Finite Group Databasebeta

← #84119⋊C3 #11921378

Subgroups — #84119⋊C3 #11921378

Order 1152.

425 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 59 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2#5812⋊C3C212yesmaximal, complemented
3#84119C311yesmaximal, characteristic, complemented
4#2884⋊C3—31nomaximal
5C3×2−1+4⋊C3—22no—
6C2×C3×#41⋊C3C411yescharacteristic, complemented
7C22×C3×SL(2,𝔽3)C411yescharacteristic, complemented
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9Dic3×SL(2,𝔽3)—22no—
10C3×#834—41nomaximal
11C2×C3×2−1+4C2×C311yescharacteristic
12C2×2−1+4⋊C3—21no—
13C2×2−1+4⋊C3D311yescharacteristic
14#5812C2×C312yescomplemented
15#519⋊C3—32no—
16#5240—31no—
17#5240—31no—
18#5240—32no—
19C2×C32×D4—41no—
20C3×#41⋊C3—21no—
21C2×C3×SL(2,𝔽3)—21no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23C2×C3×SL(2,𝔽3)D412yes—
24C3×#217—42no—
25#2884—31noSylow
26C3×2−1+4—22no—
27C3×2−1+4—62no—
28C2×C3×#41—31no—
29C2×C3×#41—31no—
30C2×C3×#41—31no—
31C2×C3×#41C3×C411yescharacteristic, complemented
32C22×C3×Q8—31no—
33C22×C3×Q8C3×C411yescharacteristic, complemented
34C22×C3×Q8C22×C311yescharacteristic
352−1+4⋊C3—24no—
362−1+4⋊C3—22no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3Dic311yescharacteristic, complemented
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)Dic311yescharacteristic, complemented
42C22×SL(2,𝔽3)C2×D311yescharacteristic
43Q8×Dic3—22no—
44Q8×Dic3—62no—
45#926—32no—
46#926—32no—
47C4×SL(2,𝔽3)—62no—
48C3×#99—121no—
49#834—32no—
50#834—31no—
51#834—32no—
52#834A411yescharacteristic
53C23×C32—41no—
54C23×C32—41no—
55C32×D4—82no—
56C2×C32×C4—41no—
57C2×C3×Dic3—82no—
58C3×SL(2,𝔽3)C22⋊C411yescharacteristic
59C2×2−1+4C3×D311yescharacteristic
60#519—32no—
61#410—91no—
62#410—92no—
63#410—91no—
64C3×#41—21no—
65C3×#41—62no—
66C3×#41—62no—
67C3×#41—62no—
68C3×#41—62no—
69C3×#41—62no—
70C3×#41—62no—
71C3×#41—61no—
72C2×C3×Q8—21no—
73C2×C3×Q8—31no—
74C2×C3×Q8—32no—
75C2×C3×Q8—31no—
76C2×C3×Q8—31no—
77C2×C3×Q8—62no—
78C2×C3×Q8—62no—
79C2×C3×Q8—61no—
80C2×C3×Q8C2×C3×C411yescharacteristic
81C2×C3×Q8C3×D412yes—
82C2×C3×D4—31no—
83C2×C3×D4—31no—
84C2×C3×D4—31no—
85C2×C3×D4—41no—
86C2×C3×D4—81no—
87C2×C3×D4C2×A411yescharacteristic
88C22×C3×C4—31no—
89C22×C3×C4—32no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92#41⋊C3—22no—
93#41⋊C3—21no—
94C2×SL(2,𝔽3)—22no—
95C2×SL(2,𝔽3)—21no—
96C2×SL(2,𝔽3)—21no—
97C2×SL(2,𝔽3)—22no—
98C2×SL(2,𝔽3)C2×Dic311yescharacteristic
99C2×SL(2,𝔽3)#6712yes—
100C3×C22⋊C4—122no—
101#217—32no—
102#217C2×A412yes—
103C4⋊Dic3—62no—
104C4⋊Dic3—62no—
105C4×Dic3—62no—
106C4×Dic3—62no—
107C22×C32—41no—
108C22×C32—42no—
109C22×C32—81no—
110C22×C32—161no—
111C22×C32—161no—
112C32×C4—81no—
113C3×Dic3—162no—
1142−1+4—22no—
1152−1+4—62no—
116C2×#41—31no—
117C2×#41—31no—
118C2×#41—31no—
119C2×#41C3×Dic311yescharacteristic, complemented
120C22×Q8—31no—
121C22×Q8C3×Dic311yescharacteristic, complemented
122C22×Q8C2×C3×D311yescharacteristic
123C4×Q8—62no—
124C4×Q8—182no—
125#117—92no—
126#117—92no—
127#99—31no—
128#99—92no—
129#99—92no—
130#99—91no—
131C23×C3—31no—
132C23×C3—41no—
133C23×C3—41no—
134C23×C3—81no—
135C23×C3—81no—
136C23×C3C4×A411yescharacteristic
137C23×C3C22×A411yescharacteristic
138C3×Q8—31no—
139C3×Q8—32no—
140C3×Q8—32no—
141C3×Q8—62no—
142C3×Q8—61no—
143C3×Q8—61no—
144C3×Q8C3×C22⋊C411yescharacteristic
145C3×D4—22no—
146C3×D4—62no—
147C3×D4—61no—
148C3×D4—61no—
149C3×D4—62no—
150C3×D4—82no—
151C3×D4—84no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—31no—
157C2×C3×C4—41no—
158C2×C3×C4—61no—
159C2×C3×C4—61no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—64no—
163C2×C3×C4—81no—
164C2×C3×C4—242no—
165C2×C3×C4C4×A411yescharacteristic
166C2×Dic3—22no—
167C2×Dic3—62no—
168SL(2,𝔽3)—21no—
169SL(2,𝔽3)#21711yescharacteristic
170C2×C32—41no—
171C2×C32—81no—
172C2×C32—82no—
173C2×C32—161no—
174#41—21no—
175#41—62no—
176#41—62no—
177#41—62no—
178#41—62no—
179#41—62no—
180#41—62no—
181#41—61no—
182C2×Q8—21no—
183C2×Q8—31no—
184C2×Q8—32no—
185C2×Q8—31no—
186C2×Q8—31no—
187C2×Q8—62no—
188C2×Q8—62no—
189C2×Q8—61no—
190C2×Q8C2×C3×Dic311yescharacteristic
191C2×Q8C3×#6712yes—
192C2×D4—31no—
193C2×D4—31no—
194C2×D4—31no—
195C2×D4D3×A411yescharacteristic
196C22×C4—31no—
197C22×C4—32no—
198C22×C4—31no—
199C22×C4—31no—
200C4⋊C4—182no—
201C4⋊C4—182no—
202C22⋊C4—32no—
203C22⋊C4—92no—
204C42—182no—
205C42—182no—
206C22×C3—21no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—42no—
211C22×C3—61no—
212C22×C3—81no—
213C22×C3—82no—
214C22×C3—81no—
215C22×C3—82no—
216C22×C3—121no—
217C22×C3—161no—
218C22×C3—161no—
219C22×C3—162no—
220C22×C3—162no—
221C22×C3C2×C4×A411yescharacteristic
222C22×C3D4×A412yes—
223C3×C4—21no—
224C3×C4—32no—
225C3×C4—31no—
226C3×C4—31no—
227C3×C4—61no—
228C3×C4—81no—
229C3×C4—82no—
230C3×C4—482no—
231Dic3—42no—
232Dic3—122no—
233C32—41noSylow
234C23—31no—
235C23Dic3×A411yescharacteristic
236C23C2×D3×A411yescharacteristic
237Q8—31no—
238Q8—32no—
239Q8—32no—
240Q8—62no—
241Q8—61no—
242Q8—61no—
243Q8C3×#21711yescharacteristic
244D4—22no—
245D4—62no—
246D4—61no—
247D4—61no—
248D4—62no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—31no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—64no—
259C2×C4—62no—
260C2×C4—182no—
261C2×C4Dic3×A411yescharacteristic
262C2×C3—22no—
263C2×C3—21no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3—82no—
267C2×C3—81no—
268C2×C3—82no—
269C2×C3—81no—
270C2×C3—121no—
271C2×C3—162no—
272C2×C3—161no—
273C2×C3—162no—
274C2×C3A4×C22⋊C411yescharacteristic
275C22—21no—
276C22—41no—
277C22—41no—
278C22—61no—
279C22—121no—
280C22C2×Dic3×A411yescharacteristic
281C22A4×#6712yes—
282C4—21no—
283C4—31no—
284C4—31no—
285C4—32no—
286C4—61no—
287C4—122no—
288C4—362no—
289C3—41no—
290C3—81no—
291C3#2884⋊C311yescharacteristic, complemented
292C2—22no—
293C2—21no—
294C2—41no—
295C2—121no—
296C2A4×#21711yescharacteristic
subgroupG/H
centerC2A4×#217
commutatorC2×C3×Q8C2×C3×C4
FrattiniC23C2×D3×A4
FittingC2×C3×2−1+4C2×C3
socleC2×C3A4×C22⋊C4
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC22×C3×Q8C22×C3
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #5812⋊C3C2semidirect
3. #84119C3semidirect
6. C2×C3×#41⋊C3C4semidirect
7. C22×C3×SL(2,𝔽3)C4semidirect
14. #5812C2×C3semidirect
31. C2×C3×#41C3×C4semidirect
33. C22×C3×Q8C3×C4semidirect
38. C2×#41⋊C3Dic3semidirect
41. C22×SL(2,𝔽3)Dic3semidirect
119. C2×#41C3×Dic3semidirect
121. C22×Q8C3×Dic3semidirect
291. C3#2884⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
#5812⋊C32
#841193
#2884⋊C33
C3×#8344