Finite Group Databasebeta

← #37581811

Subgroups — #37581811

Order 24448.

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

normal subgroups only

This group has 700 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×#20259370C211yesmaximal, characteristic
2C2×#20259353C211yesmaximal, characteristic, complemented
3#27133237C211yesmaximal, characteristic, complemented
4#27132824C214yesmaximal, complemented
5#27132817C214yesmaximal, complemented
6#27132672C211yesmaximal, characteristic, complemented
7#27132660C211yesmaximal, characteristic, complemented
8#27132629C211yesmaximal, characteristic, complemented
9C22×Q16×C191C211yesmaximal, characteristic, complemented
10#20259370—24no—
11#20259370—24no—
12C2×#15690288—21no—
13C2×#15690288C2212yes—
14C2×#15690287—22no—
15C2×#15690287C2211yescharacteristic
16C2×#15690287C2211yescharacteristic
17C2×Q8×D191—21no—
18C2×Q8×D191C2212yes—
19#20259353—24no—
20C2×#15690277C2212yes—
21C2×#15690274C2212yes—
22C2×#15690271C2211yescharacteristic
23#20259320—21no—
24#20259311—21no—
25#20259305—22no—
26#20259298C2214yes—
27#20259281C2211yescharacteristic
28#20259247C2212yes—
29#20259246C2214yes—
30#20259244C2212yes—
31(C2×D191)⋊C8C2214yes—
32C22×Q8×C191C2211yescharacteristic
33C22×Q8×C191C2211yescharacteristic
34C2×Q16×C191—24no—
35C2×Q16×C191C2214yes—
36C22×C8×C191C2211yescharacteristic
37#15690288—28no—
38#15690288—24no—
39#15690288—24no—
40#15690287—216no—
41#15690287—22no—
42#15690287—22no—
43#15690287—22no—
44#15690287D414yes—
45C2×Dic382—22no—
46C2×Dic382D411yescharacteristic
47C2×Dic382C2311yescharacteristic
48Q8×D191—28no—
49Q8×D191—24no—
50Q8×D191—24no—
51C2×D764—22no—
52C2×D764D411yescharacteristic
53C2×D764C2311yescharacteristic
54C2×C22⋊D191—21no—
55C2×C22⋊D191D412yes—
56C2×C4×D191—22no—
57C2×C4×D191—22no—
58C2×C4×D191—21no—
59C2×C4×D191C2312yes—
60#15690277—24no—
61#15690274—24no—
62#15690271—22no—
63C2×C191⋊C8C2312yes—
64C4⋊Dic191C2312yes—
65Dic191⋊C4—24no—
66C22⋊Dic191—21no—
67(C2×D191)⋊C4—24no—
68C4×Dic191—21no—
69C2×Q8×C191—21no—
70C2×Q8×C191—22no—
71C2×Q8×C191—21no—
72C2×Q8×C191—22no—
73C2×Q8×C191—22no—
74C2×Q8×C191D414yes—
75C2×Q8×C191C2312yes—
76C2×Q8×C191C2312yes—
77C22×C4×C191—21no—
78C22×C4×C191—21no—
79C22×C4×C191C2311yescharacteristic
80Q16×C191—28no—
81C2×C8×C191—22no—
82C2×C8×C191C2312yes—
83C22×D191—21no—
84C22×D191C2×D412yes—
85Dic382—28no—
86Dic382—22no—
87Dic382—21no—
88Dic382C2×D412yes—
89D764—28no—
90D764—22no—
91D764—21no—
92D764C2×D412yes—
93C22⋊D191—24no—
94C22⋊D191—24no—
95C2×Dic191—21no—
96C2×Dic191—22no—
97C2×Dic191C2×D412yes—
98C4×D191—28no—
99C4×D191—28no—
100C4×D191—24no—
101C4×D191—22no—
102C4×D191—22no—
103C191⋊C8—22no—
104C23×C191C2×D411yescharacteristic
105Q8×C191—24no—
106Q8×C191—22no—
107Q8×C191—22no—
108Q8×C191—22no—
109Q8×C191—42no—
110Q8×C191C2×D414yes—
111C2×C4×C191—24no—
112C2×C4×C191—22no—
113C2×C4×C191—22no—
114C2×C4×C191—42no—
115C2×C4×C191C2×D411yescharacteristic
116C2×C4×C191C2×D412yes—
117C2×C4×C191C2×D412yes—
118C2×C4×C191C2411yescharacteristic
119C8×C191—22no—
120C22×C191—22no—
121C22×C191#12012yes—
122C22×C191C22×D411yescharacteristic
123C2×D191—22no—
124C2×D191—22no—
125C2×D191—24no—
126C2×D191—42no—
127C4×C191—24no—
128C4×C191—42no—
129C4×C191#12012yes—
130C4×C191C22×D411yescharacteristic
131C4×C191C22×D411yescharacteristic
132Dic191—22no—
133Dic191—22no—
134Dic191—42no—
135C2×C191—22no—
136C2×C191C2×#12011yescharacteristic
137C2×C191#57812yes—
138D191—42no—
139D191—42no—
140C191#301111yescharacteristic, complemented, Sylow
141#3011—1911nomaximal, Sylow
142C2×2−1+4—1911no—
143C2×#137—1911no—
144C22×Q16D19111yescharacteristic, complemented
145#536—1911no—
146#452—1914no—
147#451—1914no—
148#419—1911no—
149#416—1911no—
150#408—1911no—
1512−1+4—3824no—
1522−1+4—3824no—
153C2×#41—3822no—
154C2×#41—3821no—
155C2×#41—1912no—
156C2×#41—1911no—
157C2×#41—1911no—
158C22×Q8—3821no—
159C22×Q8—1912no—
160C22×Q8C2×D19111yescharacteristic
161C22×Q8C2×D19111yescharacteristic
162#137—3824no—
163C2×Q16—24no—
164C2×Q16—1912no—
165C2×Q16C2×D19114yes—
166C2×SD16—1912no—
167C2×M16—1911no—
168C22×C8C2×D19111yescharacteristic
169C4⋊Q8—3821no—
170#124—3821no—
171#123—3822no—
172C22⋊Q8—1914no—
173#117—1911no—
174#103—1914no—
175#103—1912no—
176#102—1912no—
177C22⋊C8—1914no—
178#41—38216no—
179#41—3828no—
180#41—3824no—
181#41—3824no—
182#41—3822no—
183#41—3822no—
184#41—3822no—
185#41—1914no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—3822no—
192C2×Q8—3828no—
193C2×Q8—3824no—
194C2×Q8—3824no—
195C2×Q8—1911no—
196C2×Q8—1911no—
197C2×Q8C22⋊D19114yes—
198C2×Q8C22×D19112yes—
199C2×Q8C22×D19112yes—
200C2×D4—3822no—
201C2×D4—3821no—
202C2×D4—1912no—
203C2×D4—1911no—
204C2×D4—1911no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—3822no—
208C22×C4—3822no—
209C22×C4—3821no—
210C22×C4—1912no—
211C22×C4C22×D19111yescharacteristic
212Q16—28no—
213Q16—3824no—
214SD16—3824no—
215M16—3822no—
216C2×C8—22no—
217C2×C8—1912no—
218C2×C8C22×D19112yes—
219C4⋊C4—3824no—
220C4⋊C4—1912no—
221C22⋊C4—3824no—
222C22⋊C4—3821no—
223C42—3821no—
224C23—3821no—
225C23—1912no—
226C23D4×D19111yescharacteristic
227Q8—22no—
228Q8—24no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—3828no—
233Q8—3821no—
234Q8—3822no—
235Q8—1912no—
236Q8C2×C22⋊D19114yes—
237D4—3828no—
238D4—3824no—
239D4—3824no—
240D4—3821no—
241D4—3822no—
242D4—1912no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—3828no—
248C2×C4—3828no—
249C2×C4—3821no—
250C2×C4—3822no—
251C2×C4—3822no—
252C2×C4—3824no—
253C2×C4—3822no—
254C2×C4—1912no—
255C2×C4D4×D19112yes—
256C2×C4D4×D19111yescharacteristic
257C2×C4C2×C22⋊D19112yes—
258C2×C4C23×D19111yescharacteristic
259C8—22no—
260C8—3822no—
261C22—22no—
262C22—3824no—
263C22—3822no—
264C22—3822no—
265C22—7642no—
266C22#2025929012yes—
267C22C2×D4×D19111yescharacteristic
268C4—24no—
269C4—42no—
270C4—3822no—
271C4—3822no—
272C4—7642no—
273C4#2025929012yes—
274C4C2×D4×D19111yescharacteristic
275C4C22×C22⋊D19111yescharacteristic
276C2—22no—
277C2—7642no—
278C2—7642no—
279C2C2×#2025929011yescharacteristic
280C2#2713347612yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×Q16×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259370C2
hypercenterC22×Q16D191
hypocenterC191#3011
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#20259353C2semidirect
3. #27133237C2semidirect
4. #27132824C2semidirect
5. #27132817C2semidirect
6. #27132672C2semidirect
7. #27132660C2semidirect
8. #27132629C2semidirect
9. C22×Q16×C191C2semidirect
140. C191#3011semidirect
144. C22×Q16D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593702
C2×#202593532
#271332372
#271328242
#271328172
#271326722
#271326602
#271326292
C22×Q16×C1912
#3011191