Finite Group Databasebeta

← #34057405

Subgroups — #34057405

Order 19072.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 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
1Q8×D596C211yesmaximal, characteristic, complemented
2#23749984C212yesmaximal, complemented
3#23749975C211yesmaximal, characteristic, complemented
4#23749920C212yesmaximal, complemented
5#23749905C211yesmaximal, characteristic, complemented
6#23749837C212yesmaximal
7#23749827C211yesmaximal, characteristic, complemented
8#23749756C211yesmaximal, characteristic, complemented
9#23749723C212yesmaximal, complemented
10C4×#14558721C211yesmaximal, characteristic
11#556×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149—21no—
13C2×#14558723C2212yes—
14C2×#14558721C2211yescharacteristic
15#18011911C2211yescharacteristic
16#18011911C2212yes—
17#18011891—21no—
18#18011891—22no—
19C4×D596C2211yescharacteristic
20C4×D596C2212yes—
21#18011844C2212yes—
22#18011844C2211yescharacteristic
23C4⋊#11924746C2211yescharacteristic
24C4⋊#11924746C2212yes—
25#18011835C2212yes—
26#18011835C2212yes—
27#18011835C2212yes—
28#18011832C2212yes—
29#18011832C2211yescharacteristic
30#18011831C2211yescharacteristic
31#18011822C2212yes—
32C4×#11924746C2211yescharacteristic
33C4⋊Q8×C149C2211yescharacteristic
34#126×C149—22no—
35#124×C149C2212yes—
36C22⋊Q8×C149—21no—
37C4×Q8×C149C2212yes—
38C4×Q8×C149C2211yescharacteristic
39C4×D4×C149C2211yescharacteristic
40#117×C149—21no—
41C2×Dic298—22no—
42C2×Dic298—21no—
43Q8×D149—44no—
44C2×D596C2311yescharacteristic
45C2×C4×D149—22no—
46C2×C4×D149—21no—
47#14558723—24no—
48#14558721—22no—
49C2×#11924746C2311yescharacteristic
50C2×#11924746C2312yes—
51C2×#11924746C2311yescharacteristic
52C4⋊Dic149—22no—
53C4⋊Dic149—21no—
54C4⋊Dic149C2312yes—
55C4⋊Dic149C2311yescharacteristic
56#14558713—22no—
57#14558713—21no—
58C2×Q8×C149D411yescharacteristic
59C2×Q8×C149C2312yes—
60C2×D4×C149C2311yescharacteristic
61C22×C4×C149—21no—
62C4⋊C4×C149—21no—
63C4⋊C4×C149—22no—
64C4⋊C4×C149—21no—
65C4⋊C4×C149D411yescharacteristic
66C4⋊C4×C149D412yes—
67C4⋊C4×C149C2311yescharacteristic
68C22⋊C4×C149—22no—
69C22⋊C4×C149—21no—
70C22⋊C4×C149—22no—
71C42×C149—22no—
72C42×C149C2311yescharacteristic
73C42×C149C2312yes—
74C22×D149—21no—
75Dic298—44no—
76Dic298—42no—
77D596—21no—
78D596#4112yes—
79C2×Dic149—22no—
80C2×Dic149—21no—
81C4×D149—44no—
82C4×D149—42no—
83#11924746—21no—
84#11924746—22no—
85#11924746—21no—
86C23×C149—21no—
87Q8×C149—22no—
88Q8×C149—22no—
89Q8×C149—22no—
90D4×C149—21no—
91D4×C149—21no—
92C2×C4×C149—21no—
93C2×C4×C149—22no—
94C2×C4×C149—42no—
95C2×C4×C149C2×D412yes—
96C2×C4×C149C2×D412yes—
97C2×C4×C149C2×D411yescharacteristic
98C2×C4×C149C2×D411yescharacteristic
99C2×C4×C149C2411yescharacteristic
100C22×C149—41no—
101C22×C149—41no—
102C22×C149—41no—
103C22×C149C22×D411yescharacteristic
104C2×D149—22no—
105C2×D149—41no—
106C2×D149—41no—
107C4×C149—22no—
108C4×C149—22no—
109C4×C149—21no—
110C4×C149—42no—
111C4×C149—41no—
112C4×C149AGL(1,ℤ/8)12yes—
113C4×C149C2×#4111yescharacteristic
114C4×C1492−1+411yescharacteristic
115Dic149—42no—
116Dic149—41no—
117C2×C149—81no—
118C2×C149#54811yescharacteristic
119C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C149#57811yescharacteristic
121D149—42no—
122C149#336711yescharacteristic, complemented, Sylow
123#3367—1491nomaximal, Sylow
124#556D14911yescharacteristic, complemented
125D4×Q8—1491no—
126#489—1492no—
127#487—1491no—
128#477—1492no—
129#474—1491no—
130#463—1492no—
131#461—1491no—
132#446—1491no—
133#440—1492no—
134C4×D8—1491no—
135C22×Q8—2981no—
136C2×SD16—1492no—
137C2×D8—1491no—
138C4⋊Q8—1491no—
139C4⋊Q8—1492no—
140C4⋊Q8C2×D14911yescharacteristic
141#126—22no—
142#124C2×D14912yes—
143C22⋊Q8—21no—
144C22⋊Q8—2981no—
145C22⋊Q8—2982no—
146C4×Q8C2×D14912yes—
147C4×Q8C2×D14911yescharacteristic
148C4×D4—1491no—
149C4×D4—1492no—
150C4×D4C2×D14911yescharacteristic
151#117—21no—
152#107—1491no—
153#107—1492no—
154C4⋊C8—1491no—
155C4⋊C8—1492no—
156#103—1492no—
157#103—1492no—
158#103—1492no—
159#102—1491no—
160#102—1492no—
161#102—1491no—
162#97—1492no—
163C4×C8—1491no—
164C2×Q8—2982no—
165C2×Q8—2981no—
166C2×Q8—5964no—
167C2×Q8C22⋊D14911yescharacteristic
168C2×Q8C22×D14912yes—
169C2×D4—1491no—
170C2×D4C22×D14911yescharacteristic
171C22×C4—21no—
172C22×C4—2982no—
173C22×C4—2981no—
174SD16—2984no—
175D8—2982no—
176C2×C8—1491no—
177C2×C8—1491no—
178C2×C8—1492no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—2982no—
183C4⋊C4—2981no—
184C4⋊C4—1491no—
185C4⋊C4—1492no—
186C4⋊C4C22⋊D14911yescharacteristic
187C4⋊C4C22⋊D14912yes—
188C4⋊C4C22×D14911yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—2982no—
193C22⋊C4—2981no—
194C42—22no—
195C42C22×D14911yescharacteristic
196C42C22×D14912yes—
197C23—21no—
198C23—2981no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—5964no—
203Q8—5962no—
204D4—21no—
205D4—21no—
206D4—2981no—
207D4—1492no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—2982no—
212C2×C4—2981no—
213C2×C4—5964no—
214C2×C4—5962no—
215C2×C4C2×C22⋊D14912yes—
216C2×C4C2×C22⋊D14911yescharacteristic
217C2×C4C2×C22⋊D14912yes—
218C2×C4C2×C22⋊D14911yescharacteristic
219C2×C4C23×D14911yescharacteristic
220C8—2981no—
221C8—2982no—
222C8—2981no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—2982no—
227C22—5961no—
228C22—5961no—
229C22C22×C22⋊D14911yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—5962no—
236C4—5961no—
237C4#1801193612yes—
238C4#41×D14911yescharacteristic
239C4#1801195811yescharacteristic
240C2—81no—
241C2—5962no—
242C2#2375028211yescharacteristic
243C2C2×#1801193611yescharacteristic
244C2#2375044011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#556×C149C2
socleC22×C149C22×D4
prime radicalC2×#14558721C22
hypercenter#556D149
hypocenterC149#3367
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. Q8×D596C2semidirect
2. #23749984C2semidirect
3. #23749975C2semidirect
4. #23749920C2semidirect
5. #23749905C2semidirect
7. #23749827C2semidirect
8. #23749756C2semidirect
9. #23749723C2semidirect
11. #556×C149C2semidirect
122. C149#3367semidirect
124. #556D149semidirect

Maximal subgroups

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

subgroupindex
Q8×D5962
#237499842
#237499752
#237499202
#237499052
#237498372
#237498272
#237497562
#237497232
C4×#145587212
#556×C1492
#3367149