Finite Group Databasebeta

← #86767⋊C3 #11923270

Subgroups — #86767⋊C3 #11923270

Order 1152.

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

normal subgroups only

This group has 585 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2#5831⋊C3C212yesmaximal, complemented
3Dic3×#41⋊C3C212yesmaximal, complemented
4C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#5570⋊C3C211yesmaximal, characteristic, complemented
6#86767C311yesmaximal, characteristic, complemented
7#3480⋊C3—31nomaximal
8C2×C3×#41⋊C3C2212yes—
9C2×D3×SL(2,𝔽3)C2211yescharacteristic
10C3×#892—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#550C2×C311yescharacteristic, complemented
15#5831C2×C312yes—
16Dic3×#41C2×C312yescomplemented
17#5716—31no—
18#5713—31no—
19#5614—32no—
20#5600—32no—
21C4×D3×Q8C2×C311yescharacteristic
22#5570C2×C311yescharacteristic, complemented
23#5542—31no—
24C4×#237—31no—
25#557⋊C3—31no—
26C4⋊(#41⋊C3)—21no—
27C4⋊(#41⋊C3)D311yescharacteristic, complemented
28#548⋊C3—32no—
29C4×#41⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C3×C4×D3—41no—
32C3×#41⋊C3—22no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)—21no—
35C32×C4⋊C4—41no—
36C3×(C2×D3)⋊C4—42no—
37C3×C4⋊Dic3—41no—
38C3×C4×Dic3—42no—
39#3480—31noSylow
40C2×C3×#41C22×C312yes—
41C2×D3×Q8C22×C311yescharacteristic
42C2×#237—31no—
43C2×#41⋊C3—22no—
44C2×#41⋊C3C2×D312yes—
45C22×SL(2,𝔽3)—31no—
46C3×#127—31no—
47C3×#121—32no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×D4—31no—
50C3×#117—121no—
51Q8×Dic3C22×C312yes—
52Q8×Dic3C22×C311yescharacteristic
53#938—32no—
54#938—32no—
55#936—32no—
56#936—32no—
57#935—34no—
58D4×Dic3—32no—
59D4×Dic3—32no—
60D4×Dic3—32no—
61D4×Dic3—31no—
62C4×#67—32no—
63#926—32no—
64#924—32no—
65C2×C4×Dic3—32no—
66#892A411yescharacteristic
67D3×C4⋊C4—31no—
68#890—32no—
69#887—31no—
70#887—31no—
71#67⋊C4—32no—
72#879—34no—
73#877—32no—
74C42×D3—31no—
75#869—31no—
76C4×Dic6—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
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#557—31no—
91#556—91no—
92#553—92no—
93#550C3×D311yescharacteristic, complemented
94#548—32no—
95#532—91no—
96#529—92no—
97C4×#41—32no—
98C4×#41—91no—
99C2×C4×Q8—31no—
100C3×#41—22no—
101C3×#41—62no—
102C2×C3×Q8C23×C311yescharacteristic
103C2×C3×D4—32no—
104C2×C3×D4—32no—
105C2×C3×D4—31no—
106C22×C3×C4—32no—
107C22×C3×C4—121no—
108C2×#67—32no—
109C22×Dic3—32no—
110C22×Dic3—32no—
111D3×Q8—21no—
112D3×Q8—61no—
113#237—64no—
114C2×C4×D3—31no—
115C2×C4×D3—31no—
116C2×C4×D3C2×A411yescharacteristic
117C2×Dic6—31no—
118C2×Dic6—31no—
119#41⋊C3—22no—
120#41⋊C3—42no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—61no—
123C2×SL(2,𝔽3)C22×D311yescharacteristic
124C3×C4⋊C4—31no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—81no—
127C3×C4⋊C4—121no—
128C3×C4⋊C4C2×A411yescharacteristic
129C3×C22⋊C4—32no—
130C3×C22⋊C4—122no—
131C3×C42—31no—
132C3×C42—122no—
133#217—34no—
134#217—34no—
135#217—32no—
136(C2×D3)⋊C4C2×A412yes—
137C4⋊Dic3—32no—
138C4⋊Dic3—31no—
139C4⋊Dic3—32no—
140C4⋊Dic3C2×A411yescharacteristic
141Dic3⋊C4—32no—
142Dic3⋊C4—32no—
143Dic3⋊C4—32no—
144C4×Dic3—31no—
145C4×Dic3—32no—
146C4×Dic3—32no—
147C4×Dic3—31no—
148C4×Dic3—31no—
149C4×Dic3C2×A412yes—
150C22×C32—41no—
151C2×C3×D3—42no—
152C2×C3×D3—81no—
153C2×C3×D3—81no—
154C32×C4—42no—
155C32×C4—82no—
156C3×Dic3—42no—
157C3×Dic3—82no—
158C2×#41—91no—
159C2×#41C2×C3×D312yes—
160C22×Q8—31no—
161C4⋊Q8—91no—
162#127—31no—
163#126—94no—
164#125—92no—
165#124—92no—
166#124—92no—
167#123—94no—
168C22⋊Q8—92no—
169#121—32no—
170#121—92no—
171#121—92no—
172C4×Q8—31no—
173C4×Q8—32no—
174C4×Q8—91no—
175C4×Q8—91no—
176C4×Q8—91no—
177C4×Q8—91no—
178C4×Q8C2×C3×D311yescharacteristic
179C4×D4—31no—
180C4×D4—92no—
181C4×D4—92no—
182C4×D4—91no—
183C4×D4—92no—
184C4×D4—92no—
185C4×D4—92no—
186#117—31no—
187#117—92no—
188#117—92no—
189C2×C4⋊C4—91no—
190C2×C42—91no—
191C2×C42—92no—
192C23×C3—32no—
193C23×C3—121no—
194C22×D3C22×A411yescharacteristic
195C3×Q8—31no—
196C3×Q8C3×#4111yescharacteristic
197C3×D4—62no—
198C3×D4—62no—
199C3×D4—62no—
200C3×D4—122no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—81no—
208C2×C3×C4—82no—
209C2×C3×C4—124no—
210C2×C3×C4—121no—
211C2×C3×C4—122no—
212C2×C3×C4C22×A412yes—
213C2×C3×C4C22×A411yescharacteristic
214#67—64no—
215C2×Dic3—31no—
216C2×Dic3—31no—
217C2×Dic3—32no—
218C2×Dic3—64no—
219C2×Dic3—122no—
220C2×Dic3C22×A412yes—
221C2×Dic3C22×A411yescharacteristic
222C4×D3—61no—
223C4×D3—61no—
224C4×D3—121no—
225C4×D3#41⋊C314yescomplemented
226Dic6—62no—
227Dic6—61no—
228Dic6—61no—
229SL(2,𝔽3)—21no—
230SL(2,𝔽3)#23711yescharacteristic, complemented
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C3×D3—82no—
235#41—22no—
236#41—62no—
237#41—184no—
238C2×Q8—61no—
239C2×Q8—91no—
240C2×Q8—91no—
241C2×Q8—181no—
242C2×Q8C22×C3×D311yescharacteristic
243C2×D4—32no—
244C2×D4—32no—
245C2×D4—31no—
246C2×D4—92no—
247C22×C4—31no—
248C22×C4—32no—
249C22×C4—91no—
250C22×C4—92no—
251C22×C4—91no—
252C22×C4—92no—
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—92no—
256C4⋊C4—91no—
257C4⋊C4—92no—
258C4⋊C4—92no—
259C4⋊C4—92no—
260C4⋊C4—92no—
261C4⋊C4D3×A411yescharacteristic
262C22⋊C4—32no—
263C22⋊C4—32no—
264C22⋊C4—92no—
265C22⋊C4—94no—
266C22⋊C4—94no—
267C42—32no—
268C42—31no—
269C42—91no—
270C42—91no—
271C42—92no—
272C42—92no—
273C42—91no—
274C22×C3—41no—
275C22×C3—62no—
276C22×C3—62no—
277C22×C3—62no—
278C22×C3—81no—
279C22×C3—122no—
280C22×C3—241no—
281C22×C3—241no—
282C22×C3C23×A411yescharacteristic
283C2×D3—21no—
284C2×D3—21no—
285C2×D3C2×#41⋊C312yes—
286C3×C4—22no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—42no—
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—62no—
299Dic3—61no—
300Dic3—61no—
301Dic3C2×#41⋊C312yes—
302C32—41noSylow
303C23—31no—
304C23—32no—
305Q8—31no—
306Q8—181no—
307Q8—181no—
308Q8—182no—
309Q8C3×#23711yescharacteristic
310D4—62no—
311D4—62no—
312D4—62no—
313D4—122no—
314D4—184no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—34no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—91no—
323C2×C4—91no—
324C2×C4—92no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—184no—
328C2×C4—361no—
329C2×C4—362no—
330C2×C4C2×D3×A412yes—
331C2×C4C2×D3×A411yescharacteristic
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—122no—
339C2×C3—242no—
340C2×C3C22×#41⋊C311yescharacteristic
341C2×C3A4×#4111yescharacteristic
342C2×C3#594511yescharacteristic
343D3—22no—
344C22—32no—
345C22—61no—
346C22—61no—
347C22—62no—
348C22—62no—
349C22—62no—
350C22C22×D3×A411yescharacteristic
351C4—22no—
352C4—31no—
353C4—31no—
354C4—32no—
355C4—61no—
356C4—62no—
357C4—181no—
358C4—182no—
359C4—181no—
360C4D3×#41⋊C312yes—
361C3—41no—
362C3—81no—
363C3#3480⋊C311yescharacteristic, complemented
364C2—62no—
365C2—122no—
366C2A4×#23711yescharacteristic
367C2C2×D3×#41⋊C311yescharacteristic
368C2#1059662311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#550C2×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×C4⋊(#41⋊C3)C2semidirect
2. #5831⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. #5570⋊C3C2semidirect
6. #86767C3semidirect
14. C3×#550C2×C3semidirect
16. Dic3×#41C2×C3semidirect
22. #5570C2×C3semidirect
27. C4⋊(#41⋊C3)D3semidirect
88. C3×SL(2,𝔽3)#41semidirect
93. #550C3×D3semidirect
225. C4×D3#41⋊C3semidirect
230. SL(2,𝔽3)#237semidirect
363. C3#3480⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊(#41⋊C3)2
#5831⋊C32
Dic3×#41⋊C32
C4×D3×SL(2,𝔽3)2
#5570⋊C32
#867673
#3480⋊C33
C3×#8924