Finite Group Databasebeta

← #46821415

Subgroups — #46821415

Order 34432.

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

normal subgroups only

This group has 340 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
1#31771236C211yesmaximal, characteristic
2#31771233C211yesmaximal, characteristic
3#31771013C212yesmaximal, complemented
4#31771012C212yesmaximal, complemented
5#31771006C212yesmaximal, complemented
6#31771002C212yesmaximal, complemented
7#31770647C211yesmaximal, characteristic
8#31770641C211yesmaximal, characteristic
9C2×#22575384C211yesmaximal, characteristic
10#31770563C211yesmaximal, characteristic
11#428×C269C211yesmaximal, characteristic
12C22×#14524130C2211yescharacteristic
13#22575453—21no—
14#22575452—21no—
15#22575452C2212yes—
16#22575452C2212yes—
17#22575445—21no—
18#22575444—21no—
19#22575444C2212yes—
20#22575444C2212yes—
21#22575432—22no—
22#22575430—22no—
23#22575412C2211yescharacteristic
24#22575412C2211yescharacteristic
25C2×C4⋊Dic269C2211yescharacteristic
26#22575387C2212yes—
27#22575386C2211yescharacteristic
28#22575386C2211yescharacteristic
29#22575385C2212yes—
30#22575384C2212yes—
31#22575384C2212yes—
32#22575384C2211yescharacteristic
33#22575384C2211yescharacteristic
34#22575363C2212yes—
35C8×Dic269C2212yes—
36C2×M16×C269C2211yescharacteristic
37#117×C269C2211yescharacteristic
38C2×C4⋊C4×C269C2211yescharacteristic
39#107×C269C2212yes—
40#106×C269C2212yes—
41C2×Dic538—22no—
42C2×Dic538—22no—
43C22×Dic269—21no—
44C2×#14524130Q814yescomplemented
45C2×#14524130C2311yescharacteristic
46C2×#14524130C2311yescharacteristic
47C4⋊Dic269—22no—
48C4⋊Dic269—22no—
49C4⋊Dic269—21no—
50C4⋊Dic269D411yescharacteristic
51C4⋊Dic269D411yescharacteristic
52C4⋊Dic269C2312yes—
53C4⋊Dic269C2311yescharacteristic
54C4⋊Dic269C2311yescharacteristic
55#16864381—22no—
56#16864381—22no—
57#16864381—22no—
58#16864381—22no—
59C22⋊Dic269—21no—
60C22⋊Dic269D412yes—
61C4×Dic269—21no—
62C4×Dic269C2312yes—
63C22×C4×C269—21no—
64C22×C4×C269C2311yescharacteristic
65M16×C269Q814yescomplemented
66C2×C8×C269C2312yes—
67C4⋊C4×C269—21no—
68C4⋊C4×C269C2312yes—
69C4⋊C4×C269C2311yescharacteristic
70C4⋊C4×C269C2311yescharacteristic
71C22⋊C4×C269—21no—
72C42×C269—21no—
73Dic538—82no—
74Dic538—82no—
75C2×Dic269—21no—
76C2×Dic269—21no—
77C2×Dic269—22no—
78C2×Dic269—81no—
79C2×Dic269C2×D412yes—
80C2×Dic269C2×D412yes—
81#14524130C2×Q812yes—
82#14524130C2×Q812yes—
83C23×C269C2×D411yescharacteristic
84C2×C4×C269—21no—
85C2×C4×C269—21no—
86C2×C4×C269—22no—
87C2×C4×C269—81no—
88C2×C4×C269C2×D411yescharacteristic
89C2×C4×C269C2×Q812yes—
90C2×C4×C269C2×Q812yes—
91C2×C4×C269C2411yescharacteristic
92C8×C269C2×Q814yes—
93C22×C269—21no—
94C22×C269—21no—
95C22×C269C4⋊Q812yes—
96C22×C269C22×D411yescharacteristic
97C4×C269—41no—
98C4×C269—42no—
99C4×C269—41no—
100C4×C269C4⋊Q812yes—
101C4×C269C22×Q811yescharacteristic
102C4×C269C22×Q811yescharacteristic
103Dic269—22no—
104Dic269—22no—
105Dic269—41no—
106Dic269—42no—
107Dic269—41no—
108C2×C269—22no—
109C2×C269C2×C4⋊Q811yescharacteristic
110C2×C269#57711yescharacteristic
111C2×C269#57711yescharacteristic
112C269#316311yescharacteristic, complemented, Sylow
113#3163—2691nomaximal, Sylow
114#544—2691no—
115#544—2691no—
116#501—2692no—
117#501—2692no—
118#499—2692no—
119#498—2692no—
120#428—2691no—
121#428D26911yescharacteristic
122#427—2691no—
123C2×#106—2691no—
124#405—2691no—
125C2×M16C2×D26911yescharacteristic
126C22×C8—2691no—
127C4⋊Q8—2692no—
128C4⋊Q8—2692no—
129C4⋊Q8—5381no—
130C4⋊Q8—5381no—
131#125—2692no—
132#125—2692no—
133#125—5381no—
134#125—5381no—
135C22⋊Q8—5382no—
136C22⋊Q8—5382no—
137#117—2691no—
138#117—2691no—
139#117C2×D26911yescharacteristic
140C2×C4⋊C4—2691no—
141C2×C4⋊C4C2×D26911yescharacteristic
142#107—2692no—
143#107—2691no—
144#107—2691no—
145#107C2×D26912yes—
146#106—2692no—
147#106—2692no—
148#106—2692no—
149#106—2691no—
150#106—2691no—
151#106C2×D26912yes—
152#97—2692no—
153C4×C8—2692no—
154C2×Q8—5382no—
155C2×Q8—5382no—
156C22×C4—21no—
157C22×C4—5381no—
158C22×C4C22×D26911yescharacteristic
159M16Dic53814yescomplemented
160C2×C8—2694no—
161C2×C8—2691no—
162C2×C8—2691no—
163C2×C8C22×D26912yes—
164C4⋊C4—21no—
165C4⋊C4—2691no—
166C4⋊C4—2691no—
167C4⋊C4—2692no—
168C4⋊C4—2691no—
169C4⋊C4—2691no—
170C4⋊C4—5382no—
171C4⋊C4—5382no—
172C4⋊C4—5382no—
173C4⋊C4—5382no—
174C4⋊C4—5382no—
175C4⋊C4—5382no—
176C4⋊C4—5381no—
177C4⋊C4C22×D26912yes—
178C4⋊C4C22×D26911yescharacteristic
179C4⋊C4C22×D26911yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—2692no—
182C22⋊C4—5381no—
183C42—21no—
184C42—2692no—
185C42—5381no—
186C23D4×D26911yescharacteristic
187Q8—21522no—
188Q8—21522no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—2692no—
194C2×C4—2692no—
195C2×C4—5381no—
196C2×C4—5381no—
197C2×C4—5382no—
198C2×C4—21521no—
199C2×C4D4×D26911yescharacteristic
200C2×C4Q8×D26912yes—
201C2×C4C2×Dic53812yes—
202C2×C4C23×D26911yescharacteristic
203C8—2692no—
204C8—2692no—
205C8C2×Dic53814yes—
206C22—21no—
207C22—21no—
208C22#2257545212yes—
209C22C2×D4×D26911yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—5382no—
214C4—5382no—
215C4—10762no—
216C4—10761no—
217C4—10761no—
218C4#2257545212yes—
219C4C2×Q8×D26911yescharacteristic
220C4C22×Dic53811yescharacteristic
221C2—22no—
222C2C2×#2257545211yescharacteristic
223C2#3177141411yescharacteristic
224C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#428×C269C2
socleC22×C269C22×D4
prime radicalC23×C269C2×D4
hypercenter#428D269
hypocenterC269#3163
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #31771013C2semidirect
4. #31771012C2semidirect
5. #31771006C2semidirect
6. #31771002C2semidirect
44. C2×#14524130Q8semidirect
65. M16×C269Q8semidirect
112. C269#3163semidirect
159. M16Dic538semidirect

Maximal subgroups

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

subgroupindex
#317712362
#317712332
#317710132
#317710122
#317710062
#317710022
#317706472
#317706412
C2×#225753842
#317705632
#428×C2692
#3163269