Finite Group Databasebeta

← #86750⋊C3 #11923258

Subgroups — #86750⋊C3 #11923258

Order 1152.

565 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 79 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#5812⋊C3C212yesmaximal, complemented
3#5817⋊C3C212yesmaximal, complemented
4C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#86750C311yesmaximal, characteristic, complemented
7#3518⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×C3⋊(#41⋊C3)C2211yescharacteristic
10C3×#936—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×D4×Q8C2×C311yescharacteristic
15#5817C2×C312yescomplemented
16#5812C2×C312yescomplemented
17#5732—31no—
18#5728—31no—
19#5643—32no—
20#5641—32no—
21C4×#239C2×C311yescharacteristic, complemented
22Q8×Dic6C2×C311yescharacteristic, complemented
23#5563—31no—
24#5557—31no—
25Q8×SL(2,𝔽3)—31no—
26D4×SL(2,𝔽3)—21no—
27D4×SL(2,𝔽3)D311yescharacteristic, complemented
28#548⋊C3—32no—
29#519⋊C3—32no—
30C4×#41⋊C3—31no—
31C2×C32×D4—41no—
32C2×C3×Dic6—41no—
33C2×C3×SL(2,𝔽3)—22no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊(#41⋊C3)—21no—
36C3×#217—42no—
37C3×#217—42no—
38C3×C4×Dic3—41no—
39#3518—31noSylow
40C22×C3×Q8C22×C312yes—
41C2×#239C22×C311yescharacteristic
42C2×#41⋊C3—31no—
43C22×SL(2,𝔽3)—22no—
44C22×SL(2,𝔽3)C2×D312yes—
45C3×C4⋊Q8—31no—
46C3×#124—121no—
47C3×C22⋊Q8—32no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×D4—31no—
50#947—32no—
51#946—32no—
52Q8×Dic3—32no—
53Q8×Dic3—32no—
54Q8×Dic3C22×C311yescharacteristic
55Q8×Dic3C22×C312yes—
56#938—31no—
57#936A411yescharacteristic
58#930—32no—
59C4×#67—32no—
60#926—32no—
61#926—32no—
62#898—34no—
63#897—32no—
64#897—31no—
65#893—31no—
66#892—32no—
67#892—31no—
68#890—31no—
69#888—32no—
70#887—32no—
71#885—34no—
72#883—32no—
73#879—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—82no—
84C2×C32×C4—41no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—41no—
88C3×Dic6—82no—
89C3×SL(2,𝔽3)#4111yescharacteristic
90Q82—31no—
91#556—92no—
92#556—91no—
93#555—91no—
94D4×Q8C3×D311yescharacteristic
95#548—32no—
96#542—92no—
97#533—91no—
98#529—91no—
99#519—32no—
100C4×#41—31no—
101C2×C3×Q8—22no—
102C2×C3×Q8—32no—
103C2×C3×Q8—62no—
104C2×C3×Q8—121no—
105C2×C3×Q8C23×C311yescharacteristic
106C2×C3×D4—41no—
107C2×C3×D4—81no—
108C2×C3×D4C2×A411yescharacteristic
109C22×C3×C4—32no—
110C2×#67—32no—
111#239—21no—
112#239—61no—
113C2×D12—31no—
114C2×C4×D3—31no—
115C2×C4×D3—31no—
116C2×C4×D3—31no—
117C2×Dic6—32no—
118C2×Dic6C2×A411yescharacteristic
119#41⋊C3—61no—
120C2×SL(2,𝔽3)—22no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—42no—
123C2×SL(2,𝔽3)C22×D311yescharacteristic
124C3×C4⋊C4—31no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—31no—
127C3×C22⋊C4—32no—
128C3×C22⋊C4—122no—
129C3×C22⋊C4—122no—
130C3×C42—31no—
131C3×C42—121no—
132#217—32no—
133#217C2×A412yes—
134#217C2×A412yes—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—34no—
137(C2×D3)⋊C4—32no—
138C4⋊Dic3—31no—
139C4⋊Dic3—32no—
140C4⋊Dic3—32no—
141C4⋊Dic3—32no—
142C4⋊Dic3—31no—
143Dic3⋊C4—32no—
144Dic3⋊C4—32no—
145C4×Dic3—34no—
146C4×Dic3—32no—
147C4×Dic3—31no—
148C4×Dic3—31no—
149C4×Dic3C2×A411yescharacteristic
150C22×C32—41no—
151C22×C32—82no—
152C22×C32—82no—
153C22×C32—82no—
154C32×C4—42no—
155C3×Dic3—82no—
156C3×Dic3—81no—
157C3×Dic3—81no—
158C2×#41—31no—
159C22×Q8C2×C3×D312yes—
160C4⋊Q8—31no—
161C4⋊Q8—91no—
162C4⋊Q8—92no—
163#126—92no—
164#126—94no—
165#125—91no—
166#124—31no—
167#124—94no—
168#124—92no—
169#123—92no—
170C22⋊Q8—32no—
171C22⋊Q8—91no—
172C22⋊Q8—92no—
173C22⋊Q8—92no—
174#121—92no—
175#121—91no—
176C4×Q8—31no—
177C4×Q8—32no—
178C4×Q8—91no—
179C4×Q8—92no—
180C4×Q8—92no—
181C4×Q8—92no—
182C4×Q8C2×C3×D311yescharacteristic
183C4×D4—31no—
184C4×D4—91no—
185C4×D4—92no—
186C4×D4—91no—
187#117—92no—
188#117—92no—
189#117—92no—
190#117—91no—
191C2×C42—91no—
192C23×C3—42no—
193C23×C3—82no—
194C23×C3C22×A412yes—
195C22×D3—31no—
196C3×Q8—31no—
197C3×Q8—62no—
198C3×Q8—62no—
199C3×Q8—242no—
200C3×Q8C3×#4111yescharacteristic
201C3×D4—22no—
202C3×D4—82no—
203C3×D4—84no—
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×C4C22×A411yescharacteristic
215#67—122no—
216C2×Dic3—31no—
217C2×Dic3—32no—
218C2×Dic3C22×A411yescharacteristic
219C2×Dic3C22×A411yescharacteristic
220C2×Dic3C22×A412yes—
221D12—61no—
222D12—61no—
223C4×D3—61no—
224C4×D3—61no—
225C4×D3—62no—
226C4×D3—121no—
227Dic6—22no—
228Dic6—122no—
229SL(2,𝔽3)—21no—
230SL(2,𝔽3)#23711yescharacteristic
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—162no—
235#41—61no—
236#41—181no—
237C2×Q8—22no—
238C2×Q8—31no—
239C2×Q8—32no—
240C2×Q8—62no—
241C2×Q8—92no—
242C2×Q8C22×C3×D311yescharacteristic
243C2×D4—91no—
244C2×D4—92no—
245C2×D4D3×A411yescharacteristic
246C22×C4—32no—
247C22×C4—91no—
248C22×C4—91no—
249C22×C4—91no—
250C4⋊C4—31no—
251C4⋊C4—32no—
252C4⋊C4—31no—
253C4⋊C4—92no—
254C4⋊C4—92no—
255C4⋊C4—92no—
256C4⋊C4—92no—
257C4⋊C4—91no—
258C4⋊C4—92no—
259C4⋊C4—91no—
260C22⋊C4—32no—
261C22⋊C4—32no—
262C22⋊C4—32no—
263C22⋊C4—92no—
264C22⋊C4—92no—
265C22⋊C4—94no—
266C22⋊C4—92no—
267C42—31no—
268C42—31no—
269C42—91no—
270C42—91no—
271C42—92no—
272C42—94no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—82no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3—84no—
281C22×C3—81no—
282C22×C3—162no—
283C22×C3—162no—
284C22×C3C23×A411yescharacteristic
285C2×D3—61no—
286C2×D3—61no—
287C2×D3—61no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—42no—
291C3×C4—62no—
292C3×C4—61no—
293C3×C4—82no—
294C3×C4—242no—
295C3×C4—241no—
296C3×C4—241no—
297C3×C42−1+4⋊C312yes—
298Dic3—21no—
299Dic3—22no—
300Dic3—21no—
301Dic3—61no—
302Dic3—62no—
303C32—41noSylow
304C23—91no—
305C23C2×D3×A412yes—
306Q8—31no—
307Q8—62no—
308Q8—62no—
309Q8—62no—
310Q8—362no—
311Q8C3×#23711yescharacteristic
312D4—22no—
313D4—181no—
314D4—181no—
315D4—362no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—91no—
325C2×C4—92no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—182no—
329C2×C4—361no—
330C2×C4C2×D3×A411yescharacteristic
331C2×C3—42no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—162no—
339C2×C3—164no—
340C2×C3A4×#4111yescharacteristic
341C2×C3C2×2−1+4⋊C311yescharacteristic
342C2×C3#594511yescharacteristic
343D3—121no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—181no—
348C22—181no—
349C22—181no—
350C22C22×D3×A411yescharacteristic
351C4—31no—
352C4—31no—
353C4—62no—
354C4—61no—
355C4—62no—
356C4—61no—
357C4—61no—
358C4—181no—
359C4—182no—
360C4#1007⋊C312yes—
361C3—41no—
362C3—81no—
363C3#3518⋊C311yescharacteristic, complemented
364C2—42no—
365C2—361no—
366C2A4×#23711yescharacteristic
367C2C2×#1007⋊C311yescharacteristic
368C2#5968⋊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. #5812⋊C3C2semidirect
3. #5817⋊C3C2semidirect
4. C4×C3⋊(#41⋊C3)C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #86750C3semidirect
15. #5817C2×C3semidirect
16. #5812C2×C3semidirect
21. C4×#239C2×C3semidirect
22. Q8×Dic6C2×C3semidirect
27. D4×SL(2,𝔽3)D3semidirect
363. C3#3518⋊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
#5812⋊C32
#5817⋊C32
C4×C3⋊(#41⋊C3)2
SL(2,𝔽3)×Dic62
#867503
#3518⋊C33
C3×#9364