Finite Group Databasebeta

← #34054234

Subgroups — #34054234

Order 19072.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#18011931C211yesmaximal, characteristic, complemented
2#23750192C211yesmaximal, characteristic, complemented
3#23750108C211yesmaximal, characteristic, complemented
4#23749845C212yesmaximal, complemented
5#23749838C212yesmaximal, complemented
6#23749823C212yesmaximal, complemented
7#23749818C212yesmaximal, complemented
8#23749653C211yesmaximal, characteristic, complemented
9#23749627C211yesmaximal, characteristic, complemented
10C2×#18011832C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13#18011931—24no—
14C2×#14558726C2211yescharacteristic, complemented
15C2×#14558724C2211yescharacteristic, complemented
16C2×#14558723C2211yescharacteristic, complemented
17C2×#14558721C2211yescharacteristic, complemented
18C22×#11924746C2211yescharacteristic
19#18011907C2212yes—
20#18011898C2212yes—
21#18011882—21no—
22#18011882—21no—
23#18011880—22no—
24Q8×Dic149C2212yescomplemented
25D4×Dic149—22no—
26D4×Dic149C2212yescomplemented
27#18011870—21no—
28#18011870C2211yescharacteristic
29C2×C4⋊Dic149—21no—
30C2×C4⋊Dic149C2211yescharacteristic
31#18011840C2212yescomplemented
32#18011840C2212yescomplemented
33#18011836C2212yescomplemented
34#18011833C2212yescomplemented
35#18011832C2212yes—
36#18011832C2212yes—
37C2×#41×C149C2211yescharacteristic
38C22×D4×C149C2211yescharacteristic
39AGL(1,ℤ/8)×C149—24no—
40C2×SD16×C149C2212yes—
41C2×D8×C149C2212yes—
42C2×M16×C149C2211yescharacteristic
43#14558736—22no—
44#14558736—22no—
45C2×Dic298C2311yescharacteristic
46C2×D596C2311yescharacteristic
47C2×C22⋊D149—22no—
48C2×C22⋊D149—22no—
49C2×C22⋊D149—21no—
50C22×Dic149—22no—
51C22×Dic149—21no—
52C2×C4×D149—21no—
53#14558726—22no—
54#14558724—22no—
55#14558723—22no—
56#14558721—22no—
57C2×#11924746—22no—
58C2×#11924746C2311yescharacteristic
59C2×#11924746C2311yescharacteristic
60C4⋊Dic149—22no—
61C4⋊Dic149—21no—
62C4⋊Dic149D411yescharacteristic
63C4⋊Dic149D411yescharacteristic
64C4⋊Dic149C2312yes—
65C22⋊Dic149—22no—
66C22⋊Dic149—22no—
67C22⋊Dic149—22no—
68C22⋊Dic149D412yes—
69C4×Dic149—22no—
70C4×Dic149C2312yes—
71C24×C149—21no—
72#41×C149—22no—
73#41×C149D414yescomplemented
74C2×Q8×C149C2311yescharacteristic
75C2×D4×C149—21no—
76C2×D4×C149—22no—
77C2×D4×C149—21no—
78C2×D4×C149—22no—
79C2×D4×C149C2312yes—
80C2×D4×C149C2311yescharacteristic
81C22×C4×C149—21no—
82C22×C4×C149C2311yescharacteristic
83SD16×C149—24no—
84D8×C149—24no—
85M16×C149—22no—
86C2×C8×C149C2312yes—
87C22×D149—21no—
88Dic298—21no—
89Dic298—21no—
90D596—21no—
91D596—21no—
92C22⋊D149—42no—
93C22⋊D149—82no—
94C22⋊D149—82no—
95C2×Dic149—21no—
96C2×Dic149—22no—
97C2×Dic149—82no—
98C2×Dic149—81no—
99C2×Dic149C2×D412yes—
100C2×Dic149C2×D412yes—
101C4×D149—42no—
102#11924746—21no—
103#11924746—21no—
104C23×C149—21no—
105C23×C149—22no—
106C23×C149—42no—
107C23×C149—81no—
108C23×C149—81no—
109C23×C149C2×D411yescharacteristic
110Q8×C149—21no—
111Q8×C149C2×D412yes—
112D4×C149—24no—
113D4×C149—22no—
114D4×C149—21no—
115D4×C149—22no—
116D4×C149—42no—
117D4×C149C2×D412yes—
118C2×C4×C149—21no—
119C2×C4×C149—24no—
120C2×C4×C149C2×D411yescharacteristic
121C2×C4×C149C2×D412yes—
122C2×C4×C149#4112yes—
123C2×C4×C149C2411yescharacteristic
124C8×C149—22no—
125C22×C149—21no—
126C22×C149—21no—
127C22×C149—22no—
128C22×C149—41no—
129C22×C149—41no—
130C22×C149—42no—
131C22×C149—42no—
132C22×C149—42no—
133C22×C149—84no—
134C22×C149#12112yes—
135C22×C149C22×D411yescharacteristic
136C2×D149—41no—
137C2×D149—41no—
138C2×D149—41no—
139C4×C149—22no—
140C4×C149#12112yes—
141C4×C149C22×D411yescharacteristic
142C4×C149C2×#4111yescharacteristic
143Dic149—22no—
144Dic149—22no—
145Dic149—41no—
146Dic149—42no—
147C2×C149—22no—
148C2×C149—42no—
149C2×C149—82no—
150C2×C149C2×#12111yescharacteristic
151C2×C149#57611yescharacteristic
152C2×C149#57711yescharacteristic
153D149—81no—
154C149#304511yescharacteristic, complemented, Sylow
155#3045—1491nomaximal, Sylow
156C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
157C2×#135—1491no—
158#535—1491no—
159#520—1491no—
160#464—1492no—
161#463—1492no—
162#460—1492no—
163#459—1492no—
164#424—1491no—
165#417—1491no—
166C2×#102—1491no—
167C2×#41—1491no—
168C2×#41C2×D14911yescharacteristic
169C22×D4C2×D14911yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—2984no—
172C2×Q16—1491no—
173C2×SD16—1491no—
174C2×SD16—1491no—
175C2×SD16C2×D14912yes—
176C2×D8—1491no—
177C2×D8C2×D14912yes—
178C2×M16C2×D14911yescharacteristic
179C22×C8—1491no—
180#127—1492no—
181#124—1492no—
182#121—2981no—
183#121—2981no—
184#120—2982no—
185C4×Q8—1492no—
186C4×D4—2982no—
187C4×D4—1492no—
188#117—2981no—
189#117—1491no—
190C2×C4⋊C4—2981no—
191C2×C4⋊C4—1491no—
192C4⋊C8—1492no—
193C4⋊C8—1492no—
194#103—1492no—
195#102—1492no—
196#102—1492no—
197#102—1492no—
198C24—21no—
199#41—22no—
200#41—2982no—
201#41—2982no—
202#41C22⋊D14914yescomplemented
203C2×Q8—1491no—
204C2×Q8C22×D14911yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—2981no—
210C2×D4—2982no—
211C2×D4—2982no—
212C2×D4—1491no—
213C2×D4C22×D14912yes—
214C2×D4C22×D14911yescharacteristic
215C22×C4—21no—
216C22×C4—2981no—
217C22×C4—2981no—
218C22×C4—2982no—
219C22×C4C22×D14911yescharacteristic
220Q16—2982no—
221SD16—24no—
222SD16—2982no—
223SD16—2982no—
224D8—24no—
225D8—2982no—
226M16—22no—
227C2×C8—2982no—
228C2×C8—1491no—
229C2×C8—1491no—
230C2×C8C22×D14912yes—
231C4⋊C4—2982no—
232C4⋊C4—2981no—
233C4⋊C4—1492no—
234C4⋊C4—1491no—
235C4⋊C4—1491no—
236C22⋊C4—2982no—
237C22⋊C4—2982no—
238C22⋊C4—2982no—
239C22⋊C4—1492no—
240C42—2982no—
241C42—1492no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2981no—
248C23D4×D14911yescharacteristic
249Q8—21no—
250Q8—2981no—
251Q8—2981no—
252Q8C2×C22⋊D14912yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—2981no—
259D4—2981no—
260D4—5962no—
261D4—11922no—
262D4—11922no—
263D4C2×C22⋊D14912yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—2982no—
267C2×C4—2981no—
268C2×C4—5962no—
269C2×C4—1492no—
270C2×C4—1492no—
271C2×C4—11922no—
272C2×C4—11921no—
273C2×C4D4×D14911yescharacteristic
274C2×C4C2×C22⋊D14912yes—
275C2×C4#1455873512yes—
276C2×C4C23×D14911yescharacteristic
277C8—22no—
278C8—2981no—
279C8—2981no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—5961no—
290C22—5961no—
291C22—5961no—
292C22#1801188312yes—
293C22C2×D4×D14911yescharacteristic
294C4—22no—
295C4—2982no—
296C4—2982no—
297C4—5962no—
298C4—5961no—
299C4#1801188312yes—
300C4C22×C22⋊D14911yescharacteristic
301C4C2×#1455873511yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—11921no—
306C2C2×#1801188311yescharacteristic
307C2#2375042911yescharacteristic
308C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×AGL(1,ℤ/8)×C149C2
socleC22×C149C22×D4
hypercenterC2×AGL(1,ℤ/8)D149
hypocenterC149#3045
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#18011931C2semidirect
2. #23750192C2semidirect
3. #23750108C2semidirect
4. #23749845C2semidirect
5. #23749838C2semidirect
6. #23749823C2semidirect
7. #23749818C2semidirect
8. #23749653C2semidirect
9. #23749627C2semidirect
10. C2×#18011832C2semidirect
11. C2×AGL(1,ℤ/8)×C149C2semidirect
14. C2×#14558726C22semidirect
15. C2×#14558724C22semidirect
16. C2×#14558723C22semidirect
17. C2×#14558721C22semidirect
24. Q8×Dic149C22semidirect
26. D4×Dic149C22semidirect
31. #18011840C22semidirect
32. #18011840C22semidirect
33. #18011836C22semidirect
34. #18011833C22semidirect
73. #41×C149D4semidirect
154. C149#3045semidirect
156. C2×AGL(1,ℤ/8)D149semidirect
202. #41C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119312
#237501922
#237501082
#237498452
#237498382
#237498232
#237498182
#237496532
#237496272
C2×#180118322
C2×AGL(1,ℤ/8)×C1492
#3045149