Finite Group Databasebeta

← #29833351

Subgroups — #29833351

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840197C211yesmaximal, characteristic
2#21454672C211yesmaximal, characteristic, complemented
3#21447693C212yesmaximal, complemented
4#21447681C212yesmaximal, complemented
5C4×#14519720C212yesmaximal, complemented
6C4×#14519718C212yesmaximal, complemented
7#21447605C211yesmaximal, characteristic, complemented
8#21447526C211yesmaximal, characteristic, complemented
9C2×#16839562C211yesmaximal, characteristic, complemented
10#21447416C211yesmaximal, characteristic, complemented
11C61×#2951C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13#16840236—21no—
14#16840197C418yescomplemented
15C2×#14519720C2212yes—
16C2×#14519718C2212yes—
17C2×#14519717C2211yescharacteristic
18C22×#11758185C2211yescharacteristic
19#16839875—22no—
20C4×D488C2212yes—
21C4×D488C2212yes—
22#16839800C2211yescharacteristic
23#16839768—21no—
24#16839589C2212yescomplemented
25#16839587C2212yescomplemented
26#16839566C2211yescharacteristic, complemented
27#16839566C2211yescharacteristic, complemented
28#16839563C2211yescharacteristic, complemented
29#16839563C2211yescharacteristic, complemented
30#16839562C2212yes—
31#16839562C2212yes—
32#16839538C2212yescomplemented
33C4×#11758185C2212yescomplemented
34C2×#135×C61C2211yescharacteristic
35C61×#520—21no—
36C61×#440—22no—
37C4×Q16×C61C2212yescomplemented
38C4×D8×C61C2212yescomplemented
39C2×#107×C61C2211yescharacteristic
40C61×#417—21no—
41C61×#405C2211yescharacteristic
42C22×D244—21no—
43C2×D488—21no—
44C2×D488—22no—
45C2×D488C2×C414yescomplemented
46C2×D488C2312yes—
47C4×D244—22no—
48C4×D244—22no—
49#14519739—41no—
50#14519739—41no—
51#14519736—21no—
52C22⋊C4×D61—42no—
53#14519720C2×C418yescomplemented
54#14519718C2×C418yescomplemented
55#14519717C2×C414yescomplemented
56C2×#11758185—22no—
57C2×#11758185C2311yescharacteristic
58C2×#11758185C2311yescharacteristic
59C2×#11758185C2312yes—
60#14519711C2311yescharacteristic
61#14519711C2311yescharacteristic
62#14519709—21no—
63#14519699—22no—
64#14519699—22no—
65C2×#41×C61—21no—
66#135×C61—22no—
67#135×C61C2×C414yescomplemented
68C2×Q16×C61C2311yescharacteristic
69C2×SD16×C61—21no—
70C2×D8×C61C2311yescharacteristic
71C2×M16×C61D411yescharacteristic
72C22×C8×C61C2311yescharacteristic
73C4×Q8×C61—22no—
74C4×D4×C61—22no—
75C4×D4×C61—42no—
76#117×C61—41no—
77#117×C61D411yescharacteristic
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—41no—
80#107×C61—21no—
81#107×C61C2312yes—
82#103×C61—22no—
83#102×C61—22no—
84#97×C61D412yes—
85C4×C8×C61C2312yes—
86C23×D61—41no—
87#41×C61—24no—
88#41×C61—42no—
89C2×Q8×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—42no—
95C22×C4×C61C2×D411yescharacteristic
96C2×D244—24no—
97C2×D244—22no—
98C2×D244—41no—
99C2×D244—42no—
100C2×C4×D61—42no—
101C2×C4×D61—42no—
102Q16×C61—21no—
103Q16×C61C22×C412yes—
104SD16×C61—24no—
105D8×C61—21no—
106D8×C61C22×C412yes—
107M16×C61—22no—
108C2×C8×C61C22×C412yes—
109C2×C8×C61C2×D412yes—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61#4112yes—
112C2×C8×C61C2411yescharacteristic
113C4⋊C4×C61—22no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—42no—
116C4⋊C4×C61D811yescharacteristic
117C4⋊C4×C61D811yescharacteristic
118C22⋊C4×C61—42no—
119C22⋊C4×C61D812yes—
120C42×C61—42no—
121C42×C61C2×D412yes—
122C22⋊Dic61—41no—
123#11758198—42no—
124#11758198—42no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4×Dic61—41no—
128D488—24no—
129D488—22no—
130D488C22×C414yes—
131#11758185—21no—
132#11758185—21no—
133#11758185C22×C414yes—
134C23×C61—41no—
135C23×C61C2×D811yescharacteristic
136C22×D61—42no—
137C22×D61—44no—
138C22×D61—161no—
139C22×D61—161no—
140Q8×C61—22no—
141Q8×C61—41no—
142D4×C61—22no—
143D4×C61—44no—
144D4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—44no—
147C2×C4×C61—42no—
148C2×C4×C61—162no—
149C2×C4×C61—161no—
150C2×C4×C61C4×D412yes—
151C2×C4×C61C2×D812yes—
152C2×C4×C61C2×D812yes—
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61#13512yes—
155C2×C4×C61C22×D411yescharacteristic
156C2×Dic61—41no—
157C2×Dic61—41no—
158D244—24no—
159D244—42no—
160D244—44no—
161C4×D61—162no—
162C4×D61—162no—
163C8×C61—22no—
164C8×C61C4×D412yes—
165C8×C61C23×C411yescharacteristic
166C8×C61C2×#4111yescharacteristic
167#33454—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—42no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61C4×D812yes—
174C22×C61C22×D811yescharacteristic
175C2×D61—44no—
176C2×D61—82no—
177C2×D61—82no—
178C2×D61—164no—
179C4×C61—22no—
180C4×C61—22no—
181C4×C61—42no—
182C4×C61—82no—
183C4×C61C4×D812yes—
184C4×C61C2×C4×D411yescharacteristic
185C4×C61C2×#13511yescharacteristic
186Dic61—81no—
187Dic61—81no—
188C2×#509—611no—
189#2951D6111yescharacteristic, complemented
190#2948—611no—
191#2177—612no—
192#2175—612no—
193C4×SD32—612no—
194C4×D16—612no—
195#2159—611no—
196#2138—611no—
197C2×#357—611no—
198#2108—611no—
199C2×C61—22no—
200C2×C61—82no—
201C2×C61C2×C4×D811yescharacteristic
202C2×C61#341511yescharacteristic
203C2×C61#341611yescharacteristic
204D61—84no—
205C2×#135C2×D6111yescharacteristic
206C22×D8—611no—
207#520—21no—
208#518—1221no—
209#509—618no—
210C2×SD32—612no—
211C2×D16—612no—
212C2×M32—611no—
213C22×C16—611no—
214#442—1222no—
215#440—22no—
216C4×Q16C2×D6112yescomplemented
217C4×D8—612no—
218C4×D8—612no—
219C4×D8C2×D6112yescomplemented
220#427—611no—
221C2×#107C2×D6111yescharacteristic
222#418—1221no—
223#417—21no—
224#405C2×D6111yescharacteristic
225#367—612no—
226#366—612no—
227#358—611no—
228#358—611no—
229#357—611no—
230#357—611no—
231#357—612no—
232#357—612no—
233#346—612no—
234C4×C16—612no—
235C61#3345411yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×D4—1221no—
238#135—22no—
239#135C4×D6114yescomplemented
240C2×Q16C22×D6111yescharacteristic
241C2×SD16—21no—
242C2×D8—614no—
243C2×D8—612no—
244C2×D8—1222no—
245C2×D8—1221no—
246C2×D8C22×D6111yescharacteristic
247C2×M16C22⋊D6111yescharacteristic
248C22×C8C22×D6111yescharacteristic
249C4×Q8—22no—
250C4×D4—22no—
251C4×D4—42no—
252C4×D4—1222no—
253C4×D4—1222no—
254C4×D4—2441no—
255C4×D4—2441no—
256#117—41no—
257#117—1221no—
258#117C22⋊D6111yescharacteristic
259C2×C4⋊C4—21no—
260C2×C4⋊C4—41no—
261C2×C22⋊C4—2442no—
262SD32—618no—
263D16—618no—
264M32—614no—
265C2×C16—612no—
266C2×C16—611no—
267C2×C16—611no—
268C2×C16—1222no—
269#107—21no—
270#107—611no—
271#107—611no—
272#107C22×D6112yes—
273#106—1221no—
274#103—22no—
275#102—22no—
276#102—1222no—
277#102—1222no—
278#97C22⋊D6112yes—
279C4×C8C22×D6112yes—
280C24—2441no—
281#41—24no—
282#41—42no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—1222no—
287C2×D4—1224no—
288C2×D4—2441no—
289C2×D4—2442no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—2442no—
294C22×C4—2442no—
295C22×C4C2×C22⋊D6111yescharacteristic
296Q16—21no—
297Q16C2×C4×D6112yes—
298SD16—24no—
299D8—21no—
300D8—614no—
301D8—1222no—
302D8—1224no—
303D8C2×C4×D6112yes—
304M16—22no—
305C2×C8C2×C4×D6112yes—
306C2×C8#1175822212yes—
307C2×C8C2×C22⋊D6112yes—
308C2×C8C2×C22⋊D6111yescharacteristic
309C2×C8C23×D6111yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4#1175819911yescharacteristic
316C4⋊C4#1175819911yescharacteristic
317C22⋊C4—42no—
318C22⋊C4—2441no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C22⋊C4#1175819912yes—
322C42—42no—
323C42—2441no—
324C42C2×C22⋊D6112yes—
325C16—614no—
326C16—1221no—
327C16—1221no—
328C23—41no—
329C23—9761no—
330C23—9761no—
331C23—2444no—
332C23—2442no—
333C23C2×#1175819911yescharacteristic
334Q8—22no—
335Q8—41no—
336D4—22no—
337D4—44no—
338D4—41no—
339D4—1224no—
340D4—2442no—
341D4—2444no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—162no—
346C2×C4—161no—
347C2×C4—9762no—
348C2×C4—9762no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4C4×C22⋊D6112yes—
352C2×C4C2×#1175819912yes—
353C2×C4C2×#1175819912yes—
354C2×C4C2×#1175819911yescharacteristic
355C2×C4#1451979712yes—
356C2×C4C22×C22⋊D6111yescharacteristic
357C8—22no—
358C8C4×C22⋊D6112yes—
359C8C22×C4×D6111yescharacteristic
360C8C2×#1175822211yescharacteristic
361C22—21no—
362C22—21no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—9764no—
367C22—4882no—
368C22—4882no—
369C22—2444no—
370C22C4×#1175819912yes—
371C22C22×#1175819911yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—82no—
376C4—4881no—
377C4—4881no—
378C4C4×#1175819912yes—
379C4C2×C4×C22⋊D6111yescharacteristic
380C4C2×#1451979711yescharacteristic
381C2—22no—
382C2—82no—
383C2—4884no—
384C2C2×C4×#1175819911yescharacteristic
385C2#2145895811yescharacteristic
386C2#2145896311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2951C2
socleC22×C61C22×D8
prime radicalC2×#16840197C2
cosocleC2×C8×C61C24
hypercenter#2951D61
hypocenterC61#33454
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. #21454672C2semidirect
3. #21447693C2semidirect
4. #21447681C2semidirect
5. C4×#14519720C2semidirect
6. C4×#14519718C2semidirect
7. #21447605C2semidirect
8. #21447526C2semidirect
9. C2×#16839562C2semidirect
10. #21447416C2semidirect
11. C61×#2951C2semidirect
14. #16840197C4semidirect
24. #16839589C22semidirect
25. #16839587C22semidirect
26. #16839566C22semidirect
27. #16839566C22semidirect
28. #16839563C22semidirect
29. #16839563C22semidirect
32. #16839538C22semidirect
33. C4×#11758185C22semidirect
37. C4×Q16×C61C22semidirect
38. C4×D8×C61C22semidirect
45. C2×D488C2×C4semidirect
53. #14519720C2×C4semidirect
54. #14519718C2×C4semidirect
55. #14519717C2×C4semidirect
67. #135×C61C2×C4semidirect
189. #2951D61semidirect
216. C4×Q16C2×D61semidirect
219. C4×D8C2×D61semidirect
235. C61#33454semidirect
239. #135C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
#214546722
#214476932
#214476812
C4×#145197202
C4×#145197182
#214476052
#214475262
C2×#168395622
#214474162
C61×#29512
#3345461