Finite Group Databasebeta

← #34053994

Subgroups — #34053994

Order 19072.

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×#18011956C211yesmaximal, characteristic, complemented
2C2×#18011931C211yesmaximal, characteristic, complemented
3#23750203C211yesmaximal, characteristic, complemented
4#23749784C212yesmaximal, complemented
5#23749774C212yesmaximal, complemented
6#23749767C212yesmaximal, complemented
7#23749757C212yesmaximal, complemented
8#23749627C211yesmaximal, characteristic, complemented
9C2×#18011832C211yesmaximal, characteristic, complemented
10#23749593C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
12#18011956—24no—
13#18011956—24no—
14C2×#14558736C2211yescharacteristic
15C2×#14558736C2211yescharacteristic
16C2×#14558735—21no—
17C2×#14558735C2212yes—
18C22×C22⋊D149—22no—
19C2×D4×D149—21no—
20C2×D4×D149C2212yes—
21#18011931—24no—
22C2×#14558726C2211yescharacteristic, complemented
23C2×#14558724C2211yescharacteristic, complemented
24C2×#14558723C2211yescharacteristic, complemented
25C2×#14558721C2211yescharacteristic, complemented
26C22×#11924746C2211yescharacteristic, complemented
27#18011890—21no—
28#18011887C2212yescomplemented
29#18011885—21no—
30#18011883—22no—
31#18011882C2212yescomplemented
32C2×C4⋊Dic149C2211yescharacteristic, complemented
33#18011836C2212yescomplemented
34#18011833C2212yescomplemented
35#18011832C2212yescomplemented
36#18011832C2212yescomplemented
37#18011823C2212yescomplemented
38#18011823C2212yescomplemented
39C2×#41×C149C2211yescharacteristic, complemented
40C22×D4×C149C2211yescharacteristic
41AGL(1,ℤ/8)×C149—24no—
42C2×SD16×C149C2212yescomplemented
43C2×D8×C149C2212yescomplemented
44C2×M16×C149C2211yescharacteristic, complemented
45C23×D149—22no—
46#14558736—22no—
47#14558736—22no—
48#14558736—22no—
49#14558736D414yescomplemented
50#14558735—28no—
51#14558735—24no—
52#14558735—24no—
53C2×Dic298D411yescharacteristic
54C2×Dic298C2311yescharacteristic
55C2×D596D411yescharacteristic
56C2×D596C2311yescharacteristic
57C2×C22⋊D149—216no—
58C2×C22⋊D149—22no—
59C2×C22⋊D149—22no—
60C2×C22⋊D149—22no—
61C2×C22⋊D149—21no—
62C2×C22⋊D149D412yes—
63D4×D149—28no—
64D4×D149—24no—
65D4×D149—24no—
66C22×Dic149—22no—
67C22×Dic149—21no—
68C2×C4×D149—21no—
69C2×C4×D149C2312yes—
70#14558726—22no—
71#14558724—22no—
72#14558723—22no—
73#14558721—22no—
74C2×#11924746—22no—
75C2×#11924746C2311yescharacteristic
76C2×#11924746C2311yescharacteristic
77C4⋊Dic149—21no—
78C4⋊Dic149C2312yes—
79#14558715—22no—
80C22⋊Dic149—22no—
81#14558713—22no—
82C24×C149—21no—
83#41×C149—22no—
84#41×C149—22no—
85C2×Q8×C149C2311yescharacteristic
86C2×D4×C149—21no—
87C2×D4×C149—21no—
88C2×D4×C149—22no—
89C2×D4×C149D414yescomplemented
90C2×D4×C149C2311yescharacteristic
91C2×D4×C149C2312yes—
92C22×C4×C149—21no—
93C22×C4×C149C2311yescharacteristic
94SD16×C149—24no—
95D8×C149—24no—
96M16×C149—22no—
97C2×C8×C149C2312yes—
98C22×D149—21no—
99C22×D149—28no—
100C22×D149—82no—
101C22×D149—82no—
102C22×D149C2×D412yes—
103Dic298—22no—
104Dic298—21no—
105Dic298C2×D412yes—
106D596—22no—
107D596—21no—
108D596C2×D412yes—
109C22⋊D149—28no—
110C22⋊D149—28no—
111C22⋊D149—24no—
112C22⋊D149—24no—
113C22⋊D149—82no—
114C2×Dic149—28no—
115C2×Dic149—21no—
116C2×Dic149—22no—
117C2×Dic149—81no—
118C2×Dic149C2×D412yes—
119C4×D149—24no—
120C4×D149—22no—
121C4×D149—22no—
122#11924746—21no—
123#11924746—21no—
124C23×C149—22no—
125C23×C149—24no—
126C23×C149—21no—
127C23×C149—81no—
128C23×C149—81no—
129C23×C149C2×D411yescharacteristic
130Q8×C149—21no—
131Q8×C149—21no—
132D4×C149—24no—
133D4×C149—22no—
134D4×C149—21no—
135D4×C149—21no—
136D4×C149—42no—
137D4×C149C2×D414yes—
138C2×C4×C149—21no—
139C2×C4×C149—42no—
140C2×C4×C149C2×D411yescharacteristic
141C2×C4×C149C2×D412yes—
142C2×C4×C149C2×D412yes—
143C2×C4×C149C2411yescharacteristic
144C8×C149—22no—
145C22×C149—21no—
146C22×C149—21no—
147C22×C149—24no—
148C22×C149—42no—
149C22×C149—42no—
150C22×C149—41no—
151C22×C149—41no—
152C22×C149—41no—
153C22×C149—84no—
154C22×C149#12012yes—
155C22×C149C22×D411yescharacteristic
156C2×D149—22no—
157C2×D149—22no—
158C2×D149—22no—
159C2×D149—22no—
160C2×D149—41no—
161C2×D149—41no—
162C2×D149—88no—
163C4×C149—41no—
164C4×C149#12012yes—
165C4×C149C22×D411yescharacteristic
166C4×C149C22×D411yescharacteristic
167Dic149—22no—
168Dic149—22no—
169Dic149—42no—
170C2×C149—22no—
171C2×C149—44no—
172C2×C149—81no—
173C2×C149C2×#12011yescharacteristic
174C2×C149#57611yescharacteristic
175C2×C149#57711yescharacteristic
176D149—42no—
177D149—42no—
178C149#301211yescharacteristic, complemented, Sylow
179#3012—1491nomaximal, Sylow
180C2×H(5,𝔽2)—1491no—
181C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
182C2×#135—1491no—
183#537—1491no—
184#452—1492no—
185#450—1492no—
186#449—1492no—
187#447—1492no—
188#417—1491no—
189C2×#102—1491no—
190#408—1491no—
191H(5,𝔽2)—2984no—
192H(5,𝔽2)—2984no—
193C2×#41—2981no—
194C2×#41—1492no—
195C2×#41—1491no—
196C2×#41—1491no—
197C2×#41C2×D14911yescharacteristic, complemented
198C22×D4—2981no—
199C22×D4—2982no—
200C22×D4—1492no—
201C22×D4C2×D14911yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—2984no—
204C2×Q16—1491no—
205C2×SD16—1491no—
206C2×SD16—1491no—
207C2×SD16C2×D14912yescomplemented
208C2×D8—1491no—
209C2×D8C2×D14912yescomplemented
210C2×M16C2×D14911yescharacteristic, complemented
211C22×C8—1491no—
212C22⋊Q8—2981no—
213C22⋊Q8—1492no—
214#121—2981no—
215#121—2982no—
216#121—1492no—
217C2×C4⋊C4—1491no—
218#103—1492no—
219#102—1492no—
220#102—1492no—
221#102—1492no—
222C22⋊C8—1492no—
223C22⋊C8—1492no—
224C24—21no—
225C24—2982no—
226#41—22no—
227#41—22no—
228#41—2988no—
229#41—2984no—
230#41—2984no—
231#41—2982no—
232#41—2982no—
233#41—2982no—
234#41—1494no—
235C2×Q8—1491no—
236C2×Q8—1491no—
237C2×Q8C22×D14911yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—2988no—
242C2×D4—2984no—
243C2×D4—29816no—
244C2×D4—2982no—
245C2×D4—2981no—
246C2×D4—2984no—
247C2×D4—2982no—
248C2×D4—2982no—
249C2×D4—1492no—
250C2×D4—1491no—
251C2×D4—1491no—
252C2×D4C22⋊D14914yescomplemented
253C2×D4C22×D14911yescharacteristic
254C2×D4C22×D14912yes—
255C22×C4—21no—
256C22×C4—2982no—
257C22×C4—2981no—
258C22×C4—2981no—
259C22×C4—1492no—
260C22×C4C22×D14911yescharacteristic
261Q16—2982no—
262SD16—24no—
263SD16—2982no—
264SD16—2982no—
265D8—24no—
266D8—2982no—
267M16—22no—
268C2×C8—2982no—
269C2×C8—1491no—
270C2×C8—1491no—
271C2×C8C22×D14912yes—
272C4⋊C4—2982no—
273C4⋊C4—2981no—
274C4⋊C4—1492no—
275C22⋊C4—2982no—
276C22⋊C4—2982no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—2988no—
283C23—2981no—
284C23—1492no—
285C23—11922no—
286C23—11922no—
287C23D4×D14911yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2981no—
291Q8—2982no—
292Q8—1492no—
293D4—22no—
294D4—21no—
295D4—21no—
296D4—24no—
297D4—42no—
298D4—2988no—
299D4—2988no—
300D4—2984no—
301D4—2984no—
302D4—2981no—
303D4—2982no—
304D4—1492no—
305D4—11922no—
306D4C2×C22⋊D14914yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2988no—
310C2×C4—2981no—
311C2×C4—2982no—
312C2×C4—2982no—
313C2×C4—2984no—
314C2×C4—2982no—
315C2×C4—1492no—
316C2×C4—11921no—
317C2×C4D4×D14912yes—
318C2×C4D4×D14911yescharacteristic
319C2×C4C2×C22⋊D14912yes—
320C2×C4C23×D14911yescharacteristic
321C8—22no—
322C8—2981no—
323C8—2981no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—2982no—
334C22—2982no—
335C22—2982no—
336C22—2982no—
337C22—5961no—
338C22—5961no—
339C22—11928no—
340C22#1801187912yes—
341C22C2×D4×D14911yescharacteristic
342C4—41no—
343C4—2982no—
344C4—2982no—
345C4—5962no—
346C4#1801187912yes—
347C4C2×D4×D14911yescharacteristic
348C4C22×C22⋊D14911yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—5962no—
353C2—5962no—
354C2C2×#1801187911yescharacteristic
355C2#2375042911yescharacteristic
356C2#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#3012
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×#18011956C2semidirect
2. C2×#18011931C2semidirect
3. #23750203C2semidirect
4. #23749784C2semidirect
5. #23749774C2semidirect
6. #23749767C2semidirect
7. #23749757C2semidirect
8. #23749627C2semidirect
9. C2×#18011832C2semidirect
10. #23749593C2semidirect
11. C2×AGL(1,ℤ/8)×C149C2semidirect
22. C2×#14558726C22semidirect
23. C2×#14558724C22semidirect
24. C2×#14558723C22semidirect
25. C2×#14558721C22semidirect
26. C22×#11924746C22semidirect
28. #18011887C22semidirect
31. #18011882C22semidirect
32. C2×C4⋊Dic149C22semidirect
33. #18011836C22semidirect
34. #18011833C22semidirect
35. #18011832C22semidirect
36. #18011832C22semidirect
37. #18011823C22semidirect
38. #18011823C22semidirect
39. C2×#41×C149C22semidirect
42. C2×SD16×C149C22semidirect
43. C2×D8×C149C22semidirect
44. C2×M16×C149C22semidirect
49. #14558736D4semidirect
89. C2×D4×C149D4semidirect
178. C149#3012semidirect
181. C2×AGL(1,ℤ/8)D149semidirect
197. C2×#41C2×D149semidirect
207. C2×SD16C2×D149semidirect
209. C2×D8C2×D149semidirect
210. C2×M16C2×D149semidirect
252. C2×D4C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119562
C2×#180119312
#237502032
#237497842
#237497742
#237497672
#237497572
#237496272
C2×#180118322
#237495932
C2×AGL(1,ℤ/8)×C1492
#3012149