Finite Group Databasebeta

← #16857221

Subgroups — #16857221

Order 3968.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758492C211yesmaximal, characteristic, complemented
2Q8×C22⋊D31C211yesmaximal, characteristic, complemented
3#14521693C211yesmaximal, characteristic
4#14521440C211yesmaximal, characteristic, complemented
5#14521292C211yesmaximal, characteristic, complemented
6#14521290C211yesmaximal, characteristic, complemented
7#14521269C211yesmaximal, characteristic, complemented
8#14521252C211yesmaximal, characteristic, complemented
9#14521208C211yesmaximal, characteristic, complemented
10#14521205C211yesmaximal, characteristic, complemented
11#14521200C211yesmaximal, characteristic, complemented
12#14521186C211yesmaximal, characteristic, complemented
13#14521150C211yesmaximal, characteristic, complemented
14#14521120C211yesmaximal, characteristic, complemented
15C22×SD16×C31C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17C22×D4×C31C2211yescharacteristic
18C2×Q8×D31C2211yescharacteristic, complemented
19C2×#92497C2211yescharacteristic
20C2×SD16×C31—24no—
21C2×SD16×C31C2211yescharacteristic
22C2×SD16×C31C2211yescharacteristic
23C2×SD16×C31C2211yescharacteristic
24C2×SD16×C31C2211yescharacteristic
25C22×C8×C31C2211yescharacteristic
26#11758530—21no—
27C2×C22⋊Dic31—21no—
28#11758523—21no—
29#11758523C2211yescharacteristic, complemented
30Q8×Dic31C2211yescharacteristic, complemented
31#11758521—21no—
32#11758521C2211yescharacteristic, complemented
33C2×#92482C2211yescharacteristic, complemented
34C2×#92482C2211yescharacteristic, complemented
35#11758516—21no—
36#11758515C2211yescharacteristic
37#11758513C2211yescharacteristic
38#11758512—21no—
39D4×Dic31C2211yescharacteristic
40C2×#92480C2211yescharacteristic
41C2×#92480C2211yescharacteristic
42C4×C22⋊D31—21no—
43C4×C22⋊D31C2211yescharacteristic
44#11758501—21no—
45#11758501C2211yescharacteristic
46#11758501C2211yescharacteristic
47#11758492—24no—
48C2×Dic124C2211yescharacteristic, complemented
49C2×#92469C2211yescharacteristic, complemented
50#11758423C2211yescharacteristic, complemented
51#11758423C2211yescharacteristic, complemented
52#11758420C2211yescharacteristic
53#11758420C2211yescharacteristic
54C22⋊(C31⋊C8)C2211yescharacteristic
55(C2×D31)⋊C8C2211yescharacteristic, complemented
56#11758406C2211yescharacteristic, complemented
57#11758404C2211yescharacteristic, complemented
58#11758404C2211yescharacteristic, complemented
59#11758403C2211yescharacteristic, complemented
60Dic31⋊C8C2211yescharacteristic, complemented
61C24×C31—21no—
62C2×Q8×C31—21no—
63C2×Q8×C31—22no—
64C2×Q8×C31—41no—
65C2×Q8×C31C2311yescharacteristic
66C2×Q8×C31C2311yescharacteristic
67C2×D4×C31—22no—
68C2×D4×C31—21no—
69C2×D4×C31—41no—
70C2×D4×C31C2311yescharacteristic
71C2×D4×C31C2311yescharacteristic
72C22×C4×C31—21no—
73C22×C4×C31C2311yescharacteristic
74C2×C22⋊D31—21no—
75C2×C22⋊D31—21no—
76C2×C22⋊D31D411yescharacteristic
77C22×Dic31—21no—
78C22×Dic31—21no—
79Q8×D31—22no—
80Q8×D31—41no—
81#92497—22no—
82#92497—41no—
83C2×C4×D31—21no—
84C2×C4×D31C2311yescharacteristic
85C2×Dic62—21no—
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88SD16×C31—22no—
89SD16×C31—22no—
90SD16×C31—22no—
91SD16×C31D414yescomplemented
92C2×C8×C31—22no—
93C2×C8×C31C2311yescharacteristic
94C2×C8×C31C2311yescharacteristic
95C22⋊Dic31—21no—
96C22⋊Dic31—21no—
97C22⋊Dic31—21no—
98C22⋊Dic31—21no—
99C22⋊Dic31—21no—
100C22⋊Dic31—21no—
101C22⋊Dic31D411yescharacteristic
102#92482—22no—
103#92482—41no—
104#92480—22no—
105#92480—41no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4D411yescharacteristic
108C4⋊Dic31—21no—
109C4⋊Dic31C2311yescharacteristic
110C4⋊Dic31C2311yescharacteristic
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4—21no—
114Dic31⋊C4—21no—
115Dic31⋊C4—21no—
116Dic31⋊C4D411yescharacteristic
117C4×Dic31—21no—
118C4×Dic31C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120C2×C31⋊C8C2311yescharacteristic
121Dic124—22no—
122#92469—22no—
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—42no—
126C23×C31—42no—
127C23×C31—81no—
128C23×C31C2×D411yescharacteristic
129C22×D31C2×D411yescharacteristic
130Q8×C31—21no—
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—42no—
134Q8×C31C2×D412yes—
135D4×C31—21no—
136D4×C31—21no—
137D4×C31—21no—
138D4×C31—42no—
139D4×C31C2×D412yes—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—42no—
145C2×C4×C31C2×D411yescharacteristic
146C2×C4×C31C2411yescharacteristic
147C22⋊D31—22no—
148C22⋊D31—42no—
149C22⋊D31—81no—
150C2×Dic31—21no—
151C2×Dic31—21no—
152C2×Dic31—21no—
153C2×Dic31—21no—
154C2×Dic31—42no—
155C2×Dic31—81no—
156C2×Dic31C2×D411yescharacteristic
157C2×Dic31C2×D411yescharacteristic
158C2×Dic31C2×D411yescharacteristic
159C4×D31—21no—
160C4×D31—21no—
161C4×D31—42no—
162Dic62—21no—
163Dic62—21no—
164Dic62—21no—
165Dic62—21no—
166Dic62—42no—
167C8×C31—21no—
168C8×C31C2×D412yes—
169C31⋊C8—21no—
170C31⋊C8—21no—
171#3266—311nomaximal, Sylow
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—22no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31—41no—
180C22×C31—84no—
181C22×C31#13712yescomplemented
182C22×C31C22×D411yescharacteristic
183C2×D31—21no—
184C2×D31—21no—
185C2×D31—21no—
186C4×C31—22no—
187C4×C31—21no—
188C4×C31—41no—
189C4×C31C22×D411yescharacteristic
190C4×C31H(5,𝔽2)11yescharacteristic
191Dic31—21no—
192Dic31—21no—
193Dic31—21no—
194Dic31—41no—
195Dic31—41no—
196Dic31—41no—
197Dic31—41no—
198C2×#137—311no—
199C22×SD16D3111yescharacteristic, complemented
200D4×Q8—311no—
201#546—311no—
202#497—311no—
203#468—311no—
204#467—311no—
205#464—311no—
206#461—311no—
207#452—311no—
208#451—311no—
209#450—311no—
210#448—311no—
211#440—311no—
212#435—311no—
213C2×C31—22no—
214C2×C31—42no—
215C2×C31—81no—
216C2×C31D4211yescharacteristic
217C2×C31C2×#13711yescharacteristic
218C2×C31#57711yescharacteristic
219D31—41no—
220C2×#41—311no—
221C22×Q8—311no—
222C22×Q8C2×D3111yescharacteristic
223C22×D4C2×D3111yescharacteristic
224#137—624no—
225C2×Q16—311no—
226C2×Q16—311no—
227C2×Q16—311no—
228C2×SD16—24no—
229C2×SD16—311no—
230C2×SD16—311no—
231C2×SD16C2×D3111yescharacteristic
232C2×SD16C2×D3111yescharacteristic
233C2×SD16C2×D3111yescharacteristic
234C2×SD16C2×D3111yescharacteristic
235C2×M16—311no—
236C22×C8C2×D3111yescharacteristic
237C4⋊Q8—311no—
238C4⋊Q8—621no—
239#124—311no—
240#123—621no—
241C22⋊Q8—311no—
242C22⋊Q8—311no—
243C22⋊Q8—311no—
244C22⋊Q8—621no—
245C22⋊Q8—621no—
246#121—311no—
247#121—621no—
248#120—621no—
249C4×Q8—311no—
250C4×D4—311no—
251C4×D4—311no—
252C4×D4—621no—
253C2×C22⋊C4—621no—
254#107—311no—
255C4⋊C8—311no—
256#103—311no—
257#103—311no—
258#103—311no—
259#103—311no—
260#102—311no—
261#102—311no—
262C22⋊C8—311no—
263C22⋊C8—311no—
264#97—311no—
265C31#326611yescharacteristic, complemented, Sylow
266C24—21no—
267#41—622no—
268#41—1241no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—311no—
273C2×Q8—311no—
274C2×Q8—621no—
275C2×Q8—622no—
276C2×Q8—1241no—
277C2×Q8C22×D3111yescharacteristic
278C2×Q8C22×D3111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—311no—
283C2×D4—621no—
284C2×D4—621no—
285C2×D4C22×D3111yescharacteristic
286C2×D4C22×D3111yescharacteristic
287C22×C4—21no—
288C22×C4—311no—
289C22×C4—621no—
290C22×C4—621no—
291C22×C4—621no—
292C22×C4C22×D3111yescharacteristic
293Q16—622no—
294Q16—622no—
295Q16—1241no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—622no—
300SD16—1241no—
301SD16C22⋊D3114yescomplemented
302M16—622no—
303C2×C8—22no—
304C2×C8—311no—
305C2×C8—311no—
306C2×C8C22×D3111yescharacteristic
307C2×C8C22×D3111yescharacteristic
308C4⋊C4—311no—
309C4⋊C4—311no—
310C4⋊C4—311no—
311C4⋊C4—621no—
312C4⋊C4—621no—
313C4⋊C4—621no—
314C4⋊C4—621no—
315C4⋊C4—621no—
316C4⋊C4—621no—
317C22⋊C4—311no—
318C22⋊C4—311no—
319C22⋊C4—621no—
320C22⋊C4—621no—
321C22⋊C4—621no—
322C22⋊C4—621no—
323C22⋊C4—621no—
324C22⋊C4—621no—
325C22⋊C4—621no—
326C42—311no—
327C42—621no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—311no—
334C23D4×D3111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—621no—
340Q8—621no—
341Q8—621no—
342Q8—621no—
343Q8—1242no—
344Q8C2×C22⋊D3112yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—622no—
350D4—1242no—
351D4—2481no—
352D4C2×C22⋊D3112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—311no—
359C2×C4—311no—
360C2×C4—311no—
361C2×C4—621no—
362C2×C4—621no—
363C2×C4—621no—
364C2×C4—621no—
365C2×C4—621no—
366C2×C4—621no—
367C2×C4—1242no—
368C2×C4—1242no—
369C2×C4—2481no—
370C2×C4D4×D3111yescharacteristic
371C2×C4C23×D3111yescharacteristic
372C8—21no—
373C8—621no—
374C8—621no—
375C8C2×C22⋊D3112yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—621no—
386C22—621no—
387C22—621no—
388C22#1175849212yescomplemented
389C22C2×D4×D3111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—621no—
394C4—621no—
395C4—621no—
396C4—1241no—
397C4—1241no—
398C4—1241no—
399C4—1241no—
400C4#1175856111yescharacteristic
401C4C22×C22⋊D3111yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—1241no—
406C2D4×C22⋊D3111yescharacteristic
407C2C2×#1175849211yescharacteristic
408C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC22×SD16×C31C2
socleC22×C31C22×D4
prime radical#14521693C2
hypercenterC22×SD16D31
hypocenterC31#3266
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#11758492C2semidirect
2. Q8×C22⋊D31C2semidirect
4. #14521440C2semidirect
5. #14521292C2semidirect
6. #14521290C2semidirect
7. #14521269C2semidirect
8. #14521252C2semidirect
9. #14521208C2semidirect
10. #14521205C2semidirect
11. #14521200C2semidirect
12. #14521186C2semidirect
13. #14521150C2semidirect
14. #14521120C2semidirect
15. C22×SD16×C31C2semidirect
18. C2×Q8×D31C22semidirect
29. #11758523C22semidirect
30. Q8×Dic31C22semidirect
32. #11758521C22semidirect
33. C2×#92482C22semidirect
34. C2×#92482C22semidirect
48. C2×Dic124C22semidirect
49. C2×#92469C22semidirect
50. #11758423C22semidirect
51. #11758423C22semidirect
55. (C2×D31)⋊C8C22semidirect
56. #11758406C22semidirect
57. #11758404C22semidirect
58. #11758404C22semidirect
59. #11758403C22semidirect
60. Dic31⋊C8C22semidirect
91. SD16×C31D4semidirect
181. C22×C31#137semidirect
199. C22×SD16D31semidirect
265. C31#3266semidirect
301. SD16C22⋊D31semidirect
388. C22#11758492semidirect

Maximal subgroups

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

subgroupindex
C2×#117584922
Q8×C22⋊D312
#145216932
#145214402
#145212922
#145212902
#145212692
#145212522
#145212082
#145212052
#145212002
#145211862
#145211502
#145211202
C22×SD16×C312
#326631