Finite Group Databasebeta

← #87016⋊C3 #11923282

Subgroups — #87016⋊C3 #11923282

Order 1152.

583 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#557⋊C3C211yesmaximal, characteristic, complemented
2#5576⋊C3C212yesmaximal, complemented
3C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5C4×C3⋊(#41⋊C3)C212yesmaximal, complemented
6#87016C311yesmaximal, characteristic, complemented
7#3480⋊C3—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×#892—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yes—
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#557C2×C311yescharacteristic, complemented
15#5732—31no—
16#5725—31no—
17#5695—32no—
18#5691—32no—
19#5585—31no—
20#5577C2×C311yescharacteristic, complemented
21#5576C2×C312yes—
22C4×#239—31no—
23C4×#239C2×C312yescomplemented
24C4×D3×Q8C2×C311yescharacteristic
25#557⋊C3—21no—
26#557⋊C3D311yescharacteristic, complemented
27C4⋊(#41⋊C3)—31no—
28#548⋊C3—32no—
29C4×#41⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C3×C4×D3—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33D3×SL(2,𝔽3)—21no—
34C3⋊(#41⋊C3)—22no—
35C32×C4⋊C4—41no—
36C3×(C2×D3)⋊C4—42no—
37C3×C4⋊Dic3—41no—
38C3×C4×Dic3—42no—
39#3480—31noSylow
40C2×#239—31no—
41C2×#239C22×C312yes—
42C2×D3×Q8C22×C311yescharacteristic
43C2×#41⋊C3—32no—
44C22×SL(2,𝔽3)—31no—
45C3×C4⋊Q8—31no—
46C3×#125—32no—
47C3×C4×Q8—31no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×Q8C22×C312yes—
50C3×#117—121no—
51#947—32no—
52Q8×Dic3—31no—
53Q8×Dic3C22×C311yescharacteristic
54#898—34no—
55#897—32no—
56#895—32no—
57#895—32no—
58#895—32no—
59#894—34no—
60#893—32no—
61#893—32no—
62#893—31no—
63#893—32no—
64#892—32no—
65#892—32no—
66#892A411yescharacteristic
67D3×C4⋊C4—31no—
68#887—31no—
69#875—32no—
70#874—31no—
71C4×D12—32no—
72C4×D12—32no—
73C4×D12—31no—
74C42×D3—31no—
75C42×D3—32no—
76C4×Dic6—31no—
77C4×SL(2,𝔽3)—21no—
78C4×SL(2,𝔽3)—22no—
79C4×SL(2,𝔽3)—31no—
80C4×SL(2,𝔽3)C2×D311yescharacteristic
81C4×SL(2,𝔽3)C2×D312yes—
82C22×C3×D3—41no—
83C2×C32×C4—41no—
84C2×C32×C4—42no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C3×C4×D3—44no—
88C3×SL(2,𝔽3)#4111yescharacteristic, complemented
89#559—91no—
90#557C3×D311yescharacteristic, complemented
91#556—91no—
92#553—92no—
93#550—31no—
94#548—32no—
95#532—91no—
96#529—92no—
97C4×#41—32no—
98C4×#41—91no—
99C2×C4×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8C23×C311yescharacteristic
102C22×C3×C4—121no—
103#239—22no—
104#239—64no—
105#239—62no—
106D3×Q8—21no—
107D3×Q8—61no—
108C2×D12—32no—
109C2×D12—32no—
110C2×D12—31no—
111C2×D12—32no—
112C2×C4×D3—31no—
113C2×C4×D3—32no—
114C2×C4×D3—32no—
115C2×C4×D3—31no—
116C2×C4×D3—32no—
117C2×C4×D3C2×A411yescharacteristic
118C2×Dic6—31no—
119#41⋊C3—62no—
120C2×SL(2,𝔽3)—21no—
121C2×SL(2,𝔽3)—61no—
122C2×SL(2,𝔽3)C22×D311yescharacteristic
123C3×C4⋊C4—31no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—81no—
129C3×C4⋊C4—121no—
130C3×C4⋊C4C2×A411yescharacteristic
131C3×C22⋊C4—122no—
132C3×C42—31no—
133C3×C42—32no—
134C3×C42—122no—
135(C2×D3)⋊C4—34no—
136(C2×D3)⋊C4—34no—
137(C2×D3)⋊C4—32no—
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4C2×A412yes—
140C4⋊Dic3—32no—
141C4⋊Dic3—31no—
142C4⋊Dic3C2×A411yescharacteristic
143Dic3⋊C4—32no—
144C4×Dic3—31no—
145C4×Dic3—31no—
146C4×Dic3—31no—
147C4×Dic3—32no—
148C4×Dic3C2×A412yes—
149C22×C32—41no—
150C2×C3×D3—42no—
151C2×C3×D3—81no—
152C2×C3×D3—81no—
153C32×C4—42no—
154C32×C4—82no—
155C3×Dic3—42no—
156C3×Dic3—82no—
157C2×#41—32no—
158C2×#41—91no—
159C22×Q8—31no—
160C4⋊Q8—31no—
161#127—91no—
162#126—94no—
163#125—32no—
164#124—92no—
165#124—92no—
166#123—94no—
167C22⋊Q8—92no—
168#121—92no—
169#121—92no—
170#121—92no—
171C4×Q8—31no—
172C4×Q8—31no—
173C4×Q8—91no—
174C4×Q8—91no—
175C4×Q8—91no—
176C4×Q8C2×C3×D311yescharacteristic
177C4×Q8C2×C3×D312yes—
178C4×D4—92no—
179C4×D4—91no—
180C4×D4—92no—
181C4×D4—92no—
182C4×D4—92no—
183C4×D4—92no—
184C4×D4—91no—
185#117—31no—
186#117—92no—
187#117—92no—
188C2×C4⋊C4—91no—
189C2×C42—92no—
190C2×C42—91no—
191C23×C3—121no—
192C22×D3—32no—
193C22×D3C22×A411yescharacteristic
194C3×Q8—31no—
195C3×Q8—62no—
196C3×Q8C3×#4111yescharacteristic
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—32no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—81no—
203C2×C3×C4—82no—
204C2×C3×C4—124no—
205C2×C3×C4—121no—
206C2×C3×C4—122no—
207C2×C3×C4C22×A412yes—
208C2×C3×C4C22×A411yescharacteristic
209C2×Dic3—31no—
210C2×Dic3—31no—
211C2×Dic3C22×A412yes—
212C2×Dic3C22×A411yescharacteristic
213D12—62no—
214D12—62no—
215D12—64no—
216D12—62no—
217D12—122no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—61no—
221C4×D3—61no—
222C4×D3—64no—
223C4×D3—122no—
224C4×D3—121no—
225C4×D3#41⋊C314yescomplemented
226Dic6—61no—
227Dic6—61no—
228SL(2,𝔽3)—21no—
229SL(2,𝔽3)#23911yescharacteristic, complemented
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C3×D3—82no—
234#41—62no—
235#41—184no—
236#41—182no—
237C2×Q8—31no—
238C2×Q8—61no—
239C2×Q8—91no—
240C2×Q8—181no—
241C2×Q8C22×C3×D311yescharacteristic
242C2×D4—92no—
243C2×D4—92no—
244C2×D4—92no—
245C2×D4—91no—
246C22×C4—31no—
247C22×C4—92no—
248C22×C4—92no—
249C22×C4—91no—
250C22×C4—91no—
251C22×C4—92no—
252C4⋊C4—31no—
253C4⋊C4—32no—
254C4⋊C4—32no—
255C4⋊C4—31no—
256C4⋊C4—32no—
257C4⋊C4—92no—
258C4⋊C4—91no—
259C4⋊C4—92no—
260C4⋊C4D3×A411yescharacteristic
261C22⋊C4—32no—
262C22⋊C4—92no—
263C22⋊C4—94no—
264C22⋊C4—94no—
265C22⋊C4—92no—
266C42—32no—
267C42—32no—
268C42—31no—
269C42—91no—
270C42—91no—
271C42—92no—
272C42—91no—
273C22×C3—41no—
274C22×C3—81no—
275C22×C3—122no—
276C22×C3—241no—
277C22×C3—241no—
278C22×C3C23×A411yescharacteristic
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—62no—
284C2×D3C2×#41⋊C312yes—
285C3×C4—22no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—42no—
289C3×C4—62no—
290C3×C4—61no—
291C3×C4—82no—
292C3×C4—82no—
293C3×C4—122no—
294C3×C4—162no—
295C3×C4—242no—
296C3×C4C2×#41⋊C312yes—
297Dic3—22no—
298Dic3—61no—
299Dic3—61no—
300Dic3C2×#41⋊C312yes—
301C32—41noSylow
302C23—31no—
303C23—92no—
304Q8—31no—
305Q8—62no—
306Q8—181no—
307Q8—181no—
308Q8C3×#23911yescharacteristic
309D4—182no—
310D4—182no—
311D4—184no—
312D4—182no—
313D4—362no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—34no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—31no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—182no—
323C2×C4—184no—
324C2×C4—182no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—362no—
328C2×C4—361no—
329C2×C4C2×D3×A412yes—
330C2×C4C2×D3×A411yescharacteristic
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—81no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3—242no—
338C2×C3C22×#41⋊C311yescharacteristic
339C2×C3A4×#4111yescharacteristic
340C2×C3#594511yescharacteristic
341D3—22no—
342D3—122no—
343C22—32no—
344C22—61no—
345C22—61no—
346C22—182no—
347C22—182no—
348C22—182no—
349C22C22×D3×A411yescharacteristic
350C4—22no—
351C4—31no—
352C4—31no—
353C4—32no—
354C4—62no—
355C4—62no—
356C4—61no—
357C4—181no—
358C4—181no—
359C4D3×#41⋊C312yes—
360C3—41no—
361C3—81no—
362C3#3480⋊C311yescharacteristic, complemented
363C2—62no—
364C2—362no—
365C2A4×#23911yescharacteristic
366C2C2×D3×#41⋊C311yescharacteristic
367C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×C3
socleC22×C3C23×A4
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
2. #5576⋊C3C2semidirect
3. C4×D3×SL(2,𝔽3)C2semidirect
4. C4⋊(C3⋊(#41⋊C3))C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #87016C3semidirect
14. C3×#557C2×C3semidirect
20. #5577C2×C3semidirect
23. C4×#239C2×C3semidirect
26. #557⋊C3D3semidirect
88. C3×SL(2,𝔽3)#41semidirect
90. #557C3×D3semidirect
225. C4×D3#41⋊C3semidirect
229. SL(2,𝔽3)#239semidirect
362. C3#3480⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5576⋊C32
C4×D3×SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))2
C4×C3⋊(#41⋊C3)2
#870163
#3480⋊C33
C3×#8924