Finite Group Databasebeta

← #37581829

Subgroups — #37581829

Order 24448.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259367C211yesmaximal, characteristic, complemented
2C2×#20259348C211yesmaximal, characteristic, complemented
3C2×#20259344C211yesmaximal, characteristic, complemented
4#27133244C211yesmaximal, characteristic, complemented
5#27132826C212yesmaximal, complemented
6#27132815C212yesmaximal, complemented
7#27132809C212yesmaximal, complemented
8#27132798C212yesmaximal, complemented
9#27132662C211yesmaximal, characteristic, complemented
10#27132629C211yesmaximal, characteristic, complemented
11C2×#102×C191C211yesmaximal, characteristic, complemented
12#20259367—24no—
13#20259367—24no—
14C2×#15690287C2211yescharacteristic
15C2×#15690287C2211yescharacteristic
16C2×#15690287C2211yescharacteristic, complemented
17C2×#15690286—21no—
18C2×#15690286C2212yes—
19C22×C22⋊D191—22no—
20C2×D4×D191—21no—
21C2×D4×D191C2212yes—
22#20259348—24no—
23#20259344—24no—
24C2×Dic764C2211yescharacteristic, complemented
25C2×#15690276C2211yescharacteristic, complemented
26C2×#15690276C2211yescharacteristic, complemented
27C2×#15690275C2212yescomplemented
28C2×D1528C2211yescharacteristic, complemented
29C2×#15690272C2212yescomplemented
30C2×#15690271C2211yescharacteristic, complemented
31#20259302—21no—
32#20259302C2212yescomplemented
33#20259297—21no—
34#20259297C2212yescomplemented
35#20259296—22no—
36#20259245C2212yescomplemented
37#20259242C2212yescomplemented
38(C2×D191)⋊C8C2212yescomplemented
39(C2×D191)⋊C8C2212yescomplemented
40C22×D4×C191C2211yescharacteristic
41C22×C8×C191C2211yescharacteristic, complemented
42C2×C4⋊C4×C191C2211yescharacteristic, complemented
43#102×C191C2212yescomplemented
44#102×C191C2212yescomplemented
45C23×D191—22no—
46#15690287—22no—
47#15690287—22no—
48#15690287—22no—
49#15690287—22no—
50#15690287—22no—
51#15690287D414yescomplemented
52#15690286—28no—
53#15690286—24no—
54#15690286—24no—
55C2×Dic382D411yescharacteristic
56C2×Dic382C2311yescharacteristic
57C2×Dic382C2311yescharacteristic
58C2×D764—22no—
59C2×D764D411yescharacteristic
60C2×D764C2311yescharacteristic
61C2×D764C2311yescharacteristic
62C2×C22⋊D191—216no—
63C2×C22⋊D191—22no—
64C2×C22⋊D191—22no—
65C2×C22⋊D191—21no—
66C2×C22⋊D191—21no—
67C2×C22⋊D191D412yes—
68D4×D191—28no—
69D4×D191—24no—
70D4×D191—24no—
71C22×Dic191—22no—
72C2×C4×D191—21no—
73C2×C4×D191—21no—
74C2×C4×D191C2312yes—
75Dic764—22no—
76#15690276—22no—
77#15690276—22no—
78#15690275—24no—
79D1528—22no—
80#15690272—24no—
81#15690271—22no—
82C2×C191⋊C8C2312yes—
83C4⋊Dic191—22no—
84(C2×D191)⋊C4—22no—
85(C2×D191)⋊C4—22no—
86C24×C191—21no—
87C2×D4×C191—21no—
88C2×D4×C191—22no—
89C2×D4×C191D414yescomplemented
90C2×D4×C191C2312yes—
91C22×C4×C191—21no—
92C22×C4×C191C2311yescharacteristic
93C2×C8×C191—22no—
94C2×C8×C191C2311yescharacteristic
95C2×C8×C191C2311yescharacteristic
96C4⋊C4×C191—21no—
97C4⋊C4×C191C2312yes—
98C22×D191—21no—
99C22×D191—28no—
100C22×D191—21no—
101C22×D191—82no—
102C22×D191—82no—
103C22×D191C2×D412yes—
104Dic382—22no—
105Dic382—21no—
106Dic382—21no—
107Dic382—21no—
108Dic382C2×D412yes—
109D764—22no—
110D764—21no—
111D764—21no—
112D764—21no—
113D764—82no—
114D764C2×D412yes—
115C22⋊D191—28no—
116C22⋊D191—28no—
117C22⋊D191—24no—
118C22⋊D191—24no—
119C22⋊D191—42no—
120C2×Dic191—28no—
121C2×Dic191—21no—
122C2×Dic191—21no—
123C2×Dic191C2×D412yes—
124C4×D191—24no—
125C4×D191—22no—
126C4×D191—22no—
127C4×D191—42no—
128C191⋊C8—22no—
129C23×C191—22no—
130C23×C191—24no—
131C23×C191—81no—
132C23×C191—81no—
133C23×C191C2×D411yescharacteristic
134D4×C191—24no—
135D4×C191—22no—
136D4×C191C2×D414yes—
137C2×C4×C191—22no—
138C2×C4×C191—81no—
139C2×C4×C191C2×D411yescharacteristic
140C2×C4×C191C2×D412yes—
141C2×C4×C191C2×D412yes—
142C2×C4×C191C2411yescharacteristic
143C8×C191—21no—
144C8×C191—21no—
145C22×C191—21no—
146C22×C191—21no—
147C22×C191—24no—
148C22×C191—42no—
149C22×C191—42no—
150C22×C191—84no—
151C22×C191#12012yes—
152C22×C191C22×D411yescharacteristic
153C2×D191—22no—
154C2×D191—22no—
155C2×D191—22no—
156C2×D191—22no—
157C2×D191—41no—
158C2×D191—41no—
159C2×D191—41no—
160C2×D191—41no—
161C2×D191—41no—
162C2×D191—88no—
163C4×C191—42no—
164C4×C191#12012yes—
165C4×C191C22×D411yescharacteristic
166C4×C191C22×D411yescharacteristic
167Dic191—22no—
168Dic191—22no—
169Dic191—41no—
170C2×C191—22no—
171C2×C191—44no—
172C2×C191C2×#12011yescharacteristic
173C2×C191#57611yescharacteristic
174C2×C191#57711yescharacteristic
175D191—42no—
176D191—42no—
177D191—81no—
178C191#301211yescharacteristic, complemented, Sylow
179#3012—1911nomaximal, Sylow
180C2×H(5,𝔽2)—1911no—
181C2×AGL(1,ℤ/8)—1911no—
182C2×#135—1911no—
183#537—1911no—
184#452—1912no—
185#450—1912no—
186#449—1912no—
187#447—1912no—
188#417—1911no—
189C2×#102D19111yescharacteristic, complemented
190#408—1911no—
191H(5,𝔽2)—3824no—
192H(5,𝔽2)—3824no—
193C2×#41—3821no—
194C2×#41—1912no—
195C2×#41—1911no—
196C2×#41—1911no—
197C2×#41—1911no—
198C22×D4—3821no—
199C22×D4—3822no—
200C22×D4—1912no—
201C22×D4C2×D19111yescharacteristic
202AGL(1,ℤ/8)—3824no—
203#135—3824no—
204C2×Q16—1911no—
205C2×SD16—1912no—
206C2×SD16—1911no—
207C2×SD16—1911no—
208C2×D8—1912no—
209C2×D8—1911no—
210C2×M16—1911no—
211C22×C8C2×D19111yescharacteristic, complemented
212C22⋊Q8—3821no—
213C22⋊Q8—1912no—
214#121—3821no—
215#121—3822no—
216#121—1912no—
217C2×C4⋊C4C2×D19111yescharacteristic, complemented
218#103—1912no—
219#102—1912no—
220#102C2×D19112yescomplemented
221#102C2×D19112yescomplemented
222C22⋊C8—1912no—
223C22⋊C8—1912no—
224C24—21no—
225C24—3822no—
226#41—3828no—
227#41—3824no—
228#41—3824no—
229#41—3822no—
230#41—3822no—
231#41—3822no—
232#41—3822no—
233#41—3822no—
234#41—1914no—
235C2×Q8—1911no—
236C2×Q8—1911no—
237C2×Q8—1911no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—3822no—
241C2×D4—3828no—
242C2×D4—3824no—
243C2×D4—38216no—
244C2×D4—3822no—
245C2×D4—3821no—
246C2×D4—3824no—
247C2×D4—3821no—
248C2×D4—3822no—
249C2×D4—1912no—
250C2×D4—1911no—
251C2×D4—1911no—
252C2×D4—1911no—
253C2×D4D76414yescomplemented
254C2×D4C22×D19112yes—
255C22×C4—21no—
256C22×C4—3822no—
257C22×C4—3821no—
258C22×C4—3821no—
259C22×C4—1912no—
260C22×C4C22×D19111yescharacteristic
261Q16—3822no—
262SD16—3822no—
263SD16—3822no—
264SD16—3824no—
265D8—3822no—
266D8—3824no—
267M16—3822no—
268C2×C8—22no—
269C2×C8—1912no—
270C2×C8C22×D19111yescharacteristic
271C2×C8C22×D19111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3822no—
274C4⋊C4C22×D19112yes—
275C22⋊C4—3822no—
276C22⋊C4—3822no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—3828no—
282C23—3821no—
283C23—3821no—
284C23—1912no—
285C23—15282no—
286C23—15282no—
287C23D4×D19111yescharacteristic
288Q8—3821no—
289Q8—3822no—
290Q8—3821no—
291Q8—3821no—
292Q8—1912no—
293D4—22no—
294D4—24no—
295D4—3828no—
296D4—3828no—
297D4—3824no—
298D4—3824no—
299D4—3821no—
300D4—3821no—
301D4—3821no—
302D4—3822no—
303D4—1912no—
304D4—15282no—
305D4—7642no—
306D4C2×D76414yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—3828no—
310C2×C4—3821no—
311C2×C4—3822no—
312C2×C4—3822no—
313C2×C4—3824no—
314C2×C4—3821no—
315C2×C4—1912no—
316C2×C4—7642no—
317C2×C4D4×D19112yes—
318C2×C4D4×D19111yescharacteristic
319C2×C4C2×D76412yes—
320C2×C4C23×D19111yescharacteristic
321C8—21no—
322C8—21no—
323C8—3822no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—3822no—
331C22—3822no—
332C22—3822no—
333C22—3822no—
334C22—15288no—
335C22—7641no—
336C22—7641no—
337C22—7641no—
338C22—7641no—
339C22—7641no—
340C22#2025929212yes—
341C22C2×D4×D19111yescharacteristic
342C4—42no—
343C4—3822no—
344C4—3822no—
345C4—7641no—
346C4#2025929212yes—
347C4C2×D4×D19111yescharacteristic
348C4C22×D76411yescharacteristic
349C2—22no—
350C2—44no—
351C2—15281no—
352C2—7642no—
353C2—7642no—
354C2C2×#2025929211yescharacteristic
355C2#2713346411yescharacteristic
356C2#2713346911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#102×C191C2
socleC22×C191C22×D4
hypercenterC2×#102D191
hypocenterC191#3012
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#20259367C2semidirect
2. C2×#20259348C2semidirect
3. C2×#20259344C2semidirect
4. #27133244C2semidirect
5. #27132826C2semidirect
6. #27132815C2semidirect
7. #27132809C2semidirect
8. #27132798C2semidirect
9. #27132662C2semidirect
10. #27132629C2semidirect
11. C2×#102×C191C2semidirect
16. C2×#15690287C22semidirect
24. C2×Dic764C22semidirect
25. C2×#15690276C22semidirect
26. C2×#15690276C22semidirect
27. C2×#15690275C22semidirect
28. C2×D1528C22semidirect
29. C2×#15690272C22semidirect
30. C2×#15690271C22semidirect
32. #20259302C22semidirect
34. #20259297C22semidirect
36. #20259245C22semidirect
37. #20259242C22semidirect
38. (C2×D191)⋊C8C22semidirect
39. (C2×D191)⋊C8C22semidirect
41. C22×C8×C191C22semidirect
42. C2×C4⋊C4×C191C22semidirect
43. #102×C191C22semidirect
44. #102×C191C22semidirect
51. #15690287D4semidirect
89. C2×D4×C191D4semidirect
178. C191#3012semidirect
189. C2×#102D191semidirect
211. C22×C8C2×D191semidirect
217. C2×C4⋊C4C2×D191semidirect
220. #102C2×D191semidirect
221. #102C2×D191semidirect
253. C2×D4D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593672
C2×#202593482
C2×#202593442
#271332442
#271328262
#271328152
#271328092
#271327982
#271326622
#271326292
C2×#102×C1912
#3012191