Finite Group Databasebeta

← #85376

Subgroups — #85376

Order 384.

260 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 36 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×#984C211yesmaximal, characteristic, complemented
2C22×#226C211yesmaximal, characteristic, complemented
3#548⋊C3C211yesmaximal, characteristic
4#5425C211yesmaximal, characteristic, complemented
5#5419C211yesmaximal, characteristic, complemented
6C4⋊#226C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3270—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11#984—22no—
12C2×#226—22no—
13C2×#226C2211yescharacteristic
14C2×#226C2211yescharacteristic
15C2×#226C2211yescharacteristic
16#878—41nomaximal
17C4×SL(2,𝔽3)C2211yescharacteristic
18#859C2211yescharacteristic
19C2×#137—31no—
20C22×Q16—31no—
21D4×Q8—31no—
22#548D311yescharacteristic
23#497—31no—
24#470—31no—
25#465—31no—
26#464—31no—
27#462—31no—
28#452—31no—
29#451—31no—
30#451—31no—
31#448—31no—
32#441—31no—
33#435—31no—
34C22×Dic3—41no—
35C2×Dic6—41no—
36#41⋊C3—21no—
37C2×SL(2,𝔽3)—21no—
38C2×SL(2,𝔽3)C2311yescharacteristic
39#226—21no—
40#226—21no—
41#226D412yescomplemented
42C3×C22⋊C4—41no—
43#217—41no—
44C4⋊Dic3—41no—
45Dic3⋊C4—41no—
46Dic3⋊C4—41no—
47C2×#41C2×D311yescharacteristic
48C22×Q8—31no—
49C22×Q8—31no—
50C22×Q8C2×D311yescharacteristic
51#137—62no—
52#137—62no—
53C2×Q16—31no—
54C2×Q16—31no—
55C2×Q16—31no—
56C2×Q16—31no—
57C2×Q16—31no—
58C2×Q16—31no—
59C2×Q16—62no—
60C2×Q16—62no—
61C2×SD16—31no—
62C2×SD16—31no—
63C2×SD16—31no—
64C2×M16—31no—
65C22×C8—31no—
66C4⋊Q8—31no—
67C4⋊Q8—61no—
68#124—31no—
69C22⋊Q8—31no—
70C22⋊Q8—31no—
71C22⋊Q8—31no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74#121—31no—
75C4×Q8—31no—
76C4×Q8C2×D311yescharacteristic
77C4×D4—31no—
78C4×D4—61no—
79#106—31no—
80C4⋊C8—31no—
81#103—31no—
82#103—31no—
83#103—31no—
84#103—31no—
85#103—31no—
86#102—31no—
87C22⋊C8—31no—
88C22⋊C8—31no—
89#97—31no—
90C23×C3—41no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93C2×Dic3—41no—
94C2×Dic3—41no—
95C2×Dic3—41no—
96C2×Dic3—41no—
97C2×Dic3—82no—
98Dic6—82no—
99SL(2,𝔽3)C2×D411yescharacteristic
100#41—21no—
101#41—61no—
102C2×Q8—21no—
103C2×Q8—31no—
104C2×Q8—31no—
105C2×Q8—31no—
106C2×Q8—31no—
107C2×Q8—61no—
108C2×Q8—61no—
109C2×Q8—61no—
110C2×Q8—62no—
111C2×Q8—62no—
112C2×Q8—121no—
113C2×Q8—121no—
114C2×Q8C22×D311yescharacteristic
115C2×D4—31no—
116C2×D4—31no—
117C22×C4—31no—
118C22×C4—31no—
119C22×C4—61no—
120C22×C4—61no—
121Q16—32no—
122Q16—32no—
123Q16—62no—
124Q16—61no—
125Q16—62no—
126Q16—61no—
127Q16—61no—
128Q16—61no—
129Q16—121no—
130SD16—62no—
131SD16—62no—
132SD16—121no—
133M16—62no—
134C2×C8—31no—
135C2×C8—31no—
136C2×C8—31no—
137C2×C8—31no—
138C2×C8—62no—
139C4⋊C4—31no—
140C4⋊C4—31no—
141C4⋊C4—31no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—61no—
147C22⋊C4—31no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4S411yescharacteristic
152C42—31no—
153C42—61no—
154C22×C3—42no—
155C22×C3—41no—
156C22×C3—81no—
157C22×C3—81no—
158C3×C4—81no—
159C3×C4—81no—
160Dic3—42no—
161Dic3—81no—
162Dic3—81no—
163Dic3—81no—
164C23—31no—
165C23C2×S411yescharacteristic
166Q8—31no—
167Q8—32no—
168Q8—61no—
169Q8—61no—
170Q8—61no—
171Q8—61no—
172Q8—61no—
173Q8—61no—
174Q8—61no—
175Q8—122no—
176Q8—122no—
177Q8D3×D411yescharacteristic
178D4—61no—
179D4—61no—
180D4—62no—
181C2×C4—31no—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—122no—
193C2×C4—122no—
194C2×C4C2×S411yescharacteristic
195C2×C4C2×S411yescharacteristic
196C8—32no—
197C8—61no—
198C8—61no—
199C8—61no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—82no—
204C22—21no—
205C22—21no—
206C22—61no—
207C22—61no—
208C22—61no—
209C22#98412yescomplemented
210C22C22×S411yescharacteristic
211C4—21no—
212C4—21no—
213C4—31no—
214C4—31no—
215C4—61no—
216C4—62no—
217C4—61no—
218C4—121no—
219C4—121no—
220C4—121no—
221C3—41noSylow
222C2—22no—
223C2—121no—
224C2D4×S411yescharacteristic
225C2C2×#98411yescharacteristic
226C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#548D3
prime radical#548⋊C3C2
hypercenterC22⋊C4S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#984C2semidirect
2. C22×#226C2semidirect
4. #5425C2semidirect
5. #5419C2semidirect
6. C4⋊#226C2semidirect
7. #5388C2semidirect
41. #226D4semidirect
209. C22#984semidirect

Maximal subgroups

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

subgroupindex
C2×#9842
C22×#2262
#548⋊C32
#54252
#54192
C4⋊#2262
#53882
#32703
#8784