Finite Group Databasebeta

← #16855072

Subgroups — #16855072

Order 3968.

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

normal subgroups only

This group has 698 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
12−1+4×D31C211yesmaximal, characteristic, complemented
2#14521845C212yesmaximal, complemented
3#14521619C211yesmaximal, characteristic, complemented
4#14521234C211yesmaximal, characteristic, complemented
5#14521233C212yesmaximal, complemented
6#14521232C211yesmaximal, characteristic, complemented
7#14521221C211yesmaximal, characteristic, complemented
8#14521220C212yesmaximal, complemented
9#14521064C212yesmaximal, complemented
10#101×D31C211yesmaximal, characteristic, complemented
11C31×#454C211yesmaximal, characteristic, complemented
12C31×2−1+4C2211yescharacteristic
13#11758567—22no—
14#11758567—22no—
15#11758567C2212yes—
16#41×D31—22no—
17#41×D31—22no—
18#41×D31C2212yes—
19#11758564—22no—
20#11758564C2211yescharacteristic
21C2×Q8×D31—22no—
22C2×Q8×D31C2211yescharacteristic
23C2×#92497C2211yescharacteristic
24C31×AGL(1,ℤ/8)C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758528C2212yescomplemented
27#11758521C2211yescharacteristic, complemented
28#11758513C2211yescharacteristic, complemented
29#11758509C2212yescomplemented
30#11758493—22no—
31#11758492—22no—
32#11758489—22no—
33#11758488—22no—
34#11758486C2212yescomplemented
35M16×D31C2212yes—
36#11758460—22no—
37#11758455—22no—
38#11758449C2211yescharacteristic
39#11758446C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#101C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758424C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758412C2212yescomplemented
46#11758393C2212yescomplemented
47#41×C31—22no—
48#41×C31—22no—
49#41×C31C2312yes—
50C2×Q8×C31—22no—
51C2×Q8×C31C2311yescharacteristic
52C2×D4×C31C2311yescharacteristic
53C2×C22⋊D31—21no—
54C22×Dic31—21no—
55#92499—22no—
56#92499—22no—
57#92499D412yes—
58Q8×D31—22no—
59Q8×D31—24no—
60Q8×D31—22no—
61Q8×D31—22no—
62Q8×D31—22no—
63Q8×D31—22no—
64Q8×D31D412yes—
65#92497—22no—
66#92497—24no—
67#92497—22no—
68#92497—24no—
69#92497—22no—
70#92497—22no—
71#92497—22no—
72#92497D412yes—
73D4×D31—22no—
74D4×D31—22no—
75D4×D31D412yes—
76#92495—24no—
77#92495—24no—
78#92495—22no—
79#92495—22no—
80#92495—22no—
81#92495C2312yes—
82C2×C4×D31—24no—
83C2×C4×D31—22no—
84C2×C4×D31D412yes—
85C2×C4×D31C2311yescharacteristic
86C2×Dic62—24no—
87C2×Dic62—22no—
88C2×Dic62D412yes—
89C2×Dic62C2311yescharacteristic
90C2×Dic62C2311yescharacteristic
91SD16×C31—22no—
92D8×C31—22no—
93M16×C31C2312yes—
94C22⋊C4×C31—22no—
95C42×C31C2311yescharacteristic
96C22⋊Dic31—22no—
97#92482—22no—
98#92480—22no—
99#92480—22no—
100#92479—22no—
101(C2×D31)⋊C4—21no—
102C4⋊Dic31—22no—
103Dic31⋊C4—22no—
104Dic31⋊C4—21no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107Dic124—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—21no—
112C22×D31—22no—
113C22×D31C2×D411yescharacteristic
114Q8×C31—22no—
115Q8×C31—22no—
116Q8×C31C2×D412yes—
117D4×C31—22no—
118D4×C31—22no—
119D4×C31—22no—
120D4×C31C2×D412yes—
121C2×C4×C31—24no—
122C2×C4×C31—22no—
123C2×C4×C31—21no—
124C2×C4×C31C2×D412yes—
125C2×C4×C31C2411yescharacteristic
126C22⋊D31—24no—
127C22⋊D31—22no—
128C22⋊D31—22no—
129C22⋊D31—22no—
130C22⋊D31—41no—
131C22⋊D31—41no—
132C2×Dic31—24no—
133C2×Dic31—22no—
134C2×Dic31—21no—
135C2×Dic31—21no—
136C2×Dic31—42no—
137C2×Dic31C2×D412yes—
138C2×Dic31C2×D411yescharacteristic
139D124—22no—
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×D31C2×D412yes—
150C4×D31C2×D412yes—
151Dic62—24no—
152Dic62—22no—
153Dic62—22no—
154Dic62—22no—
155Dic62—24no—
156Dic62—22no—
157Dic62C2×D412yes—
158C8×C31—22no—
159C31⋊C8—22no—
160#3020—311nomaximal, Sylow
161C22×C31—22no—
162C22×C31—42no—
163C22×C31—41no—
164C22×C31C22×D411yescharacteristic
165C2×D31—22no—
166C2×D31—41no—
167C2×D31—44no—
168C2×D31#12011yescharacteristic
169C2×D31#12011yescharacteristic
170C4×C31—22no—
171C4×C31—22no—
172C4×C31—41no—
173C4×C31C22×D412yes—
174Dic31—22no—
175Dic31—22no—
176Dic31—41no—
177Dic31—41no—
178Dic31#12012yes—
179C2×2−1+4—311no—
180#575—312no—
181#536—311no—
182#456—311no—
183#456—312no—
184#456—311no—
185#454—312no—
186#454—311no—
187#454D3111yescharacteristic, complemented
188#421—312no—
189C2×#101—311no—
190C2×C31—21no—
191C2×C31—42no—
192C2×C31—81no—
193C2×C31C2×#12011yescharacteristic
194D31—21no—
195D31—21no—
196D31—42no—
1972−1+4—312no—
1982−1+4—311no—
1992−1+4—622no—
2002−1+4—622no—
2012−1+4—622no—
2022−1+4C2×D3111yescharacteristic
203C2×#41—312no—
204C2×#41—311no—
205C2×#41—622no—
206C2×#41—622no—
207C22×Q8—311no—
208C22×Q8—622no—
209#137—312no—
210#137—312no—
211#137—622no—
212AGL(1,ℤ/8)—312no—
213AGL(1,ℤ/8)—622no—
214AGL(1,ℤ/8)C2×D3112yescomplemented
215#135—622no—
216#135—622no—
217C2×M16—312no—
218C4⋊Q8—311no—
219C4⋊Q8—311no—
220#124—311no—
221#124C2×D3111yescharacteristic, complemented
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#101—312no—
231#101C2×D3111yescharacteristic, complemented
232C31#302011yescharacteristic, 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—624no—
244#41—622no—
245#41—622no—
246#41—622no—
247#41—624no—
248#41—622no—
249#41—622no—
250#41—622no—
251#41—622no—
252#41C22×D3112yes—
253C2×Q8—22no—
254C2×Q8—312no—
255C2×Q8—311no—
256C2×Q8—312no—
257C2×Q8—311no—
258C2×Q8—622no—
259C2×Q8—624no—
260C2×Q8—622no—
261C2×Q8—622no—
262C2×Q8—622no—
263C2×Q8—622no—
264C2×Q8—624no—
265C2×Q8—622no—
266C2×Q8C22×D3111yescharacteristic
267C2×D4—312no—
268C2×D4—622no—
269C2×D4—622no—
270C2×D4—621no—
271C2×D4C22×D3111yescharacteristic
272C22×C4—311no—
273C22×C4—312no—
274C22×C4—622no—
275C22×C4—624no—
276C22×C4—621no—
277Q16—622no—
278Q16—622no—
279SD16—22no—
280SD16—622no—
281SD16—622no—
282SD16—622no—
283D8—22no—
284D8—622no—
285M16—312no—
286M16—622no—
287M16C22×D3112yes—
288C2×C8—622no—
289C4⋊C4—621no—
290C4⋊C4—622no—
291C4⋊C4—622no—
292C22⋊C4—22no—
293C22⋊C4—621no—
294C22⋊C4—622no—
295C42—311no—
296C42C22×D3111yescharacteristic
297C23—21no—
298C23—311no—
299C23—622no—
300Q8—22no—
301Q8—22no—
302Q8—312no—
303Q8—622no—
304Q8—624no—
305Q8—624no—
306Q8—622no—
307Q8—622no—
308Q8—622no—
309Q8D4×D3112yes—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—312no—
314D4—622no—
315D4—622no—
316D4—622no—
317D4—622no—
318D4—624no—
319D4—622no—
320D4—1241no—
321D4—1241no—
322D4D4×D3112yes—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—311no—
327C2×C4—312no—
328C2×C4—312no—
329C2×C4—312no—
330C2×C4—622no—
331C2×C4—622no—
332C2×C4—622no—
333C2×C4—622no—
334C2×C4—624no—
335C2×C4—624no—
336C2×C4—622no—
337C2×C4—624no—
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—42no—
348C22—41no—
349C22—311no—
350C22—311no—
351C22—622no—
352C22—1241no—
353C22—1244no—
354C22C2×D4×D3111yescharacteristic
355C4—22no—
356C4—22no—
357C4—41no—
358C4—312no—
359C4—622no—
360C4—622no—
361C4—1241no—
362C4—1241no—
363C4C2×D4×D3112yes—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—621no—
368C2—621no—
369C2—1242no—
370C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#454C2
socleC2×C31C2×#120
hypercenter#454D31
hypocenterC31#3020
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. 2−1+4×D31C2semidirect
2. #14521845C2semidirect
3. #14521619C2semidirect
4. #14521234C2semidirect
5. #14521233C2semidirect
6. #14521232C2semidirect
7. #14521221C2semidirect
8. #14521220C2semidirect
9. #14521064C2semidirect
10. #101×D31C2semidirect
11. C31×#454C2semidirect
24. C31×AGL(1,ℤ/8)C22semidirect
25. C31×#124C22semidirect
26. #11758528C22semidirect
27. #11758521C22semidirect
28. #11758513C22semidirect
29. #11758509C22semidirect
34. #11758486C22semidirect
39. #11758446C22semidirect
40. C31×#104C22semidirect
41. C31×#101C22semidirect
42. #11758425C22semidirect
43. #11758424C22semidirect
44. #11758413C22semidirect
45. #11758412C22semidirect
46. #11758393C22semidirect
187. #454D31semidirect
214. AGL(1,ℤ/8)C2×D31semidirect
221. #124C2×D31semidirect
228. #104C2×D31semidirect
231. #101C2×D31semidirect
232. C31#3020semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D312
#145218452
#145216192
#145212342
#145212332
#145212322
#145212212
#145212202
#145210642
#101×D312
C31×#4542
#302031