Finite Group Databasebeta

← #15682774

Subgroups — #15682774

Order 2944.

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#13093254C211yesmaximal, characteristic
2#137×D23C212yesmaximal, complemented
3#13092992C211yesmaximal, characteristic, complemented
4#13092602C211yesmaximal, characteristic, complemented
5#13092600C211yesmaximal, characteristic, complemented
6#13092599C211yesmaximal, characteristic, complemented
7#13092596C212yesmaximal, complemented
8#13092593C212yesmaximal, complemented
9#13092432C212yesmaximal, complemented
10#13092400C211yesmaximal, characteristic, complemented
11C23×#456C211yesmaximal, characteristic, complemented
12C23×2−1+4C2211yescharacteristic
13#10625160—22no—
14#10625160—22no—
15#10625160C2212yes—
16#41×D23—22no—
17#41×D23—22no—
18#41×D23C2212yes—
19#10625158—22no—
20#10625158C2211yescharacteristic
21C2×#67292—22no—
22C2×#67292C2211yescharacteristic
23C2×Q8×D23C2211yescharacteristic, complemented
24C23×#137C2212yescomplemented
25C23×C4⋊Q8C2211yescharacteristic, complemented
26#10625122C2212yescomplemented
27#10625118C2211yescharacteristic, complemented
28#10625115C2211yescharacteristic, complemented
29#10625113C2212yescomplemented
30#10625089—22no—
31Q16×D23—22no—
32#10625086—22no—
33SD16×D23—22no—
34#10625080C2212yescomplemented
35M16×D23C2212yescomplemented
36#10625067—22no—
37#10625065—22no—
38#10625046C2211yescharacteristic, complemented
39#10625043C2211yescharacteristic, complemented
40C23×#104C2212yescomplemented
41C23×#101C2211yescharacteristic, complemented
42#10625019C2212yescomplemented
43#10625018C2211yescharacteristic, complemented
44#10625007C2212yescomplemented
45#10625005C2212yescomplemented
46#10624987C2212yescomplemented
47#41×C23—22no—
48#41×C23—22no—
49#41×C23C2312yes—
50C2×Q8×C23—22no—
51C2×Q8×C23C2311yescharacteristic
52C2×Q8×C23C2311yescharacteristic
53#67292—22no—
54#67292—24no—
55#67292—22no—
56#67292—22no—
57#67292—22no—
58#67292—22no—
59#67292D412yes—
60Q8×D23—22no—
61Q8×D23—22no—
62Q8×D23—22no—
63Q8×D23—22no—
64Q8×D23D412yes—
65#67290—22no—
66#67290—22no—
67#67290D412yes—
68D4×D23—24no—
69D4×D23—22no—
70D4×D23—24no—
71D4×D23—22no—
72D4×D23—22no—
73D4×D23D412yes—
74#67288—24no—
75#67288—24no—
76#67288—22no—
77#67288—22no—
78#67288—22no—
79#67288C2312yes—
80C2×D92—24no—
81C2×D92—22no—
82C2×D92D412yes—
83C2×D92C2311yescharacteristic
84C2×C4×D23—24no—
85C2×C4×D23—22no—
86C2×C4×D23—21no—
87C2×C4×D23D412yes—
88C2×C4×D23C2311yescharacteristic
89C2×Dic46—21no—
90C2×Dic46C2311yescharacteristic
91Q16×C23—22no—
92SD16×C23—22no—
93M16×C23C2312yes—
94C4⋊C4×C23—22no—
95C42×C23C2311yescharacteristic
96#67275—22no—
97#67275—22no—
98#67274—22no—
99#67273—22no—
100(C2×D23)⋊C4—21no—
101(C2×D23)⋊C4—22no—
102(C2×D23)⋊C4—22no—
103Dic23⋊C4—22no—
104Dic23⋊C4—21no—
105C4×Dic23C2311yescharacteristic
106#67267C2312yes—
107Dic92—22no—
108#67263—22no—
109#67262—22no—
110C8×D23—22no—
111C22×D23—24no—
112C22×D23—22no—
113C22×D23C2×D412yes—
114C22×D23C2×D411yescharacteristic
115Q8×C23—22no—
116Q8×C23—22no—
117Q8×C23—22no—
118Q8×C23C2×D412yes—
119D4×C23—22no—
120D4×C23—22no—
121D4×C23C2×D412yes—
122C2×C4×C23—24no—
123C2×C4×C23—22no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23C2×D412yes—
127C2×C4×C23C2411yescharacteristic
128C22⋊D23—24no—
129C22⋊D23—22no—
130C22⋊D23—22no—
131C22⋊D23—22no—
132C2×Dic23—22no—
133C2×Dic23—21no—
134C2×Dic23—21no—
135C2×Dic23C2×D411yescharacteristic
136D92—24no—
137D92—22no—
138D92—22no—
139D92—24no—
140D92—22no—
141D92C2×D412yes—
142C4×D23—24no—
143C4×D23—24no—
144C4×D23—22no—
145C4×D23—22no—
146C4×D23—22no—
147C4×D23—22no—
148C4×D23—22no—
149C4×D23—42no—
150C4×D23C2×D412yes—
151C4×D23C2×D412yes—
152Dic46—22no—
153Dic46—22no—
154Dic46—22no—
155Dic46—41no—
156Dic46—41no—
157Dic46C2×D412yes—
158C8×C23—22no—
159C23⋊C8—22no—
160#3022—231nomaximal, Sylow
161C22×C23—22no—
162C22×C23C22×D411yescharacteristic
163C2×D23—22no—
164C2×D23—22no—
165C2×D23—41no—
166C2×D23—42no—
167C2×D23—82no—
168C2×D23—84no—
169C2×D23#12012yes—
170C4×C23—22no—
171C4×C23—22no—
172C4×C23—41no—
173C4×C23—41no—
174C4×C23C22×D412yes—
175Dic23—22no—
176Dic23—41no—
177Dic23—41no—
178Dic23#12011yescharacteristic
179Dic23#12011yescharacteristic
180#585—231no—
181C2×#137—232no—
182#536—231no—
183#456—231no—
184#456—231no—
185#456—231no—
186#456D2311yescharacteristic, complemented
187#455—232no—
188#455—232no—
189#421—232no—
190#413—231no—
191C2×C23—21no—
192C2×C23—42no—
193C2×C23C2×#12011yescharacteristic
194D23—22no—
195D23—42no—
196D23—42no—
1972−1+4—231no—
1982−1+4—462no—
1992−1+4C2×D2311yescharacteristic
200H(5,𝔽2)—232no—
201H(5,𝔽2)—462no—
202H(5,𝔽2)—462no—
203C2×#41—231no—
204C2×#41—232no—
205C2×#41—462no—
206C2×#41—462no—
207C2×#41—462no—
208C22×Q8—231no—
209#137—232no—
210#137—232no—
211#137—232no—
212#137—462no—
213#137—462no—
214#137C2×D2312yescomplemented
215C2×Q16—462no—
216C2×SD16—462no—
217C2×M16—232no—
218C4⋊Q8—231no—
219C4⋊Q8C2×D2311yescharacteristic, complemented
220#124—231no—
221#124—231no—
222#123—462no—
223C22⋊Q8—462no—
224#117—231no—
225#104—232no—
226#104—232no—
227#104—232no—
228#104C2×D2312yescomplemented
229#101—231no—
230#101C2×D2311yescharacteristic, complemented
231#100—232no—
232C23#302211yescharacteristic, complemented, Sylow
233#41—22no—
234#41—22no—
235#41—232no—
236#41—232no—
237#41—232no—
238#41—462no—
239#41—462no—
240#41—464no—
241#41—464no—
242#41—462no—
243#41—462no—
244#41—462no—
245#41—462no—
246#41—462no—
247#41—462no—
248#41—464no—
249#41—462no—
250#41—462no—
251#41C22×D2312yes—
252C2×Q8—22no—
253C2×Q8—232no—
254C2×Q8—231no—
255C2×Q8—461no—
256C2×Q8—462no—
257C2×Q8—462no—
258C2×Q8—462no—
259C2×Q8—462no—
260C2×Q8C22×D2311yescharacteristic
261C2×Q8C22×D2311yescharacteristic
262C2×D4—232no—
263C2×D4—232no—
264C2×D4—231no—
265C2×D4—462no—
266C2×D4—464no—
267C2×D4—462no—
268C2×D4—464no—
269C2×D4—462no—
270C2×D4—464no—
271C2×D4—462no—
272C22×C4—231no—
273C22×C4—232no—
274C22×C4—462no—
275C22×C4—464no—
276C22×C4—461no—
277Q16—22no—
278Q16—462no—
279Q16—462no—
280Q16—462no—
281SD16—22no—
282SD16—462no—
283SD16—462no—
284SD16—462no—
285M16—232no—
286M16—462no—
287M16C22×D2312yes—
288C2×C8—462no—
289C4⋊C4—22no—
290C4⋊C4—461no—
291C4⋊C4—462no—
292C22⋊C4—461no—
293C22⋊C4—462no—
294C22⋊C4—462no—
295C42—231no—
296C42C22×D2311yescharacteristic
297C23—231no—
298C23—232no—
299C23—462no—
300C23—464no—
301Q8—22no—
302Q8—22no—
303Q8—22no—
304Q8—232no—
305Q8—462no—
306Q8—462no—
307Q8—462no—
308Q8—921no—
309Q8—921no—
310Q8D4×D2312yes—
311D4—22no—
312D4—22no—
313D4—232no—
314D4—464no—
315D4—462no—
316D4—464no—
317D4—462no—
318D4—462no—
319D4—462no—
320D4—464no—
321D4—462no—
322D4—462no—
323D4D4×D2312yes—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—231no—
329C2×C4—232no—
330C2×C4—232no—
331C2×C4—462no—
332C2×C4—462no—
333C2×C4—462no—
334C2×C4—462no—
335C2×C4—464no—
336C2×C4—464no—
337C2×C4—462no—
338C2×C4—462no—
339C2×C4—461no—
340C2×C4—461no—
341C2×C4—922no—
342C2×C4D4×D2312yes—
343C2×C4C23×D2311yescharacteristic
344C8—22no—
345C8—462no—
346C22—22no—
347C22—232no—
348C22—462no—
349C22—462no—
350C22—921no—
351C22—922no—
352C22—1842no—
353C22—1844no—
354C22C2×D4×D2311yescharacteristic
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—231no—
360C4—231no—
361C4—462no—
362C4—921no—
363C4—921no—
364C4C2×D4×D2312yes—
365C2—21no—
366C2—42no—
367C2—462no—
368C2—922no—
369C2—922no—
370C2#120×D2311yescharacteristic
subgroupG/H
centerC2#120×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#456C2
socleC2×C23C2×#120
prime radical#13093254C2
hypercenter#456D23
hypocenterC23#3022
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D23C2semidirect
3. #13092992C2semidirect
4. #13092602C2semidirect
5. #13092600C2semidirect
6. #13092599C2semidirect
7. #13092596C2semidirect
8. #13092593C2semidirect
9. #13092432C2semidirect
10. #13092400C2semidirect
11. C23×#456C2semidirect
23. C2×Q8×D23C22semidirect
24. C23×#137C22semidirect
25. C23×C4⋊Q8C22semidirect
26. #10625122C22semidirect
27. #10625118C22semidirect
28. #10625115C22semidirect
29. #10625113C22semidirect
34. #10625080C22semidirect
35. M16×D23C22semidirect
38. #10625046C22semidirect
39. #10625043C22semidirect
40. C23×#104C22semidirect
41. C23×#101C22semidirect
42. #10625019C22semidirect
43. #10625018C22semidirect
44. #10625007C22semidirect
45. #10625005C22semidirect
46. #10624987C22semidirect
186. #456D23semidirect
214. #137C2×D23semidirect
219. C4⋊Q8C2×D23semidirect
228. #104C2×D23semidirect
230. #101C2×D23semidirect
232. C23#3022semidirect

Maximal subgroups

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

subgroupindex
#130932542
#137×D232
#130929922
#130926022
#130926002
#130925992
#130925962
#130925932
#130924322
#130924002
C23×#4562
#302223