Finite Group Databasebeta

← #37581879

Subgroups — #37581879

Order 24448.

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×#20259368C211yesmaximal, characteristic, complemented
2#27133463C211yesmaximal, characteristic, complemented
3#27133461C211yesmaximal, characteristic, complemented
4#27133232C211yesmaximal, characteristic, complemented
5#27132842C214yesmaximal, complemented
6#27132832C214yesmaximal, complemented
7#27132681C211yesmaximal, characteristic, complemented
8C2×#20259238C211yesmaximal, characteristic, complemented
9#421×C191C211yesmaximal, characteristic, complemented
10#20259368—22no—
11#20259368—22no—
12#20259368—22no—
13#20259368C2214yes—
14C2×#15690288—21no—
15C2×#15690287—21no—
16C2×#15690287C2211yescharacteristic
17C2×#15690287C2211yescharacteristic, complemented
18C22×D764—21no—
19C22×D764C2211yescharacteristic
20C2×D4×D191—22no—
21C2×D4×D191—21no—
22#20259355—21no—
23#20259355C2212yescomplemented
24#20259353—21no—
25#20259352C2212yescomplemented
26#20259350—21no—
27#20259350C2212yescomplemented
28#20259348—21no—
29#20259347C2212yescomplemented
30#20259344—22no—
31#20259342—22no—
32C2×#15690271C2211yescharacteristic, complemented
33#20259319C2212yescomplemented
34#20259310C2212yescomplemented
35#20259297—22no—
36#20259292—22no—
37#20259250C2214yescomplemented
38#20259238C2214yescomplemented
39C2×#41×C191C2211yescharacteristic
40C2×M16×C191C2211yescharacteristic, complemented
41#117×C191C2211yescharacteristic, complemented
42#104×C191C2214yescomplemented
43C23×D191—22no—
44C23×D191D412yes—
45#15690288—24no—
46#15690288—24no—
47#15690287—24no—
48#15690287—24no—
49#15690287—21no—
50#15690287—21no—
51#15690287—21no—
52#15690287—21no—
53#15690287—21no—
54#15690287—21no—
55#15690287D412yescomplemented
56#15690287C2312yes—
57C2×Dic382D411yescharacteristic, complemented
58C2×Dic382C2311yescharacteristic, complemented
59C2×D764—22no—
60C2×D764—24no—
61C2×D764—24no—
62C2×D764—24no—
63C2×D764—21no—
64C2×D764—21no—
65C2×D764—21no—
66C2×D764—21no—
67C2×D764D412yes—
68C2×D764D411yescharacteristic, complemented
69C2×D764C2312yes—
70C2×D764C2311yescharacteristic, complemented
71C2×C22⋊D191—22no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74D4×D191—28no—
75D4×D191—24no—
76D4×D191—28no—
77D4×D191—24no—
78C2×C4×D191—22no—
79C2×C4×D191—21no—
80C2×C4×D191—21no—
81Dic764—22no—
82#15690277—22no—
83#15690276—22no—
84#15690276—22no—
85#15690275—22no—
86#15690274—22no—
87D1528—22no—
88#15690272—22no—
89#15690271—21no—
90#15690271C2312yescomplemented
91C2×C191⋊C8—21no—
92(C2×D191)⋊C4—24no—
93#41×C191—21no—
94#41×C191—21no—
95#41×C191D412yescomplemented
96#41×C191C2312yes—
97C2×Q8×C191D411yescharacteristic, complemented
98C2×D4×C191—21no—
99C2×D4×C191D411yescharacteristic, complemented
100C22×C4×C191—21no—
101C22×C4×C191C2311yescharacteristic
102M16×C191—21no—
103M16×C191C2312yescomplemented
104C2×C8×C191—21no—
105C4⋊C4×C191—21no—
106C22⋊C4×C191—21no—
107C42×C191C2312yescomplemented
108C22×D191—22no—
109C22×D191—21no—
110C22×D191—28no—
111C22×D191—22no—
112C22×D191—21no—
113C22×D191—82no—
114C22×D191—84no—
115C22×D191C2×D414yes—
116Dic382—21no—
117Dic382—21no—
118Dic382—21no—
119Dic382C2×D412yes—
120D764—28no—
121D764—22no—
122D764—22no—
123D764—21no—
124D764—22no—
125D764—22no—
126D764—21no—
127D764—21no—
128D764—84no—
129D764C2×D412yes—
130C22⋊D191—28no—
131C22⋊D191—22no—
132C22⋊D191—22no—
133C22⋊D191—41no—
134C22⋊D191—41no—
135C2×Dic191—21no—
136C2×Dic191—21no—
137C4×D191—28no—
138C4×D191—22no—
139C4×D191—22no—
140C4×D191—41no—
141C4×D191—41no—
142C191⋊C8—22no—
143C23×C191—21no—
144C23×C191C2×D411yescharacteristic
145Q8×C191—21no—
146Q8×C191C2×D412yes—
147D4×C191—22no—
148D4×C191—22no—
149D4×C191—21no—
150D4×C191C2×D412yes—
151C2×C4×C191—21no—
152C2×C4×C191—22no—
153C2×C4×C191—22no—
154C2×C4×C191—22no—
155C2×C4×C191C2×D411yescharacteristic
156C2×C4×C191C2×D411yescharacteristic
157C2×C4×C191C2×D411yescharacteristic
158C2×C4×C191C2×D411yescharacteristic
159C2×C4×C191C2×D411yescharacteristic
160C2×C4×C191C2411yescharacteristic
161C8×C191—22no—
162C22×C191—22no—
163C22×C191—41no—
164C22×C191—42no—
165C22×C191#12011yescharacteristic
166C22×C191#12011yescharacteristic
167C22×C191C22×D411yescharacteristic
168C2×D191—24no—
169C2×D191—22no—
170C2×D191—44no—
171C2×D191—42no—
172C2×D191—44no—
173C2×D191—42no—
174C2×D191—41no—
175C2×D191—82no—
176C2×D191—88no—
177C4×C191—22no—
178C4×C191—42no—
179C4×C191#12011yescharacteristic
180C4×C191#12011yescharacteristic
181C4×C191C22×D411yescharacteristic
182C4×C191C22×D411yescharacteristic
183Dic191—22no—
184Dic191—41no—
185C2×C191—21no—
186C2×C191—21no—
187C2×C191—21no—
188C2×C191—42no—
189C2×C191C2×#12011yescharacteristic
190D191—44no—
191D191—42no—
192D191—81no—
193C191#301911yescharacteristic, complemented, Sylow
194#3019—1911nomaximal, Sylow
195C2×H(5,𝔽2)—1911no—
196#575—1911no—
197#575—1911no—
198#535—1911no—
199#455—1914no—
200#453—1914no—
201#421—1911no—
202#421D19111yescharacteristic, complemented
203C2×#100—1911no—
204H(5,𝔽2)—3822no—
205H(5,𝔽2)—3822no—
206H(5,𝔽2)—3822no—
207H(5,𝔽2)—1914no—
208C2×#41—3821no—
209C2×#41—3821no—
210C2×#41—1911no—
211C2×#41—1911no—
212C2×#41C2×D19111yescharacteristic
213C22×D4—3821no—
214C22×D4—3822no—
215C22×D4—3821no—
216C22×D4—1911no—
217#137—3821no—
218#137—3821no—
219#137—1912no—
220#137—1912no—
221AGL(1,ℤ/8)—3821no—
222AGL(1,ℤ/8)—3821no—
223AGL(1,ℤ/8)—1912no—
224AGL(1,ℤ/8)—1912no—
225#135—3822no—
226#135—3822no—
227C2×M16—1911no—
228C2×M16C2×D19111yescharacteristic, complemented
229#127—1912no—
230#124—1912no—
231#121—3822no—
232#120—3822no—
233#117C2×D19111yescharacteristic, complemented
234#104—1914no—
235#104C2×D19114yescomplemented
236#100—1914no—
237C24—3822no—
238C24—1912no—
239#41—21no—
240#41—21no—
241#41—3824no—
242#41—3824no—
243#41—3824no—
244#41—3824no—
245#41—3821no—
246#41—3821no—
247#41—3821no—
248#41—3821no—
249#41—3821no—
250#41—3821no—
251#41—1912no—
252#41—1912no—
253#41D76412yescomplemented
254#41C22×D19112yes—
255C2×Q8—1911no—
256C2×Q8—1911no—
257C2×Q8D76411yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—3822no—
260C2×D4—3824no—
261C2×D4—3824no—
262C2×D4—3824no—
263C2×D4—3828no—
264C2×D4—3824no—
265C2×D4—3828no—
266C2×D4—3822no—
267C2×D4—3821no—
268C2×D4—3824no—
269C2×D4—3821no—
270C2×D4—3821no—
271C2×D4—3821no—
272C2×D4—3821no—
273C2×D4—3821no—
274C2×D4—1912no—
275C2×D4—1911no—
276C2×D4—1911no—
277C2×D4—1912no—
278C2×D4D76411yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—3822no—
281C22×C4—3821no—
282C22×C4—3821no—
283C22×C4C22×D19111yescharacteristic
284Q16—3822no—
285Q16—3822no—
286SD16—3822no—
287SD16—3822no—
288SD16—3822no—
289SD16—3822no—
290D8—3822no—
291D8—3822no—
292M16—21no—
293M16—3821no—
294M16—1912no—
295M16C22×D19112yescomplemented
296C2×C8—21no—
297C2×C8—3821no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3824no—
301C42C22×D19112yescomplemented
302C23—21no—
303C23—3828no—
304C23—3822no—
305C23—3822no—
306C23—3821no—
307C23—3821no—
308C23—1914no—
309C23—15284no—
310C23—15282no—
311C23D4×D19111yescharacteristic
312Q8—21no—
313Q8—3821no—
314Q8—3821no—
315Q8—3821no—
316Q8—1912no—
317Q8C2×D76412yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—3828no—
322D4—3828no—
323D4—3822no—
324D4—3822no—
325D4—3822no—
326D4—3821no—
327D4—3822no—
328D4—3821no—
329D4—3822no—
330D4—3821no—
331D4—3822no—
332D4—1912no—
333D4—15284no—
334D4—7641no—
335D4—7641no—
336D4C2×D76412yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—3828no—
342C2×C4—3821no—
343C2×C4—3822no—
344C2×C4—3822no—
345C2×C4—3821no—
346C2×C4—7641no—
347C2×C4—7641no—
348C2×C4D4×D19111yescharacteristic
349C2×C4D4×D19111yescharacteristic
350C2×C4D4×D19111yescharacteristic
351C2×C4C2×D76411yescharacteristic
352C2×C4C2×D76411yescharacteristic
353C2×C4C23×D19111yescharacteristic
354C8—22no—
355C8—3822no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—3824no—
360C22—3822no—
361C22—15282no—
362C22—15288no—
363C22—7642no—
364C22—7644no—
365C22—7642no—
366C22—7644no—
367C22—7641no—
368C22#2025929211yescharacteristic
369C22#2025929211yescharacteristic
370C22C2×D4×D19111yescharacteristic
371C4—22no—
372C4—42no—
373C4—3822no—
374C4—7641no—
375C4#2025929211yescharacteristic
376C4#2025929211yescharacteristic
377C4C2×D4×D19111yescharacteristic
378C4C22×D76411yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—15281no—
384C2—7642no—
385C2—7644no—
386C2C2×#2025929211yescharacteristic
subgroupG/H
centerC2C2×#20259292
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#421×C191C2
socleC2×C191C2×#120
hypercenter#421D191
hypocenterC191#3019
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#20259368C2semidirect
2. #27133463C2semidirect
3. #27133461C2semidirect
4. #27133232C2semidirect
5. #27132842C2semidirect
6. #27132832C2semidirect
7. #27132681C2semidirect
8. C2×#20259238C2semidirect
9. #421×C191C2semidirect
17. C2×#15690287C22semidirect
23. #20259355C22semidirect
25. #20259352C22semidirect
27. #20259350C22semidirect
29. #20259347C22semidirect
32. C2×#15690271C22semidirect
33. #20259319C22semidirect
34. #20259310C22semidirect
37. #20259250C22semidirect
38. #20259238C22semidirect
40. C2×M16×C191C22semidirect
41. #117×C191C22semidirect
42. #104×C191C22semidirect
55. #15690287D4semidirect
57. C2×Dic382D4semidirect
58. C2×Dic382C23semidirect
68. C2×D764D4semidirect
70. C2×D764C23semidirect
90. #15690271C23semidirect
95. #41×C191D4semidirect
97. C2×Q8×C191D4semidirect
99. C2×D4×C191D4semidirect
103. M16×C191C23semidirect
107. C42×C191C23semidirect
193. C191#3019semidirect
202. #421D191semidirect
228. C2×M16C2×D191semidirect
233. #117C2×D191semidirect
235. #104C2×D191semidirect
253. #41D764semidirect
257. C2×Q8D764semidirect
278. C2×D4D764semidirect
295. M16C22×D191semidirect
301. C42C22×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593682
#271334632
#271334612
#271332322
#271328422
#271328322
#271326812
C2×#202592382
#421×C1912
#3019191