Finite Group Databasebeta

← #87017⋊C3 #11923283

Subgroups — #87017⋊C3 #11923283

Order 1152.

607 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 89 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
3SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
4C4×D3×SL(2,𝔽3)C212yesmaximal, complemented
5C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
6#87017C311yesmaximal, characteristic, complemented
7#3482⋊C3—31nomaximal
8C2×D3×SL(2,𝔽3)C2212yes—
9C2×C3⋊(#41⋊C3)C2211yescharacteristic
10C3×#893—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#557C2×C311yescharacteristic, complemented
15#5728—31no—
16#5726—31no—
17#5694—32no—
18#5691—32no—
19#5584—31no—
20#5576C2×C312yescomplemented
21Q8×D12C2×C311yescharacteristic
22C4×#239—31no—
23C4×#239C2×C311yescharacteristic, complemented
24C4×D3×Q8C2×C312yescomplemented
25#557⋊C3—21no—
26#557⋊C3D311yescharacteristic, complemented
27D4×SL(2,𝔽3)—31no—
28#548⋊C3—32no—
29C4×#41⋊C3—31no—
30C2×C4×SL(2,𝔽3)—32no—
31C2×C3×D12—41no—
32C2×C3×C4×D3—42no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)—22no—
35C3⋊(#41⋊C3)—21no—
36C32×C4⋊C4—41no—
37C3×(C2×D3)⋊C4—42no—
38C3×C4×Dic3—41no—
39#3482—31noSylow
40C2×#239—31no—
41C2×#239C22×C311yescharacteristic
42C2×D3×Q8C22×C312yes—
43C2×#41⋊C3—31no—
44C22×SL(2,𝔽3)—32no—
45C3×C4⋊Q8—31no—
46C3×#125—32no—
47C3×C4×Q8—31no—
48C3×C4×Q8C22×C312yes—
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×D4—121no—
51#946—32no—
52Q8×Dic3—31no—
53Q8×Dic3C22×C311yescharacteristic
54#898—34no—
55#897—32no—
56#897—32no—
57#895—32no—
58#894—34no—
59#893—32no—
60#893—31no—
61#893A411yescharacteristic
62#892—32no—
63#892—31no—
64#892—32no—
65#892—31no—
66D3×C4⋊C4—32no—
67#890—32no—
68#887—32no—
69#875—32no—
70C4×D12—31no—
71C4×D12—31no—
72C4×D12—32no—
73C42×D3—32no—
74C42×D3—31no—
75#869—31no—
76C4×Dic6—32no—
77C4×SL(2,𝔽3)—21no—
78C4×SL(2,𝔽3)—22no—
79C4×SL(2,𝔽3)—31no—
80C4×SL(2,𝔽3)C2×D312yes—
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C22×C3×D3—42no—
83C2×C32×C4—41no—
84C2×C32×C4—42no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—41no—
87C3×D12—44no—
88C3×C4×D3—84no—
89C3×SL(2,𝔽3)#4111yescharacteristic, complemented
90#557C3×D311yescharacteristic, complemented
91#556—91no—
92#552—92no—
93D4×Q8—31no—
94#548—32no—
95#547—91no—
96#533—91no—
97#529—92no—
98C4×#41—31no—
99C4×#41—91no—
100C2×C4×Q8—32no—
101C2×C3×Q8—31no—
102C2×C3×Q8C23×C311yescharacteristic
103C2×C3×D4—121no—
104C22×C3×C4—122no—
105#239—21no—
106#239—64no—
107#239—61no—
108D3×Q8—22no—
109D3×Q8—62no—
110C2×D12—31no—
111C2×D12—32no—
112C2×D12C2×A411yescharacteristic
113C2×C4×D3—32no—
114C2×C4×D3—32no—
115C2×C4×D3—31no—
116C2×C4×D3—31no—
117C2×C4×D3—31no—
118C2×C4×D3C2×A412yes—
119C2×Dic6—32no—
120#41⋊C3—61no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—62no—
123C2×SL(2,𝔽3)C22×D311yescharacteristic
124C3×C4⋊C4—32no—
125C3×C4⋊C4—31no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—32no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—81no—
130C3×C4⋊C4C2×A411yescharacteristic
131C3×C22⋊C4—122no—
132C3×C42—32no—
133C3×C42—31no—
134C3×C42—121no—
135(C2×D3)⋊C4—32no—
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⋊Dic3—32no—
143C4⋊Dic3—31no—
144Dic3⋊C4—34no—
145C4×Dic3—32no—
146C4×Dic3—31no—
147C4×Dic3—32no—
148C4×Dic3—31no—
149C4×Dic3C2×A411yescharacteristic
150C22×C32—41no—
151C2×C3×D3—44no—
152C2×C3×D3—82no—
153C2×C3×D3—82no—
154C32×C4—42no—
155C32×C4—82no—
156C3×Dic3—42no—
157C3×Dic3—81no—
158C2×#41—31no—
159C2×#41—91no—
160C22×Q8—32no—
161C4⋊Q8—31no—
162C4⋊Q8—91no—
163#126—94no—
164#125—32no—
165#125—92no—
166#124—92no—
167#123—94no—
168C22⋊Q8—92no—
169C22⋊Q8—92no—
170C22⋊Q8—92no—
171#121—92no—
172C4×Q8—31no—
173C4×Q8—31no—
174C4×Q8—91no—
175C4×Q8—92no—
176C4×Q8—92no—
177C4×Q8C2×C3×D312yes—
178C4×Q8C2×C3×D311yescharacteristic
179C4×D4—31no—
180C4×D4—92no—
181C4×D4—91no—
182C4×D4—91no—
183C4×D4—92no—
184C4×D4—91no—
185#117—92no—
186#117—92no—
187#117—91no—
188#117—91no—
189C2×C4⋊C4—92no—
190C2×C42—91no—
191C2×C42—92no—
192C23×C3—122no—
193C22×D3—31no—
194C22×D3C22×A412yes—
195C3×Q8—31no—
196C3×Q8—62no—
197C3×Q8C3×#4111yescharacteristic
198C3×D4—124no—
199C2×C3×C4—31no—
200C2×C3×C4—32no—
201C2×C3×C4—31no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—81no—
205C2×C3×C4—82no—
206C2×C3×C4—121no—
207C2×C3×C4—121no—
208C2×C3×C4—244no—
209C2×C3×C4C22×A412yes—
210C2×C3×C4C22×A411yescharacteristic
211C2×Dic3—31no—
212C2×Dic3—32no—
213C2×Dic3C22×A411yescharacteristic
214C2×Dic3C22×A411yescharacteristic
215D12—61no—
216D12—61no—
217D12—64no—
218D12#41⋊C314yescomplemented
219C4×D3—24no—
220C4×D3—61no—
221C4×D3—61no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—64no—
225C4×D3—62no—
226C4×D3—62no—
227Dic6—62no—
228Dic6—62no—
229SL(2,𝔽3)—21no—
230SL(2,𝔽3)#23911yescharacteristic, complemented
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C3×D3—84no—
235#41—61no—
236#41—184no—
237#41—181no—
238C2×Q8—31no—
239C2×Q8—62no—
240C2×Q8—92no—
241C2×Q8—182no—
242C2×Q8C22×C3×D311yescharacteristic
243C2×D4—31no—
244C2×D4—92no—
245C2×D4—91no—
246C22×C4—32no—
247C22×C4—91no—
248C22×C4—91no—
249C22×C4—92no—
250C22×C4—92no—
251C22×C4—91no—
252C4⋊C4—32no—
253C4⋊C4—31no—
254C4⋊C4—32no—
255C4⋊C4—32no—
256C4⋊C4—92no—
257C4⋊C4—91no—
258C4⋊C4—94no—
259C4⋊C4—92no—
260C4⋊C4—91no—
261C4⋊C4D3×A411yescharacteristic
262C22⋊C4—32no—
263C22⋊C4—92no—
264C22⋊C4—92no—
265C22⋊C4—94no—
266C22⋊C4—92no—
267C42—31no—
268C42—31no—
269C42—32no—
270C42—91no—
271C42—91no—
272C42—92no—
273C42—92no—
274C22×C3—41no—
275C22×C3—81no—
276C22×C3—124no—
277C22×C3—242no—
278C22×C3—242no—
279C22×C3C23×A411yescharacteristic
280C2×D3—22no—
281C2×D3—22no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—61no—
285C2×D3C2×#41⋊C314yes—
286C3×C4—22no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—42no—
290C3×C4—62no—
291C3×C4—61no—
292C3×C4—82no—
293C3×C4—82no—
294C3×C4—122no—
295C3×C4—162no—
296C3×C4—241no—
297C3×C4C2×#41⋊C312yes—
298Dic3—21no—
299Dic3—61no—
300Dic3—62no—
301Dic32−1+4⋊C312yes—
302C32—41noSylow
303C23—32no—
304C23—91no—
305Q8—31no—
306Q8—62no—
307Q8—182no—
308Q8—182no—
309Q8C3×#23911yescharacteristic
310D4—34no—
311D4—181no—
312D4—181no—
313D4—184no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—32no—
319C2×C4—64no—
320C2×C4—91no—
321C2×C4—92no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—182no—
325C2×C4—182no—
326C2×C4—182no—
327C2×C4—184no—
328C2×C4—182no—
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—244no—
338C2×C3C22×#41⋊C311yescharacteristic
339C2×C3A4×#4111yescharacteristic
340C2×C3C2×2−1+4⋊C311yescharacteristic
341D3—24no—
342D3—121no—
343C22—34no—
344C22—62no—
345C22—62no—
346C22—181no—
347C22—181no—
348C22—181no—
349C22C22×D3×A411yescharacteristic
350C4—22no—
351C4—31no—
352C4—31no—
353C4—32no—
354C4—61no—
355C4—62no—
356C4—61no—
357C4—182no—
358C4—181no—
359C4D3×#41⋊C312yes—
360C3—41no—
361C3—81no—
362C3#3482⋊C311yescharacteristic, complemented
363C2—64no—
364C2—361no—
365C2A4×#23911yescharacteristic
366C2C2×D3×#41⋊C311yescharacteristic
367C2D3×2−1+4⋊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. SL(2,𝔽3)×D12C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #87017C3semidirect
14. C3×#557C2×C3semidirect
20. #5576C2×C3semidirect
23. C4×#239C2×C3semidirect
24. C4×D3×Q8C2×C3semidirect
26. #557⋊C3D3semidirect
89. C3×SL(2,𝔽3)#41semidirect
90. #557C3×D3semidirect
218. D12#41⋊C3semidirect
230. SL(2,𝔽3)#239semidirect
362. C3#3482⋊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
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
C4×C3⋊(#41⋊C3)2
#870173
#3482⋊C33
C3×#8934