Finite Group Databasebeta

← #82461⋊C3 #11921267

Subgroups — #82461⋊C3 #11921267

Order 1152.

389 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 79 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
4#82461C311yesmaximal, characteristic, complemented
5#2351⋊C3—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C3×C4×SL(2,𝔽3)—21no—
8Dic3×SL(2,𝔽3)—21no—
9Dic3×SL(2,𝔽3)C412yescomplemented
10C3×C3⋊#95—41nomaximal
11C2×C3×C4×Q8C2×C311yescharacteristic
12C2×Q8×Dic3C2×C311yescharacteristic
13C2×Q8×Dic3C2×C311yescharacteristic
14C2×C4×SL(2,𝔽3)—21no—
15C2×C4×SL(2,𝔽3)—31no—
16C2×C4×SL(2,𝔽3)—31no—
17C2×C4×SL(2,𝔽3)D311yescharacteristic
18#4977—31no—
19#4972—31no—
20C4×Dic3⋊C4—31no—
21#4928—31no—
22C22×C32×C4—41no—
23C22×C3×Dic3—41no—
24C22×C3×Dic3—41no—
25C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
26C2×C3×SL(2,𝔽3)D411yescharacteristic
27C2×C3×SL(2,𝔽3)Q811yescharacteristic
28#2351—31noSylow
29C22×C3×Q8C22×C311yescharacteristic
30C22×SL(2,𝔽3)—21no—
31C22×SL(2,𝔽3)C2×D311yescharacteristic
32C3×C4×Q8—21no—
33C3×C4×Q8—61no—
34C2×C3×C4⋊C4—31no—
35C2×C3×C42—31no—
36Q8×Dic3—21no—
37Q8×Dic3—32no—
38Q8×Dic3—61no—
39Q8×Dic3C3×C412yescomplemented
40C2×C4⋊Dic3—31no—
41C2×C4⋊Dic3—31no—
42C2×Dic3⋊C4—31no—
43C2×Dic3⋊C4—31no—
44C2×C4×Dic3—31no—
45C2×C4×Dic3—31no—
46C4×SL(2,𝔽3)—22no—
47C4×SL(2,𝔽3)—21no—
48C4×SL(2,𝔽3)—32no—
49C4×SL(2,𝔽3)—61no—
50C3×#95—121no—
51C3⋊#95—32no—
52C3⋊#95—31no—
53C3⋊#95A411yescharacteristic
54C23×C32—41no—
55C2×C32×C4—42no—
56C2×C32×C4—81no—
57C2×C32×C4—81no—
58C2×C3×Dic3—42no—
59C2×C3×Dic3—42no—
60C2×C3×Dic3—81no—
61C2×C3×Dic3—81no—
62C2×C3×Dic3—81no—
63C2×C3×Dic3—81no—
64C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
65C2×C4×Q8—31no—
66C2×C4×Q8—31no—
67C2×C4×Q8C3×D311yescharacteristic
68#389—91no—
69#389—91no—
70#382—91no—
71C4×C4⋊C4—91no—
72C2×C3×Q8—31no—
73C2×C3×Q8—31no—
74C2×C3×Q8—31no—
75C2×C3×Q8C2×C3×C411yescharacteristic
76C2×C3×Q8C3×D411yescharacteristic
77C2×C3×Q8C3×Q811yescharacteristic
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—41no—
81C22×C3×C4—81no—
82C22×C3×C4—121no—
83C22×C3×C4—121no—
84C22×C3×C4C2×A411yescharacteristic
85C22×Dic3—31no—
86C22×Dic3—31no—
87C22×Dic3C2×A411yescharacteristic
88C22×Dic3C2×A411yescharacteristic
89C2×SL(2,𝔽3)—21no—
90C2×SL(2,𝔽3)—21no—
91C2×SL(2,𝔽3)—21no—
92C2×SL(2,𝔽3)Dic611yescharacteristic
93C2×SL(2,𝔽3)C4×D311yescharacteristic
94C2×SL(2,𝔽3)#6711yescharacteristic
95C3×C4⋊C4—61no—
96C3×C4⋊C4—61no—
97C3×C42—61no—
98C3×C42—61no—
99C4⋊Dic3—32no—
100C4⋊Dic3—32no—
101C4⋊Dic3—61no—
102C4⋊Dic3—61no—
103Dic3⋊C4—62no—
104Dic3⋊C4—62no—
105C4×Dic3—32no—
106C4×Dic3—32no—
107C4×Dic3—61no—
108C4×Dic3—61no—
109C22×C32—41no—
110C22×C32—41no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32—41no—
114C22×C32—41no—
115C22×C32—41no—
116C32×C4—82no—
117C3×Dic3—82no—
118C3×Dic3—82no—
119C22×Q8C2×C3×D311yescharacteristic
120C4×Q8—21no—
121C4×Q8—32no—
122C4×Q8—61no—
123C4×Q8—61no—
124C4×Q8—92no—
125C4×Q8—181no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—91no—
128C2×C4⋊C4—91no—
129C2×C4⋊C4—91no—
130C2×C4⋊C4—91no—
131C2×C42—31no—
132C2×C42—91no—
133C2×C42—91no—
134#95—31no—
135#95—91no—
136#95—92no—
137C23×C3—41no—
138C23×C3—81no—
139C23×C3C22×A411yescharacteristic
140C3×Q8—31no—
141C3×Q8—31no—
142C3×Q8—31no—
143C3×Q8—32no—
144C3×Q8C3×C4⋊C411yescharacteristic
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—42no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×C3×C4—81no—
158C2×C3×C4—81no—
159C2×C3×C4—82no—
160C2×C3×C4—82no—
161C2×C3×C4—82no—
162C2×C3×C4—122no—
163C2×C3×C4—122no—
164C2×C3×C4—241no—
165C2×C3×C4—241no—
166C2×C3×C4—241no—
167C2×C3×C4—241no—
168C2×C3×C4#41⋊C312yes—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3—21no—
173C2×Dic3—32no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3C4×A412yes—
180C2×Dic3#41⋊C312yes—
181SL(2,𝔽3)—21no—
182SL(2,𝔽3)Dic3⋊C411yescharacteristic, complemented
183C2×C32—41no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32—41no—
187C2×C32—41no—
188C2×C32—41no—
189C2×C32—41no—
190C2×Q8—31no—
191C2×Q8—31no—
192C2×Q8—31no—
193C2×Q8C3×Dic611yescharacteristic
194C2×Q8C3×C4×D311yescharacteristic
195C2×Q8C3×#6711yescharacteristic
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—91no—
201C22×C4—91no—
202C22×C4D3×A411yescharacteristic
203C4⋊C4—61no—
204C4⋊C4—61no—
205C4⋊C4—92no—
206C4⋊C4—92no—
207C4⋊C4—182no—
208C4⋊C4—181no—
209C4⋊C4—181no—
210C4⋊C4—182no—
211C42—61no—
212C42—61no—
213C42—92no—
214C42—92no—
215C42—181no—
216C42—181no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—81no—
225C22×C3—81no—
226C22×C3—81no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3—81no—
230C22×C3—81no—
231C22×C3C2×C4×A411yescharacteristic
232C22×C3D4×A411yescharacteristic
233C22×C3Q8×A411yescharacteristic
234C22×C3C2×#41⋊C311yescharacteristic
235C22×C3C2×#41⋊C311yescharacteristic
236C22×C32−1+4⋊C311yescharacteristic
237C22×C32−1+4⋊C311yescharacteristic
238C3×C4—22no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—82no—
244C3×C4—84no—
245C3×C4—121no—
246C3×C4—242no—
247C3×C4—242no—
248Dic3—22no—
249Dic3—22no—
250Dic3—62no—
251Dic3—121no—
252C32—41noSylow
253C23C2×D3×A411yescharacteristic
254Q8—31no—
255Q8—31no—
256Q8—31no—
257Q8—32no—
258Q8C3×Dic3⋊C411yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—32no—
268C2×C4—32no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—92no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4C3⋊(#41⋊C3)12yes—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—81no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3—81no—
297C2×C3A4×C4⋊C411yescharacteristic
298C2×C3C4×#41⋊C311yescharacteristic
299C2×C3#519⋊C311yescharacteristic
300C2×C3#548⋊C311yescharacteristic
301C2×C3#548⋊C311yescharacteristic
302C2×C3#557⋊C311yescharacteristic
303C2×C3#557⋊C311yescharacteristic
304C22A4×Dic611yescharacteristic
305C22C4×D3×A411yescharacteristic
306C22C2×C3⋊(#41⋊C3)11yescharacteristic
307C22#1007⋊C311yescharacteristic
308C22D3×#41⋊C311yescharacteristic
309C22#1010⋊C311yescharacteristic
310C22A4×#6711yescharacteristic
311C4—22no—
312C4—31no—
313C4—31no—
314C4—31no—
315C4—31no—
316C4—62no—
317C4—62no—
318C4—121no—
319C4—182no—
320C4—361no—
321C3—41no—
322C3—81no—
323C3#2351⋊C311yescharacteristic, complemented
324C2C4×C3⋊(#41⋊C3)11yescharacteristic
325C2#5569⋊C311yescharacteristic
326C2#5570⋊C311yescharacteristic
327C2#5572⋊C311yescharacteristic
328C2A4×Dic3⋊C411yescharacteristic
329C2#5817⋊C311yescharacteristic
330C2#5831⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3×SL(2,𝔽3)C22
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C4⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #82461C3semidirect
9. Dic3×SL(2,𝔽3)C4semidirect
39. Q8×Dic3C3×C4semidirect
64. C3×SL(2,𝔽3)C4⋊C4semidirect
182. SL(2,𝔽3)Dic3⋊C4semidirect
323. C3#2351⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
#824613
#2351⋊C33
C3×C3⋊#954