Finite Group Databasebeta

← #29663118

Subgroups — #29663118

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456743C212yesmaximal
2#500×D61C211yesmaximal, characteristic, complemented
3#21454526C211yesmaximal, characteristic
4#21445331C212yesmaximal, complemented
5#21445330C212yesmaximal, complemented
6#21445209C211yesmaximal, characteristic, complemented
7#21445208C211yesmaximal, characteristic, complemented
8#21445205C211yesmaximal, characteristic, complemented
9#21444566C211yesmaximal, characteristic, complemented
10#21444502C212yesmaximal, complemented
11C61×#1673C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840309C2211yescharacteristic
14C4⋊Q8×D61C2212yes—
15#16840167C2211yescharacteristic
16#16840166—21no—
17#16840166—21no—
18#16840165C2212yes—
19#16840133—22no—
20#16840111—22no—
21#16839968C2212yes—
22#16839967C2212yes—
23#16839964C2212yes—
24#16839902—22no—
25#16839899—22no—
26#16839840C2211yescharacteristic
27#16839839C2211yescharacteristic
28#16839837—21no—
29C8×D244C2211yescharacteristic
30C8×C22⋊D61—21no—
31#16839814C2212yes—
32#107×D61—21no—
33#107×D61—21no—
34#16839784C2211yescharacteristic
35#16839727—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839509C2212yes—
40#16839508C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#461C2212yes—
44C8×D4×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758222—21no—
48C2×Q8×D61—22no—
49C2×#11758190—22no—
50C2×#11758200—22no—
51C2×#11758201—22no—
52#14519783—44no—
53C2×#11758194—21no—
54C2×#11758189—21no—
55C2×#11758189—21no—
56C2×C8×D61—21no—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61D411yescharacteristic
59#14519777C2312yes—
60#14519776—22no—
61#14519776—22no—
62#14519776D411yescharacteristic
63#14519775—22no—
64#14519773D411yescharacteristic
65#14519763—21no—
66#14519752D411yescharacteristic
67#14519750—21no—
68#14519748D411yescharacteristic
69C4×D244C2311yescharacteristic
70C4×C22⋊D61—21no—
71C4⋊C4×D61—22no—
72C4⋊C4×D61—22no—
73C42×D61C2311yescharacteristic
74#14519711—21no—
75#14519711—21no—
76#14519710—22no—
77#14519710—22no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2311yescharacteristic
80#14519706—21no—
81#14519702—22no—
82#14519701—22no—
83#14519700—22no—
84C22⋊#92418—21no—
85#14519689—22no—
86#14519689—21no—
87#14519689—21no—
88C4×#92418C2311yescharacteristic
89C8×Dic61—22no—
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×SD16×C61—22no—
93C22×C8×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×D4×C61C2311yescharacteristic
96#107×C61—21no—
97#107×C61—21no—
98C4⋊C8×C61C2312yes—
99C4⋊C8×C61C2311yescharacteristic
100#103×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×D4×C61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109Q8×D61—82no—
110Q8×D61—82no—
111#11758222—42no—
112#11758222—42no—
113C2×D244—21no—
114C2×D244C2×D411yescharacteristic
115C2×C4×D61—21no—
116C2×C4×D61—42no—
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×D411yescharacteristic
119C2×C4×D61#4111yescharacteristic
120C2×Dic122—21no—
121C2×Dic122—22no—
122C2×Dic122—42no—
123SD16×C61—82no—
124C2×C8×C61—21no—
125C2×C8×C61—22no—
126C2×C8×C61—42no—
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61C2×D411yescharacteristic
132C22⋊C4×C61—21no—
133C42×C61C2411yescharacteristic
134C22⋊Dic61—21no—
135#11758201—82no—
136#11758200—82no—
137#11758198—21no—
138#11758198—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61C2×D411yescharacteristic
142#11758196—21no—
143#11758196—42no—
144#11758196—42no—
145C4×Dic61C2×D411yescharacteristic
146C4×Dic61C2×D411yescharacteristic
147C4×Dic61#4111yescharacteristic
148#11758194—42no—
149C2×#92418—22no—
150C2×#92418—21no—
151C2×#92418C2×D411yescharacteristic
152C2×#92418C2×D411yescharacteristic
153#11758190—82no—
154#11758189—42no—
155#11758189—42no—
156C8×D61—22no—
157C8×D61—42no—
158C8×D61SD1614yescomplemented
159C23×C61—21no—
160C22×D61—21no—
161C22×D61#12111yescharacteristic
162Q8×C61—42no—
163Q8×C61—42no—
164D4×C61—21no—
165D4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C22⋊D61—42no—
174C22⋊D61—42no—
175C22⋊D61—42no—
176C2×Dic61—21no—
177C2×Dic61—42no—
178C2×Dic61#12111yescharacteristic
179C2×Dic61#12111yescharacteristic
180C2×Dic61#12111yescharacteristic
181D244—21no—
182D244—21no—
183D244—42no—
184C4×D61—21no—
185C4×D61—21no—
186C4×D61—22no—
187C4×D61—41no—
188C4×D61—41no—
189C4×D61—84no—
190C4×D61C2×SD1612yes—
191C4×D61#13712yes—
192Dic122—42no—
193Dic122—42no—
194Dic122—42no—
195Dic122—84no—
196C8×C61—21no—
197C8×C61—41no—
198C8×C61—42no—
199C8×C61C2×SD1612yes—
200#92418—21no—
201#92418—41no—
202#92418—42no—
203#92418C2×SD1612yes—
204#19192—611nomaximal, Sylow
205C22×C61—41no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61C2×#12111yescharacteristic
209C2×D61—21no—
210C2×D61—21no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—41no—
214C2×D61#46112yes—
215C4×C61—21no—
216C4×C61—41no—
217C4×C61—82no—
218C4×C61C22×SD1611yescharacteristic
219C4×C61C2×#13711yescharacteristic
220C4×C61#57611yescharacteristic
221C4×C61#57611yescharacteristic
222Dic61—21no—
223Dic61—41no—
224Dic61—82no—
225Dic61#46112yes—
226Dic61#47312yes—
227#3206—612no—
228C2×#500—611no—
229#2931—611no—
230#1693—612no—
231#1693—612no—
232#1673—611no—
233#1673—611no—
234#1673—611no—
235#1673D6111yescharacteristic, complemented
236#1564—611no—
237#1549—612no—
238C2×C61—81no—
239C2×C61C2×#46111yescharacteristic
240C2×C61#305811yescharacteristic
241C2×C61C2×#47311yescharacteristic
242D61—22no—
243D61—81no—
244C2×#131—1221no—
245#532—611no—
246C2×C4⋊Q8—612no—
247#500—612no—
248#500—611no—
249#500—1221no—
250#500—1221no—
251#500C2×D6111yescharacteristic
252#492—1222no—
253#487—1222no—
254#461—612no—
255#461—612no—
256#461—612no—
257#461C2×D6112yes—
258#448—1222no—
259#448—1222no—
260#436—611no—
261#436—611no—
262#436—1221no—
263C8×D4—611no—
264C8×D4—1221no—
265C8×D4C2×D6111yescharacteristic
266#431—612no—
267C2×#107—1221no—
268C2×#107—1221no—
269#423—611no—
270#408—1221no—
271C2×C4×C8—611no—
272#335—611no—
273#335C2×D6111yescharacteristic
274#334—612no—
275#332—612no—
276#332—612no—
277#332—612no—
278#332C2×D6112yes—
279C61#1919211yescharacteristic, complemented, Sylow
280C2×#41—1221no—
281C22×Q8—1222no—
282C2×SD16—22no—
283C2×SD16—1222no—
284C2×SD16—1222no—
285C2×SD16—1222no—
286#131—2444no—
287C2×M16—1221no—
288C2×M16—1221no—
289C2×M16—1221no—
290C22×C8—21no—
291C22×C8—611no—
292C22×C8—611no—
293C22×C8—1221no—
294C4⋊Q8—611no—
295C4⋊Q8—612no—
296C4⋊Q8—1222no—
297C4⋊Q8—1222no—
298C4⋊Q8—1222no—
299C4⋊Q8C22×D6112yes—
300#127—611no—
301#124—1221no—
302#121—611no—
303#121—611no—
304#121—1221no—
305C4×D4—611no—
306C4×D4—1221no—
307C4×D4C22×D6111yescharacteristic
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1222no—
310C2×C42—611no—
311#107—21no—
312#107—21no—
313#107—1222no—
314#107—1222no—
315#107—1221no—
316#107—1221no—
317C4⋊C8—611no—
318C4⋊C8—612no—
319C4⋊C8—1221no—
320C4⋊C8C22×D6111yescharacteristic
321C4⋊C8C22×D6112yes—
322#103—22no—
323#103—1222no—
324#103—1222no—
325#103—1222no—
326C22⋊C8—21no—
327C22⋊C8—1221no—
328C22⋊C8—1222no—
329C22⋊C8—1221no—
330C22⋊C8—1221no—
331C4×C8—611no—
332C4×C8—611no—
333C4×C8—611no—
334C4×C8—1222no—
335C4×C8C22×D6111yescharacteristic
336#41—2442no—
337#41—2442no—
338C2×Q8—22no—
339C2×Q8—1221no—
340C2×Q8—1222no—
341C2×Q8—4882no—
342C2×Q8—4882no—
343C2×Q8—2442no—
344C2×D4—611no—
345C2×D4—1221no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4D4×D6111yescharacteristic
350C22×C4—21no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—1221no—
355C22×C4—2442no—
356SD16—82no—
357SD16—4882no—
358SD16—4882no—
359SD16—4882no—
360M16—2442no—
361M16—2442no—
362M16—2442no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—42no—
366C2×C8—611no—
367C2×C8—614no—
368C2×C8—611no—
369C2×C8—1222no—
370C2×C8—1222no—
371C2×C8—1221no—
372C2×C8—2442no—
373C2×C8D4×D6111yescharacteristic
374C2×C8D4×D6111yescharacteristic
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—611no—
378C4⋊C4—1221no—
379C4⋊C4—1222no—
380C4⋊C4—1222no—
381C4⋊C4—2442no—
382C4⋊C4—2442no—
383C4⋊C4D4×D6111yescharacteristic
384C22⋊C4—21no—
385C22⋊C4—1221no—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C42—611no—
389C42—611no—
390C42—611no—
391C42C23×D6111yescharacteristic
392C23—21no—
393C23—611no—
394C23—1221no—
395Q8—42no—
396Q8—42no—
397Q8—4884no—
398Q8—2442no—
399Q8—2442no—
400Q8—2442no—
401D4—21no—
402D4—21no—
403D4—1221no—
404D4—1221no—
405D4—2442no—
406D4—2442no—
407D4—2442no—
408D4—2442no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—611no—
414C2×C4—611no—
415C2×C4—612no—
416C2×C4—612no—
417C2×C4—611no—
418C2×C4—1222no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—4884no—
423C2×C4—2442no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4C2×D4×D6111yescharacteristic
427C2×C4C2×D4×D6111yescharacteristic
428C2×C4#41×D6111yescharacteristic
429C8—21no—
430C8—41no—
431C8—42no—
432C8—612no—
433C8—1221no—
434C8—2441no—
435C8—2442no—
436C8SD16×D6112yes—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—612no—
441C22—1221no—
442C22—1221no—
443C22—2441no—
444C22—2441no—
445C22—2441no—
446C22#121×D6111yescharacteristic
447C4—21no—
448C4—41no—
449C4—82no—
450C4—612no—
451C4—612no—
452C4—1221no—
453C4—4882no—
454C4—2441no—
455C4C2×SD16×D6111yescharacteristic
456C4#137×D6111yescharacteristic
457C4#1684056311yescharacteristic
458C4#1684056411yescharacteristic
459C2—81no—
460C2—1222no—
461C2—4881no—
462C2#461×D6111yescharacteristic
463C2#2145546211yescharacteristic
464C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1673C2
socleC22×C61C2×#121
prime radical#16840309C22
hypercenter#1673D61
hypocenterC61#19192
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1673

Complements

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

NH ≅ G/Ntype
2. #500×D61C2semidirect
4. #21445331C2semidirect
5. #21445330C2semidirect
6. #21445209C2semidirect
7. #21445208C2semidirect
8. #21445205C2semidirect
9. #21444566C2semidirect
10. #21444502C2semidirect
11. C61×#1673C2semidirect
158. C8×D61SD16semidirect
235. #1673D61semidirect
279. C61#19192semidirect

Maximal subgroups

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

subgroupindex
#214567432
#500×D612
#214545262
#214453312
#214453302
#214452092
#214452082
#214452052
#214445662
#214445022
C61×#16732
#1919261