Finite Group Databasebeta

← #15682756

Subgroups — #15682756

Order 2944.

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

normal subgroups only

This group has 762 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
1C2×#10625160C211yesmaximal, characteristic, complemented
2#13093217C211yesmaximal, characteristic, complemented
3#13093215C211yesmaximal, characteristic, complemented
4#13092986C211yesmaximal, characteristic, complemented
5#13092596C214yesmaximal, complemented
6#13092586C214yesmaximal, complemented
7#13092435C211yesmaximal, characteristic, complemented
8C2×#10625005C211yesmaximal, characteristic, complemented
9C23×#421C211yesmaximal, characteristic, complemented
10C2×#41×C23C2211yescharacteristic
11#10625160—22no—
12#10625160—22no—
13#10625160—22no—
14#10625160C2214yes—
15C2×#67292—21no—
16C2×D4×D23—22no—
17C2×D4×D23—21no—
18C2×#67288—21no—
19C2×#67288C2211yescharacteristic
20C2×#67288C2211yescharacteristic, complemented
21C22×D92—21no—
22C22×D92C2211yescharacteristic
23C2×M16×C23C2211yescharacteristic, complemented
24C23×#117C2211yescharacteristic, complemented
25#10625122C2212yescomplemented
26#10625121—22no—
27#10625120C2212yescomplemented
28#10625113—21no—
29#10625103—21no—
30C2×#67267C2211yescharacteristic, complemented
31#10625080—21no—
32#10625080C2212yescomplemented
33#10625079—21no—
34#10625079C2212yescomplemented
35#10625075—22no—
36#10625066—22no—
37#10625053—22no—
38#10625046C2212yescomplemented
39#10625045C2212yescomplemented
40C23×#104C2214yescomplemented
41#10625005C2214yescomplemented
42#10624987C2214yescomplemented
43C23×D23—22no—
44C23×D23D412yes—
45#41×C23—21no—
46#41×C23—21no—
47#41×C23D412yescomplemented
48#41×C23C2312yes—
49C2×Q8×C23D411yescharacteristic, complemented
50C2×D4×C23—21no—
51C2×D4×C23D411yescharacteristic, complemented
52C22×C4×C23—21no—
53C22×C4×C23C2311yescharacteristic
54C2×C22⋊D23—22no—
55C2×C22⋊D23—21no—
56C2×C22⋊D23—21no—
57#67292—24no—
58#67292—24no—
59D4×D23—28no—
60D4×D23—24no—
61D4×D23—28no—
62D4×D23—24no—
63#67288—24no—
64#67288—21no—
65#67288—21no—
66#67288—21no—
67#67288—21no—
68#67288—21no—
69#67288—21no—
70#67288—24no—
71#67288D412yescomplemented
72#67288C2312yes—
73C2×D92—22no—
74C2×D92—24no—
75C2×D92—21no—
76C2×D92—21no—
77C2×D92—21no—
78C2×D92—21no—
79C2×D92—24no—
80C2×D92—24no—
81C2×D92D412yes—
82C2×D92D411yescharacteristic, complemented
83C2×D92C2312yes—
84C2×D92C2311yescharacteristic, complemented
85C2×C4×D23—22no—
86C2×C4×D23—21no—
87C2×C4×D23—21no—
88C2×Dic46D411yescharacteristic, complemented
89C2×Dic46C2311yescharacteristic, complemented
90M16×C23—21no—
91M16×C23C2312yescomplemented
92C2×C8×C23—21no—
93C4⋊C4×C23—21no—
94C22⋊C4×C23—21no—
95C42×C23C2312yescomplemented
96#67275—22no—
97#67274—22no—
98#67273—22no—
99#67272—22no—
100(C2×D23)⋊C4—24no—
101#67267—21no—
102#67267C2312yescomplemented
103C2×C23⋊C8—21no—
104Dic92—22no—
105D184—22no—
106#67263—22no—
107#67263—22no—
108C23×C23—21no—
109C23×C23C2×D411yescharacteristic
110C22×D23—22no—
111C22×D23—21no—
112C22×D23—28no—
113C22×D23—22no—
114C22×D23—21no—
115C22×D23—82no—
116C22×D23—84no—
117C22×D23C2×D414yes—
118Q8×C23—21no—
119Q8×C23C2×D412yes—
120D4×C23—22no—
121D4×C23—22no—
122D4×C23—21no—
123D4×C23C2×D412yes—
124C2×C4×C23—21no—
125C2×C4×C23—22no—
126C2×C4×C23—22no—
127C2×C4×C23—22no—
128C2×C4×C23C2×D411yescharacteristic
129C2×C4×C23C2×D411yescharacteristic
130C2×C4×C23C2×D411yescharacteristic
131C2×C4×C23C2×D411yescharacteristic
132C2×C4×C23C2×D411yescharacteristic
133C2×C4×C23C2411yescharacteristic
134C22⋊D23—28no—
135C22⋊D23—22no—
136C22⋊D23—22no—
137C22⋊D23—41no—
138C22⋊D23—41no—
139C2×Dic23—21no—
140C2×Dic23—21no—
141D92—28no—
142D92—22no—
143D92—21no—
144D92—22no—
145D92—22no—
146D92—22no—
147D92—21no—
148D92—21no—
149D92—84no—
150D92C2×D412yes—
151C4×D23—28no—
152C4×D23—22no—
153C4×D23—22no—
154C4×D23—41no—
155C4×D23—41no—
156Dic46—21no—
157Dic46—21no—
158Dic46—21no—
159Dic46C2×D412yes—
160C8×C23—22no—
161C23⋊C8—22no—
162#3019—231nomaximal, Sylow
163C22×C23—22no—
164C22×C23—41no—
165C22×C23—42no—
166C22×C23#12011yescharacteristic
167C22×C23#12011yescharacteristic
168C22×C23C22×D411yescharacteristic
169C2×D23—24no—
170C2×D23—22no—
171C2×D23—44no—
172C2×D23—42no—
173C2×D23—44no—
174C2×D23—42no—
175C2×D23—41no—
176C2×D23—82no—
177C2×D23—88no—
178C4×C23—22no—
179C4×C23—42no—
180C4×C23#12011yescharacteristic
181C4×C23#12011yescharacteristic
182C4×C23C22×D411yescharacteristic
183C4×C23C22×D411yescharacteristic
184Dic23—22no—
185Dic23—41no—
186C2×H(5,𝔽2)—231no—
187#575—231no—
188#575—231no—
189#535—231no—
190#455—234no—
191#453—234no—
192#421—231no—
193#421D2311yescharacteristic, complemented
194C2×#100—231no—
195C2×C23—21no—
196C2×C23—21no—
197C2×C23—21no—
198C2×C23—42no—
199C2×C23C2×#12011yescharacteristic
200D23—44no—
201D23—42no—
202D23—81no—
203H(5,𝔽2)—234no—
204H(5,𝔽2)—462no—
205H(5,𝔽2)—462no—
206H(5,𝔽2)—462no—
207C2×#41—231no—
208C2×#41—231no—
209C2×#41—461no—
210C2×#41—461no—
211C2×#41C2×D2311yescharacteristic
212C22×D4—231no—
213C22×D4—461no—
214C22×D4—462no—
215C22×D4—461no—
216#137—232no—
217#137—232no—
218#137—461no—
219#137—461no—
220AGL(1,ℤ/8)—232no—
221AGL(1,ℤ/8)—232no—
222AGL(1,ℤ/8)—461no—
223AGL(1,ℤ/8)—461no—
224#135—462no—
225#135—462no—
226C2×M16—231no—
227C2×M16C2×D2311yescharacteristic, complemented
228#127—232no—
229#124—232no—
230#121—462no—
231#120—462no—
232#117C2×D2311yescharacteristic, complemented
233#104—234no—
234#104C2×D2314yescomplemented
235#100—234no—
236C23#301911yescharacteristic, complemented, Sylow
237C24—232no—
238C24—462no—
239#41—21no—
240#41—21no—
241#41—232no—
242#41—232no—
243#41—464no—
244#41—464no—
245#41—464no—
246#41—464no—
247#41—461no—
248#41—461no—
249#41—461no—
250#41—461no—
251#41—461no—
252#41—461no—
253#41D9212yescomplemented
254#41C22×D2312yes—
255C2×Q8—231no—
256C2×Q8—231no—
257C2×Q8D9211yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—231no—
260C2×D4—232no—
261C2×D4—232no—
262C2×D4—231no—
263C2×D4—464no—
264C2×D4—462no—
265C2×D4—464no—
266C2×D4—464no—
267C2×D4—468no—
268C2×D4—468no—
269C2×D4—464no—
270C2×D4—462no—
271C2×D4—461no—
272C2×D4—464no—
273C2×D4—461no—
274C2×D4—461no—
275C2×D4—461no—
276C2×D4—461no—
277C2×D4—461no—
278C2×D4D9211yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—462no—
281C22×C4—461no—
282C22×C4—461no—
283C22×C4C22×D2311yescharacteristic
284Q16—462no—
285Q16—462no—
286SD16—462no—
287SD16—462no—
288SD16—462no—
289SD16—462no—
290D8—462no—
291D8—462no—
292M16—21no—
293M16—232no—
294M16—461no—
295M16C22×D2312yescomplemented
296C2×C8—21no—
297C2×C8—461no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—464no—
301C42C22×D2312yescomplemented
302C23—21no—
303C23—234no—
304C23—468no—
305C23—461no—
306C23—462no—
307C23—462no—
308C23—461no—
309C23—1842no—
310C23—1844no—
311C23D4×D2311yescharacteristic
312Q8—21no—
313Q8—232no—
314Q8—461no—
315Q8—461no—
316Q8—461no—
317Q8C2×D9212yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—232no—
322D4—468no—
323D4—468no—
324D4—462no—
325D4—462no—
326D4—461no—
327D4—462no—
328D4—462no—
329D4—462no—
330D4—461no—
331D4—462no—
332D4—461no—
333D4—921no—
334D4—921no—
335D4—1844no—
336D4C2×D9212yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—468no—
342C2×C4—461no—
343C2×C4—462no—
344C2×C4—462no—
345C2×C4—461no—
346C2×C4—921no—
347C2×C4—921no—
348C2×C4C2×D9211yescharacteristic
349C2×C4C2×D9211yescharacteristic
350C2×C4D4×D2311yescharacteristic
351C2×C4D4×D2311yescharacteristic
352C2×C4D4×D2311yescharacteristic
353C2×C4C23×D2311yescharacteristic
354C8—22no—
355C8—462no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—462no—
360C22—464no—
361C22—924no—
362C22—924no—
363C22—922no—
364C22—922no—
365C22—921no—
366C22—1842no—
367C22—1848no—
368C22#1062505311yescharacteristic
369C22#1062505311yescharacteristic
370C22C2×D4×D2311yescharacteristic
371C4—22no—
372C4—42no—
373C4—462no—
374C4—921no—
375C4#1062505311yescharacteristic
376C4#1062505311yescharacteristic
377C4C22×D9211yescharacteristic
378C4C2×D4×D2311yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—924no—
384C2—922no—
385C2—1841no—
386C2C2×#1062505311yescharacteristic
subgroupG/H
centerC2C2×#10625053
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#421C2
socleC2×C23C2×#120
hypercenter#421D23
hypocenterC23#3019
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#10625160C2semidirect
2. #13093217C2semidirect
3. #13093215C2semidirect
4. #13092986C2semidirect
5. #13092596C2semidirect
6. #13092586C2semidirect
7. #13092435C2semidirect
8. C2×#10625005C2semidirect
9. C23×#421C2semidirect
20. C2×#67288C22semidirect
23. C2×M16×C23C22semidirect
24. C23×#117C22semidirect
25. #10625122C22semidirect
27. #10625120C22semidirect
30. C2×#67267C22semidirect
32. #10625080C22semidirect
34. #10625079C22semidirect
38. #10625046C22semidirect
39. #10625045C22semidirect
40. C23×#104C22semidirect
41. #10625005C22semidirect
42. #10624987C22semidirect
47. #41×C23D4semidirect
49. C2×Q8×C23D4semidirect
51. C2×D4×C23D4semidirect
71. #67288D4semidirect
82. C2×D92D4semidirect
84. C2×D92C23semidirect
88. C2×Dic46D4semidirect
89. C2×Dic46C23semidirect
91. M16×C23C23semidirect
95. C42×C23C23semidirect
102. #67267C23semidirect
193. #421D23semidirect
227. C2×M16C2×D23semidirect
232. #117C2×D23semidirect
234. #104C2×D23semidirect
236. C23#3019semidirect
253. #41D92semidirect
257. C2×Q8D92semidirect
278. C2×D4D92semidirect
295. M16C22×D23semidirect
301. C42C22×D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106251602
#130932172
#130932152
#130929862
#130925962
#130925862
#130924352
C2×#106250052
C23×#4212
#301923