Finite Group Databasebeta

← #16855084

Subgroups — #16855084

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×#11758485C211yesmaximal, characteristic, complemented
3C2×#11758526C211yesmaximal, characteristic, complemented
4#14521618C211yesmaximal, characteristic, complemented
5#14521223C214yesmaximal, complemented
6#14521218C214yesmaximal, complemented
7#14521067C211yesmaximal, characteristic, complemented
8#14521031C211yesmaximal, characteristic, complemented
9C31×#421C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11#11758567—21no—
12#11758567C2212yescomplemented
13#11758566—21no—
14#11758566C2212yescomplemented
15#41×D31—22no—
16#41×D31—22no—
17#11758564—21no—
18#11758561—21no—
19#11758561—22no—
20C2×#92495—22no—
21C2×#92495C2211yescharacteristic
22C2×#92495C2211yescharacteristic
23C22×D124C2211yescharacteristic, complemented
24C2×M16×C31C2211yescharacteristic, complemented
25C31×#117C2211yescharacteristic, complemented
26#11758526—22no—
27#11758526C2214yescomplemented
28C2×#92481—21no—
29C2×#92479—21no—
30C2×#92474C2211yescharacteristic, complemented
31#11758485—22no—
32#11758485C2214yescomplemented
33C2×D248—21no—
34C2×#92470—21no—
35#11758472—22no—
36#11758459—22no—
37#11758452C2212yescomplemented
38#11758451C2212yescomplemented
39C31×#104C2214yescomplemented
40#11758412C2212yescomplemented
41#11758411C2212yescomplemented
42#11758393C2214yescomplemented
43C23×D31—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×C31D411yescharacteristic, complemented
51C22×C4×C31—21no—
52C22×C4×C31C2311yescharacteristic
53C2×C22⋊D31—24no—
54C2×C22⋊D31—21no—
55C2×C22⋊D31D412yes—
56#92499—22no—
57#92499—22no—
58Q8×D31—22no—
59Q8×D31—22no—
60#92497—24no—
61#92497—22no—
62#92497—22no—
63#92497—24no—
64D4×D31—24no—
65D4×D31—22no—
66D4×D31—24no—
67D4×D31—22no—
68#92495—24no—
69#92495—24no—
70#92495—21no—
71#92495—21no—
72#92495—21no—
73#92495—21no—
74#92495—21no—
75#92495—21no—
76#92495—22no—
77#92495—22no—
78#92495D412yescomplemented
79#92495C2312yes—
80C2×D124—22no—
81C2×D124—21no—
82C2×D124—21no—
83C2×D124—21no—
84C2×D124—21no—
85C2×D124—21no—
86C2×D124—24no—
87C2×D124D411yescharacteristic, complemented
88C2×D124C2312yes—
89C2×D124C2311yescharacteristic
90C2×C4×D31—24no—
91C2×C4×D31—21no—
92C2×C4×D31D412yes—
93C2×Dic62—22no—
94C2×Dic62D411yescharacteristic, complemented
95C2×Dic62C2311yescharacteristic
96M16×C31—21no—
97M16×C31C2312yes—
98C2×C8×C31—21no—
99C4⋊C4×C31—21no—
100C22⋊C4×C31—21no—
101C42×C31C2312yes—
102#92481—24no—
103#92479—24no—
104(C2×D31)⋊C4—24no—
105#92474—21no—
106#92474C2312yes—
107C2×C31⋊C8—21no—
108D248—24no—
109#92470—24no—
110C23×C31—21no—
111C23×C31C2×D411yescharacteristic
112C22×D31—24no—
113C22×D31—21no—
114C22×D31—22no—
115C22×D31—41no—
116C22×D31—41no—
117C22×D31—82no—
118C22×D31C2×D412yes—
119Q8×C31—21no—
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×C31C2×D411yescharacteristic
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2411yescharacteristic
135C22⋊D31—24no—
136C22⋊D31—24no—
137C22⋊D31—22no—
138C22⋊D31—22no—
139C22⋊D31—22no—
140C22⋊D31—22no—
141C2×Dic31—24no—
142C2×Dic31—21no—
143C2×Dic31C2×D412yes—
144D124—24no—
145D124—21no—
146D124—21no—
147D124—21no—
148D124—22no—
149D124—22no—
150D124—41no—
151D124—41no—
152D124—84no—
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—81no—
171C22×C31#12011yescharacteristic
172C22×C31#12011yescharacteristic
173C22×C31C22×D411yescharacteristic
174C2×D31—22no—
175C2×D31—22no—
176C2×D31—42no—
177C2×D31—44no—
178C2×D31—42no—
179C2×D31—81no—
180C2×D31—84no—
181C2×D31—82no—
182C2×D31—82no—
183C4×C31—22no—
184C4×C31—42no—
185C4×C31#12011yescharacteristic
186C4×C31#12011yescharacteristic
187C4×C31C22×D411yescharacteristic
188C4×C31C22×D411yescharacteristic
189Dic31—22no—
190Dic31—22no—
191#585—311no—
192C2×AGL(1,ℤ/8)—311no—
193C2×AGL(1,ℤ/8)—311no—
194#535—311no—
195#454—314no—
196#453—314no—
197#421—311no—
198#421D3111yescharacteristic, complemented
199#413—311no—
200C2×C31—21no—
201C2×C31—21no—
202C2×C31—21no—
203C2×C31—42no—
204C2×C31C2×#12011yescharacteristic
205D31—42no—
206D31—42no—
207D31—82no—
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×D4—311no—
222AGL(1,ℤ/8)—314no—
223AGL(1,ℤ/8)—314no—
224AGL(1,ℤ/8)—622no—
225AGL(1,ℤ/8)—622no—
226C2×SD16—621no—
227C2×SD16—621no—
228C2×D8—621no—
229C2×D8—621no—
230C2×M16—311no—
231C2×M16C2×D3111yescharacteristic, complemented
232#127—312no—
233#124—312no—
234#121—622no—
235#120—622no—
236#117C2×D3111yescharacteristic, complemented
237#104—314no—
238#104C2×D3114yescomplemented
239#101—312no—
240#100—312no—
241C31#302111yescharacteristic, complemented, Sylow
242C24—621no—
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#41D12412yescomplemented
264#41C22×D3112yes—
265C2×Q8—311no—
266C2×Q8—311no—
267C2×Q8—622no—
268C2×Q8—622no—
269C2×Q8—622no—
270C2×Q8D12411yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—312no—
273C2×D4—311no—
274C2×D4—311no—
275C2×D4—312no—
276C2×D4—622no—
277C2×D4—624no—
278C2×D4—624no—
279C2×D4—624no—
280C2×D4—622no—
281C2×D4—624no—
282C2×D4—621no—
283C2×D4—622no—
284C2×D4—621no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—621no—
288C2×D4—621no—
289C2×D4D12411yescharacteristic, complemented
290C22×C4—21no—
291C22×C4—312no—
292C22×C4—624no—
293C22×C4—621no—
294C22×C4C22×D3111yescharacteristic
295SD16—624no—
296SD16—624no—
297D8—624no—
298D8—624no—
299M16—21no—
300M16—312no—
301M16—621no—
302M16C22×D3112yes—
303C2×C8—21no—
304C2×C8—621no—
305C4⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—624no—
308C42C22×D3112yes—
309C23—21no—
310C23—312no—
311C23—624no—
312C23—621no—
313C23—622no—
314C23—1241no—
315C23—1241no—
316C23—2482no—
317C23D4×D3111yescharacteristic
318Q8—21no—
319Q8—312no—
320Q8—624no—
321Q8—621no—
322Q8—621no—
323Q8—621no—
324Q8C2×D12412yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—312no—
329D4—624no—
330D4—624no—
331D4—624no—
332D4—622no—
333D4—622no—
334D4—622no—
335D4—622no—
336D4—621no—
337D4—621no—
338D4—622no—
339D4—621no—
340D4—622no—
341D4—1241no—
342D4—1241no—
343D4—2484no—
344D4C2×D12412yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—312no—
350C2×C4—624no—
351C2×C4—624no—
352C2×C4—624no—
353C2×C4—621no—
354C2×C4—622no—
355C2×C4—622no—
356C2×C4—622no—
357C2×C4—622no—
358C2×C4C2×D12411yescharacteristic
359C2×C4C2×D12411yescharacteristic
360C2×C4D4×D3111yescharacteristic
361C2×C4D4×D3111yescharacteristic
362C2×C4D4×D3111yescharacteristic
363C2×C4C23×D3111yescharacteristic
364C8—22no—
365C8—622no—
366C22—22no—
367C22—41no—
368C22—81no—
369C22—622no—
370C22—622no—
371C22—1242no—
372C22—1244no—
373C22—1242no—
374C22—2481no—
375C22—2482no—
376C22—2482no—
377C22—2484no—
378C22#1175845911yescharacteristic
379C22#1175845911yescharacteristic
380C22C2×D4×D3111yescharacteristic
381C4—22no—
382C4—42no—
383C4—622no—
384C4—622no—
385C4#1175845911yescharacteristic
386C4#1175845911yescharacteristic
387C4C22×D12411yescharacteristic
388C4C2×D4×D3111yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—1242no—
394C2—1242no—
395C2—2482no—
396C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#421C2
socleC2×C31C2×#120
hypercenter#421D31
hypocenterC31#3021
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. C2×#11758485C2semidirect
3. C2×#11758526C2semidirect
4. #14521618C2semidirect
5. #14521223C2semidirect
6. #14521218C2semidirect
7. #14521067C2semidirect
8. #14521031C2semidirect
9. C31×#421C2semidirect
12. #11758567C22semidirect
14. #11758566C22semidirect
23. C22×D124C22semidirect
24. C2×M16×C31C22semidirect
25. C31×#117C22semidirect
27. #11758526C22semidirect
30. C2×#92474C22semidirect
32. #11758485C22semidirect
37. #11758452C22semidirect
38. #11758451C22semidirect
39. C31×#104C22semidirect
40. #11758412C22semidirect
41. #11758411C22semidirect
42. #11758393C22semidirect
46. #41×C31D4semidirect
48. C2×Q8×C31D4semidirect
50. C2×D4×C31D4semidirect
78. #92495D4semidirect
87. C2×D124D4semidirect
94. C2×Dic62D4semidirect
198. #421D31semidirect
231. C2×M16C2×D31semidirect
236. #117C2×D31semidirect
238. #104C2×D31semidirect
241. C31#3021semidirect
263. #41D124semidirect
270. C2×Q8D124semidirect
289. C2×D4D124semidirect

Maximal subgroups

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

subgroupindex
#145218852
C2×#117584852
C2×#117585262
#145216182
#145212232
#145212182
#145210672
#145210312
C31×#4212
#302131