Finite Group Databasebeta

← #16855087

Subgroups — #16855087

Order 3968.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521885C211yesmaximal, characteristic
2C2×#11758519C211yesmaximal, characteristic, complemented
3#14521623C211yesmaximal, characteristic, complemented
4#14521230C214yesmaximal, complemented
5#14521225C214yesmaximal, complemented
6#14521066C211yesmaximal, characteristic, complemented
7#14521065C211yesmaximal, characteristic, complemented
8#14521031C211yesmaximal, characteristic, complemented
9C2×C31×#137C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×Q8×C31C2211yescharacteristic, complemented
12#11758567—21no—
13#11758567C2212yes—
14#11758566—21no—
15#11758566C2212yes—
16#41×D31—22no—
17#41×D31—22no—
18#11758564—21no—
19#11758561—21no—
20#11758561—22no—
21C2×#92495—22no—
22C2×#92495C2211yescharacteristic
23C2×#92495C2211yescharacteristic
24C31×#137—22no—
25C31×#137C2214yescomplemented
26C2×Q16×C31—21no—
27C2×SD16×C31—21no—
28C2×M16×C31C2211yescharacteristic, complemented
29#11758524C2212yescomplemented
30#11758523—22no—
31#11758521C2212yescomplemented
32C2×#92482—21no—
33#11758519—22no—
34#11758519C2214yescomplemented
35C2×#92481—21no—
36#11758506—22no—
37#11758503C2211yescharacteristic, complemented
38C2×#92474C2211yescharacteristic, complemented
39#11758425C2214yescomplemented
40#11758413C2214yescomplemented
41#11758412C2212yescomplemented
42#11758411C2212yescomplemented
43#41×C31—21no—
44#41×C31—21no—
45#41×C31D412yes—
46#41×C31C2312yes—
47C2×Q8×C31—21no—
48C2×Q8×C31—21no—
49C2×Q8×C31—21no—
50C2×Q8×C31—21no—
51C2×Q8×C31—21no—
52C2×Q8×C31D411yescharacteristic
53C2×Q8×C31C2312yes—
54C2×D4×C31—21no—
55C2×D4×C31D411yescharacteristic
56C22×C4×C31—21no—
57C22×C4×C31—21no—
58C22×C4×C31C2311yescharacteristic
59C2×C22⋊D31—24no—
60C2×C22⋊D31—21no—
61C2×C22⋊D31D412yes—
62#92499—22no—
63#92499—22no—
64Q8×D31—22no—
65Q8×D31—22no—
66#92497—24no—
67#92497—22no—
68#92497—22no—
69#92497—24no—
70D4×D31—24no—
71D4×D31—22no—
72D4×D31—24no—
73D4×D31—22no—
74#92495—24no—
75#92495—24no—
76#92495—21no—
77#92495—21no—
78#92495—21no—
79#92495—21no—
80#92495—21no—
81#92495—21no—
82#92495—22no—
83#92495—22no—
84#92495D412yes—
85#92495C2312yes—
86C2×D124—22no—
87C2×D124D411yescharacteristic
88C2×D124C2311yescharacteristic
89C2×C4×D31—24no—
90C2×C4×D31—21no—
91C2×C4×D31D412yes—
92C2×Dic62—22no—
93C2×Dic62D411yescharacteristic
94C2×Dic62C2311yescharacteristic
95Q16×C31—24no—
96SD16×C31—24no—
97M16×C31—21no—
98M16×C31C2312yes—
99C2×C8×C31—21no—
100C22⋊Dic31—21no—
101#92482—24no—
102#92481—24no—
103(C2×D31)⋊C4—24no—
104C4⋊Dic31—21no—
105Dic31⋊C4—24no—
106C4×Dic31C2312yes—
107#92474—21no—
108#92474C2312yes—
109C2×C31⋊C8—21no—
110C23×C31—21no—
111C23×C31C2×D411yescharacteristic
112C22×D31—24no—
113C22×D31—21no—
114C22×D31C2×D412yes—
115Q8×C31—21no—
116Q8×C31—22no—
117Q8×C31—22no—
118Q8×C31—41no—
119Q8×C31—41no—
120Q8×C31C2×D412yes—
121D4×C31—22no—
122D4×C31—22no—
123D4×C31—21no—
124D4×C31C2×D412yes—
125C2×C4×C31—21no—
126C2×C4×C31—22no—
127C2×C4×C31—22no—
128C2×C4×C31—22no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2×D411yescharacteristic
135C2×C4×C31C2×D411yescharacteristic
136C2×C4×C31C2411yescharacteristic
137C22⋊D31—24no—
138C22⋊D31—24no—
139C22⋊D31—22no—
140C22⋊D31—22no—
141C22⋊D31—22no—
142C22⋊D31—22no—
143C2×Dic31—24no—
144C2×Dic31—21no—
145C2×Dic31—22no—
146C2×Dic31C2×D412yes—
147D124—24no—
148D124—21no—
149D124—21no—
150D124—21no—
151D124C2×D412yes—
152C4×D31—24no—
153C4×D31—24no—
154C4×D31—22no—
155C4×D31—22no—
156C4×D31—22no—
157C4×D31—22no—
158Dic62—24no—
159Dic62—21no—
160Dic62—21no—
161Dic62—21no—
162Dic62C2×D412yes—
163C8×C31—22no—
164C31⋊C8—22no—
165#3022—311nomaximal, Sylow
166C22×C31—22no—
167C22×C31—41no—
168C22×C31—81no—
169C22×C31#12011yescharacteristic
170C22×C31#12011yescharacteristic
171C22×C31C22×D411yescharacteristic
172C2×D31—22no—
173C2×D31—22no—
174C2×D31—42no—
175C2×D31—81no—
176C2×D31—84no—
177C4×C31—22no—
178C4×C31—42no—
179C4×C31#12011yescharacteristic
180C4×C31#12011yescharacteristic
181C4×C31C22×D411yescharacteristic
182C4×C31C22×D411yescharacteristic
183Dic31—22no—
184Dic31—22no—
185Dic31—42no—
186#585—311no—
187C2×#137—311no—
188C2×#137D3111yescharacteristic, complemented
189#536—311no—
190#456—314no—
191#455—314no—
192#421—311no—
193#421—311no—
194#413—311no—
195C2×C31—21no—
196C2×C31—21no—
197C2×C31—21no—
198C2×C31—42no—
199C2×C31C2×#12011yescharacteristic
200D31—42no—
201D31—42no—
2022−1+4—312no—
2032−1+4—621no—
2042−1+4—621no—
205H(5,𝔽2)—312no—
206H(5,𝔽2)—622no—
207H(5,𝔽2)—621no—
208H(5,𝔽2)—621no—
209C2×#41—311no—
210C2×#41—311no—
211C2×#41—622no—
212C2×#41—622no—
213C2×#41—622no—
214C2×#41C2×D3111yescharacteristic
215C22×Q8C2×D3111yescharacteristic, complemented
216#137—22no—
217#137—314no—
218#137—622no—
219#137C2×D3114yescomplemented
220C2×Q16—21no—
221C2×Q16—621no—
222C2×SD16—21no—
223C2×SD16—621no—
224C2×M16—311no—
225C2×M16C2×D3111yescharacteristic, complemented
226C4⋊Q8—312no—
227#124—312no—
228#123—622no—
229C22⋊Q8—622no—
230#117—311no—
231#104—314no—
232#104—314no—
233#101—312no—
234#100—312no—
235C31#302211yescharacteristic, complemented, Sylow
236#41—21no—
237#41—21no—
238#41—312no—
239#41—312no—
240#41—622no—
241#41—622no—
242#41—624no—
243#41—624no—
244#41—624no—
245#41—624no—
246#41—622no—
247#41—622no—
248#41—622no—
249#41—622no—
250#41—621no—
251#41—621no—
252#41—621no—
253#41—621no—
254#41—621no—
255#41—621no—
256#41C22⋊D3112yes—
257#41C22×D3112yes—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—311no—
264C2×Q8—311no—
265C2×Q8—622no—
266C2×Q8—622no—
267C2×Q8—622no—
268C2×Q8C22⋊D3111yescharacteristic
269C2×Q8C22×D3112yes—
270C2×D4—21no—
271C2×D4—312no—
272C2×D4—311no—
273C2×D4—311no—
274C2×D4—622no—
275C2×D4—624no—
276C2×D4—624no—
277C2×D4—622no—
278C2×D4—624no—
279C2×D4—621no—
280C2×D4—622no—
281C2×D4C22⋊D3111yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—312no—
285C22×C4—624no—
286C22×C4—621no—
287C22×C4C22×D3111yescharacteristic
288Q16—24no—
289Q16—624no—
290SD16—24no—
291SD16—624no—
292M16—21no—
293M16—312no—
294M16—621no—
295M16C22×D3112yes—
296C2×C8—21no—
297C2×C8—621no—
298C4⋊C4—624no—
299C4⋊C4—621no—
300C22⋊C4—624no—
301C22⋊C4—621no—
302C42—312no—
303C23—21no—
304C23—312no—
305C23—624no—
306C23—621no—
307C23D4×D3111yescharacteristic
308Q8—21no—
309Q8—22no—
310Q8—22no—
311Q8—41no—
312Q8—41no—
313Q8—312no—
314Q8—624no—
315Q8—621no—
316Q8—621no—
317Q8—621no—
318Q8C2×C22⋊D3112yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—312no—
323D4—624no—
324D4—624no—
325D4—624no—
326D4—622no—
327D4—622no—
328D4—622no—
329D4—622no—
330D4—621no—
331D4—621no—
332D4—621no—
333D4C2×C22⋊D3112yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—312no—
341C2×C4—624no—
342C2×C4—624no—
343C2×C4—624no—
344C2×C4—621no—
345C2×C4—622no—
346C2×C4—622no—
347C2×C4—622no—
348C2×C4—622no—
349C2×C4—622no—
350C2×C4D4×D3111yescharacteristic
351C2×C4D4×D3111yescharacteristic
352C2×C4D4×D3111yescharacteristic
353C2×C4C2×C22⋊D3111yescharacteristic
354C2×C4C2×C22⋊D3111yescharacteristic
355C2×C4C23×D3111yescharacteristic
356C8—22no—
357C8—622no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—622no—
362C22—622no—
363C22—1242no—
364C22—2481no—
365C22—2484no—
366C22#1175851411yescharacteristic
367C22#1175851411yescharacteristic
368C22C2×D4×D3111yescharacteristic
369C4—22no—
370C4—42no—
371C4—622no—
372C4—622no—
373C4—1242no—
374C4#1175851411yescharacteristic
375C4#1175851411yescharacteristic
376C4C2×D4×D3111yescharacteristic
377C4C22×C22⋊D3111yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—1242no—
383C2—1242no—
384C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#137C2
socleC2×C31C2×#120
prime radical#14521885C2
hypercenterC2×#137D31
hypocenterC31#3022
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. #14521623C2semidirect
4. #14521230C2semidirect
5. #14521225C2semidirect
6. #14521066C2semidirect
7. #14521065C2semidirect
8. #14521031C2semidirect
9. C2×C31×#137C2semidirect
11. C22×Q8×C31C22semidirect
25. C31×#137C22semidirect
28. C2×M16×C31C22semidirect
29. #11758524C22semidirect
31. #11758521C22semidirect
34. #11758519C22semidirect
37. #11758503C22semidirect
38. C2×#92474C22semidirect
39. #11758425C22semidirect
40. #11758413C22semidirect
41. #11758412C22semidirect
42. #11758411C22semidirect
188. C2×#137D31semidirect
215. C22×Q8C2×D31semidirect
219. #137C2×D31semidirect
225. C2×M16C2×D31semidirect
235. C31#3022semidirect

Maximal subgroups

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

subgroupindex
#145218852
C2×#117585192
#145216232
#145212302
#145212252
#145210662
#145210652
#145210312
C2×C31×#1372
#302231