Finite Group Databasebeta

← #84075⋊C3 #11921368

Subgroups — #84075⋊C3 #11921368

Order 1152.

441 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3⋊(#131⋊C3)C212yesmaximal, complemented
3#84075C311yesmaximal, characteristic, complemented
4#2880⋊C3—31nomaximal
5C2×C3×#41⋊C3—22no—
6C2×C3×#41⋊C3C411yescharacteristic
7C2×C3×#41⋊C3C2211yescharacteristic
8C22×C3×SL(2,𝔽3)C411yescharacteristic
9C3⋊(#131⋊C3)—22no—
10C3×#830—41nomaximal
11C22×C3×#41C2×C311yescharacteristic
12C22×#41⋊C3—21no—
13C22×#41⋊C3D311yescharacteristic
14C2×#948C2×C312yescomplemented
15C2×#131⋊C3—32no—
16#5234—32no—
17#5207—31no—
18C2×#830—31no—
19C22×C32×C4—41no—
20C3×#41⋊C3—21no—
21C3×#41⋊C3D412yes—
22C2×C3×SL(2,𝔽3)—21no—
23C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
24C2×C3×C3⋊C8—42no—
25#2880—31noSylow
26C2×C3×#41—22no—
27C2×C3×#41—31no—
28C2×C3×#41—31no—
29C2×C3×#41—62no—
30C2×C3×#41C3×C411yescharacteristic
31C2×C3×#41C22×C311yescharacteristic
32C22×C3×Q8C3×C411yescharacteristic
33C22×C3×D4—31no—
34C23×C3×C4—31no—
35C2×#41⋊C3—24no—
36C2×#41⋊C3—22no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3—21no—
39C2×#41⋊C3Dic311yescharacteristic
40C2×#41⋊C3C2×D311yescharacteristic
41C22×SL(2,𝔽3)—21no—
42C22×SL(2,𝔽3)Dic311yescharacteristic
43#948—22no—
44#948—62no—
45C2×#208—32no—
46C22×C3⋊C8—32no—
47#131⋊C3—62no—
48C3×C22⋊C8—121no—
49#830—32no—
50#830—31no—
51#830—32no—
52#830A411yescharacteristic
53C23×C32—41no—
54C2×C32×C4—41no—
55C2×C32×C4—41no—
56C2×C32×C4—82no—
57C3×SL(2,𝔽3)C22⋊C411yescharacteristic
58C3×C3⋊C8—82no—
59C22×#41C3×D311yescharacteristic
60C2×#131—32no—
61#408—92no—
62#407—91no—
63C2×C22⋊C8—91no—
64C24×C3—31no—
65C3×#41—21no—
66C3×#41—32no—
67C3×#41—62no—
68C3×#41—62no—
69C3×#41—62no—
70C3×#41—62no—
71C3×#41—61no—
72C3×#41C3×D412yes—
73C2×C3×Q8—21no—
74C2×C3×Q8—31no—
75C2×C3×Q8—61no—
76C2×C3×Q8C2×C3×C411yescharacteristic
77C2×C3×D4—31no—
78C2×C3×D4—31no—
79C2×C3×D4—32no—
80C2×C3×D4—62no—
81C2×C3×D4—62no—
82C22×C3×C4—31no—
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—41no—
88C22×C3×C4—62no—
89C22×C3×C4—62no—
90C22×C3×C4—81no—
91C22×C3×C4C2×A411yescharacteristic
92#41⋊C3—22no—
93#41⋊C3—21no—
94#41⋊C3—22no—
95#41⋊C3#6712yes—
96C2×SL(2,𝔽3)—22no—
97C2×SL(2,𝔽3)—21no—
98C2×SL(2,𝔽3)—21no—
99C2×SL(2,𝔽3)C2×Dic311yescharacteristic
100C2×C3×C8—122no—
101#208—62no—
102#208—62no—
103C2×C3⋊C8—32no—
104C2×C3⋊C8—62no—
105C2×C3⋊C8—62no—
106C2×C3⋊C8C2×A412yes—
107C22×C32—42no—
108C22×C32—41no—
109C22×C32—81no—
110C22×C32—81no—
111C32×C4—42no—
112C32×C4—81no—
113C2×#41—22no—
114C2×#41—31no—
115C2×#41—31no—
116C2×#41—62no—
117C2×#41C3×Dic311yescharacteristic
118C2×#41C2×C3×D311yescharacteristic
119C22×Q8C3×Dic311yescharacteristic
120C22×D4—31no—
121C23×C4—31no—
122#131—62no—
123#131—182no—
124C2×M16—92no—
125C22×C8—92no—
126C22⋊C8—31no—
127C22⋊C8—92no—
128C22⋊C8—92no—
129C22⋊C8—91no—
130C23×C3—31no—
131C23×C3—31no—
132C23×C3—41no—
133C23×C3—62no—
134C23×C3—62no—
135C23×C3—81no—
136C23×C3—121no—
137C23×C3C4×A411yescharacteristic
138C3×Q8—31no—
139C3×Q8—61no—
140C3×Q8—61no—
141C3×Q8C3×C22⋊C411yescharacteristic
142C3×D4—32no—
143C3×D4—32no—
144C3×D4—61no—
145C3×D4—61no—
146C3×D4—64no—
147C2×C3×C4—22no—
148C2×C3×C4—31no—
149C2×C3×C4—32no—
150C2×C3×C4—32no—
151C2×C3×C4—31no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—62no—
157C2×C3×C4—61no—
158C2×C3×C4—61no—
159C2×C3×C4—62no—
160C2×C3×C4—82no—
161C2×C3×C4—84no—
162C2×C3×C4—81no—
163C2×C3×C4—81no—
164C2×C3×C4C4×A411yescharacteristic
165C2×C3×C4C22×A411yescharacteristic
166SL(2,𝔽3)—21no—
167SL(2,𝔽3)#21711yescharacteristic
168C3×C8—242no—
169C3⋊C8—22no—
170C3⋊C8—62no—
171C2×C32—41no—
172C2×C32—41no—
173C2×C32—41no—
174C2×C32—82no—
175C24—31no—
176#41—21no—
177#41—32no—
178#41—62no—
179#41—62no—
180#41—62no—
181#41—62no—
182#41—61no—
183#41C3×#6712yes—
184C2×Q8—21no—
185C2×Q8—31no—
186C2×Q8—61no—
187C2×Q8C2×C3×Dic311yescharacteristic
188C2×D4—31no—
189C2×D4—31no—
190C2×D4—32no—
191C2×D4—62no—
192C2×D4—62no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4D3×A411yescharacteristic
201M16—182no—
202M16—182no—
203C2×C8—32no—
204C2×C8—92no—
205C2×C8—182no—
206C2×C8—182no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—32no—
210C22×C3—42no—
211C22×C3—41no—
212C22×C3—61no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—61no—
216C22×C3—61no—
217C22×C3—81no—
218C22×C3—81no—
219C22×C3—82no—
220C22×C3—82no—
221C22×C3—82no—
222C22×C3—81no—
223C22×C3—124no—
224C22×C3#131⋊C312yescomplemented
225C22×C3C2×C4×A411yescharacteristic
226C3×C4—21no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—42no—
230C3×C4—61no—
231C3×C4—81no—
232C3×C4—82no—
233C3×C4—82no—
234C3×C4D4×A412yes—
235C32—41noSylow
236C23—31no—
237C23—31no—
238C23—62no—
239C23—62no—
240C23—121no—
241C23Dic3×A411yescharacteristic
242Q8—31no—
243Q8—61no—
244Q8—61no—
245Q8C3×#21711yescharacteristic
246D4—32no—
247D4—32no—
248D4—61no—
249D4—61no—
250D4—64no—
251C2×C4—22no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—31no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4Dic3×A411yescharacteristic
263C2×C4C2×D3×A411yescharacteristic
264C8—62no—
265C8—182no—
266C2×C3—22no—
267C2×C3—41no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—62no—
271C2×C3—82no—
272C2×C3—84no—
273C2×C3—81no—
274C2×C3—81no—
275C2×C3—81no—
276C2×C3—121no—
277C2×C3A4×C22⋊C411yescharacteristic
278C2×C3C2×#131⋊C311yescharacteristic
279C2×C3#568⋊C311yescharacteristic
280C22—21no—
281C22—21no—
282C22—32no—
283C22—61no—
284C22—61no—
285C22—61no—
286C22—61no—
287C22—61no—
288C22—124no—
289C22C3⋊(#131⋊C3)12yescomplemented
290C22C2×Dic3×A411yescharacteristic
291C4—21no—
292C4—31no—
293C4—31no—
294C4—61no—
295C4A4×#6712yes—
296C3—41no—
297C3—81no—
298C3#2880⋊C311yescharacteristic, complemented
299C2—22no—
300C2—62no—
301C2—121no—
302C2C2×C3⋊(#131⋊C3)11yescharacteristic
303C2#5819⋊C311yescharacteristic
304C2A4×#21711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC22×C3×#41C2×C3
socleC22×C3C2×C4×A4
prime radicalC22×C3×#41⋊C3C2
cosocleC2×C3×#41C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(#131⋊C3)C2semidirect
3. #84075C3semidirect
14. C2×#948C2×C3semidirect
224. C22×C3#131⋊C3semidirect
289. C22C3⋊(#131⋊C3)semidirect
298. C3#2880⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3⋊(#131⋊C3)2
#840753
#2880⋊C33
C3×#8304