Finite Group Databasebeta

← #86768⋊C3 #11923264

Subgroups — #86768⋊C3 #11923264

Order 1152.

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

normal subgroups only

This group has 605 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2#5834⋊C3C212yesmaximal, complemented
3Dic3×#41⋊C3C212yesmaximal, complemented
4C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#86768C311yesmaximal, characteristic, complemented
7#85604—31nomaximal
8C2×C3×#41⋊C3C2212yes—
9C2×C3⋊(#41⋊C3)C2211yescharacteristic
10C3×#887—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#550C2×C311yescharacteristic
15#5834C2×C312yes—
16Dic3×#41C2×C312yes—
17#5718—31no—
18#5716—31no—
19#5616—32no—
20#5596—32no—
21C4×#239C2×C311yescharacteristic
22Q8×Dic6C2×C311yescharacteristic
23#5563—31no—
24C4×#237—31no—
25Q8×SL(2,𝔽3)—31no—
26C4⋊(#41⋊C3)—21no—
27C4⋊(#41⋊C3)—32no—
28C4⋊(#41⋊C3)D311yescharacteristic, complemented
29C4×#41⋊C3—32no—
30C4×#41⋊C3—31no—
31C2×C3×Dic6—41no—
32C3×#41⋊C3—22no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊(#41⋊C3)—21no—
35C32×C4⋊C4—41no—
36C3×Dic3⋊C4—42no—
37C3×C4×Dic3—41no—
38C3×C4×Dic3—42no—
39#3483—31noSylow
40C2×C3×#41C22×C312yes—
41C2×#239C22×C311yescharacteristic
42C2×#237—32no—
43C2×#237—31no—
44C2×#41⋊C3—22no—
45C2×#41⋊C3—31no—
46C2×#41⋊C3C2×D312yes—
47C3×#127—31no—
48C3×#121—32no—
49C3×C4×Q8—121no—
50C3×C4×Q8C22×C311yescharacteristic
51C3×C4×D4—31no—
52Q8×Dic3C22×C312yes—
53Q8×Dic3C22×C311yescharacteristic
54#940—32no—
55#939—34no—
56#938—32no—
57#938—32no—
58#936—32no—
59D4×Dic3—32no—
60D4×Dic3—32no—
61D4×Dic3—31no—
62#930—32no—
63C4×#67—32no—
64#926—32no—
65C2×C4×Dic3—32no—
66#893—31no—
67#892—31no—
68#892—31no—
69#888—32no—
70#887A411yescharacteristic
71#885—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
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—41no—
87C3×Dic6—44no—
88C3×SL(2,𝔽3)#4111yescharacteristic, complemented
89Q82—31no—
90#555—91no—
91#555—92no—
92#550—32no—
93#550—91no—
94#550C3×D311yescharacteristic
95#532—92no—
96#532—91no—
97C4×#41—32no—
98C4×#41—31no—
99C4×#41—91no—
100C3×#41—22no—
101C3×#41—62no—
102C2×C3×Q8—121no—
103C2×C3×Q8C23×C311yescharacteristic
104C2×C3×D4—32no—
105C2×C3×D4—32no—
106C2×C3×D4—31no—
107C22×C3×C4—32no—
108C2×#67—34no—
109C2×#67—32no—
110C22×Dic3—32no—
111C22×Dic3—32no—
112#239—21no—
113#239—61no—
114#237—64no—
115#237—124no—
116C2×D12—31no—
117C2×C4×D3—31no—
118C2×C4×D3—31no—
119C2×C4×D3—31no—
120C2×Dic6—32no—
121C2×Dic6C2×A411yescharacteristic
122#41⋊C3—22no—
123#41⋊C3—42no—
124#41⋊C3—61no—
125C2×SL(2,𝔽3)—21no—
126C2×SL(2,𝔽3)C22×D311yescharacteristic
127C3×C4⋊C4—31no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—81no—
130C3×C4⋊C4—122no—
131C3×C4⋊C4C2×A411yescharacteristic
132C3×C22⋊C4—32no—
133C3×C42—31no—
134C3×C42—121no—
135C3×C42—122no—
136#217—34no—
137#217—32no—
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4—32no—
140C4⋊Dic3—31no—
141C4⋊Dic3—32no—
142C4⋊Dic3—31no—
143Dic3⋊C4—32no—
144Dic3⋊C4C2×A412yes—
145C4×Dic3—32no—
146C4×Dic3—32no—
147C4×Dic3—31no—
148C4×Dic3—31no—
149C4×Dic3C2×A411yescharacteristic
150C4×Dic3C2×A412yes—
151C22×C32—41no—
152C32×C4—42no—
153C32×C4—82no—
154C3×Dic3—44no—
155C3×Dic3—42no—
156C3×Dic3—81no—
157C2×#41—31no—
158C2×#41—91no—
159C2×#41—92no—
160C2×#41C2×C3×D312yes—
161C4⋊Q8—91no—
162C4⋊Q8—92no—
163#127—31no—
164#127—92no—
165#124—94no—
166#124—92no—
167#121—32no—
168#121—92no—
169#121—94no—
170#121—92no—
171#121—92no—
172C4×Q8—31no—
173C4×Q8—31no—
174C4×Q8—32no—
175C4×Q8—91no—
176C4×Q8C2×C3×D311yescharacteristic
177C4×D4—31no—
178C4×D4—92no—
179C4×D4—91no—
180C4×D4—91no—
181C4×D4—92no—
182C4×D4—91no—
183C4×D4—92no—
184C4×D4—92no—
185#117—92no—
186#117—91no—
187#117—91no—
188#117—92no—
189C2×C42—91no—
190C2×C42—92no—
191C23×C3—32no—
192C22×D3—31no—
193C3×Q8—31no—
194C3×Q8—124no—
195C3×Q8C3×#4111yescharacteristic
196C3×D4—62no—
197C3×D4—62no—
198C3×D4—62no—
199C3×D4—122no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—81no—
207C2×C3×C4—82no—
208C2×C3×C4—121no—
209C2×C3×C4—122no—
210C2×C3×C4—121no—
211C2×C3×C4C22×A412yes—
212C2×C3×C4C22×A411yescharacteristic
213#67—64no—
214#67—124no—
215C2×Dic3—31no—
216C2×Dic3—32no—
217C2×Dic3—64no—
218C2×Dic3—64no—
219C2×Dic3C22×A411yescharacteristic
220C2×Dic3C22×A412yes—
221C2×Dic3C22×A411yescharacteristic
222D12—61no—
223D12—61no—
224C4×D3—61no—
225C4×D3—61no—
226C4×D3—62no—
227C4×D3—62no—
228Dic6—122no—
229Dic6#41⋊C314yescomplemented
230SL(2,𝔽3)—21no—
231SL(2,𝔽3)#23711yescharacteristic, complemented
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235#41—22no—
236#41—62no—
237#41—61no—
238#41—184no—
239#41—181no—
240#41—364no—
241C2×Q8—31no—
242C2×Q8—92no—
243C2×Q8C22×C3×D311yescharacteristic
244C2×D4—32no—
245C2×D4—32no—
246C2×D4—31no—
247C2×D4—92no—
248C2×D4—91no—
249C2×D4—94no—
250C22×C4—32no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4—92no—
254C22×C4—92no—
255C22×C4—91no—
256C4⋊C4—31no—
257C4⋊C4—32no—
258C4⋊C4—92no—
259C4⋊C4—91no—
260C4⋊C4—92no—
261C4⋊C4—91no—
262C4⋊C4D3×A411yescharacteristic
263C22⋊C4—32no—
264C22⋊C4—92no—
265C22⋊C4—92no—
266C22⋊C4—94no—
267C22⋊C4—92no—
268C42—31no—
269C42—32no—
270C42—31no—
271C42—91no—
272C42—91no—
273C42—92no—
274C42—92no—
275C22×C3—41no—
276C22×C3—62no—
277C22×C3—62no—
278C22×C3—62no—
279C22×C3—81no—
280C22×C3C23×A411yescharacteristic
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—61no—
284C3×C4—22no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—42no—
288C3×C4—61no—
289C3×C4—82no—
290C3×C4—82no—
291C3×C4—124no—
292C3×C4—122no—
293C3×C4—162no—
294C3×C4—241no—
295C3×C4C2×#41⋊C312yes—
296Dic3—21no—
297Dic3—61no—
298Dic3—62no—
299Dic3C2×#41⋊C314yes—
300Dic3#99412yes—
301C32—41noSylow
302C23—32no—
303C23—91no—
304Q8—31no—
305Q8—34no—
306Q8—362no—
307Q8C3×#23711yescharacteristic
308D4—62no—
309D4—62no—
310D4—62no—
311D4—122no—
312D4—184no—
313D4—181no—
314D4—181no—
315D4—364no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—32no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—91no—
324C2×C4—92no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—182no—
328C2×C4—184no—
329C2×C4—184no—
330C2×C4—182no—
331C2×C4C2×D3×A412yes—
332C2×C4C2×D3×A411yescharacteristic
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—81no—
339C2×C3—122no—
340C2×C3C22×#41⋊C311yescharacteristic
341C2×C3A4×#4111yescharacteristic
342C2×C3C2×#99411yescharacteristic
343D3—121no—
344C22—62no—
345C22—62no—
346C22—62no—
347C22—181no—
348C22—181no—
349C22—181no—
350C22C22×D3×A411yescharacteristic
351C4—22no—
352C4—31no—
353C4—31no—
354C4—34no—
355C4—32no—
356C4—61no—
357C4—61no—
358C4—181no—
359C4—182no—
360C4D3×#41⋊C312yes—
361C3—41no—
362C3—81no—
363C3#8560411yescharacteristic, complemented
364C2—122no—
365C2—361no—
366C2A4×#23711yescharacteristic
367C2C2×D3×#41⋊C311yescharacteristic
368C2D3×#99411yescharacteristic
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. #5834⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. C4×C3⋊(#41⋊C3)C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #86768C3semidirect
28. C4⋊(#41⋊C3)D3semidirect
88. C3×SL(2,𝔽3)#41semidirect
229. Dic6#41⋊C3semidirect
231. SL(2,𝔽3)#237semidirect
363. C3#85604semidirect

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
#5834⋊C32
Dic3×#41⋊C32
C4×C3⋊(#41⋊C3)2
SL(2,𝔽3)×Dic62
#867683
#856043
C3×#8874