Finite Group Databasebeta

← #29834825

Subgroups — #29834825

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455401C211yesmaximal, characteristic
2C2×#107×D61C211yesmaximal, characteristic, complemented
3#21448023C212yesmaximal, complemented
4#21448018C212yesmaximal, complemented
5#21448006C212yesmaximal, complemented
6#21448001C212yesmaximal, complemented
7#21447608C211yesmaximal, characteristic, complemented
8#21447532C211yesmaximal, characteristic, complemented
9#21447529C211yesmaximal, characteristic, complemented
10#21447429C211yesmaximal, characteristic, complemented
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840274—21no—
15C2×C4⋊C4×D61—21no—
16C2×#14519717C2211yescharacteristic
17#16840004C2211yescharacteristic
18#16840004C2211yescharacteristic
19#16839999C2211yescharacteristic
20#16839999C2211yescharacteristic
21#16839992—21no—
22#16839992—21no—
23C2×#14519711C2211yescharacteristic
24C2×#14519710—21no—
25#107×D61—21no—
26#107×D61—21no—
27#107×D61C2212yes—
28#107×D61C2212yes—
29#16839756—21no—
30#16839589C2212yescomplemented
31#16839587C2212yescomplemented
32#16839567C2212yescomplemented
33#16839565C2212yescomplemented
34#16839564C2212yescomplemented
35#16839562C2212yescomplemented
36#16839541C2212yes—
37#16839541C2212yes—
38C2×M32×C61C2211yescharacteristic
39C61×#427C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#367C2212yescomplemented
42C61×#366C2212yescomplemented
43C2×#11758222—21no—
44C22×C4×D61—41no—
45C22×C4×D61D411yescharacteristic
46#14519799—22no—
47#14519799—22no—
48C2×Dic244C2311yescharacteristic
49C2×#11758190—21no—
50C2×D488C2311yescharacteristic
51C22×#92418D411yescharacteristic
52C2×C8×D61—22no—
53C2×C8×D61—22no—
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56C2×C8×D61C2311yescharacteristic
57C2×C8×D61C2311yescharacteristic
58#14519761—41no—
59#14519757—21no—
60#14519757—21no—
61#14519752—21no—
62#14519752—21no—
63#14519747—41no—
64C4×D244—41no—
65C4×D244—41no—
66C2×C4⋊Dic61—21no—
67C2×#11758196—41no—
68C4⋊C4×D61—22no—
69C4⋊C4×D61—22no—
70C4⋊C4×D61—41no—
71C4⋊C4×D61—41no—
72#14519717—22no—
73C2×#11758185C2312yes—
74#14519711—21no—
75#14519711C2312yes—
76#14519710—22no—
77#14519710—22no—
78#14519700—21no—
79#14519700—21no—
80#14519697—21no—
81#14519697—21no—
82C22×C8×C61C2311yescharacteristic
83#117×C61—21no—
84C2×C4⋊C4×C61—21no—
85M32×C61—22no—
86C2×C16×C61C2312yes—
87#107×C61—21no—
88#107×C61C2311yescharacteristic
89#107×C61C2311yescharacteristic
90#107×C61C2312yes—
91#106×C61—21no—
92C23×D61D811yescharacteristic
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C2×C22⋊D61—41no—
96C22×Dic61—41no—
97C22×Dic61D811yescharacteristic
98#11758222—42no—
99#11758222—42no—
100C2×D244—21no—
101C2×D244—41no—
102C2×D244—41no—
103C2×C4×D61—22no—
104C2×C4×D61—22no—
105C2×C4×D61—42no—
106C2×C4×D61—42no—
107C2×C4×D61—41no—
108C2×C4×D61—84no—
109C2×C4×D61D811yescharacteristic
110C2×C4×D61D811yescharacteristic
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61C2×D411yescharacteristic
113C2×Dic122—21no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61#4112yes—
116C2×C8×C61#4112yes—
117C2×C8×C61C2411yescharacteristic
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C42×C61—41no—
124#11758198—41no—
125#11758198—41no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131C4⋊Dic61—41no—
132#11758196—42no—
133#11758196—42no—
134C2×#92418—22no—
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137Dic244—21no—
138Dic244—21no—
139D488—21no—
140D488—21no—
141#11758190—42no—
142C8×D61—22no—
143C8×D61—22no—
144C8×D61—21no—
145C8×D61—21no—
146C8×D61—21no—
147C8×D61—21no—
148C16×C61—22no—
149#11758185—22no—
150C23×C61C2×D811yescharacteristic
151C22×D61—22no—
152C22×D61—22no—
153C22×D61—22no—
154C22×D61—41no—
155C22×D61C2×D811yescharacteristic
156C22×D61C2×D811yescharacteristic
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—82no—
161C2×C4×C61#12312yes—
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61#13712yes—
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—82no—
166C2×Dic61—22no—
167C2×Dic61—42no—
168C2×Dic61—41no—
169C2×Dic61—82no—
170C2×Dic61C2×D811yescharacteristic
171C2×Dic61C2×D811yescharacteristic
172D244—41no—
173D244—41no—
174D244—82no—
175D244—82no—
176C4×D61—21no—
177C4×D61—21no—
178C4×D61—21no—
179C4×D61—21no—
180C4×D61—22no—
181C4×D61—22no—
182C4×D61—84no—
183C4×D61—84no—
184C4×D61—82no—
185Dic122—41no—
186Dic122—41no—
187C8×C61#12312yes—
188C8×C61C2×#4111yescharacteristic
189C8×C61C2×#4111yescharacteristic
190#92418—21no—
191#92418—21no—
192#33646—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61#48012yes—
196C22×C61C22×D811yescharacteristic
197C2×D61—28no—
198C2×D61—21no—
199C2×D61—21no—
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C2×D61#50912yes—
206C2×D61#51012yes—
207C4×C61—82no—
208C4×C61—81no—
209C4×C61—81no—
210C4×C61#48012yes—
211C4×C61C2×#12311yescharacteristic
212C4×C61C2×#13711yescharacteristic
213Dic61—21no—
214Dic61—21no—
215Dic61—81no—
216Dic61—82no—
217#3050—611no—
218C22×#107—611no—
219#2236—612no—
220#2235—612no—
221#2233—612no—
222#2232—612no—
223#2159—611no—
224#2159D6111yescharacteristic, complemented
225#2139—611no—
226#2139—611no—
227#2112—611no—
228C2×C61—22no—
229C2×C61C2×#48011yescharacteristic
230C2×C61C2×#50911yescharacteristic
231C2×C61C2×#51011yescharacteristic
232D61—22no—
233D61—22no—
234D61—161no—
235C2×#135—611no—
236C23×C8—611no—
237#525—1221no—
238C22×C4⋊C4—1221no—
239C2×M32—611no—
240C2×M32C2×D6111yescharacteristic
241#467—611no—
242#467—611no—
243#466—611no—
244#466—611no—
245#465—1221no—
246#465—1221no—
247#427C2×D6111yescharacteristic
248C2×#107—611no—
249C2×#107—612no—
250C2×#107—612no—
251C2×#107—1221no—
252C2×#107—1221no—
253C2×#107—1221no—
254C2×#107C2×D6111yescharacteristic
255#416—1221no—
256#367—612no—
257#367C2×D6112yescomplemented
258#366—612no—
259#366C2×D6112yescomplemented
260#358—612no—
261#358—612no—
262#357—612no—
263#357—612no—
264C22⋊C16—612no—
265C22⋊C16—612no—
266C61#3364611yescharacteristic, complemented, Sylow
267C2×#41—1221no—
268C23×C4—611no—
269C23×C4—2441no—
270#135—1222no—
271#135—1222no—
272C2×Q16—611no—
273C2×SD16—1221no—
274C2×D8—611no—
275C22×C8—611no—
276C22×C8—611no—
277C22×C8—611no—
278C22×C8—611no—
279C22×C8—611no—
280C22×C8—1222no—
281C22×C8—1222no—
282C22×C8C22×D6111yescharacteristic
283#123—2441no—
284C22⋊Q8—1221no—
285C22⋊Q8—1221no—
286#121—1221no—
287#121—1221no—
288#120—2441no—
289C4×D4—2441no—
290C4×D4—2441no—
291#117—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—1222no—
294C2×C4⋊C4—1222no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—2441no—
297C2×C4⋊C4—2441no—
298C2×C4⋊C4—2441no—
299M32—22no—
300M32—1222no—
301C2×C16—612no—
302C2×C16C22×D6112yes—
303#107—21no—
304#107—612no—
305#107—1222no—
306#107—1222no—
307#107—1221no—
308#107C22×D6112yes—
309#107C22×D6111yescharacteristic
310#107C22×D6111yescharacteristic
311#106—21no—
312#103—1221no—
313#103—1221no—
314#102—1221no—
315#102—1221no—
316C24—611no—
317#41—2442no—
318#41—2442no—
319C2×Q8—1221no—
320C2×D4—1221no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C22×C4—41no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—1222no—
331C22×C4—1222no—
332C22×C4—4884no—
333C22×C4—2442no—
334C22×C4—2442no—
335C22×C4—2441no—
336C22×C4—2441no—
337C22×C4D4×D6111yescharacteristic
338Q16—1221no—
339Q16—1221no—
340SD16—2442no—
341D8—1221no—
342D8—1221no—
343C2×C8—611no—
344C2×C8—611no—
345C2×C8—1222no—
346C2×C8—1221no—
347C2×C8—1221no—
348C2×C8—1221no—
349C2×C8—1221no—
350C2×C8—1222no—
351C2×C8—1222no—
352C2×C8#1175822212yes—
353C2×C8D4×D6111yescharacteristic
354C2×C8#1175822612yes—
355C2×C8C23×D6111yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—41no—
360C4⋊C4—1222no—
361C4⋊C4—2442no—
362C4⋊C4—2442no—
363C4⋊C4—2441no—
364C4⋊C4—2441no—
365C4⋊C4—2441no—
366C22⋊C4—41no—
367C22⋊C4—2441no—
368C22⋊C4—2441no—
369C22⋊C4—2441no—
370C22⋊C4—2441no—
371C42—41no—
372C16—22no—
373C16—1222no—
374C23—611no—
375C23—611no—
376C23—1222no—
377C23—1222no—
378C23—1222no—
379C23—2441no—
380C23D8×D6111yescharacteristic
381Q8—2441no—
382Q8—2441no—
383D4—4882no—
384D4—4882no—
385D4—4882no—
386D4—2441no—
387D4—2441no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—611no—
393C2×C4—611no—
394C2×C4—1222no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1222no—
400C2×C4—1222no—
401C2×C4—4884no—
402C2×C4—4884no—
403C2×C4—4882no—
404C2×C4—4882no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4#1451976212yes—
408C2×C4D8×D6111yescharacteristic
409C2×C4#1451980912yes—
410C2×C4C2×D4×D6111yescharacteristic
411C8—1221no—
412C8—1221no—
413C8#1451976212yes—
414C8C2×#1175822211yescharacteristic
415C8C2×#1175822611yescharacteristic
416C22—21no—
417C22—21no—
418C22—612no—
419C22—612no—
420C22—1228no—
421C22—1221no—
422C22—1221no—
423C22—1221no—
424C22—1221no—
425C22—4881no—
426C22—4881no—
427C22—4881no—
428C22#1684007712yes—
429C22C2×D8×D6111yescharacteristic
430C4—81no—
431C4—82no—
432C4—81no—
433C4—1221no—
434C4—1221no—
435C4—4881no—
436C4—4882no—
437C4#1684007712yes—
438C4C2×#1451976211yescharacteristic
439C4C2×#1451980911yescharacteristic
440C2—22no—
441C2—1222no—
442C2—1222no—
443C2—9761no—
444C2C2×#1684007711yescharacteristic
445C2#509×D6111yescharacteristic
446C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radical#21455401C2
hypercenter#2159D61
hypocenterC61#33646
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
2. C2×#107×D61C2semidirect
3. #21448023C2semidirect
4. #21448018C2semidirect
5. #21448006C2semidirect
6. #21448001C2semidirect
7. #21447608C2semidirect
8. #21447532C2semidirect
9. #21447529C2semidirect
10. #21447429C2semidirect
11. C61×#2159C2semidirect
30. #16839589C22semidirect
31. #16839587C22semidirect
32. #16839567C22semidirect
33. #16839565C22semidirect
34. #16839564C22semidirect
35. #16839562C22semidirect
41. C61×#367C22semidirect
42. C61×#366C22semidirect
224. #2159D61semidirect
257. #367C2×D61semidirect
259. #366C2×D61semidirect
266. C61#33646semidirect

Maximal subgroups

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

subgroupindex
#214554012
C2×#107×D612
#214480232
#214480182
#214480062
#214480012
#214476082
#214475322
#214475292
#214474292
C61×#21592
#3364661