Finite Group Databasebeta

← #16855088

Subgroups — #16855088

Order 3968.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 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#14521886C211yesmaximal, characteristic
2#137×D31C212yesmaximal, complemented
3#14521624C211yesmaximal, characteristic, complemented
4#14521234C211yesmaximal, characteristic, complemented
5#14521232C211yesmaximal, characteristic, complemented
6#14521231C211yesmaximal, characteristic, complemented
7#14521228C212yesmaximal, complemented
8#14521225C212yesmaximal, complemented
9#14521064C212yesmaximal, complemented
10#14521032C211yesmaximal, characteristic, complemented
11C31×#456C211yesmaximal, characteristic, complemented
12C31×2−1+4C2211yescharacteristic
13#11758566—22no—
14#11758566—22no—
15#11758566C2212yes—
16#41×D31—22no—
17#41×D31—22no—
18#41×D31C2212yes—
19#11758564—22no—
20#11758564C2211yescharacteristic
21C2×#92499—22no—
22C2×#92499C2211yescharacteristic
23C2×Q8×D31C2211yescharacteristic, complemented
24C31×#137C2212yescomplemented
25C31×C4⋊Q8C2211yescharacteristic, complemented
26#11758528C2212yescomplemented
27#11758524C2211yescharacteristic, complemented
28#11758521C2211yescharacteristic, complemented
29#11758519C2212yescomplemented
30#11758495—22no—
31Q16×D31—22no—
32#11758492—22no—
33SD16×D31—22no—
34#11758486C2212yescomplemented
35M16×D31C2212yescomplemented
36#11758473—22no—
37#11758471—22no—
38#11758452C2211yescharacteristic, complemented
39#11758449C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#101C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758424C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758411C2212yescomplemented
46#11758393C2212yescomplemented
47#41×C31—22no—
48#41×C31—22no—
49#41×C31C2312yes—
50C2×Q8×C31—22no—
51C2×Q8×C31C2311yescharacteristic
52C2×Q8×C31C2311yescharacteristic
53#92499—22no—
54#92499—24no—
55#92499—22no—
56#92499—22no—
57#92499—22no—
58#92499—22no—
59#92499D412yes—
60Q8×D31—22no—
61Q8×D31—22no—
62Q8×D31—22no—
63Q8×D31—22no—
64Q8×D31D412yes—
65#92497—22no—
66#92497—22no—
67#92497D412yes—
68D4×D31—24no—
69D4×D31—22no—
70D4×D31—24no—
71D4×D31—22no—
72D4×D31—22no—
73D4×D31D412yes—
74#92495—24no—
75#92495—24no—
76#92495—22no—
77#92495—22no—
78#92495—22no—
79#92495C2312yes—
80C2×D124—24no—
81C2×D124—22no—
82C2×D124D412yes—
83C2×D124C2311yescharacteristic
84C2×C4×D31—24no—
85C2×C4×D31—22no—
86C2×C4×D31—21no—
87C2×C4×D31D412yes—
88C2×C4×D31C2311yescharacteristic
89C2×Dic62—21no—
90C2×Dic62C2311yescharacteristic
91Q16×C31—22no—
92SD16×C31—22no—
93M16×C31C2312yes—
94C4⋊C4×C31—22no—
95C42×C31C2311yescharacteristic
96#92482—22no—
97#92482—22no—
98#92481—22no—
99#92480—22no—
100(C2×D31)⋊C4—21no—
101(C2×D31)⋊C4—22no—
102(C2×D31)⋊C4—22no—
103Dic31⋊C4—22no—
104Dic31⋊C4—21no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107Dic124—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C22×D31—24no—
112C22×D31—22no—
113C22×D31C2×D412yes—
114C22×D31C2×D411yescharacteristic
115Q8×C31—22no—
116Q8×C31—22no—
117Q8×C31—22no—
118Q8×C31C2×D412yes—
119D4×C31—22no—
120D4×C31—22no—
121D4×C31C2×D412yes—
122C2×C4×C31—24no—
123C2×C4×C31—22no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31C2×D412yes—
127C2×C4×C31C2411yescharacteristic
128C22⋊D31—24no—
129C22⋊D31—22no—
130C22⋊D31—22no—
131C22⋊D31—22no—
132C2×Dic31—22no—
133C2×Dic31—21no—
134C2×Dic31—21no—
135C2×Dic31C2×D411yescharacteristic
136D124—24no—
137D124—22no—
138D124—22no—
139D124—24no—
140D124—22no—
141D124C2×D412yes—
142C4×D31—24no—
143C4×D31—24no—
144C4×D31—22no—
145C4×D31—22no—
146C4×D31—22no—
147C4×D31—22no—
148C4×D31—22no—
149C4×D31—42no—
150C4×D31C2×D412yes—
151C4×D31C2×D412yes—
152Dic62—22no—
153Dic62—22no—
154Dic62—22no—
155Dic62—41no—
156Dic62—41no—
157Dic62C2×D412yes—
158C8×C31—22no—
159C31⋊C8—22no—
160#3022—311nomaximal, Sylow
161C22×C31—22no—
162C22×C31C22×D411yescharacteristic
163C2×D31—22no—
164C2×D31—22no—
165C2×D31—41no—
166C2×D31—42no—
167C2×D31—82no—
168C2×D31—84no—
169C2×D31#12012yes—
170C4×C31—22no—
171C4×C31—22no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31C22×D412yes—
175Dic31—22no—
176Dic31—41no—
177Dic31—41no—
178Dic31#12011yescharacteristic
179Dic31#12011yescharacteristic
180#585—311no—
181C2×#137—312no—
182#536—311no—
183#456—311no—
184#456—311no—
185#456—311no—
186#456D3111yescharacteristic, complemented
187#455—312no—
188#455—312no—
189#421—312no—
190#413—311no—
191C2×C31—21no—
192C2×C31—42no—
193C2×C31C2×#12011yescharacteristic
194D31—22no—
195D31—42no—
196D31—42no—
1972−1+4—311no—
1982−1+4—622no—
1992−1+4C2×D3111yescharacteristic
200H(5,𝔽2)—312no—
201H(5,𝔽2)—622no—
202H(5,𝔽2)—622no—
203C2×#41—311no—
204C2×#41—312no—
205C2×#41—622no—
206C2×#41—622no—
207C2×#41—622no—
208C22×Q8—311no—
209#137—312no—
210#137—312no—
211#137—312no—
212#137—622no—
213#137—622no—
214#137C2×D3112yescomplemented
215C2×Q16—622no—
216C2×SD16—622no—
217C2×M16—312no—
218C4⋊Q8—311no—
219C4⋊Q8C2×D3111yescharacteristic, complemented
220#124—311no—
221#124—311no—
222#123—622no—
223C22⋊Q8—622no—
224#117—311no—
225#104—312no—
226#104—312no—
227#104—312no—
228#104C2×D3112yescomplemented
229#101—311no—
230#101C2×D3111yescharacteristic, complemented
231#100—312no—
232C31#302211yescharacteristic, complemented, Sylow
233#41—22no—
234#41—22no—
235#41—312no—
236#41—312no—
237#41—312no—
238#41—622no—
239#41—622no—
240#41—624no—
241#41—624no—
242#41—622no—
243#41—622no—
244#41—622no—
245#41—622no—
246#41—622no—
247#41—622no—
248#41—624no—
249#41—622no—
250#41—622no—
251#41C22×D3112yes—
252C2×Q8—22no—
253C2×Q8—312no—
254C2×Q8—311no—
255C2×Q8—621no—
256C2×Q8—622no—
257C2×Q8—622no—
258C2×Q8—622no—
259C2×Q8—622no—
260C2×Q8C22×D3111yescharacteristic
261C2×Q8C22×D3111yescharacteristic
262C2×D4—312no—
263C2×D4—312no—
264C2×D4—311no—
265C2×D4—622no—
266C2×D4—624no—
267C2×D4—622no—
268C2×D4—624no—
269C2×D4—622no—
270C2×D4—624no—
271C2×D4—622no—
272C22×C4—311no—
273C22×C4—312no—
274C22×C4—622no—
275C22×C4—624no—
276C22×C4—621no—
277Q16—22no—
278Q16—622no—
279Q16—622no—
280Q16—622no—
281SD16—22no—
282SD16—622no—
283SD16—622no—
284SD16—622no—
285M16—312no—
286M16—622no—
287M16C22×D3112yes—
288C2×C8—622no—
289C4⋊C4—22no—
290C4⋊C4—621no—
291C4⋊C4—622no—
292C22⋊C4—621no—
293C22⋊C4—622no—
294C22⋊C4—622no—
295C42—311no—
296C42C22×D3111yescharacteristic
297C23—311no—
298C23—312no—
299C23—622no—
300C23—624no—
301Q8—22no—
302Q8—22no—
303Q8—22no—
304Q8—312no—
305Q8—622no—
306Q8—622no—
307Q8—622no—
308Q8—1241no—
309Q8—1241no—
310Q8D4×D3112yes—
311D4—22no—
312D4—22no—
313D4—312no—
314D4—624no—
315D4—622no—
316D4—624no—
317D4—622no—
318D4—622no—
319D4—622no—
320D4—624no—
321D4—622no—
322D4—622no—
323D4D4×D3112yes—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—311no—
329C2×C4—312no—
330C2×C4—312no—
331C2×C4—622no—
332C2×C4—622no—
333C2×C4—622no—
334C2×C4—622no—
335C2×C4—624no—
336C2×C4—624no—
337C2×C4—622no—
338C2×C4—622no—
339C2×C4—621no—
340C2×C4—621no—
341C2×C4—1242no—
342C2×C4D4×D3112yes—
343C2×C4C23×D3111yescharacteristic
344C8—22no—
345C8—622no—
346C22—22no—
347C22—312no—
348C22—622no—
349C22—622no—
350C22—1241no—
351C22—1242no—
352C22—2482no—
353C22—2484no—
354C22C2×D4×D3111yescharacteristic
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—311no—
360C4—311no—
361C4—622no—
362C4—1241no—
363C4—1241no—
364C4C2×D4×D3112yes—
365C2—21no—
366C2—42no—
367C2—622no—
368C2—1242no—
369C2—1242no—
370C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#456C2
socleC2×C31C2×#120
prime radical#14521886C2
hypercenter#456D31
hypocenterC31#3022
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D31C2semidirect
3. #14521624C2semidirect
4. #14521234C2semidirect
5. #14521232C2semidirect
6. #14521231C2semidirect
7. #14521228C2semidirect
8. #14521225C2semidirect
9. #14521064C2semidirect
10. #14521032C2semidirect
11. C31×#456C2semidirect
23. C2×Q8×D31C22semidirect
24. C31×#137C22semidirect
25. C31×C4⋊Q8C22semidirect
26. #11758528C22semidirect
27. #11758524C22semidirect
28. #11758521C22semidirect
29. #11758519C22semidirect
34. #11758486C22semidirect
35. M16×D31C22semidirect
38. #11758452C22semidirect
39. #11758449C22semidirect
40. C31×#104C22semidirect
41. C31×#101C22semidirect
42. #11758425C22semidirect
43. #11758424C22semidirect
44. #11758413C22semidirect
45. #11758411C22semidirect
46. #11758393C22semidirect
186. #456D31semidirect
214. #137C2×D31semidirect
219. C4⋊Q8C2×D31semidirect
228. #104C2×D31semidirect
230. #101C2×D31semidirect
232. C31#3022semidirect

Maximal subgroups

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

subgroupindex
#145218862
#137×D312
#145216242
#145212342
#145212322
#145212312
#145212282
#145212252
#145210642
#145210322
C31×#4562
#302231