Finite Group Databasebeta

← #29835376

Subgroups — #29835376

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456330C211yesmaximal, characteristic, complemented
2#495×D61C211yesmaximal, characteristic
3#21448058C211yesmaximal, characteristic, complemented
4#21448056C211yesmaximal, characteristic, complemented
5#21448055C211yesmaximal, characteristic, complemented
6#21448052C212yesmaximal, complemented
7#21448049C212yesmaximal, complemented
8#21447537C212yesmaximal, complemented
9#21447533C212yesmaximal
10#21447412C211yesmaximal, characteristic, complemented
11C61×#2243C211yesmaximal, characteristic, complemented
12C2×Q16×D61C2211yescharacteristic
13C2×SD16×D61—21no—
14C2×D8×D61C2211yescharacteristic
15#16840317—21no—
16#124×D61—21no—
17C2×#14519716C2212yes—
18#16840167C2211yescharacteristic, complemented
19#16840163—21no—
20#16840163C2212yescomplemented
21#16840158—21no—
22#16840147C2211yescharacteristic
23#16840146C2211yescharacteristic
24#16840145C2211yescharacteristic
25#16840144—21no—
26#16840143—21no—
27#16839799—21no—
28#16839799C2212yes—
29C4×C8×D61C2211yescharacteristic
30#16839567C2212yescomplemented
31#16839566C2212yescomplemented
32#16839565C2212yescomplemented
33#16839564C2212yes—
34#16839563C2212yes—
35#16839562C2212yes—
36#16839538C2211yescharacteristic, complemented
37C16×Dic61C2212yescomplemented
38C61×#500C2211yescharacteristic, complemented
39C61×#495C2211yescharacteristic
40C61×#358C2212yescomplemented
41C61×#357C2212yes—
42C61×#346C2211yescharacteristic, complemented
43C2×Q8×D61—21no—
44C2×D4×D61—21no—
45C2×Dic244C2311yescharacteristic
46C2×#11758202—21no—
47Q16×D61—24no—
48C2×#11758190—21no—
49C2×#11758200—21no—
50C2×#11758201—21no—
51SD16×D61—44no—
52C2×D488C2311yescharacteristic
53C2×#11758199—21no—
54D8×D61—24no—
55C2×C8×D61D411yescharacteristic
56C2×C8×D61C2311yescharacteristic
57#14519777—21no—
58#14519768—22no—
59#14519766—21no—
60#14519765—21no—
61#14519764—21no—
62#14519763—41no—
63#14519763—41no—
64#14519757—41no—
65#14519753—41no—
66Q8×Dic61—41no—
67#14519742—41no—
68#14519735—22no—
69C22⋊C4×D61—41no—
70C22⋊C4×D61—41no—
71C42×D61D411yescharacteristic
72#14519716—24no—
73C2×#11758185C2312yes—
74#14519711—21no—
75#14519711C2312yes—
76#14519708—22no—
77#14519708—22no—
78C4×#92418D411yescharacteristic
79C8×Dic61D411yescharacteristic
80C8×Dic61C2311yescharacteristic
81C2×Q16×C61C2311yescharacteristic
82C2×SD16×C61—21no—
83C2×D8×C61C2311yescharacteristic
84C4⋊Q8×C61—21no—
85#124×C61—21no—
86C2×C16×C61C2312yes—
87#107×C61—21no—
88#107×C61C2312yes—
89C4×C8×C61C2311yescharacteristic
90C23×D61—41no—
91C2×Q8×C61—21no—
92C2×Q8×C61—41no—
93C2×D4×C61—21no—
94C2×C22⋊D61—41no—
95Q8×D61—42no—
96Q8×D61—42no—
97D4×D61—42no—
98D4×D61—42no—
99C2×D244—21no—
100C2×C4×D61—41no—
101C2×C4×D61D811yescharacteristic
102C2×C4×D61D811yescharacteristic
103C2×C4×D61C2×D411yescharacteristic
104C2×Dic122—21no—
105C2×Dic122—41no—
106C2×Dic122—41no—
107Q16×C61—22no—
108SD16×C61—42no—
109D8×C61—22no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2411yescharacteristic
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C42×C61C2×D411yescharacteristic
117C22⋊Dic61—41no—
118C22⋊Dic61—41no—
119#11758202—42no—
120#11758201—42no—
121#11758200—42no—
122#11758199—42no—
123#11758198—42no—
124#11758198—41no—
125#11758198—41no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130C4⋊Dic61—42no—
131#11758196—42no—
132C4×Dic61—42no—
133C4×Dic61D811yescharacteristic
134C4×Dic61D811yescharacteristic
135C4×Dic61C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138Dic244—22no—
139D488—22no—
140#11758190—42no—
141C8×D61SD1614yes—
142C8×D61#4114yes—
143C16×C61—22no—
144#11758185—22no—
145C23×C61—41no—
146C22×D61—41no—
147C22×D61—82no—
148C22×D61—82no—
149C22×D61—82no—
150C22×D61C2×D811yescharacteristic
151Q8×C61—41no—
152Q8×C61—41no—
153Q8×C61—82no—
154D4×C61—41no—
155D4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61C2×D811yescharacteristic
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C22⋊D61—82no—
162C2×Dic61—42no—
163C2×Dic61—41no—
164C2×Dic61C2×D811yescharacteristic
165C2×Dic61C2×D811yescharacteristic
166C2×Dic61C2×D811yescharacteristic
167D244—41no—
168D244—41no—
169C4×D61—41no—
170C4×D61—41no—
171C4×D61—82no—
172C4×D61#12412yes—
173C4×D61C2×SD1612yes—
174Dic122—41no—
175Dic122—41no—
176Dic122—82no—
177Dic122—82no—
178C8×C61C2×SD1612yes—
179C8×C61C2×#4112yes—
180#92418#12412yes—
181#92418C2×SD1612yes—
182#33706—611nomaximal, Sylow
183C22×C61—81no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61C22×D811yescharacteristic
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—164no—
193C2×D61#48612yes—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—81no—
197C4×C61—82no—
198C4×C61C2×#12411yescharacteristic
199C4×C61C22×SD1611yescharacteristic
200Dic61—21no—
201Dic61—21no—
202Dic61—81no—
203Dic61—82no—
204Dic61#48612yes—
205#3160—611no—
206C2×#495—611no—
207#2243—611no—
208#2243—611no—
209#2243—611no—
210#2243D6111yescharacteristic, complemented
211#2242—612no—
212#2242—612no—
213#2141—612no—
214#2140—612no—
215C4×M32—611no—
216C2×C61—161no—
217C2×C61C2×#48611yescharacteristic
218C2×C61#341411yescharacteristic
219C2×C61#341411yescharacteristic
220D61—22no—
221D61—161no—
222C22×Q16—611no—
223C22×SD16—1221no—
224C22×D8—611no—
225#533—1221no—
226C2×#124—1221no—
227C2×M32—612no—
228#500—611no—
229#500C2×D6111yescharacteristic, complemented
230#499—612no—
231#499—1221no—
232#498—1221no—
233#495—611no—
234#495—611no—
235#495—611no—
236#495—1221no—
237#495—1221no—
238#495C2×D6111yescharacteristic
239#427—612no—
240#427—1221no—
241C2×C4×C8—611no—
242#358—612no—
243#358—612no—
244#358—612no—
245#358C2×D6112yescomplemented
246#357—612no—
247#357—612no—
248#357—612no—
249#357C2×D6112yes—
250#346—611no—
251#346C2×D6111yescharacteristic, complemented
252C4×C16—612no—
253C61#3370611yescharacteristic, complemented, Sylow
254C22×Q8—1221no—
255C22×D4—1221no—
256C2×Q16—611no—
257C2×Q16—1224no—
258C2×Q16—1221no—
259C2×Q16C22×D6111yescharacteristic
260C2×SD16—21no—
261C2×SD16—1221no—
262C2×SD16—1221no—
263C2×SD16—1221no—
264C2×SD16—2444no—
265C2×D8—611no—
266C2×D8—1224no—
267C2×D8—1221no—
268C2×D8C22×D6111yescharacteristic
269C22×C8—611no—
270C22×C8—611no—
271C4⋊Q8—21no—
272C4⋊Q8—1221no—
273#125—1222no—
274#124—21no—
275#124—1221no—
276#124—1221no—
277#124—1221no—
278#124—2441no—
279#124—2441no—
280C22⋊Q8—2441no—
281C22⋊Q8—2441no—
282C4×Q8—2441no—
283C4×Q8—2441no—
284#117—1222no—
285C2×C22⋊C4—2441no—
286C2×C22⋊C4—2441no—
287C2×C42—611no—
288M32—1224no—
289C2×C16—612no—
290C2×C16C22×D6112yes—
291#107—21no—
292#107—612no—
293#107—1221no—
294#107C22×D6112yes—
295#106—1222no—
296#106—1222no—
297C4×C8—611no—
298C4×C8—611no—
299C4×C8—611no—
300C4×C8C22×D6111yescharacteristic
301C24—2441no—
302C2×Q8—21no—
303C2×Q8—41no—
304C2×Q8—1221no—
305C2×Q8—2441no—
306C2×Q8—2442no—
307C2×Q8—2442no—
308C2×Q8—2441no—
309C2×D4—21no—
310C2×D4—1221no—
311C2×D4—2442no—
312C2×D4—2442no—
313C2×D4—2441no—
314C22×C4—611no—
315C22×C4—611no—
316C22×C4—611no—
317C22×C4—2441no—
318Q16—22no—
319Q16—1222no—
320Q16—2442no—
321SD16—42no—
322SD16—2442no—
323SD16—2442no—
324SD16—2442no—
325D8—22no—
326D8—1222no—
327D8—2442no—
328C2×C8—611no—
329C2×C8—611no—
330C2×C8—614no—
331C2×C8—614no—
332C2×C8D4×D6111yescharacteristic
333C2×C8C23×D6111yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—1222no—
337C4⋊C4—2442no—
338C4⋊C4—2442no—
339C4⋊C4—2441no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C22⋊C4—2442no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C42—611no—
350C42—611no—
351C42—611no—
352C42—2442no—
353C42D4×D6111yescharacteristic
354C16—22no—
355C16—1222no—
356C23—41no—
357C23—611no—
358C23—4882no—
359C23—4882no—
360C23—4882no—
361C23—2441no—
362Q8—41no—
363Q8—41no—
364Q8—82no—
365Q8—4882no—
366Q8—4882no—
367Q8—2441no—
368Q8—2441no—
369D4—41no—
370D4—41no—
371D4—4882no—
372D4—2441no—
373D4—2441no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—611no—
377C2×C4—612no—
378C2×C4—612no—
379C2×C4—611no—
380C2×C4—611no—
381C2×C4—4882no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2442no—
385C2×C4—2441no—
386C2×C4D8×D6111yescharacteristic
387C2×C4D8×D6111yescharacteristic
388C2×C4C2×D4×D6111yescharacteristic
389C8—612no—
390C8—612no—
391C8SD16×D6112yes—
392C8#41×D6112yes—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—612no—
397C22—1221no—
398C22—1221no—
399C22—9764no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22C2×D8×D6111yescharacteristic
404C4—21no—
405C4—21no—
406C4—82no—
407C4—81no—
408C4—612no—
409C4—1221no—
410C4—1221no—
411C4—4882no—
412C4—4881no—
413C4#124×D6111yescharacteristic
414C4C2×SD16×D6111yescharacteristic
415C2—161no—
416C2—1222no—
417C2—9761no—
418C2#486×D6111yescharacteristic
419C2#2145895111yescharacteristic
420C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2243C2
socleC22×C61C22×D8
prime radicalC2×D8×D61C22
hypercenter#2243D61
hypocenterC61#33706
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2243

Complements

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

NH ≅ G/Ntype
1. #21456330C2semidirect
3. #21448058C2semidirect
4. #21448056C2semidirect
5. #21448055C2semidirect
6. #21448052C2semidirect
7. #21448049C2semidirect
8. #21447537C2semidirect
10. #21447412C2semidirect
11. C61×#2243C2semidirect
18. #16840167C22semidirect
20. #16840163C22semidirect
30. #16839567C22semidirect
31. #16839566C22semidirect
32. #16839565C22semidirect
36. #16839538C22semidirect
37. C16×Dic61C22semidirect
38. C61×#500C22semidirect
40. C61×#358C22semidirect
42. C61×#346C22semidirect
210. #2243D61semidirect
229. #500C2×D61semidirect
245. #358C2×D61semidirect
251. #346C2×D61semidirect
253. C61#33706semidirect

Maximal subgroups

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

subgroupindex
#214563302
#495×D612
#214480582
#214480562
#214480552
#214480522
#214480492
#214475372
#214475332
#214474122
C61×#22432
#3370661