Finite Group Databasebeta

← #86753⋊C3 #11923263

Subgroups — #86753⋊C3 #11923263

Order 1152.

589 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 81 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
2SL(2,𝔽3)×#67C212yesmaximal, complemented
3#5812⋊C3C212yesmaximal, complemented
4C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#5570⋊C3C211yesmaximal, characteristic, complemented
6#86753C311yesmaximal, characteristic, complemented
7#3523⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×D3×SL(2,𝔽3)C2211yescharacteristic
10C3×#938—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×D4×Q8C2×C311yescharacteristic, complemented
15Q8×#67C2×C312yescomplemented
16#5812C2×C312yescomplemented
17#5728—31no—
18#5721—31no—
19#5639—32no—
20#5625—32no—
21C4×D3×Q8C2×C311yescharacteristic, complemented
22#5570C2×C311yescharacteristic, complemented
23#5557—31no—
24#5542—31no—
25#557⋊C3—31no—
26D4×SL(2,𝔽3)—21no—
27D4×SL(2,𝔽3)—32no—
28D4×SL(2,𝔽3)D311yescharacteristic, complemented
29#519⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C32×D4—41no—
32C2×C3×#67—42no—
33C2×C3×C4×D3—41no—
34C2×C3×SL(2,𝔽3)—22no—
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36D3×SL(2,𝔽3)—21no—
37C3×#217—42no—
38C3×C4⋊Dic3—41no—
39#3523—31noSylow
40C22×C3×Q8C22×C312yes—
41C2×D3×Q8C22×C311yescharacteristic
42C22×SL(2,𝔽3)—22no—
43C22×SL(2,𝔽3)—31no—
44C22×SL(2,𝔽3)C2×D312yes—
45C3×C4⋊Q8—31no—
46C3×C22⋊Q8—32no—
47C3×#121—121no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×D4—31no—
50#946—32no—
51Q8×Dic3—32no—
52Q8×Dic3—32no—
53Q8×Dic3C22×C311yescharacteristic
54Q8×Dic3C22×C312yes—
55#944—32no—
56#938A411yescharacteristic
57#936—31no—
58C4×#67—32no—
59#926—32no—
60#926—32no—
61#924—32no—
62#897—31no—
63#892—32no—
64D3×C4⋊C4—31no—
65#890—32no—
66#890—31no—
67#890—32no—
68#889—34no—
69#887—32no—
70#887—31no—
71#883—32no—
72#879—34no—
73#879—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
82C23×C32—42no—
83C22×C3×D3—41no—
84C32×D4—82no—
85C2×C32×C4—41no—
86C3×#67—84no—
87C2×C3×Dic3—42no—
88C2×C3×Dic3—41no—
89C3×C4×D3—82no—
90C3×SL(2,𝔽3)#4111yescharacteristic
91#557—31no—
92#557—91no—
93#556—91no—
94#552—92no—
95D4×Q8—32no—
96D4×Q8C3×D311yescharacteristic, complemented
97#541—92no—
98#533—91no—
99#529—91no—
100#519—32no—
101C2×C4×Q8—31no—
102C2×C3×Q8—22no—
103C2×C3×Q8—32no—
104C2×C3×Q8—62no—
105C2×C3×Q8C23×C311yescharacteristic
106C2×C3×D4—41no—
107C2×C3×D4—81no—
108C2×C3×D4—122no—
109C2×C3×D4C2×A411yescharacteristic
110C22×C3×C4—32no—
111C22×C3×C4—121no—
112C2×#67C2×A412yes—
113D3×Q8—21no—
114D3×Q8—61no—
115C2×C4×D3—31no—
116C2×C4×D3—31no—
117C2×C4×D3C2×A411yescharacteristic
118C2×Dic6—31no—
119C2×Dic6—31no—
120C2×SL(2,𝔽3)—22no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—42no—
123C2×SL(2,𝔽3)—61no—
124C2×SL(2,𝔽3)C22×D311yescharacteristic
125C3×C4⋊C4—31no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4—121no—
129C3×C22⋊C4—32no—
130C3×C22⋊C4—122no—
131C3×C42—31no—
132#217—32no—
133#217—32no—
134#217C2×A412yes—
135(C2×D3)⋊C4—32no—
136C4⋊Dic3—32no—
137C4⋊Dic3—32no—
138C4⋊Dic3—32no—
139C4⋊Dic3—31no—
140C4⋊Dic3—32no—
141C4⋊Dic3C2×A411yescharacteristic
142Dic3⋊C4—32no—
143Dic3⋊C4—34no—
144Dic3⋊C4—32no—
145Dic3⋊C4—32no—
146C4×Dic3—34no—
147C4×Dic3—31no—
148C4×Dic3—32no—
149C4×Dic3—31no—
150C4×Dic3—31no—
151C22×C32—41no—
152C22×C32—82no—
153C22×C32—82no—
154C22×C32—82no—
155C2×C3×D3—42no—
156C2×C3×D3—81no—
157C2×C3×D3—81no—
158C32×C4—42no—
159C3×Dic3—81no—
160C3×Dic3—82no—
161C22×Q8—31no—
162C22×Q8C2×C3×D312yes—
163C4⋊Q8—31no—
164C4⋊Q8—91no—
165C4⋊Q8—92no—
166#126—92no—
167#126—94no—
168#125—91no—
169#125—92no—
170#125—92no—
171#125—94no—
172#124—91no—
173#123—92no—
174C22⋊Q8—32no—
175C22⋊Q8—91no—
176C22⋊Q8—92no—
177C22⋊Q8—92no—
178#121—31no—
179C4×Q8—31no—
180C4×Q8—32no—
181C4×Q8—91no—
182C4×Q8—91no—
183C4×Q8—92no—
184C4×Q8—92no—
185C4×Q8—92no—
186C4×Q8—91no—
187C4×Q8C2×C3×D311yescharacteristic
188C4×D4—31no—
189C4×D4—92no—
190#117—92no—
191#117—92no—
192#117—92no—
193C2×C4⋊C4—91no—
194C2×C42—91no—
195C23×C3—42no—
196C23×C3—82no—
197C23×C3—121no—
198C23×C3C22×A412yes—
199C22×D3C22×A411yescharacteristic
200C3×Q8—31no—
201C3×Q8—62no—
202C3×Q8—62no—
203C3×Q8C3×#4111yescharacteristic
204C3×D4—22no—
205C3×D4—82no—
206C3×D4—84no—
207C3×D4—244no—
208C2×C3×C4—31no—
209C2×C3×C4—32no—
210C2×C3×C4—31no—
211C2×C3×C4—41no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—81no—
215C2×C3×C4—121no—
216C2×C3×C4—122no—
217C2×C3×C4—242no—
218C2×C3×C4C22×A411yescharacteristic
219#67—24no—
220C2×Dic3—31no—
221C2×Dic3—32no—
222C2×Dic3—31no—
223C2×Dic3C22×A411yescharacteristic
224C2×Dic3C22×A412yes—
225C4×D3—22no—
226C4×D3—61no—
227C4×D3—61no—
228C4×D3—62no—
229Dic6—62no—
230Dic6—61no—
231Dic6—61no—
232SL(2,𝔽3)—21no—
233SL(2,𝔽3)#23711yescharacteristic
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—162no—
238C3×D3—82no—
239C2×Q8—22no—
240C2×Q8—32no—
241C2×Q8—62no—
242C2×Q8—61no—
243C2×Q8—91no—
244C2×Q8—91no—
245C2×Q8—181no—
246C2×Q8C22×C3×D311yescharacteristic
247C2×D4—32no—
248C2×D4D3×A411yescharacteristic
249C22×C4—32no—
250C22×C4—31no—
251C22×C4—91no—
252C22×C4—91no—
253C4⋊C4—31no—
254C4⋊C4—32no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—92no—
258C4⋊C4—92no—
259C4⋊C4—92no—
260C4⋊C4—92no—
261C4⋊C4—91no—
262C4⋊C4—92no—
263C4⋊C4—92no—
264C4⋊C4—92no—
265C4⋊C4—94no—
266C22⋊C4—32no—
267C22⋊C4—32no—
268C22⋊C4—92no—
269C22⋊C4—92no—
270C22⋊C4—92no—
271C42—31no—
272C42—91no—
273C42—91no—
274C42—92no—
275C42—94no—
276C42—91no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—82no—
282C22×C3—82no—
283C22×C3—82no—
284C22×C3—84no—
285C22×C3—81no—
286C22×C3—122no—
287C22×C3—162no—
288C22×C3—162no—
289C22×C3—241no—
290C22×C3—241no—
291C22×C3C23×A411yescharacteristic
292C2×D3—21no—
293C2×D3—21no—
294C2×D32−1+4⋊C312yes—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—42no—
298C3×C4—62no—
299C3×C4—61no—
300C3×C4—82no—
301C3×C4—241no—
302C3×C4—242no—
303C3×C42−1+4⋊C312yes—
304Dic3—21no—
305Dic3—22no—
306Dic3—61no—
307Dic3—62no—
308Dic3—61no—
309C32—41noSylow
310C23—31no—
311C23C2×D3×A412yes—
312Q8—31no—
313Q8—62no—
314Q8—62no—
315Q8—181no—
316Q8—181no—
317Q8—182no—
318Q8C3×#23711yescharacteristic
319D4—22no—
320D4—64no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—182no—
335C2×C4C2×D3×A411yescharacteristic
336C2×C3—42no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3—81no—
343C2×C3—162no—
344C2×C3—164no—
345C2×C3—242no—
346C2×C3A4×#4111yescharacteristic
347C2×C3C2×2−1+4⋊C311yescharacteristic
348C2×C3C2×2−1+4⋊C311yescharacteristic
349D3—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—32no—
354C22—61no—
355C22—61no—
356C22C22×D3×A411yescharacteristic
357C4—31no—
358C4—31no—
359C4—62no—
360C4—61no—
361C4—61no—
362C4—62no—
363C4—181no—
364C4—182no—
365C4—181no—
366C4#1007⋊C312yes—
367C3—41no—
368C3—81no—
369C3#3523⋊C311yescharacteristic, complemented
370C2—42no—
371C2—62no—
372C2A4×#23711yescharacteristic
373C2C2×#1007⋊C311yescharacteristic
374C2D3×2−1+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. SL(2,𝔽3)×#67C2semidirect
3. #5812⋊C3C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. #5570⋊C3C2semidirect
6. #86753C3semidirect
14. C3×D4×Q8C2×C3semidirect
15. Q8×#67C2×C3semidirect
16. #5812C2×C3semidirect
21. C4×D3×Q8C2×C3semidirect
22. #5570C2×C3semidirect
28. D4×SL(2,𝔽3)D3semidirect
96. D4×Q8C3×D3semidirect
369. C3#3523⋊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
SL(2,𝔽3)×#672
#5812⋊C32
C4×D3×SL(2,𝔽3)2
#5570⋊C32
#867533
#3523⋊C33
C3×#9384