Finite Group Databasebeta

← #87292⋊C3 #11923397

Subgroups — #87292⋊C3 #11923397

Order 1152.

805 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 805 conjugacy classes of subgroups — too many to draw. Only its 91 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×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×#1010⋊C3C211yesmaximal, characteristic, complemented
3Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
4SL(2,𝔽3)×#67C214yesmaximal, complemented
5#87292C311yesmaximal, characteristic, complemented
6#3499⋊C3—31nomaximal
7C3×2−1+4⋊C3—22no—
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2212yes—
10#1010⋊C3—22no—
11C2×D3×SL(2,𝔽3)C2212yes—
12C3×#940—41nomaximal
13Dic3×SL(2,𝔽3)C2212yes—
14C2×C3×2−1+4C2×C311yescharacteristic, complemented
15C2×#1010C2×C311yescharacteristic, complemented
16C2×2−1+4⋊C3—21no—
17C2×2−1+4⋊C3—31no—
18C2×2−1+4⋊C3D311yescharacteristic, complemented
19#5832—31no—
20Dic3×#41C2×C311yescharacteristic, complemented
21#5825—31no—
22#5824—34no—
23#5816—31no—
24Q8×#67C2×C314yescomplemented
25C2×#944—31no—
26D4×SL(2,𝔽3)—34no—
27C4×#41⋊C3—31no—
28C2×C32×D4—41no—
29C2×C3×#67—44no—
30C2×C3×D12—41no—
31C3×#41⋊C3D412yescomplemented
32C2×C3×SL(2,𝔽3)—22no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)—22no—
35C3×C4×Dic3—41no—
36#3499—31noSylow
37C3×2−1+4—22no—
38C3×2−1+4—62no—
39C2×C3×#41—31no—
40C2×C3×#41—32no—
41C2×C3×#41C22×C311yescharacteristic
42C22×C3×Q8—31no—
43C22×C3×Q8C22×C312yes—
44#1010—22no—
45#1010—62no—
46C2×D3×Q8C22×C312yes—
47C2×#237—32no—
48C2×#235—31no—
49C22×Dic6—31no—
502−1+4⋊C3—24no—
512−1+4⋊C3—22no—
522−1+4⋊C3—62no—
53C2×#41⋊C3—21no—
54C2×#41⋊C3C2×D311yescharacteristic
55C22×SL(2,𝔽3)—22no—
56C22×SL(2,𝔽3)—32no—
57C22×SL(2,𝔽3)C2×D312yes—
58C3×#127—121no—
59#947—31no—
60#946—34no—
61Q8×Dic3C22×C312yes—
62#944—32no—
63#944—34no—
64#944—31no—
65#940A411yescharacteristic
66#939—34no—
67#936—31no—
68#935—34no—
69D4×Dic3—32no—
70#929—34no—
71C4×#67—34no—
72#926—31no—
73#924—34no—
74C2×Dic3⋊C4—34no—
75C2×C4×Dic3—31no—
76C4×SL(2,𝔽3)—32no—
77C23×C32—42no—
78C22×C3×D3—42no—
79C32×D4—82no—
80C2×C32×C4—41no—
81C3×#67—88no—
82C2×C3×Dic3—42no—
83C3×D12—82no—
84C3×SL(2,𝔽3)C2×D411yescharacteristic
85C2×2−1+4—31no—
86C2×2−1+4C3×D311yescharacteristic, complemented
87D4×Q8—34no—
88#547—94no—
89#536—91no—
90#536—91no—
91#532—91no—
92C2×C4⋊Q8—91no—
93C4×#41—31no—
94C3×#41—32no—
95C3×#41—62no—
96C3×#41—62no—
97C3×#41—64no—
98C3×#41—64no—
99C3×#41C3×D412yescomplemented
100C2×C3×Q8—22no—
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8—62no—
104C2×C3×Q8—62no—
105C2×C3×Q8—62no—
106C2×C3×Q8C23×C311yescharacteristic
107C2×C3×D4—31no—
108C2×C3×D4—32no—
109C2×C3×D4—41no—
110C2×C3×D4—81no—
111C2×C3×D4—121no—
112C2×C3×D4—124no—
113C2×C3×D4C2×A411yescharacteristic
114C22×C3×C4—31no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C2×#67—32no—
118C2×#67C2×A414yes—
119C22×Dic3—32no—
120C22×Dic3—32no—
121D3×Q8—22no—
122D3×Q8—62no—
123#237—64no—
124#237—64no—
125#235—62no—
126#235—62no—
127C2×D12C2×A411yescharacteristic
128C2×C4×D3—32no—
129C2×Dic6—31no—
130C2×Dic6—32no—
131C2×Dic6—62no—
132C2×Dic6—62no—
133#41⋊C3—22no—
134#41⋊C3#6712yescomplemented
135C2×SL(2,𝔽3)—24no—
136C2×SL(2,𝔽3)—22no—
137C2×SL(2,𝔽3)—21no—
138C2×SL(2,𝔽3)—62no—
139C2×SL(2,𝔽3)C22×D311yescharacteristic
140C3×C42—121no—
141#217—32no—
142#217—34no—
143(C2×D3)⋊C4—34no—
144C4⋊Dic3—32no—
145Dic3⋊C4—38no—
146Dic3⋊C4—34no—
147C4×Dic3—31no—
148C4×Dic3—32no—
149C4×Dic3C2×A411yescharacteristic
150C22×C32—41no—
151C22×C32—82no—
152C22×C32—84no—
153C2×C3×D3—82no—
154C2×C3×D3—84no—
155C32×C4—42no—
156C3×Dic3—44no—
1572−1+4—22no—
1582−1+4—62no—
1592−1+4—62no—
1602−1+4—182no—
161C2×#41—31no—
162C2×#41—32no—
163C2×#41—91no—
164C2×#41—92no—
165C2×#41C2×C3×D311yescharacteristic
166C22×Q8—31no—
167C22×Q8—32no—
168C22×Q8—91no—
169C22×Q8C2×C3×D312yes—
170C4⋊Q8—91no—
171C4⋊Q8—92no—
172C4⋊Q8—94no—
173#127—31no—
174#124—91no—
175#124—91no—
176#123—94no—
177#123—94no—
178C22⋊Q8—94no—
179C22⋊Q8—94no—
180#121—94no—
181C4×Q8—32no—
182C4×D4—94no—
183C4×D4—92no—
184#117—91no—
185C2×C4⋊C4—94no—
186C2×C42—91no—
187C23×C3—31no—
188C23×C3—42no—
189C23×C3—82no—
190C23×C3—122no—
191C23×C3C22×A412yes—
192C22×D3C22×A412yes—
193C3×Q8—31no—
194C3×Q8—62no—
195C3×Q8—62no—
196C3×Q8—62no—
197C3×Q8C2×C3×D411yescharacteristic
198C3×D4—22no—
199C3×D4—32no—
200C3×D4—32no—
201C3×D4—64no—
202C3×D4—82no—
203C3×D4—84no—
204C3×D4—242no—
205C3×D4—248no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—41no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—64no—
214C2×C3×C4—81no—
215C2×C3×C4—122no—
216C2×C3×C4C22×A411yescharacteristic
217#67—28no—
218#67—64no—
219C2×Dic3—32no—
220C2×Dic3—32no—
221C2×Dic3—64no—
222C2×Dic3—64no—
223C2×Dic3C22×A412yes—
224D12—22no—
225C4×D3—62no—
226C4×D3—62no—
227Dic6—62no—
228Dic6—62no—
229Dic6—62no—
230SL(2,𝔽3)—21no—
231SL(2,𝔽3)C2×#6711yescharacteristic
232C2×C32—41no—
233C2×C32—42no—
234C2×C32—162no—
235C3×D3—162no—
236#41—32no—
237#41—62no—
238#41—62no—
239#41—64no—
240#41—64no—
241#41—182no—
242#41—182no—
243#41—184no—
244#41—184no—
245#41C3×#6712yescomplemented
246C2×Q8—22no—
247C2×Q8—31no—
248C2×Q8—32no—
249C2×Q8—62no—
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8—92no—
254C2×Q8—91no—
255C2×Q8—182no—
256C2×Q8—182no—
257C2×Q8—182no—
258C2×Q8C22×C3×D311yescharacteristic
259C2×D4—31no—
260C2×D4—34no—
261C2×D4—32no—
262C2×D4—31no—
263C2×D4—92no—
264C2×D4D3×A411yescharacteristic
265C22×C4—31no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—92no—
269C22×C4—92no—
270C22×C4—92no—
271C4⋊C4—94no—
272C4⋊C4—92no—
273C4⋊C4—98no—
274C22⋊C4—94no—
275C22⋊C4—92no—
276C22⋊C4—94no—
277C42—31no—
278C42—92no—
279C42—91no—
280C22×C3—22no—
281C22×C3—24no—
282C22×C3—32no—
283C22×C3—41no—
284C22×C3—62no—
285C22×C3—82no—
286C22×C3—84no—
287C22×C3—84no—
288C22×C3—81no—
289C22×C3—164no—
290C22×C3—244no—
291C22×C3—242no—
292C22×C3C23×A411yescharacteristic
293C2×D3—22no—
294C2×D3—24no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—42no—
298C3×C4—62no—
299C3×C4—82no—
300C3×C4—124no—
301C3×C4D4×A412yes—
302Dic3—62no—
303Dic3—62no—
304Dic32−1+4⋊C314yescomplemented
305C32—41noSylow
306C23—31no—
307C23—32no—
308C23C2×D3×A412yes—
309Q8—31no—
310Q8—62no—
311Q8—62no—
312Q8—62no—
313Q8—182no—
314Q8—182no—
315Q8—182no—
316Q8C2×C3×#6711yescharacteristic
317D4—22no—
318D4—32no—
319D4—32no—
320D4—68no—
321D4—64no—
322D4—62no—
323D4—184no—
324C2×C4—32no—
325C2×C4—31no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—64no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—184no—
337C2×C4—184no—
338C2×C4C2×D3×A411yescharacteristic
339C2×C3—42no—
340C2×C3—42no—
341C2×C3—41no—
342C2×C3—62no—
343C2×C3—82no—
344C2×C3—81no—
345C2×C3—162no—
346C2×C3—164no—
347C2×C3—482no—
348C2×C3C2×D4×A411yescharacteristic
349C2×C3C2×2−1+4⋊C312yes—
350D3—42no—
351C22—24no—
352C22—22no—
353C22—32no—
354C22—62no—
355C22—64no—
356C22—62no—
357C22C22×D3×A411yescharacteristic
358C4—34no—
359C4—31no—
360C4—31no—
361C4—62no—
362C4—182no—
363C4—182no—
364C4A4×#6712yes—
365C3—41no—
366C3—81no—
367C3#3499⋊C311yescharacteristic, complemented
368C2—42no—
369C2—62no—
370C2—122no—
371C2D3×2−1+4⋊C312yes—
372C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×2−1+4C2×C3
socleC22×C3C23×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C2×C3×D4
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. C2×C3×2−1+4⋊C3C2semidirect
2. C2×#1010⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. SL(2,𝔽3)×#67C2semidirect
5. #87292C3semidirect
14. C2×C3×2−1+4C2×C3semidirect
15. C2×#1010C2×C3semidirect
18. C2×2−1+4⋊C3D3semidirect
20. Dic3×#41C2×C3semidirect
24. Q8×#67C2×C3semidirect
31. C3×#41⋊C3D4semidirect
86. C2×2−1+4C3×D3semidirect
99. C3×#41C3×D4semidirect
134. #41⋊C3#67semidirect
245. #41C3×#67semidirect
304. Dic32−1+4⋊C3semidirect
367. C3#3499⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×#1010⋊C32
Dic3×#41⋊C32
SL(2,𝔽3)×#672
#872923
#3499⋊C33
C3×#9404