Finite Group Databasebeta

← #85847⋊C3 #11923205

Subgroups — #85847⋊C3 #11923205

Order 1152.

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

normal subgroups only

This group has 753 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×#1010⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊#994C211yesmaximal, characteristic, complemented
3C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C214yesmaximal, complemented
5#85847C311yesmaximal, characteristic, complemented
6#3500⋊C3—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8#1010⋊C3—22no—
9C3⋊#994—22no—
10C2×D3×SL(2,𝔽3)C2212yescomplemented
11C2×C3⋊(#41⋊C3)C2212yes—
12C3×#875—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14C2×#1010C2×C311yescharacteristic
15C2×#1009C2×C311yescharacteristic, complemented
16C2×2−1+4⋊C3—31no—
17C2×#994—31no—
18C3×C4×#41C2×C311yescharacteristic, complemented
19#5576C2×C314yescomplemented
20#5554—34no—
21#5526—31no—
22#5525—31no—
23#5477—31no—
24C2×#875—31no—
25#548⋊C3—34no—
26C4×#41⋊C3—21no—
27C4×#41⋊C3D311yescharacteristic, complemented
28C2×C3×D12—41no—
29C2×C3×Dic6—41no—
30C3×#41⋊C3D412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32D3×SL(2,𝔽3)—22no—
33C3⋊(#41⋊C3)—22no—
34C32×C42—41no—
35C3×(C2×D3)⋊C4—44no—
36#3500—31noSylow
37C2×C3×#41C22×C311yescharacteristic
38#1010—22no—
39#1010—62no—
40#1009—22no—
41#1009—62no—
42C2×#239C22×C312yes—
43C2×D3×Q8C22×C312yes—
44C2×#237—32no—
45C2×D3×D4—32no—
46C2×#235—31no—
47C2×#235—31no—
48C22×D12—31no—
49C22×Dic6—31no—
502−1+4⋊C3—62no—
51#994—62no—
52C2×#41⋊C3—21no—
53C2×#41⋊C3—32no—
54C2×#41⋊C3C2×D311yescharacteristic
55C22×SL(2,𝔽3)—32no—
56C3×#124—121no—
57C3×C4×Q8C22×C312yes—
58C3×C4×D4—32no—
59C3×#117—31no—
60C2×C3×C42—31no—
61#930—34no—
62C2×(C2×D3)⋊C4—34no—
63#897—34no—
64#895—34no—
65#886—34no—
66#882—34no—
67#875—32no—
68#875—31no—
69#875—34no—
70#875A411yescharacteristic
71#874—31no—
72C4×D12—34no—
73#869—31no—
74C4×SL(2,𝔽3)—22no—
75C4×SL(2,𝔽3)C2×D312yescomplemented
76C22×C3×D3—42no—
77C2×C32×C4—41no—
78C2×C32×C4—42no—
79C2×C3×Dic3—42no—
80C3×D12—82no—
81C3×Dic6—82no—
82C3×SL(2,𝔽3)C2×D411yescharacteristic
83C2×2−1+4—31no—
84C2×H(5,𝔽2)—31no—
85#548—34no—
86#546—94no—
87#536—91no—
88#535—91no—
89#532—91no—
90C2×#124—91no—
91C4×#41C3×D311yescharacteristic, complemented
92C23×D3—32no—
93C3×#41—32no—
94C3×#41C3×D412yes—
95C2×C3×Q8—121no—
96C2×C3×Q8C23×C311yescharacteristic
97C2×C3×D4—31no—
98C2×C3×D4—121no—
99C22×C3×C4—31no—
100C22×C3×C4—32no—
101C2×#67—32no—
102C2×#67—32no—
103C22×Dic3—32no—
104#239—22no—
105#239—62no—
106D3×Q8—22no—
107D3×Q8—62no—
108#237—64no—
109#237—64no—
110D3×D4—64no—
111D3×D4—64no—
112#235—62no—
113#235—62no—
114#235—62no—
115#235—62no—
116C2×D12—32no—
117C2×D12—31no—
118C2×D12—34no—
119C2×D12—62no—
120C2×D12—62no—
121C2×D12C2×A411yescharacteristic
122C2×C4×D3—32no—
123C2×C4×D3—32no—
124C2×Dic6—31no—
125C2×Dic6—32no—
126C2×Dic6—62no—
127C2×Dic6—62no—
128C2×Dic6C2×A411yescharacteristic
129#41⋊C3—22no—
130#41⋊C3—62no—
131#41⋊C3D1212yescomplemented
132C2×SL(2,𝔽3)—21no—
133C2×SL(2,𝔽3)—62no—
134C2×SL(2,𝔽3)C22×D311yescharacteristic
135C3×C4⋊C4—32no—
136C3×C22⋊C4—32no—
137C3×C22⋊C4—124no—
138C3×C42—31no—
139C3×C42—32no—
140C3×C42—41no—
141C3×C42—81no—
142C3×C42C2×A411yescharacteristic
143(C2×D3)⋊C4—38no—
144(C2×D3)⋊C4—34no—
145(C2×D3)⋊C4C2×A414yes—
146C4⋊Dic3—34no—
147C22×C32—41no—
148C2×C3×D3—82no—
149C2×C3×D3—84no—
150C32×C4—42no—
151C32×C4—82no—
152C3×Dic3—82no—
1532−1+4—62no—
1542−1+4—182no—
155H(5,𝔽2)—62no—
156H(5,𝔽2)—182no—
157C2×#41—32no—
158C2×#41—91no—
159C2×#41—91no—
160C2×#41—92no—
161C2×#41C2×C3×D311yescharacteristic
162C22×Q8—32no—
163C22×Q8—91no—
164C22×D4—91no—
165C22×D4—92no—
166C4⋊Q8—91no—
167#127—91no—
168#124—31no—
169#124—92no—
170#124—94no—
171#124—91no—
172#123—94no—
173C22⋊Q8—94no—
174#121—94no—
175#121—94no—
176#120—94no—
177C4×Q8C2×C3×D312yes—
178C4×D4—32no—
179C4×D4—94no—
180#117—31no—
181C2×C22⋊C4—94no—
182C2×C42—31no—
183C23×C3—31no—
184C23×C3—122no—
185C22×D3—32no—
186C22×D3—64no—
187C22×D3—124no—
188C22×D3C22×A412yes—
189C3×Q8—31no—
190C3×Q8—242no—
191C3×Q8C2×C3×D411yescharacteristic
192C3×D4—32no—
193C3×D4—32no—
194C3×D4—242no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—82no—
202C2×C3×C4—81no—
203C2×C3×C4—122no—
204C2×C3×C4—122no—
205C2×C3×C4C22×A412yes—
206C2×C3×C4C22×A411yescharacteristic
207#67—64no—
208#67—64no—
209C2×Dic3—32no—
210C2×Dic3—64no—
211C2×Dic3C22×A412yes—
212D12—22no—
213D12—62no—
214D12—62no—
215D12—62no—
216D12—124no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—62no—
221Dic6—22no—
222Dic6—62no—
223Dic6—62no—
224Dic6—62no—
225SL(2,𝔽3)—21no—
226SL(2,𝔽3)C2×D1211yescharacteristic
227C2×C32—41no—
228C2×C32—42no—
229C3×D3—162no—
230C24—92no—
231#41—32no—
232#41—62no—
233#41—182no—
234#41—182no—
235#41—182no—
236#41—182no—
237#41—184no—
238#41—184no—
239#41—182no—
240#41C3×D1212yes—
241C2×Q8—31no—
242C2×Q8—62no—
243C2×Q8—92no—
244C2×Q8—91no—
245C2×Q8—182no—
246C2×Q8—182no—
247C2×Q8—182no—
248C2×Q8C22×C3×D311yescharacteristic
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—92no—
252C2×D4—92no—
253C2×D4—94no—
254C2×D4—92no—
255C2×D4—91no—
256C2×D4—182no—
257C2×D4—182no—
258C2×D4—184no—
259C2×D4—184no—
260C22×C4—31no—
261C22×C4—32no—
262C22×C4—92no—
263C22×C4—92no—
264C22×C4—92no—
265C4⋊C4—32no—
266C4⋊C4—94no—
267C22⋊C4—32no—
268C22⋊C4—34no—
269C22⋊C4—98no—
270C22⋊C4—94no—
271C42—31no—
272C42—32no—
273C42D3×A411yescharacteristic
274C22×C3—32no—
275C22×C3—41no—
276C22×C3—62no—
277C22×C3—81no—
278C22×C3—242no—
279C22×C3—244no—
280C22×C3C23×A411yescharacteristic
281C2×D3—22no—
282C2×D3—24no—
283C2×D3—62no—
284C2×D3—64no—
285C2×D3—244no—
286C3×C4—22no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—42no—
290C3×C4—62no—
291C3×C4—82no—
292C3×C4—84no—
293C3×C4—82no—
294C3×C4—242no—
295C3×C4D4×A412yes—
296Dic3—22no—
297Dic3—62no—
298C32—41noSylow
299C23—31no—
300C23—32no—
301C23—92no—
302C23—184no—
303C23—364no—
304Q8—31no—
305Q8—62no—
306Q8—182no—
307Q8—182no—
308Q8—182no—
309Q8C2×C3×D1211yescharacteristic
310D4—32no—
311D4—32no—
312D4—62no—
313D4—182no—
314D4—182no—
315D4—184no—
316D4—184no—
317D4—182no—
318D4—364no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—32no—
324C2×C4—92no—
325C2×C4—122no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4—184no—
331C2×C4C2×D3×A412yes—
332C2×C4C2×D3×A411yescharacteristic
333C2×C3—41no—
334C2×C3—42no—
335C2×C3—62no—
336C2×C3—81no—
337C2×C3—82no—
338C2×C3—482no—
339C2×C3C2×D4×A411yescharacteristic
340C2×C3#594512yes—
341D3—42no—
342D3—122no—
343C22—32no—
344C22—62no—
345C22—62no—
346C22—64no—
347C22—182no—
348C22—184no—
349C22—724no—
350C22C22×D3×A411yescharacteristic
351C4—22no—
352C4—31no—
353C4—31no—
354C4—62no—
355C4—62no—
356C4—182no—
357C4A4×D1212yes—
358C3—41no—
359C3—81no—
360C3#3500⋊C311yescharacteristic, complemented
361C2—62no—
362C2—122no—
363C2—362no—
364C2C2×A4×D1211yescharacteristic
365C2#5964⋊C312yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
hypercenterC42D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#1010⋊C3C2semidirect
2. C2×C3⋊#994C2semidirect
3. C3×C4×#41⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. #85847C3semidirect
10. C2×D3×SL(2,𝔽3)C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
15. C2×#1009C2×C3semidirect
18. C3×C4×#41C2×C3semidirect
19. #5576C2×C3semidirect
27. C4×#41⋊C3D3semidirect
30. C3×#41⋊C3D4semidirect
75. C4×SL(2,𝔽3)C2×D3semidirect
91. C4×#41C3×D3semidirect
131. #41⋊C3D12semidirect
360. C3#3500⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1010⋊C32
C2×C3⋊#9942
C3×C4×#41⋊C32
#5576⋊C32
#858473
#3500⋊C33
C3×#8754