Finite Group Databasebeta

← #86744⋊C3 #11923255

Subgroups — #86744⋊C3 #11923255

Order 1152.

623 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 97 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5817⋊C3C212yesmaximal, complemented
3C2×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
4#5570⋊C3C211yesmaximal, characteristic, complemented
5C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
6#86744C311yesmaximal, characteristic, complemented
7#3482⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×C3⋊(#41⋊C3)C2211yescharacteristic
10C3×D4×Dic3—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×D4×Q8C2×C311yescharacteristic
15#5817C2×C312yescomplemented
16C2×Q8×Dic3C2×C312yescomplemented
17#5732—31no—
18#5728—31no—
19#5642—32no—
20#5624—32no—
21C4×#239C2×C311yescharacteristic, complemented
22#5570C2×C311yescharacteristic, complemented
23#5555—31no—
24C4×#237—31no—
25#557⋊C3—31no—
26D4×SL(2,𝔽3)—21no—
27D4×SL(2,𝔽3)D311yescharacteristic, complemented
28#548⋊C3—32no—
29C4×#41⋊C3—31no—
30C2×C4×SL(2,𝔽3)—32no—
31C2×C32×D4—41no—
32C22×C3×Dic3—42no—
33C2×C3×SL(2,𝔽3)—22no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊(#41⋊C3)—21no—
36C3×#217—42no—
37C3×C4⋊Dic3—41no—
38C3×C4×Dic3—41no—
39#3482—31noSylow
40C22×C3×Q8C22×C312yes—
41C2×#239C22×C311yescharacteristic
42C2×#237—31no—
43C2×#41⋊C3—31no—
44C22×SL(2,𝔽3)—22no—
45C22×SL(2,𝔽3)C2×D312yes—
46C3×C4⋊Q8—31no—
47C3×C22⋊Q8—32no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×D4—31no—
50C3×C4×D4—121no—
51#947—32no—
52#946—32no—
53Q8×Dic3—32no—
54Q8×Dic3—32no—
55Q8×Dic3C22×C311yescharacteristic
56Q8×Dic3C22×C312yes—
57D4×Dic3A411yescharacteristic
58#930—32no—
59C4×#67—32no—
60C2×C4⋊Dic3—32no—
61C2×C4×Dic3—32no—
62#898—34no—
63#897—32no—
64#893—31no—
65#892—32no—
66#892—31no—
67#892—31no—
68#890—32no—
69#890—32no—
70#887—31no—
71#886—34no—
72#67⋊C4—32no—
73#877—32no—
74C4×D12—31no—
75C42×D3—31no—
76#869—31no—
77C4×Dic6—31no—
78C4×SL(2,𝔽3)—21no—
79C4×SL(2,𝔽3)—31no—
80C4×SL(2,𝔽3)—32no—
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C23×C32—42no—
83C32×D4—44no—
84C2×C32×C4—41no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—41no—
88C2×C3×Dic3—84no—
89C3×SL(2,𝔽3)#4111yescharacteristic, complemented
90#557—31no—
91#556—91no—
92#552—92no—
93D4×Q8C3×D311yescharacteristic
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—22no—
102C2×C3×Q8—32no—
103C2×C3×Q8—62no—
104C2×C3×Q8C23×C311yescharacteristic
105C2×C3×D4—41no—
106C2×C3×D4—81no—
107C2×C3×D4C2×A411yescharacteristic
108C22×C3×C4—32no—
109C22×C3×C4—122no—
110C2×#67—32no—
111C22×Dic3—32no—
112C22×Dic3C2×A412yes—
113#239—21no—
114#239—61no—
115#237—64no—
116C2×D12—31no—
117C2×C4×D3—31no—
118C2×C4×D3—31no—
119C2×C4×D3—31no—
120C2×Dic6—31no—
121#41⋊C3—61no—
122C2×SL(2,𝔽3)—22no—
123C2×SL(2,𝔽3)—21no—
124C2×SL(2,𝔽3)—42no—
125C2×SL(2,𝔽3)C22×D311yescharacteristic
126C3×C4⋊C4—31no—
127C3×C4⋊C4—32no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—121no—
130C3×C22⋊C4—32no—
131C3×C22⋊C4—122no—
132C3×C42—31no—
133C3×C42—121no—
134#217C2×A412yes—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—34no—
137(C2×D3)⋊C4—32no—
138C4⋊Dic3—32no—
139C4⋊Dic3—34no—
140C4⋊Dic3—32no—
141C4⋊Dic3—31no—
142C4⋊Dic3C2×A411yescharacteristic
143Dic3⋊C4—32no—
144Dic3⋊C4—32no—
145C4×Dic3—32no—
146C4×Dic3—31no—
147C4×Dic3—31no—
148C4×Dic3—32no—
149C4×Dic3—32no—
150C4×Dic3C2×A411yescharacteristic
151C22×C32—44no—
152C22×C32—41no—
153C22×C32—82no—
154C22×C32—82no—
155C32×C4—42no—
156C3×Dic3—42no—
157C3×Dic3—82no—
158C3×Dic3—81no—
159C2×#41—31no—
160C2×#41—91no—
161C22×Q8C2×C3×D312yes—
162C4⋊Q8—31no—
163C4⋊Q8—91no—
164#126—94no—
165#125—92no—
166#125—92no—
167#124—92no—
168#123—94no—
169C22⋊Q8—32no—
170C22⋊Q8—92no—
171C22⋊Q8—92no—
172#121—92no—
173C4×Q8—31no—
174C4×Q8—32no—
175C4×Q8—91no—
176C4×Q8—91no—
177C4×Q8—92no—
178C4×Q8—92no—
179C4×Q8C2×C3×D311yescharacteristic
180C4×D4—31no—
181C4×D4—31no—
182C4×D4—91no—
183C4×D4—92no—
184C4×D4—92no—
185C4×D4—91no—
186#117—92no—
187#117—91no—
188#117—92no—
189#117—91no—
190C2×C4⋊C4—92no—
191C2×C42—91no—
192C2×C42—92no—
193C23×C3—42no—
194C23×C3—82no—
195C23×C3C22×A412yes—
196C22×D3—31no—
197C3×Q8—31no—
198C3×Q8—62no—
199C3×Q8—62no—
200C3×Q8C3×#4111yescharacteristic
201C3×D4—44no—
202C3×D4—84no—
203C3×D4#41⋊C314yescomplemented
204C2×C3×C4—31no—
205C2×C3×C4—32no—
206C2×C3×C4—31no—
207C2×C3×C4—41no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—81no—
211C2×C3×C4—121no—
212C2×C3×C4—122no—
213C2×C3×C4—121no—
214C2×C3×C4—244no—
215C2×C3×C4C22×A411yescharacteristic
216#67—64no—
217C2×Dic3—24no—
218C2×Dic3—31no—
219C2×Dic3—32no—
220C2×Dic3—64no—
221C2×Dic3C22×A412yes—
222C2×Dic3C22×A411yescharacteristic
223C2×Dic3C22×A411yescharacteristic
224D12—61no—
225D12—61no—
226C4×D3—61no—
227C4×D3—61no—
228C4×D3—62no—
229C4×D3—62no—
230Dic6—62no—
231SL(2,𝔽3)—21no—
232SL(2,𝔽3)#23711yescharacteristic, complemented
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—84no—
237#41—61no—
238#41—184no—
239#41—181no—
240C2×Q8—22no—
241C2×Q8—32no—
242C2×Q8—62no—
243C2×Q8—91no—
244C2×Q8C22×C3×D311yescharacteristic
245C2×D4—91no—
246C2×D4—92no—
247C2×D4D3×A411yescharacteristic
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—92no—
253C22×C4—91no—
254C4⋊C4—31no—
255C4⋊C4—32no—
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—92no—
259C4⋊C4—92no—
260C4⋊C4—91no—
261C4⋊C4—92no—
262C4⋊C4—94no—
263C4⋊C4—92no—
264C22⋊C4—32no—
265C22⋊C4—32no—
266C22⋊C4—92no—
267C22⋊C4—92no—
268C22⋊C4—94no—
269C42—31no—
270C42—31no—
271C42—91no—
272C42—91no—
273C42—92no—
274C42—92no—
275C42—92no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—44no—
279C22×C3—41no—
280C22×C3—82no—
281C22×C3—82no—
282C22×C3—84no—
283C22×C3—81no—
284C22×C3—162no—
285C22×C3—162no—
286C22×C3C2×#41⋊C314yes—
287C22×C3C23×A411yescharacteristic
288C2×D3—61no—
289C2×D3—61no—
290C2×D3—61no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—42no—
294C3×C4—62no—
295C3×C4—61no—
296C3×C4—82no—
297C3×C4—122no—
298C3×C4—242no—
299C3×C4—241no—
300C3×C4C2×#41⋊C312yes—
301Dic3—21no—
302Dic3—22no—
303Dic3—62no—
304Dic3—61no—
305Dic32−1+4⋊C312yes—
306C32—41noSylow
307C23—91no—
308C23C2×D3×A412yes—
309Q8—31no—
310Q8—62no—
311Q8—62no—
312Q8—182no—
313Q8C3×#23711yescharacteristic
314D4—181no—
315D4—181no—
316D4—184no—
317D4C3⋊(#41⋊C3)14yescomplemented
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—32no—
323C2×C4—32no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—64no—
327C2×C4—91no—
328C2×C4—92no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—182no—
332C2×C4—184no—
333C2×C4—182no—
334C2×C4C2×D3×A411yescharacteristic
335C2×C3—24no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—84no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3—81no—
343C2×C3—164no—
344C2×C3C22×#41⋊C311yescharacteristic
345C2×C3A4×#4111yescharacteristic
346C2×C3C2×2−1+4⋊C311yescharacteristic
347D3—121no—
348C22—22no—
349C22—22no—
350C22—181no—
351C22—181no—
352C22—181no—
353C22C2×C3⋊(#41⋊C3)14yes—
354C22C22×D3×A411yescharacteristic
355C4—31no—
356C4—31no—
357C4—32no—
358C4—62no—
359C4—61no—
360C4—62no—
361C4—61no—
362C4—181no—
363C4—182no—
364C4C2×C3⋊(#41⋊C3)12yes—
365C3—41no—
366C3—81no—
367C3#3482⋊C311yescharacteristic, complemented
368C2—24no—
369C2—361no—
370C2A4×#23711yescharacteristic
371C2C22×C3⋊(#41⋊C3)11yescharacteristic
372C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×D4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×SL(2,𝔽3)C2semidirect
2. #5817⋊C3C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. #5570⋊C3C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #86744C3semidirect
15. #5817C2×C3semidirect
16. C2×Q8×Dic3C2×C3semidirect
21. C4×#239C2×C3semidirect
22. #5570C2×C3semidirect
27. D4×SL(2,𝔽3)D3semidirect
89. C3×SL(2,𝔽3)#41semidirect
203. C3×D4#41⋊C3semidirect
232. SL(2,𝔽3)#237semidirect
317. D4C3⋊(#41⋊C3)semidirect
367. C3#3482⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
#5817⋊C32
C2×Dic3×SL(2,𝔽3)2
#5570⋊C32
C4×C3⋊(#41⋊C3)2
#867443
#3482⋊C33
C3×D4×Dic34