Finite Group Databasebeta

← #84099⋊C3 #11921373

Subgroups — #84099⋊C3 #11921373

Order 1152.

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

normal subgroups only

This group has 421 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×#994C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#84099C311yesmaximal, characteristic, complemented
5#2897⋊C3—31nomaximal
6C3×#994—21no—
7C3×#994C412yescomplemented
8C2×C3×#41⋊C3—21no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C3⋊(#131⋊C3)—21no—
11Dic3×SL(2,𝔽3)—21no—
12C3×#835—41nomaximal
13C2×C3×H(5,𝔽2)C2×C311yescharacteristic
14C2×#994—21no—
15C2×#994D311yescharacteristic
16#5827C2×C311yescharacteristic
17C2×#948C2×C311yescharacteristic
18C2×#131⋊C3—31no—
19#520⋊C3—31no—
20#5233—31no—
21#5232—31no—
22#5215—31no—
23C2×#832—31no—
24C2×C32×Q8—41no—
25C3×#41⋊C3—22no—
26C3×#41⋊C3C2×C411yescharacteristic
27C3×#41⋊C3D411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C3×C4⋊Dic3—41no—
30C2×C3×C3⋊C8—41no—
31#2897—31noSylow
32C3×H(5,𝔽2)—21no—
33C3×H(5,𝔽2)—32no—
34C3×H(5,𝔽2)—61no—
35C3×H(5,𝔽2)C3×C412yescomplemented
36C2×C3×#41—21no—
37C2×C3×#41—31no—
38C2×C3×#41C22×C311yescharacteristic
39C22×C3×D4—31no—
40C22×C3×D4—61no—
41#994—22no—
42#994—21no—
43#994—22no—
44#994Dic312yescomplemented
45C2×#41⋊C3—22no—
46C2×#41⋊C3—21no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3C2×D311yescharacteristic
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×#103—121no—
60#835—31no—
61#835A411yescharacteristic
62#832—32no—
63#832—31no—
64#832—31no—
65C32×Q8—42no—
66C32×Q8—81no—
67C2×C32×C4—41no—
68C2×C32×C4—81no—
69C2×C3×Dic3—81no—
70C3×SL(2,𝔽3)C22⋊C411yescharacteristic
71C3×C3⋊C8—81no—
72C2×H(5,𝔽2)C3×D311yescharacteristic
73C2×#131—31no—
74#520—31no—
75#418—91no—
76#417—91no—
77#416—91no—
78C2×#102—91no—
79C24×C3—61no—
80C3×#41—22no—
81C3×#41—31no—
82C3×#41—31no—
83C3×#41—32no—
84C3×#41—32no—
85C3×#41—61no—
86C3×#41—61no—
87C3×#41—62no—
88C3×#41C2×C3×C411yescharacteristic
89C3×#41C3×D411yescharacteristic
90C2×C3×Q8—41no—
91C2×C3×Q8—81no—
92C2×C3×Q8C3×D411yescharacteristic
93C2×C3×Q8C2×A411yescharacteristic
94C2×C3×D4—31no—
95C2×C3×D4—32no—
96C2×C3×D4—32no—
97C2×C3×D4—31no—
98C2×C3×D4—61no—
99C2×C3×D4—61no—
100C2×C3×D4—61no—
101C2×C3×D4—61no—
102C2×C3×D4—64no—
103C2×C3×D4—62no—
104C2×C3×D4—62no—
105C22×C3×C4—31no—
106C22×C3×C4—61no—
107C22×Dic3—61no—
108#41⋊C3—24no—
109#41⋊C3—22no—
110#41⋊C3—21no—
111#41⋊C3—21no—
112#41⋊C3C2×Dic311yescharacteristic
113#41⋊C3#6711yescharacteristic
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—
130C32×C4—41no—
131C32×C4—41no—
132C32×C4—82no—
133C3×Dic3—161no—
134H(5,𝔽2)—21no—
135H(5,𝔽2)—32no—
136H(5,𝔽2)—61no—
137H(5,𝔽2)C3×Dic312yescomplemented
138C2×#41—21no—
139C2×#41—31no—
140C2×#41C2×C3×D311yescharacteristic
141C22×D4—31no—
142C22×D4—61no—
143#131—61no—
144#131—181no—
145C2×M16—91no—
146C22×C8—91no—
147C4×Q8—61no—
148C4×D4—181no—
149#117—91no—
150C2×C4⋊C4—91no—
151#103—31no—
152#103—91no—
153#102—91no—
154#102—91no—
155#102—92no—
156C23×C3—31no—
157C23×C3—62no—
158C23×C3—62no—
159C23×C3—61no—
160C23×C3—241no—
161C23×C3—241no—
162C3×Q8—21no—
163C3×Q8—31no—
164C3×Q8—42no—
165C3×Q8—81no—
166C3×Q8—82no—
167C3×Q8—82no—
168C3×Q8C3×C22⋊C411yescharacteristic
169C3×Q8C4×A412yes—
170C3×D4—31no—
171C3×D4—31no—
172C3×D4—32no—
173C3×D4—32no—
174C3×D4—62no—
175C3×D4—62no—
176C3×D4—62no—
177C3×D4—62no—
178C3×D4—61no—
179C2×C3×C4—21no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—41no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—81no—
189C2×C3×C4—82no—
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—
204C24—61no—
205#41—22no—
206#41—31no—
207#41—31no—
208#41—32no—
209#41—32no—
210#41—61no—
211#41—61no—
212#41—62no—
213#41C2×C3×Dic311yescharacteristic
214#41C3×#6711yescharacteristic
215C2×Q8C3×#6711yescharacteristic
216C2×Q8D3×A411yescharacteristic
217C2×D4—31no—
218C2×D4—32no—
219C2×D4—32no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—64no—
226C2×D4—62no—
227C2×D4—62no—
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—181no—
231M16—181no—
232M16—181no—
233C2×C8—31no—
234C2×C8—91no—
235C2×C8—181no—
236C2×C8—181no—
237C4⋊C4—31no—
238C4⋊C4—91no—
239C4⋊C4—181no—
240C22⋊C4—181no—
241C42—181no—
242C22×C3—31no—
243C22×C3—31no—
244C22×C3—41no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—62no—
248C22×C3—81no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3—242no—
252C22×C3—242no—
253C22×C3D4×A411yescharacteristic
254C3×C4—22no—
255C3×C4—31no—
256C3×C4—31no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—82no—
260C3×C4—84no—
261C3×C4—81no—
262C3×C4—81no—
263C3×C4—481no—
264C3×C4C2×C4×A411yescharacteristic
265C3×C4D4×A411yescharacteristic
266Dic3—41no—
267Dic3—121no—
268C32—41noSylow
269C23—31no—
270C23—62no—
271C23—62no—
272C23—61no—
273C23—241no—
274C23—241no—
275Q8—21no—
276Q8—31no—
277Q8C3×#21711yescharacteristic
278Q8Dic3×A412yes—
279D4—31no—
280D4—31no—
281D4—32no—
282D4—32no—
283D4—62no—
284D4—62no—
285D4—62no—
286D4—62no—
287D4—61no—
288C2×C4—21no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—61no—
297C2×C4—181no—
298C2×C4—721no—
299C2×C4C2×D3×A411yescharacteristic
300C8—61no—
301C8—181no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—81no—
308C2×C3—81no—
309C2×C3—81no—
310C2×C3—122no—
311C2×C3A4×C22⋊C411yescharacteristic
312C2×C3#576⋊C311yescharacteristic
313C2×C3#577⋊C311yescharacteristic
314C22—31no—
315C22—31no—
316C22—61no—
317C22—61no—
318C22—62no—
319C22—121no—
320C22—121no—
321C22—242no—
322C22—242no—
323C22A4×#6711yescharacteristic
324C4—22no—
325C4—31no—
326C4—31no—
327C4—121no—
328C4—361no—
329C4C2×Dic3×A411yescharacteristic
330C4A4×#6711yescharacteristic
331C3—41no—
332C3—81no—
333C3#2897⋊C311yescharacteristic, complemented
334C2—61no—
335C2—61no—
336C2—122no—
337C2#5836⋊C311yescharacteristic
338C2#5835⋊C311yescharacteristic
339C2A4×#21711yescharacteristic
subgroupG/H
centerC22A4×#67
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC2×C3×H(5,𝔽2)C2×C3
socleC22×C3D4×A4
prime radicalC2×C3×#994C2
cosocleC2×C3×#41C22×C3
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. C2×C3⋊(#131⋊C3)C2semidirect
4. #84099C3semidirect
7. C3×#994C4semidirect
35. C3×H(5,𝔽2)C3×C4semidirect
44. #994Dic3semidirect
137. H(5,𝔽2)C3×Dic3semidirect
333. C3#2897⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9942
#5827⋊C32
C2×C3⋊(#131⋊C3)2
#840993
#2897⋊C33
C3×#8354