Finite Group Databasebeta

← #85828⋊C3 #11923195

Subgroups — #85828⋊C3 #11923195

Order 1152.

553 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 553 conjugacy classes of subgroups — too many to draw. Only its 95 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×C4×#41⋊C3C211yesmaximal, characteristic
2#5827⋊C3C212yesmaximal
3#5570⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×Dic6C212yesmaximal, complemented
5#85828C311yesmaximal, characteristic, complemented
6#3503⋊C3—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8C3×#869—41nomaximal
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11Dic3×SL(2,𝔽3)C2214yes—
12C3×C4×#41C2×C311yescharacteristic
13#5827C2×C312yes—
14#5570C2×C312yes—
15Q8×Dic6C2×C312yes—
16#5543—32no—
17D4×Dic6—32no—
18#5519—32no—
19#5470—31no—
20C2×#869—31no—
21Q8×SL(2,𝔽3)—32no—
22#557⋊C3—32no—
23#520⋊C3—32no—
24C4×#41⋊C3—21no—
25C4×#41⋊C3D311yescharacteristic
26C2×C3×Dic6—42no—
27C3×#41⋊C3Q812yescomplemented
28C2×C3×SL(2,𝔽3)C2311yescharacteristic
29C32×C42—41no—
30C3×C4⋊Dic3—42no—
31C3×C4⋊Dic3—42no—
32#3503—31noSylow
33C2×C3×#41C22×C311yescharacteristic
34C22×Dic6—32no—
35C2×#41⋊C3—21no—
36C2×#41⋊C3C2×D311yescharacteristic
37C3×C4⋊Q8—121no—
38C3×C4×Q8C22×C311yescharacteristic
39C3×C4×Q8C22×C311yescharacteristic
40C3×C4×D4—31no—
41C3×C4×D4—31no—
42C3×#117—31no—
43C2×C3×C42—31no—
44Q8×Dic3C22×C314yes—
45D4×Dic3—34no—
46#926—32no—
47C2×C4⋊Dic3—32no—
48C2×C4⋊Dic3—32no—
49#924—32no—
50#924—32no—
51#890—34no—
52#888—34no—
53#878—38no—
54#870—32no—
55#869—32no—
56#869—32no—
57#869—31no—
58#869—32no—
59#869A411yescharacteristic
60C4×Dic6—32no—
61C4×Dic6—32no—
62C4×SL(2,𝔽3)—21no—
63C4×SL(2,𝔽3)—21no—
64C4×SL(2,𝔽3)—34no—
65C4×SL(2,𝔽3)C2×D311yescharacteristic
66C4×SL(2,𝔽3)C2×D311yescharacteristic
67C2×C32×C4—41no—
68C2×C32×C4—41no—
69C2×C32×C4—41no—
70C2×C3×Dic3—44no—
71C3×Dic6—84no—
72C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
73Q82—32no—
74#557—32no—
75#554—92no—
76D4×Q8—92no—
77#544—92no—
78#533—91no—
79C2×C4⋊Q8—91no—
80#520—32no—
81C4×#41C3×D311yescharacteristic
82C3×#41—32no—
83C3×#41C3×Q812yescomplemented
84C2×C3×Q8—122no—
85C2×C3×Q8C23×C311yescharacteristic
86C2×C3×D4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×Dic3—34no—
91C2×Dic6—32no—
92C2×Dic6—34no—
93C2×Dic6—124no—
94C2×Dic6C2×A412yes—
95#41⋊C3—22no—
96#41⋊C3Dic612yescomplemented
97C2×SL(2,𝔽3)—21no—
98C2×SL(2,𝔽3)C22×D311yescharacteristic
99C3×C4⋊C4—31no—
100C3×C4⋊C4—31no—
101C3×C4⋊C4—122no—
102C3×C4⋊C4—122no—
103C3×C22⋊C4—32no—
104C3×C42—31no—
105C3×C42—31no—
106C3×C42—31no—
107C3×C42—41no—
108C3×C42—81no—
109C3×C42C2×A411yescharacteristic
110#217—34no—
111C4⋊Dic3—34no—
112C4⋊Dic3—32no—
113C4⋊Dic3—32no—
114C4⋊Dic3—34no—
115C4⋊Dic3C2×A412yes—
116C4⋊Dic3C2×A412yes—
117Dic3⋊C4—34no—
118Dic3⋊C4—34no—
119C4×Dic3—34no—
120C22×C32—41no—
121C32×C4—42no—
122C32×C4—42no—
123C32×C4—42no—
124C3×Dic3—84no—
125C2×#41C2×C3×D311yescharacteristic
126C22×Q8—92no—
127C4⋊Q8—31no—
128C4⋊Q8—92no—
129C4⋊Q8—92no—
130C4⋊Q8—92no—
131C4⋊Q8—94no—
132C4⋊Q8—91no—
133#125—92no—
134#125—94no—
135C22⋊Q8—98no—
136C22⋊Q8—92no—
137C22⋊Q8—92no—
138C4×Q8—34no—
139C4×Q8—92no—
140C4×Q8—92no—
141C4×Q8C2×C3×D311yescharacteristic
142C4×Q8C2×C3×D311yescharacteristic
143C4×D4—31no—
144C4×D4—31no—
145C4×D4—94no—
146#117—31no—
147#117—92no—
148C2×C4⋊C4—92no—
149C2×C4⋊C4—92no—
150C2×C42—31no—
151C23×C3—31no—
152C3×Q8—31no—
153C3×Q8—244no—
154C3×Q8C2×C3×Q811yescharacteristic
155C3×D4—32no—
156C3×D4—32no—
157C2×C3×C4—31no—
158C2×C3×C4—32no—
159C2×C3×C4—32no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—62no—
166C2×C3×C4—62no—
167C2×C3×C4—81no—
168C2×C3×C4—81no—
169C2×C3×C4—81no—
170C2×C3×C4—124no—
171C2×C3×C4C22×A411yescharacteristic
172C2×C3×C4C22×A411yescharacteristic
173C2×C3×C4C22×A411yescharacteristic
174C2×Dic3—34no—
175C2×Dic3—124no—
176C2×Dic3C22×A414yes—
177Dic6—24no—
178Dic6—64no—
179Dic6—124no—
180SL(2,𝔽3)—21no—
181SL(2,𝔽3)C2×Dic611yescharacteristic, complemented
182C2×C32—41no—
183C2×C32—41no—
184C2×C32—41no—
185#41—32no—
186#41C3×Dic612yescomplemented
187C2×Q8—32no—
188C2×Q8—94no—
189C2×Q8—92no—
190C2×Q8—364no—
191C2×Q8C22×C3×D311yescharacteristic
192C2×D4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—94no—
197C4⋊C4—31no—
198C4⋊C4—31no—
199C4⋊C4—32no—
200C4⋊C4—32no—
201C4⋊C4—94no—
202C4⋊C4—94no—
203C4⋊C4—94no—
204C4⋊C4—94no—
205C4⋊C4—92no—
206C4⋊C4—92no—
207C22⋊C4—32no—
208C22⋊C4—94no—
209C42—31no—
210C42—31no—
211C42—31no—
212C42—94no—
213C42D3×A411yescharacteristic
214C22×C3—32no—
215C22×C3—41no—
216C22×C3—61no—
217C22×C3—61no—
218C22×C3—81no—
219C22×C3C23×A411yescharacteristic
220C3×C4—31no—
221C3×C4—31no—
222C3×C4—42no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—61no—
226C3×C4—61no—
227C3×C4—82no—
228C3×C4—82no—
229C3×C4—82no—
230C3×C4—244no—
231C3×C4Q8×A412yes—
232C3×C4#99412yes—
233C3×C42−1+4⋊C312yes—
234Dic3—24no—
235Dic3—64no—
236C32—41noSylow
237C23—31no—
238Q8—31no—
239Q8—64no—
240Q8—184no—
241Q8—364no—
242Q8C2×C3×Dic611yescharacteristic
243D4—32no—
244D4—32no—
245C2×C4—31no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—34no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—94no—
254C2×C4—364no—
255C2×C4C2×D3×A411yescharacteristic
256C2×C4C2×D3×A411yescharacteristic
257C2×C4C2×D3×A411yescharacteristic
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—41no—
261C2×C3—62no—
262C2×C3—81no—
263C2×C3—81no—
264C2×C3—81no—
265C2×C3C2×Q8×A411yescharacteristic
266C2×C3C2×#99411yescharacteristic
267C2×C3C2×2−1+4⋊C311yescharacteristic
268C22—32no—
269C22—61no—
270C22—61no—
271C22C22×D3×A411yescharacteristic
272C4—31no—
273C4—31no—
274C4—61no—
275C4—61no—
276C4—64no—
277C4—184no—
278C4A4×Dic612yes—
279C4C3⋊#99412yes—
280C4#1010⋊C312yes—
281C3—41no—
282C3—81no—
283C3#3503⋊C311yescharacteristic, complemented
284C2—62no—
285C2C2×A4×Dic611yescharacteristic
286C2C2×C3⋊#99411yescharacteristic
287C2C2×#1010⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
prime radicalC2×C3×#41⋊C3C22
hypercenterC42D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #5570⋊C3C2semidirect
4. SL(2,𝔽3)×Dic6C2semidirect
5. #85828C3semidirect
27. C3×#41⋊C3Q8semidirect
72. C3×SL(2,𝔽3)C2×Q8semidirect
83. C3×#41C3×Q8semidirect
96. #41⋊C3Dic6semidirect
181. SL(2,𝔽3)C2×Dic6semidirect
186. #41C3×Dic6semidirect
283. C3#3503⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#5827⋊C32
#5570⋊C32
SL(2,𝔽3)×Dic62
#858283
#3503⋊C33
C3×#8694