Finite Group Databasebeta

← #87283⋊C3 #11923394

Subgroups — #87283⋊C3 #11923394

Order 1152.

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

normal subgroups only

This group has 765 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
1C2×C3×#994C211yesmaximal, characteristic, complemented
2C2×#1010⋊C3C211yesmaximal, characteristic, complemented
3#5831⋊C3C214yesmaximal, complemented
4Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
5#87283C311yesmaximal, characteristic, complemented
6#3500⋊C3—31nomaximal
7C3×#994—22no—
8C2×C3×#41⋊C3C2212yes—
9C2×C3×#41⋊C3C2211yescharacteristic
10#1010⋊C3—22no—
11C2×D3×SL(2,𝔽3)C2212yescomplemented
12C3×#947—41nomaximal
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
15C2×#1010C2×C311yescharacteristic
16C2×2−1+4⋊C3—31no—
17C2×#994—21no—
18C2×#994D311yescharacteristic, complemented
19#5832—31no—
20#5831C2×C314yescomplemented
21Dic3×#41C2×C311yescharacteristic, complemented
22#5825—31no—
23#5806—34no—
24#5805—31no—
25C2×#936—31no—
26#548⋊C3—34no—
27C4×#41⋊C3—31no—
28C2×C32×Q8—41no—
29C2×C3×D12—41no—
30C3×#41⋊C3—22no—
31C3×#41⋊C3D412yescomplemented
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33D3×SL(2,𝔽3)—22no—
34C3×(C2×D3)⋊C4—44no—
35C3×C4×Dic3—41no—
36#3500—31noSylow
37C3×H(5,𝔽2)—22no—
38C3×H(5,𝔽2)—62no—
39C2×C3×#41—31no—
40C2×C3×#41C22×C312yes—
41C2×C3×#41C22×C311yescharacteristic
42C22×C3×D4—31no—
43C22×C3×D4—32no—
44#1010—22no—
45#1010—62no—
46C2×D3×Q8C22×C312yes—
47C2×#237—32no—
48C2×#235—31no—
49C22×Dic6—31no—
502−1+4⋊C3—62no—
51#994—24no—
52#994—22no—
53C2×#41⋊C3—22no—
54C2×#41⋊C3—21no—
55C2×#41⋊C3C2×D312yes—
56C2×#41⋊C3C2×D311yescharacteristic
57C22×SL(2,𝔽3)—32no—
58C3×#124—121no—
59#953—34no—
60C2×#217—34no—
61#947A411yescharacteristic
62Q8×Dic3C22×C312yes—
63#944—31no—
64#940—31no—
65#939—34no—
66#938—34no—
67#936—32no—
68#936—34no—
69#936—31no—
70#935—34no—
71D4×Dic3—32no—
72C4×#67—34no—
73#926—31no—
74#924—34no—
75C2×C4×Dic3—31no—
76C4×SL(2,𝔽3)—32no—
77C22×C3×D3—42no—
78C32×Q8—82no—
79C2×C32×C4—41no—
80C2×C32×C4—42no—
81C2×C3×Dic3—42no—
82C3×D12—82no—
83C3×SL(2,𝔽3)C2×D411yescharacteristic
84C2×2−1+4—31no—
85C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
86#548—34no—
87#546—94no—
88#536—91no—
89#535—91no—
90#532—91no—
91C2×#124—91no—
92C4×#41—31no—
93C24×C3—32no—
94C3×#41—22no—
95C3×#41—32no—
96C3×#41—62no—
97C3×#41—62no—
98C3×#41—62no—
99C3×#41C3×D412yes—
100C2×C3×Q8—41no—
101C2×C3×Q8—81no—
102C2×C3×Q8C2×A411yescharacteristic
103C2×C3×Q8C23×C311yescharacteristic
104C2×C3×D4—31no—
105C2×C3×D4—32no—
106C2×C3×D4—32no—
107C2×C3×D4—31no—
108C2×C3×D4—62no—
109C2×C3×D4—62no—
110C2×C3×D4—64no—
111C2×C3×D4—64no—
112C2×C3×D4—121no—
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C2×#67—32no—
116C2×#67—34no—
117C22×Dic3—32no—
118C22×Dic3—32no—
119D3×Q8—22no—
120D3×Q8—62no—
121#237—64no—
122#237—64no—
123#235—62no—
124#235—62no—
125C2×D12C2×A411yescharacteristic
126C2×C4×D3—32no—
127C2×Dic6—31no—
128C2×Dic6—32no—
129C2×Dic6—62no—
130C2×Dic6—62no—
131#41⋊C3—24no—
132#41⋊C3—22no—
133#41⋊C3—22no—
134#41⋊C3#6712yescomplemented
135C2×SL(2,𝔽3)—21no—
136C2×SL(2,𝔽3)—62no—
137C2×SL(2,𝔽3)C22×D311yescharacteristic
138C3×C22⋊C4—124no—
139C3×C42—121no—
140#217—38no—
141#217—32no—
142#217—34no—
143(C2×D3)⋊C4C2×A414yes—
144C4⋊Dic3—32no—
145Dic3⋊C4—34no—
146C4×Dic3—31no—
147C4×Dic3—32no—
148C4×Dic3C2×A411yescharacteristic
149C22×C32—41no—
150C2×C3×D3—82no—
151C2×C3×D3—84no—
152C32×C4—42no—
153C32×C4—82no—
154C3×Dic3—82no—
1552−1+4—62no—
1562−1+4—182no—
157H(5,𝔽2)—22no—
158H(5,𝔽2)—62no—
159C2×#41—31no—
160C2×#41—91no—
161C2×#41—92no—
162C2×#41C2×C3×D312yes—
163C2×#41C2×C3×D311yescharacteristic
164C22×Q8—32no—
165C22×Q8—91no—
166C22×D4—31no—
167C22×D4—32no—
168C4⋊Q8—91no—
169#127—91no—
170#124—31no—
171#124—91no—
172#124—92no—
173#124—94no—
174#123—94no—
175C22⋊Q8—94no—
176#121—94no—
177#121—94no—
178#120—94no—
179C4×Q8—32no—
180C4×D4—94no—
181C4×D4—92no—
182#117—91no—
183C2×C22⋊C4—94no—
184C2×C42—91no—
185C23×C3—31no—
186C23×C3—32no—
187C23×C3—64no—
188C23×C3—122no—
189C23×C3—124no—
190C22×D3C22×A412yes—
191C3×Q8—22no—
192C3×Q8—31no—
193C3×Q8—82no—
194C3×Q8—84no—
195C3×Q8C2×C3×D411yescharacteristic
196C3×D4—32no—
197C3×D4—32no—
198C3×D4—64no—
199C3×D4—62no—
200C3×D4—62no—
201C3×D4—62no—
202C3×D4—242no—
203C2×C3×C4—31no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—82no—
211C2×C3×C4—81no—
212C2×C3×C4—122no—
213C2×C3×C4C22×A412yes—
214C2×C3×C4C22×A411yescharacteristic
215#67—64no—
216#67—124no—
217C2×Dic3—32no—
218C2×Dic3—32no—
219C2×Dic3—64no—
220C2×Dic3—122no—
221C2×Dic3C22×A412yes—
222D12—22no—
223C4×D3—62no—
224C4×D3—62no—
225Dic6—62no—
226Dic6—62no—
227Dic6—62no—
228SL(2,𝔽3)—21no—
229SL(2,𝔽3)C2×#6711yescharacteristic
230C2×C32—41no—
231C2×C32—42no—
232C3×D3—162no—
233C24—32no—
234#41—22no—
235#41—32no—
236#41—62no—
237#41—62no—
238#41—62no—
239#41—182no—
240#41—182no—
241#41—184no—
242#41—184no—
243#41C3×#6712yes—
244C2×Q8—62no—
245C2×Q8—92no—
246C2×Q8—91no—
247C2×Q8—182no—
248C2×Q8—182no—
249C2×Q8—182no—
250C2×Q8D3×A411yescharacteristic
251C2×Q8C22×C3×D311yescharacteristic
252C2×D4—31no—
253C2×D4—32no—
254C2×D4—32no—
255C2×D4—31no—
256C2×D4—31no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—64no—
260C2×D4—64no—
261C2×D4—92no—
262C2×D4—94no—
263C22×C4—31no—
264C22×C4—32no—
265C22×C4—92no—
266C22×C4—92no—
267C22×C4—92no—
268C4⋊C4—92no—
269C4⋊C4—94no—
270C22⋊C4—34no—
271C22⋊C4—94no—
272C22⋊C4—98no—
273C22⋊C4—92no—
274C42—31no—
275C42—92no—
276C42—91no—
277C22×C3—32no—
278C22×C3—41no—
279C22×C3—62no—
280C22×C3—62no—
281C22×C3—64no—
282C22×C3—81no—
283C22×C3—242no—
284C22×C3—244no—
285C22×C3—244no—
286C22×C3C23×A411yescharacteristic
287C2×D3—22no—
288C2×D3—24no—
289C3×C4—22no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—42no—
293C3×C4—82no—
294C3×C4—84no—
295C3×C4—82no—
296C3×C4—242no—
297C3×C4D4×A412yes—
298Dic3—22no—
299Dic3—62no—
300Dic3—62no—
301C32—41noSylow
302C23—31no—
303C23—32no—
304C23—32no—
305C23—64no—
306C23—124no—
307Q8—22no—
308Q8—31no—
309Q8—182no—
310Q8—182no—
311Q8—182no—
312Q8C2×C3×#6711yescharacteristic
313D4—32no—
314D4—32no—
315D4—64no—
316D4—62no—
317D4—62no—
318D4—62no—
319D4—62no—
320D4—184no—
321D4—364no—
322C2×C4—31no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—92no—
329C2×C4—92no—
330C2×C4—182no—
331C2×C4—182no—
332C2×C4—184no—
333C2×C4—362no—
334C2×C4C2×D3×A412yes—
335C2×C4C2×D3×A411yescharacteristic
336C2×C3—42no—
337C2×C3—41no—
338C2×C3—62no—
339C2×C3—82no—
340C2×C3—81no—
341C2×C3—122no—
342C2×C3—482no—
343C2×C3C2×D4×A411yescharacteristic
344C2×C3#594512yes—
345D3—42no—
346C22—32no—
347C22—62no—
348C22—64no—
349C22—64no—
350C22—62no—
351C22—62no—
352C22—244no—
353C22C22×D3×A411yescharacteristic
354C4—22no—
355C4—31no—
356C4—31no—
357C4—62no—
358C4—182no—
359C4—182no—
360C4A4×#6712yes—
361C3—41no—
362C3—81no—
363C3#3500⋊C311yescharacteristic, complemented
364C2—62no—
365C2—122no—
366C2—122no—
367C2#1059662312yes—
368C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×H(5,𝔽2)C2×C3
socleC22×C3C23×A4
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#994C2semidirect
2. C2×#1010⋊C3C2semidirect
3. #5831⋊C3C2semidirect
4. Dic3×#41⋊C3C2semidirect
5. #87283C3semidirect
11. C2×D3×SL(2,𝔽3)C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
14. C2×C3×H(5,𝔽2)C2×C3semidirect
18. C2×#994D3semidirect
20. #5831C2×C3semidirect
21. Dic3×#41C2×C3semidirect
31. C3×#41⋊C3D4semidirect
85. C2×H(5,𝔽2)C3×D3semidirect
134. #41⋊C3#67semidirect
363. C3#3500⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C2×#1010⋊C32
#5831⋊C32
Dic3×#41⋊C32
#872833
#3500⋊C33
C3×#9474