
388D8—2442no—
389M16—42no—
390C2×C8—22no—
391C2×C8—612no—
392C2×C8—612no—
393C2×C8—1222no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—4882no—
397C2×C8—4881no—
398C2×C8—2442no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4C2×C22⋊D6111yescharacteristic
408C4⋊C4C2×C22⋊D6111yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—21no—
422C23—2441no—
423C23#1451974911yescharacteristic
424Q8—21no—
425Q8—42no—
426Q8—42no—
427Q8—82no—
428Q8—2441no—
429Q8—2441no—
430Q8C2×#1175820112yes—
431D4—24no—
432D4—21no—
433D4—4881no—
434D4—4882no—
435D4—2441no—
436D4—2441no—
437D4—2441no—
438D4—2441no—
439D4C2×#1175820112yes—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—4882no—
453C2×C4—2441no—
454C2×C4#1451974911yescharacteristic
455C2×C4#1451974911yescharacteristic
456C2×C4#1451974911yescharacteristic
457C2×C4C2×#1175820112yes—
458C2×C4#1451979712yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C2×#1175822411yescharacteristic
462C8—42no—
463C8—1222no—
464C8—1222no—
465C8—2441no—
466C8—2441no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—4881no—
474C22—4881no—
475C22—4881no—
476C22#1683999312yes—
477C22C2×#1451974911yescharacteristic
478C4—21no—
479C4—22no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—82no—
485C4—4881no—
486C4#1683999312yes—
487C4C22×#1175820111yescharacteristic
488C4C2×#1451979711yescharacteristic
489C4#1684056611yescharacteristic
490C4#1684057211yescharacteristic
491C2—22no—
492C2—42no—
493C2—9761no—
494C2#2145536211yescharacteristic
495C2C2×#1683999311yescharacteristic
496C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3079C2
socleC22×C61C2×#121
prime radical#21455274C2
hypercenter#3079D61
hypocenterC61#19408
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. #21456744C2semidirect
3. #21454930C2semidirect
4. #21445247C2semidirect
5. #21445240C2semidirect
6. #21445169C2semidirect
7. #21445162C2semidirect
8. #21444591C2semidirect
9. #21444497C2semidirect
10. C2×#16839508C2semidirect
11. C61×#3079C2semidirect
111. #41×C61SD16semidirect
244. #3079D61semidirect
297. C61#19408semidirect
362. #41#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214552742
#214549302
#214452472
#214452402
#214451692
#214451622
#214445912
#214444972
C2×#168395082
C61×#30792
#1940861