Finite Group Databasebeta

← #16855080

Subgroups — #16855080

Order 3968.

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

normal subgroups only

This group has 736 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, complemented
2C2×#11758509C211yesmaximal, characteristic, complemented
3#14521616C211yesmaximal, characteristic, complemented
4#14521220C214yesmaximal, complemented
5#14521215C214yesmaximal, complemented
6#14521066C211yesmaximal, characteristic, complemented
7#14521065C211yesmaximal, characteristic, complemented
8#14521031C211yesmaximal, characteristic, complemented
9C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic, complemented
12#11758567—21no—
13#11758567C2212yescomplemented
14#11758566—21no—
15#11758566C2212yescomplemented
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×AGL(1,ℤ/8)—22no—
25C31×AGL(1,ℤ/8)C2214yescomplemented
26C2×SD16×C31—21no—
27C2×D8×C31—21no—
28C2×M16×C31C2211yescharacteristic, complemented
29#11758530—22no—
30#11758517C2212yescomplemented
31#11758515—22no—
32#11758513C2212yescomplemented
33C2×#92480—21no—
34#11758509—22no—
35#11758509C2214yescomplemented
36C2×#92479—21no—
37#11758503C2211yescharacteristic, complemented
38C2×#92474C2211yescharacteristic, complemented
39#11758425C2214yescomplemented
40#11758413C2214yescomplemented
41#11758412C2212yescomplemented
42#11758411C2212yescomplemented
43C24×C31—21no—
44#41×C31—21no—
45#41×C31—21no—
46#41×C31D412yescomplemented
47#41×C31C2312yes—
48C2×Q8×C31D411yescharacteristic, complemented
49C2×D4×C31—21no—
50C2×D4×C31—21no—
51C2×D4×C31—21no—
52C2×D4×C31—21no—
53C2×D4×C31—21no—
54C2×D4×C31—21no—
55C2×D4×C31D411yescharacteristic, complemented
56C2×D4×C31C2312yes—
57C22×C4×C31—21no—
58C22×C4×C31C2311yescharacteristic
59C2×C22⋊D31—24no—
60C2×C22⋊D31—21no—
61C2×C22⋊D31—24no—
62C2×C22⋊D31D412yes—
63#92499—22no—
64#92499—22no—
65Q8×D31—22no—
66Q8×D31—22no—
67#92497—24no—
68#92497—22no—
69#92497—22no—
70#92497—24no—
71D4×D31—24no—
72D4×D31—22no—
73D4×D31—24no—
74D4×D31—22no—
75#92495—24no—
76#92495—24no—
77#92495—21no—
78#92495—21no—
79#92495—21no—
80#92495—21no—
81#92495—21no—
82#92495—21no—
83#92495—22no—
84#92495—22no—
85#92495D412yescomplemented
86#92495C2312yes—
87C2×D124—22no—
88C2×D124D411yescharacteristic, complemented
89C2×D124C2311yescharacteristic
90C2×C4×D31—24no—
91C2×C4×D31—21no—
92C2×C4×D31D412yes—
93C2×Dic62—22no—
94C2×Dic62D411yescharacteristic, complemented
95C2×Dic62C2311yescharacteristic
96SD16×C31—24no—
97D8×C31—24no—
98M16×C31—21no—
99M16×C31C2312yes—
100C2×C8×C31—21no—
101C22⋊Dic31—21no—
102C22⋊Dic31—24no—
103#92480—24no—
104#92479—24no—
105C4⋊Dic31—21no—
106C4×Dic31C2312yes—
107#92474—21no—
108#92474C2312yes—
109C2×C31⋊C8—21no—
110C23×C31—21no—
111C23×C31—22no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—82no—
115C23×C31C2×D411yescharacteristic
116C22×D31—24no—
117C22×D31—21no—
118C22×D31C2×D412yes—
119Q8×C31—21no—
120Q8×C31C2×D412yes—
121D4×C31—22no—
122D4×C31—22no—
123D4×C31—21no—
124D4×C31—22no—
125D4×C31—22no—
126D4×C31—41no—
127D4×C31—41no—
128D4×C31C2×D412yes—
129C2×C4×C31—21no—
130C2×C4×C31—22no—
131C2×C4×C31—22no—
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2×D411yescharacteristic
135C2×C4×C31C2×D411yescharacteristic
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2411yescharacteristic
138C22⋊D31—24no—
139C22⋊D31—24no—
140C22⋊D31—22no—
141C22⋊D31—22no—
142C22⋊D31—22no—
143C22⋊D31—22no—
144C22⋊D31—84no—
145C2×Dic31—24no—
146C2×Dic31—21no—
147C2×Dic31—22no—
148C2×Dic31C2×D412yes—
149D124—24no—
150D124—21no—
151D124—21no—
152D124—21no—
153D124C2×D412yes—
154C4×D31—24no—
155C4×D31—24no—
156C4×D31—22no—
157C4×D31—22no—
158C4×D31—22no—
159C4×D31—22no—
160Dic62—24no—
161Dic62—21no—
162Dic62—21no—
163Dic62—21no—
164Dic62C2×D412yes—
165C8×C31—22no—
166C31⋊C8—22no—
167#3021—311nomaximal, Sylow
168C22×C31—22no—
169C22×C31—41no—
170C22×C31—44no—
171C22×C31—42no—
172C22×C31—81no—
173C22×C31—82no—
174C22×C31—82no—
175C22×C31#12011yescharacteristic
176C22×C31#12011yescharacteristic
177C22×C31C22×D411yescharacteristic
178C2×D31—22no—
179C2×D31—22no—
180C2×D31—42no—
181C2×D31—81no—
182C2×D31—84no—
183C4×C31—22no—
184C4×C31#12011yescharacteristic
185C4×C31#12011yescharacteristic
186C4×C31C22×D411yescharacteristic
187C4×C31C22×D411yescharacteristic
188Dic31—22no—
189Dic31—22no—
190Dic31—42no—
191#585—311no—
192C2×AGL(1,ℤ/8)—311no—
193C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
194#535—311no—
195#454—314no—
196#453—314no—
197#421—311no—
198#421—311no—
199#413—311no—
200C2×C31—21no—
201C2×C31—21no—
202C2×C31—21no—
203C2×C31—42no—
204C2×C31—82no—
205C2×C31C2×#12011yescharacteristic
206D31—42no—
207D31—42no—
2082−1+4—312no—
2092−1+4—621no—
2102−1+4—621no—
211H(5,𝔽2)—312no—
212H(5,𝔽2)—622no—
213H(5,𝔽2)—621no—
214H(5,𝔽2)—621no—
215C2×#41—311no—
216C2×#41—311no—
217C2×#41—622no—
218C2×#41—622no—
219C2×#41—622no—
220C2×#41C2×D3111yescharacteristic
221C22×D4C2×D3111yescharacteristic, complemented
222AGL(1,ℤ/8)—22no—
223AGL(1,ℤ/8)—314no—
224AGL(1,ℤ/8)—622no—
225AGL(1,ℤ/8)C2×D3114yescomplemented
226C2×SD16—21no—
227C2×SD16—621no—
228C2×D8—21no—
229C2×D8—621no—
230C2×M16—311no—
231C2×M16C2×D3111yescharacteristic, complemented
232#127—312no—
233#124—312no—
234#121—622no—
235#120—622no—
236#117—311no—
237#104—314no—
238#104—314no—
239#101—312no—
240#100—312no—
241C31#302111yescharacteristic, complemented, Sylow
242C24—21no—
243#41—21no—
244#41—21no—
245#41—312no—
246#41—312no—
247#41—622no—
248#41—622no—
249#41—624no—
250#41—624no—
251#41—624no—
252#41—624no—
253#41—622no—
254#41—622no—
255#41—622no—
256#41—622no—
257#41—621no—
258#41—621no—
259#41—621no—
260#41—621no—
261#41—621no—
262#41—621no—
263#41C22⋊D3112yescomplemented
264#41C22×D3112yes—
265C2×Q8—311no—
266C2×Q8—311no—
267C2×Q8—622no—
268C2×Q8—622no—
269C2×Q8—622no—
270C2×Q8C22⋊D3111yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—312no—
278C2×D4—311no—
279C2×D4—311no—
280C2×D4—622no—
281C2×D4—624no—
282C2×D4—624no—
283C2×D4—622no—
284C2×D4—624no—
285C2×D4—621no—
286C2×D4—622no—
287C2×D4—624no—
288C2×D4C22⋊D3111yescharacteristic, complemented
289C2×D4C22×D3112yes—
290C22×C4—21no—
291C22×C4—312no—
292C22×C4—624no—
293C22×C4—621no—
294C22×C4C22×D3111yescharacteristic
295SD16—24no—
296SD16—624no—
297D8—24no—
298D8—624no—
299M16—21no—
300M16—312no—
301M16—621no—
302M16C22×D3112yes—
303C2×C8—21no—
304C2×C8—621no—
305C4⋊C4—621no—
306C22⋊C4—624no—
307C22⋊C4—621no—
308C42—312no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—312no—
315C23—624no—
316C23—621no—
317C23D4×D3111yescharacteristic
318Q8—21no—
319Q8—312no—
320Q8—624no—
321Q8—621no—
322Q8—621no—
323Q8—621no—
324Q8C2×C22⋊D3112yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—22no—
329D4—22no—
330D4—41no—
331D4—41no—
332D4—312no—
333D4—624no—
334D4—624no—
335D4—624no—
336D4—622no—
337D4—622no—
338D4—622no—
339D4—622no—
340D4—621no—
341D4—621no—
342D4—621no—
343D4—2484no—
344D4C2×C22⋊D3112yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—312no—
349C2×C4—624no—
350C2×C4—624no—
351C2×C4—624no—
352C2×C4—621no—
353C2×C4—622no—
354C2×C4—622no—
355C2×C4—622no—
356C2×C4—622no—
357C2×C4—622no—
358C2×C4D4×D3111yescharacteristic
359C2×C4D4×D3111yescharacteristic
360C2×C4D4×D3111yescharacteristic
361C2×C4C2×C22⋊D3111yescharacteristic
362C2×C4C2×C22⋊D3111yescharacteristic
363C2×C4C23×D3111yescharacteristic
364C8—22no—
365C8—622no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—622no—
374C22—622no—
375C22—1242no—
376C22—2481no—
377C22—2484no—
378C22#1175851411yescharacteristic
379C22#1175851411yescharacteristic
380C22C2×D4×D3111yescharacteristic
381C4—22no—
382C4—622no—
383C4—622no—
384C4—1242no—
385C4#1175851411yescharacteristic
386C4#1175851411yescharacteristic
387C4C2×D4×D3111yescharacteristic
388C4C22×C22⋊D3111yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—1242no—
395C2—1242no—
396C2C2×#1175851411yescharacteristic
subgroupG/H
centerC2C2×#11758514
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC2×C31C2×#120
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#3021
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. C2×#11758509C2semidirect
3. #14521616C2semidirect
4. #14521220C2semidirect
5. #14521215C2semidirect
6. #14521066C2semidirect
7. #14521065C2semidirect
8. #14521031C2semidirect
9. C2×C31×AGL(1,ℤ/8)C2semidirect
11. C22×D4×C31C22semidirect
13. #11758567C22semidirect
15. #11758566C22semidirect
25. C31×AGL(1,ℤ/8)C22semidirect
28. C2×M16×C31C22semidirect
30. #11758517C22semidirect
32. #11758513C22semidirect
35. #11758509C22semidirect
37. #11758503C22semidirect
38. C2×#92474C22semidirect
39. #11758425C22semidirect
40. #11758413C22semidirect
41. #11758412C22semidirect
42. #11758411C22semidirect
46. #41×C31D4semidirect
48. C2×Q8×C31D4semidirect
55. C2×D4×C31D4semidirect
85. #92495D4semidirect
88. C2×D124D4semidirect
94. C2×Dic62D4semidirect
193. C2×AGL(1,ℤ/8)D31semidirect
221. C22×D4C2×D31semidirect
225. AGL(1,ℤ/8)C2×D31semidirect
231. C2×M16C2×D31semidirect
241. C31#3021semidirect
263. #41C22⋊D31semidirect
270. C2×Q8C22⋊D31semidirect
288. C2×D4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218852
C2×#117585092
#145216162
#145212202
#145212152
#145210662
#145210652
#145210312
C2×C31×AGL(1,ℤ/8)2
#302131