Finite Group Databasebeta

← D4×#14558723 #34056387

Subgroups — D4×#14558723 #34056387

Order 19072.

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

normal subgroups only

This group has 546 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
1C22×#14558723C212yesmaximal, complemented
2D4×D596C211yesmaximal, characteristic
3#23749999C211yesmaximal, characteristic, complemented
4#23749856C212yesmaximal, complemented
5#23749827C211yesmaximal, characteristic, complemented
6#23749825C211yesmaximal, characteristic, complemented
7#23749775C212yesmaximal, complemented
8#23749764C212yesmaximal, complemented
9C4×#14558723C211yesmaximal, characteristic, complemented
10D4×#11924746C211yesmaximal, characteristic, complemented
11D4×Q8×C149C211yesmaximal, characteristic, complemented
12C22×D596C2212yes—
13C2×D4×D149—21no—
14C2×#14558723—28no—
15C2×#14558723C2212yes—
16C2×#14558723C2211yescharacteristic
17C2×#14558723C2212yescomplemented
18C2×#14558723C2212yescomplemented
19C2×#14558723C2212yescomplemented
20C22×#11924746C2212yescomplemented
21#18011908C2211yescharacteristic
22#18011886—21no—
23#18011885C2212yes—
24#18011881—22no—
25C4×D596C2211yescharacteristic
26#18011842C2211yescharacteristic, complemented
27C4⋊#11924746C2211yescharacteristic, complemented
28#18011835C2211yescharacteristic, complemented
29#18011835C2212yescomplemented
30#18011831C2211yescharacteristic
31#18011831C2212yescomplemented
32C22⋊#11924746C2212yescomplemented
33C4×#11924746C2211yescharacteristic, complemented
34C22×Q8×C149C2212yescomplemented
35C4⋊Q8×C149—21no—
36C4⋊Q8×C149C2211yescharacteristic, complemented
37C22⋊Q8×C149—22no—
38C22⋊Q8×C149C2212yescomplemented
39C4×Q8×C149C2211yescharacteristic, complemented
40C4×D4×C149—21no—
41C4×D4×C149C2211yescharacteristic
42C23×D149—22no—
43C2×D596—24no—
44C2×D596—21no—
45C2×D596—22no—
46C2×D596—21no—
47C2×D596—42no—
48C2×D596C2312yes—
49C2×D596C2311yescharacteristic
50C2×C22⋊D149—22no—
51D4×D149—44no—
52C2×C4×D149—21no—
53#14558723—24no—
54#14558723—24no—
55#14558723—24no—
56#14558723—42no—
57#14558723D414yesdirect product factor
58C2×#11924746—24no—
59C2×#11924746C2312yes—
60C2×#11924746C2311yescharacteristic
61C2×#11924746C2311yescharacteristic
62C4⋊Dic149C2311yescharacteristic
63#14558713—21no—
64#14558713—22no—
65C2×Q8×C149—22no—
66C2×Q8×C149—24no—
67C2×Q8×C149—42no—
68C2×Q8×C149C2312yes—
69C2×Q8×C149C2311yescharacteristic
70C2×D4×C149D411yescharacteristic
71C22×C4×C149—22no—
72C22×C4×C149C2312yes—
73C4⋊C4×C149—21no—
74C4⋊C4×C149—21no—
75C4⋊C4×C149—21no—
76C4⋊C4×C149—22no—
77C4⋊C4×C149D411yescharacteristic
78C4⋊C4×C149C2311yescharacteristic
79C22⋊C4×C149—22no—
80C22⋊C4×C149D412yes—
81C42×C149—21no—
82C42×C149C2311yescharacteristic
83C22×D149—21no—
84C22×D149—22no—
85C22×D149—44no—
86C22×D149—44no—
87C22×D149—82no—
88D596—22no—
89D596—21no—
90D596—22no—
91D596—44no—
92D596—42no—
93D596—81no—
94D596C2×D412yes—
95C22⋊D149—44no—
96C2×Dic149—21no—
97C4×D149—42no—
98#11924746—22no—
99#11924746—21no—
100#11924746C2×D412yescomplemented
101C23×C149C2×D412yes—
102Q8×C149—21no—
103Q8×C149—22no—
104Q8×C149—22no—
105Q8×C149—44no—
106Q8×C149C2×D412yescomplemented
107D4×C149SD1614yescomplemented
108C2×C4×C149—21no—
109C2×C4×C149—22no—
110C2×C4×C149—22no—
111C2×C4×C149—21no—
112C2×C4×C149—22no—
113C2×C4×C149—44no—
114C2×C4×C149C2×D411yescharacteristic
115C2×C4×C149C2×D412yes—
116C2×C4×C149C2×D411yescharacteristic
117C2×C4×C149C2411yescharacteristic
118C22×C149—22no—
119C22×C149—22no—
120C22×C149C2×SD1614yescomplemented
121C22×C149C22×D411yescharacteristic
122C2×D149—22no—
123C2×D149—41no—
124C2×D149—41no—
125C2×D149—42no—
126C2×D149—42no—
127C2×D149—42no—
128C2×D149—88no—
129C4×C149—22no—
130C4×C149—22no—
131C4×C149—21no—
132C4×C149—42no—
133C4×C149—41no—
134C4×C149C2×SD1612yescomplemented
135C4×C149C22×D411yescharacteristic
136C4×C149H(5,𝔽2)11yescharacteristic
137Dic149—41no—
138C2×C149—24no—
139C2×C149D4211yescharacteristic
140C2×C149C22×SD1611yescharacteristic
141C2×C149#57711yescharacteristic
142D149—42no—
143D149—82no—
144C149D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1491nomaximal, Sylow
146C22×SD16—1492no—
147D4×Q8D14911yescharacteristic, complemented
148D42—1491no—
149#492—1491no—
150#465—1492no—
151#461—1491no—
152#460—1491no—
153#450—1492no—
154#448—1492no—
155C4×SD16—1491no—
156C8×D4—1491no—
157C22×Q8C2×D14912yescomplemented
158C22×D4—2981no—
159C22×D4—1492no—
160C2×SD16—2988no—
161C2×SD16—1492no—
162C2×SD16—1492no—
163C2×SD16—1492no—
164C2×SD16—1492no—
165C2×SD16—1491no—
166C22×C8—1492no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D14911yescharacteristic, complemented
169#127—1491no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D14912yescomplemented
172#121—2981no—
173#121—1492no—
174#120—2982no—
175C4×Q8C2×D14911yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—1491no—
178C4×D4C2×D14911yescharacteristic
179#106—1491no—
180C4⋊C8—1491no—
181#103—1491no—
182#103—1492no—
183#102—1491no—
184#102—1492no—
185C22⋊C8—1492no—
186C4×C8—1491no—
187C24—2982no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D14912yes—
192C2×Q8C22×D14911yescharacteristic
193C2×D4—2984no—
194C2×D4—2982no—
195C2×D4—2982no—
196C2×D4—2981no—
197C2×D4—2981no—
198C2×D4—5962no—
199C2×D4—5964no—
200C2×D4—1492no—
201C2×D4—1491no—
202C2×D4C22⋊D14911yescharacteristic
203C22×C4—22no—
204C22×C4—2981no—
205C22×C4C22×D14912yes—
206SD16—2984no—
207SD16—2984no—
208SD16—2984no—
209SD16—5962no—
210SD16—1494no—
211C2×C8—2984no—
212C2×C8—1491no—
213C2×C8—1492no—
214C2×C8—1491no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—1491no—
220C4⋊C4C22⋊D14911yescharacteristic
221C4⋊C4C22×D14911yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—2981no—
224C22⋊C4—2982no—
225C22⋊C4C22⋊D14912yes—
226C42—21no—
227C42C22×D14911yescharacteristic
228C23—2981no—
229C23—2982no—
230C23—5964no—
231C23—5964no—
232C23—11922no—
233C23C2×C22⋊D14912yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D14912yescomplemented
239D4—2982no—
240D4—2981no—
241D4—2982no—
242D4—5964no—
243D4—5964no—
244D4—5962no—
245D4—1492no—
246D4—11921no—
247D4#1455872314yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—2981no—
255C2×C4—5962no—
256C2×C4C2×C22⋊D14911yescharacteristic
257C2×C4C2×C22⋊D14912yes—
258C2×C4C2×C22⋊D14911yescharacteristic
259C2×C4C23×D14911yescharacteristic
260C8—2982no—
261C8—2981no—
262C8—1492no—
263C22—22no—
264C22—22no—
265C22—2982no—
266C22—5961no—
267C22—5961no—
268C22—5962no—
269C22—5962no—
270C22—5962no—
271C22—11928no—
272C22C2×#1455872314yescomplemented
273C22C22×C22⋊D14911yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—5961no—
280C4C2×#1455872312yescomplemented
281C4C2×D4×D14911yescharacteristic
282C4#1801195611yescharacteristic
283C2—24no—
284C2—5962no—
285C2—11922no—
286C2D4×C22⋊D14911yescharacteristic
287C2C22×#1455872311yescharacteristic
288C2#2375043311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingD4×Q8×C149C2
socleC22×C149C22×D4
prime radicalD4×D596C2
hypercenterD4×Q8D149
hypocenterC149D4×SD16
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14558723C2semidirect
3. #23749999C2semidirect
4. #23749856C2semidirect
5. #23749827C2semidirect
6. #23749825C2semidirect
7. #23749775C2semidirect
8. #23749764C2semidirect
9. C4×#14558723C2semidirect
10. D4×#11924746C2semidirect
11. D4×Q8×C149C2semidirect
17. C2×#14558723C22semidirect
18. C2×#14558723C22semidirect
19. C2×#14558723C22semidirect
20. C22×#11924746C22semidirect
26. #18011842C22semidirect
27. C4⋊#11924746C22semidirect
28. #18011835C22semidirect
29. #18011835C22semidirect
31. #18011831C22semidirect
32. C22⋊#11924746C22semidirect
33. C4×#11924746C22semidirect
34. C22×Q8×C149C22semidirect
36. C4⋊Q8×C149C22semidirect
38. C22⋊Q8×C149C22semidirect
39. C4×Q8×C149C22semidirect
57. #14558723D4direct (N × H)
100. #11924746C2×D4semidirect
106. Q8×C149C2×D4semidirect
107. D4×C149SD16semidirect
120. C22×C149C2×SD16semidirect
134. C4×C149C2×SD16semidirect
144. C149D4×SD16semidirect
147. D4×Q8D149semidirect
157. C22×Q8C2×D149semidirect
168. C4⋊Q8C2×D149semidirect
171. C22⋊Q8C2×D149semidirect
175. C4×Q8C2×D149semidirect
238. Q8D4×D149semidirect
247. D4#14558723direct (N × H)
272. C22C2×#14558723semidirect
280. C4C2×#14558723semidirect

Maximal subgroups

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

subgroupindex
C22×#145587232
D4×D5962
#237499992
#237498562
#237498272
#237498252
#237497752
#237497642
C4×#145587232
D4×#119247462
D4×Q8×C1492
D4×SD16149