Finite Group Databasebeta

← #83414

Subgroups — #83414

Order 384.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C3×#536C211yesmaximal, characteristic
2#5518C211yesmaximal, characteristic, complemented
3#5239C212yesmaximal
4#5228C212yesmaximal, complemented
5#5092C211yesmaximal, characteristic, complemented
6#5090C212yesmaximal, complemented
7#5082C211yesmaximal, characteristic, complemented
8#5060C211yesmaximal, characteristic, complemented
9#5056C211yesmaximal, characteristic, complemented
10#5054C212yesmaximal, complemented
11#5000C211yesmaximal, characteristic, complemented
12#3223—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4⋊Q8C2211yescharacteristic
17C3×#124C2211yescharacteristic
18C3×#124C2211yescharacteristic
19C3×#123—22no—
20C3×C22⋊Q8—22no—
21C3×#117C2211yescharacteristic
22C2×#216C2212yes—
23C2×#216C2212yes—
24C2×#216C2212yes—
25C2×#214C2212yes—
26#924C2212yes—
27#889—22no—
28#879—22no—
29#870C2211yescharacteristic
30#869C2211yescharacteristic
31C4×Dic6C2212yes—
32#835C2212yes—
33#835C2212yes—
34#835C2212yes—
35#832C2212yes—
36#830C2212yes—
37C4⋊(C3⋊C8)C2212yes—
38#803C2211yescharacteristic
39C4×C3⋊C8C2211yescharacteristic
40#541—31no—
41#536D311yescharacteristic
42#497—31no—
43#494—31no—
44#491—31no—
45#488—31no—
46#464—32no—
47#462—32no—
48#452—32no—
49#451—32no—
50#433—31no—
51C3×#41—42no—
52C2×C3×Q8—21no—
53C2×C3×Q8—42no—
54C2×C3×Q8C2311yescharacteristic
55C2×C3×Q8C2311yescharacteristic
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C2×Dic6C2312yes—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4D412yes—
66C3×C22⋊C4—22no—
67C3×C22⋊C4—22no—
68C3×C22⋊C4D412yes—
69C3×C42C2311yescharacteristic
70C3×C42C2311yescharacteristic
71#217—22no—
72#216—42no—
73#216—42no—
74#216—42no—
75#214—42no—
76C4⋊Dic3—22no—
77C4⋊Dic3C2312yes—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81C4×Dic3—22no—
82C2×C3⋊C8C2312yes—
83C2×C3⋊C8C2312yes—
84C2×#41C2×D311yescharacteristic
85C22×Q8C2×D311yescharacteristic
86C2×Q16—32no—
87C2×Q16—32no—
88C2×Q16—32no—
89C2×SD16—32no—
90C4⋊Q8—31no—
91C4⋊Q8C2×D311yescharacteristic
92C4⋊Q8C2×D311yescharacteristic
93#126—62no—
94#125—31no—
95#125—62no—
96#124C2×D311yescharacteristic
97#124C2×D311yescharacteristic
98#123—22no—
99C22⋊Q8—22no—
100C22⋊Q8—32no—
101C4×Q8—32no—
102#117C2×D311yescharacteristic
103C4⋊C8—32no—
104#103—32no—
105#103—32no—
106#103—32no—
107#102—32no—
108C22⋊C8—32no—
109#97—31no—
110C4×C8—31no—
111C23×C3—21no—
112C23×C3C2×D411yescharacteristic
113C3×Q8—22no—
114C3×Q8—22no—
115C3×Q8—22no—
116C3×Q8—42no—
117C3×D4—22no—
118C3×D4—42no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
125C2×C3×C4C2×D412yes—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D412yes—
128C2×C3×C4C2411yescharacteristic
129C2×Dic3—22no—
130C2×Dic3—22no—
131Dic6—22no—
132Dic6—22no—
133C3⋊C8—22no—
134C3⋊C8—22no—
135#41—42no—
136C2×Q8—21no—
137C2×Q8—32no—
138C2×Q8—42no—
139C2×Q8C22×D311yescharacteristic
140C2×Q8C22×D311yescharacteristic
141C2×Q8C22×D311yescharacteristic
142C2×D4—21no—
143C2×D4C22×D311yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C22×D311yescharacteristic
147Q16—122no—
148Q16—122no—
149Q16—122no—
150SD16—122no—
151C2×C8—32no—
152C2×C8—32no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—32no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4#6712yes—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—62no—
164C22⋊C4#6712yes—
165C42—62no—
166C42C22×D311yescharacteristic
167C42C22×D311yescharacteristic
168C22×C3—21no—
169C22×C3—22no—
170C22×C3—41no—
171C22×C3—42no—
172C22×C3C22×D411yescharacteristic
173C3×C4—21no—
174C3×C4—22no—
175C3×C4—22no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4H(5,𝔽2)12yes—
180Dic3—42no—
181Dic3—42no—
182C23—21no—
183C23C2×#6711yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—22no—
187Q8—42no—
188Q8—62no—
189Q8—62no—
190D4—22no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4C2×#6712yes—
201C2×C4C2×#6712yes—
202C2×C4C2×#6711yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—62no—
205C8—62no—
206C2×C3—41no—
207C2×C3—81no—
208C2×C3#53511yescharacteristic
209C2×C3#57812yes—
210C22—21no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22C22×#6711yescharacteristic
215C4—21no—
216C4—22no—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—122no—
222C4—122no—
223C4#100412yes—
224C3#322311yescharacteristic, complemented, Sylow
225C2—41no—
226C2—81no—
227C2#580511yescharacteristic
228C2#583812yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#536D3
hypocenterC3#3223
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #5518C2semidirect
4. #5228C2semidirect
5. #5092C2semidirect
6. #5090C2semidirect
7. #5082C2semidirect
8. #5060C2semidirect
9. #5056C2semidirect
10. #5054C2semidirect
11. #5000C2semidirect
224. C3#3223semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#55182
#52392
#52282
#50922
#50902
#50822
#50602
#50562
#50542
#50002
#32233