Finite Group Databasebeta

← #11920721

Subgroups — #11920721

Order 1152.

439 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 49 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
2C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#5426C211yesmaximal, characteristic, complemented
4#73153—31nomaximal
5#70305—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3⋊#994—21no—
8C3⋊#994C412yescomplemented
9C2×C3⋊(#41⋊C3)—21no—
10C3⋊GU(2,𝔽3)—21no—
11C3×#859—21no—
12#63565—41nomaximal
13C2×#1009D311yescharacteristic
14C2×#994—31no—
15#5426D311yescharacteristic, complemented
16C2×GU(2,𝔽3)—31no—
17C2×GU(2,𝔽3)—61no—
18C3×#417—31no—
19C2×#837—31no—
20C2×#825—31no—
21C2×#823—31no—
22#4988—31no—
23#4966—31no—
24C2×C3×Dic6—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C3⋊(#41⋊C3)—22no—
29C2×C3⋊(C3⋊C8)—41no—
30C3×C4⋊Dic3—41no—
31#1792—91noSylow
32C2×C3×#41C2×D311yescharacteristic
33#1009—21no—
34#1009—32no—
35#1009—61no—
36#1009Dic312yescomplemented
37C2×#239—21no—
38C2×D3×D4—61no—
39C2×#235—31no—
40C22×D12—31no—
41#994—32no—
42#994—61no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3—61no—
45C2×#41⋊C3C2×D311yescharacteristic
46C2×C3×M16—31no—
47C2×C3×C4⋊C4—31no—
48C2×(C2×D3)⋊C4—62no—
49C2×C4×Dic3—31no—
50C2×#208—31no—
51GU(2,𝔽3)—62no—
52GU(2,𝔽3)—61no—
53#859—21no—
54C3×#103—61no—
55C3×#102—61no—
56#837—61no—
57#837—61no—
58#835—121no—
59#825—62no—
60#823—62no—
61#810—61no—
62#809—61no—
63C2×C32×C4—41no—
64C2×C3×Dic3—81no—
65C2×C3×Dic3—81no—
66C3×Dic6—42no—
67C3×Dic6—81no—
68C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
69C3⋊(C3⋊C8)—81no—
70C2×H(5,𝔽2)—31no—
71C2×#104—91no—
72C2×#104—91no—
73#417—31no—
74#417—91no—
75C2×#100—91no—
76#390—91no—
77C23×D3—32no—
78C3×#41—31no—
79C3×#41—31no—
80C3×#41C2×Dic311yescharacteristic
81C3×#41#6711yescharacteristic
82C2×C3×Q8—121no—
83C2×C3×Q8#6711yescharacteristic
84C2×C3×D4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87C2×#67—61no—
88C22×Dic3—61no—
89#239—22no—
90#239—62no—
91D3×D4—64no—
92D3×D4—62no—
93D3×D4—62no—
94#235—32no—
95#235—32no—
96#235—61no—
97#235—61no—
98C2×D12—31no—
99C2×D12—32no—
100C2×D12—32no—
101C2×D12—61no—
102C2×D12—61no—
103C2×D12—61no—
104C2×C4×D3—61no—
105C2×Dic6S411yescharacteristic
106#41⋊C3—21no—
107#41⋊C3—21no—
108#41⋊C3—62no—
109#41⋊C3C4×D311yescharacteristic
110#41⋊C3D1211yescharacteristic
111C2×SL(2,𝔽3)—21no—
112C2×SL(2,𝔽3)#6711yescharacteristic
113C3×M16—61no—
114C3×M16—61no—
115C2×C3×C8—61no—
116C3×C4⋊C4—61no—
117C3×C4⋊C4—61no—
118(C2×D3)⋊C4—124no—
119C4⋊Dic3—41no—
120C4×Dic3—61no—
121C4×Dic3—61no—
122#208—61no—
123#208—61no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8—121no—
126C2×C3⋊C8—241no—
127C22×C32—41no—
128C32×C4—41no—
129C32×C4—41no—
130C3×Dic3—82no—
131C3×Dic3—161no—
132H(5,𝔽2)—32no—
133H(5,𝔽2)—61no—
134H(5,𝔽2)—92no—
135H(5,𝔽2)—181no—
136C2×#41—61no—
137C2×#41—91no—
138C2×#41D3211yescharacteristic
139C22×D4—91no—
140C22×D4—181no—
141C2×M16—31no—
142C2×M16—91no—
143C2×C4⋊C4—31no—
144C2×C22⋊C4—182no—
145C2×C42—91no—
146#104—182no—
147#104—181no—
148#104—181no—
149#103—61no—
150#103—181no—
151#102—61no—
152#102—181no—
153#100—182no—
154C23×C3—31no—
155C22×D3—34no—
156C22×D3—61no—
157C22×D3—62no—
158C22×D3—122no—
159C22×D3—122no—
160C3×Q8—31no—
161C3×Q8—122no—
162C3×Q8—241no—
163C3×Q8#21711yescharacteristic
164C3×D4—31no—
165C3×D4—31no—
166C3×D4—61no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—41no—
173C2×C3×C4—81no—
174C2×C3×C4—122no—
175C2×C3×C4—121no—
176C2×C3×C4—241no—
177C2×C3×C4C2×S411yescharacteristic
178#67—62no—
179#67—62no—
180C2×Dic3—21no—
181C2×Dic3—62no—
182C2×Dic3—81no—
183C2×Dic3—121no—
184C2×Dic3—121no—
185D12—32no—
186D12—62no—
187D12—62no—
188D12—61no—
189C4×D3—62no—
190C4×D3—62no—
191Dic6—21no—
192Dic6SL(2,ℤ/4)12yes—
193SL(2,𝔽3)—21no—
194SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic, complemented
195C3×C8—121no—
196C3⋊C8—121no—
197C3⋊C8—241no—
198C3⋊C8—242no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C24—92no—
203#41—31no—
204#41—31no—
205#41—62no—
206#41—92no—
207#41—92no—
208#41—181no—
209#41—181no—
210#41—182no—
211#41D3×Dic311yescharacteristic
212#41#62611yescharacteristic
213C2×Q8—31no—
214C2×Q8#62511yescharacteristic
215C2×D4—31no—
216C2×D4—92no—
217C2×D4—92no—
218C2×D4—91no—
219C2×D4—181no—
220C2×D4—181no—
221C2×D4—181no—
222C2×D4—181no—
223C2×D4—182no—
224C2×D4—182no—
225C2×D4—184no—
226C22×C4—31no—
227C22×C4—61no—
228C22×C4—181no—
229C22×C4—181no—
230M16—61no—
231M16—61no—
232M16—181no—
233M16—181no—
234C2×C8—61no—
235C2×C8—181no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C22⋊C4—364no—
239C42—181no—
240C42—181no—
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—41no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—81no—
247C22×C3GL(2,ℤ/4)11yescharacteristic
248C2×D3—62no—
249C2×D3—121no—
250C2×D3—121no—
251C2×D3—124no—
252C2×D3—242no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—81no—
258C3×C4—81no—
259C3×C4—242no—
260C3×C4—241no—
261C3×C4C2×SL(2,ℤ/4)11yescharacteristic
262C3×C4GL(2,ℤ/4)11yescharacteristic
263Dic3—22no—
264Dic3—122no—
265Dic3—161no—
266C32—41noSylow
267C23—31no—
268C23—94no—
269C23—182no—
270C23—181no—
271C23—362no—
272C23—362no—
273Q8—31no—
274Q8—32no—
275Q8—61no—
276Q8(C2×D3)⋊Dic311yescharacteristic
277D4—31no—
278D4—31no—
279D4—61no—
280D4—92no—
281D4—182no—
282D4—182no—
283D4—182no—
284D4—182no—
285D4—181no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—61no—
292C2×C4—122no—
293C2×C4—121no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4—361no—
298C2×C4—361no—
299C2×C4D3×S411yescharacteristic
300C8—121no—
301C8—361no—
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#542811yescharacteristic
311C2×C3#542911yescharacteristic
312C2×C3#543211yescharacteristic
313D3—122no—
314C22—31no—
315C22—31no—
316C22—61no—
317C22—61no—
318C22—182no—
319C22—364no—
320C22—361no—
321C22—361no—
322C22—722no—
323C22#6420911yescharacteristic
324C4—31no—
325C4—31no—
326C4—62no—
327C4—241no—
328C4—362no—
329C4D3×SL(2,ℤ/4)11yescharacteristic
330C4#6421011yescharacteristic
331C3—41no—
332C3—81no—
333C3#7315311yescharacteristic, complemented
334C2—61no—
335C2—61no—
336C2—362no—
337C2#1059323311yescharacteristic
338C2#1059323511yescharacteristic
339C2(C2×D3)⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22#64209
commutatorC3×#41⋊C3C2×C4
Frattini, hypercenterC2×C4D3×S4
FittingC2×C3×#41C2×D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×C3⋊#994C2
cosocleC2×C3×#41⋊C3C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊GU(2,𝔽3)C2semidirect
3. C3×#5426C2semidirect
8. C3⋊#994C4semidirect
15. #5426D3semidirect
36. #1009Dic3semidirect
68. C3×SL(2,𝔽3)C22⋊C4semidirect
194. SL(2,𝔽3)(C2×D3)⋊C4semidirect
333. C3#73153semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
C2×C3⋊GU(2,𝔽3)2
C3×#54262
#731533
#703053
#635654