Finite Group Databasebeta

← #84111⋊C3 #11921379

Subgroups — #84111⋊C3 #11921379

Order 1152.

425 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 63 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#5827⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#84111C311yesmaximal, characteristic, complemented
5#2898⋊C3—31nomaximal
6C3×2−1+4⋊C3—21no—
7C3×2−1+4⋊C3C412yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)—21no—
10C3⋊(#131⋊C3)—21no—
11Dic3×SL(2,𝔽3)—21no—
12C3×#832—41nomaximal
13C2×C3×2−1+4C2×C311yescharacteristic
14C2×2−1+4⋊C3—21no—
15C2×2−1+4⋊C3D311yescharacteristic
16#5827C2×C311yescharacteristic, complemented
17C2×#948C2×C311yescharacteristic, complemented
18C2×#131⋊C3—31no—
19#520⋊C3—31no—
20#5233—31no—
21#5232—31no—
22#5225—31no—
23C2×#835—31no—
24C2×C32×D4—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)—22no—
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C3×C4⋊Dic3—41no—
30C2×C3×C3⋊C8—41no—
31#2898—31noSylow
32C3×2−1+4—21no—
33C3×2−1+4—32no—
34C3×2−1+4—61no—
35C3×2−1+4C3×C412yescomplemented
36C2×C3×#41—31no—
37C2×C3×#41—61no—
38C2×C3×#41C22×C311yescharacteristic
39C22×C3×Q8—21no—
40C22×C3×Q8—31no—
412−1+4⋊C3—22no—
422−1+4⋊C3—21no—
432−1+4⋊C3—22no—
442−1+4⋊C3Dic312yescomplemented
45C2×#41⋊C3—21no—
46C2×#41⋊C3C2×D311yescharacteristic
47C22×SL(2,𝔽3)—22no—
48C22×SL(2,𝔽3)—21no—
49#948—21no—
50#948—61no—
51Q8×Dic3—21no—
52D4×Dic3—61no—
53#926—31no—
54C2×C4⋊Dic3—31no—
55C2×#208—31no—
56C22×C3⋊C8—31no—
57#131⋊C3—61no—
58C4×SL(2,𝔽3)—61no—
59C3×#102—121no—
60#835—32no—
61#835—31no—
62#835—31no—
63#832—31no—
64#832A411yescharacteristic
65C23×C32—81no—
66C32×D4—42no—
67C32×D4—81no—
68C2×C32×C4—41no—
69C2×C3×Dic3—81no—
70C3×SL(2,𝔽3)C22⋊C411yescharacteristic
71C3×C3⋊C8—81no—
72C2×2−1+4C3×D311yescharacteristic
73C2×#131—31no—
74#520—31no—
75#419—91no—
76#417—91no—
77#416—91no—
78C2×#103—91no—
79C3×#41—31no—
80C3×#41—31no—
81C3×#41—32no—
82C3×#41—32no—
83C3×#41—61no—
84C3×#41—61no—
85C3×#41—64no—
86C3×#41—62no—
87C3×#41—62no—
88C3×#41C2×C3×C411yescharacteristic
89C3×#41C3×D411yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—32no—
92C2×C3×Q8—32no—
93C2×C3×Q8—31no—
94C2×C3×Q8—61no—
95C2×C3×Q8—61no—
96C2×C3×Q8—61no—
97C2×C3×Q8—62no—
98C2×C3×Q8C3×D411yescharacteristic
99C2×C3×D4—31no—
100C2×C3×D4—41no—
101C2×C3×D4—61no—
102C2×C3×D4—81no—
103C2×C3×D4C2×A411yescharacteristic
104C22×C3×C4—31no—
105C22×C3×C4—61no—
106C22×C3×C4—61no—
107C22×Dic3—61no—
108#41⋊C3—21no—
109#41⋊C3—21no—
110#41⋊C3C2×Dic311yescharacteristic
111#41⋊C3#6711yescharacteristic
112C2×SL(2,𝔽3)—24no—
113C2×SL(2,𝔽3)—22no—
114C2×SL(2,𝔽3)—21no—
115C2×SL(2,𝔽3)#6711yescharacteristic
116C2×C3×C8—121no—
117C3×C4⋊C4—121no—
118#217—61no—
119C4⋊Dic3—31no—
120C4⋊Dic3—61no—
121C4⋊Dic3C2×A411yescharacteristic
122C4×Dic3—61no—
123#208—61no—
124#208—61no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—61no—
127C2×C3⋊C8—61no—
128C2×C3⋊C8C2×A411yescharacteristic
129C22×C32—41no—
130C22×C32—82no—
131C22×C32—161no—
132C22×C32—161no—
133C32×C4—41no—
134C32×C4—41no—
135C3×Dic3—161no—
1362−1+4—21no—
1372−1+4—32no—
1382−1+4—61no—
1392−1+4C3×Dic312yescomplemented
140C2×#41—31no—
141C2×#41—61no—
142C2×#41C2×C3×D311yescharacteristic
143C22×Q8—21no—
144C22×Q8—31no—
145#131—61no—
146#131—181no—
147C2×M16—91no—
148C22×C8—91no—
149C4×Q8—61no—
150C4×D4—181no—
151#117—91no—
152C2×C4⋊C4—91no—
153#103—91no—
154#103—91no—
155#103—92no—
156#102—31no—
157#102—91no—
158C23×C3—21no—
159C23×C3—31no—
160C23×C3—81no—
161C23×C3—82no—
162C3×Q8—31no—
163C3×Q8—32no—
164C3×Q8—62no—
165C3×Q8—62no—
166C3×Q8—61no—
167C3×Q8C3×C22⋊C411yescharacteristic
168C3×D4—21no—
169C3×D4—31no—
170C3×D4—31no—
171C3×D4—32no—
172C3×D4—42no—
173C3×D4—62no—
174C3×D4—62no—
175C3×D4—81no—
176C3×D4—82no—
177C3×D4—82no—
178C3×D4C4×A412yes—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—41no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—61no—
189C2×C3×C4—62no—
190C2×C3×C4—81no—
191C2×C3×C4—241no—
192C2×C3×C4C22×A411yescharacteristic
193C2×Dic3—21no—
194C2×Dic3—61no—
195C2×Dic3—241no—
196SL(2,𝔽3)—21no—
197SL(2,𝔽3)#21711yescharacteristic
198C3×C8—241no—
199C3⋊C8—21no—
200C3⋊C8—61no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—162no—
205#41—31no—
206#41—31no—
207#41—32no—
208#41—32no—
209#41—61no—
210#41—61no—
211#41—64no—
212#41—62no—
213#41—62no—
214#41C2×C3×Dic311yescharacteristic
215#41C3×#6711yescharacteristic
216C2×Q8—22no—
217C2×Q8—32no—
218C2×Q8—32no—
219C2×Q8—31no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—62no—
224C2×Q8C3×#6711yescharacteristic
225C2×D4—31no—
226C2×D4—61no—
227C2×D4D3×A411yescharacteristic
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—181no—
232M16—181no—
233M16—181no—
234C2×C8—31no—
235C2×C8—91no—
236C2×C8—181no—
237C2×C8—181no—
238C4⋊C4—31no—
239C4⋊C4—91no—
240C4⋊C4—181no—
241C22⋊C4—181no—
242C42—181no—
243C22×C3—22no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—82no—
252C22×C3—84no—
253C22×C3—81no—
254C22×C3—161no—
255C22×C3—161no—
256C22×C3—162no—
257C22×C3—162no—
258C22×C3D4×A411yescharacteristic
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—62no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4—481no—
267C3×C4C2×C4×A411yescharacteristic
268C3×C4D4×A411yescharacteristic
269Dic3—41no—
270Dic3—121no—
271C32—41noSylow
272C23—21no—
273C23—31no—
274Q8—31no—
275Q8—32no—
276Q8—62no—
277Q8—62no—
278Q8—61no—
279Q8C3×#21711yescharacteristic
280D4—21no—
281D4—31no—
282D4—31no—
283D4—32no—
284D4—62no—
285D4—62no—
286D4Dic3×A412yes—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—61no—
296C2×C4—62no—
297C2×C4—61no—
298C2×C4—181no—
299C2×C4—721no—
300C2×C4C2×D3×A411yescharacteristic
301C8—61no—
302C8—181no—
303C2×C3—42no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—81no—
310C2×C3—81no—
311C2×C3—81no—
312C2×C3—162no—
313C2×C3—164no—
314C2×C3A4×C22⋊C411yescharacteristic
315C2×C3#577⋊C311yescharacteristic
316C2×C3#578⋊C311yescharacteristic
317C22—22no—
318C22—31no—
319C22—31no—
320C22—41no—
321C22—41no—
322C22—61no—
323C22—61no—
324C22A4×#6711yescharacteristic
325C4—31no—
326C4—31no—
327C4—62no—
328C4—121no—
329C4—361no—
330C4C2×Dic3×A411yescharacteristic
331C4A4×#6711yescharacteristic
332C3—41no—
333C3—81no—
334C3#2898⋊C311yescharacteristic, complemented
335C2—42no—
336C2—61no—
337C2—61no—
338C2#5838⋊C311yescharacteristic
339C2#5837⋊C311yescharacteristic
340C2A4×#21711yescharacteristic
subgroupG/H
centerC22A4×#67
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC2×C3×2−1+4C2×C3
socleC22×C3D4×A4
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×C3×#41C22×C3
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. C2×C3⋊(#131⋊C3)C2semidirect
4. #84111C3semidirect
7. C3×2−1+4⋊C3C4semidirect
16. #5827C2×C3semidirect
17. C2×#948C2×C3semidirect
35. C3×2−1+4C3×C4semidirect
44. 2−1+4⋊C3Dic3semidirect
139. 2−1+4C3×Dic3semidirect
334. C3#2898⋊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
#5827⋊C32
C2×C3⋊(#131⋊C3)2
#841113
#2898⋊C33
C3×#8324