Finite Group Databasebeta

← #84125⋊C3 #11921383

Subgroups — #84125⋊C3 #11921383

Order 1152.

360 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 27 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×#5950C211yesmaximal, characteristic, complemented
2C3⋊(B(4,𝔽2)⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#458⋊C3)C211yesmaximal, characteristic, complemented
4#84125C311yesmaximal, characteristic, complemented
5#3025⋊C3—31nomaximal
6C3×#997C2211yescharacteristic
7C4×D3×A4—41nomaximal
8C3×#585C2×C311yescharacteristic, complemented
9#5950—21no—
10#5950D311yescharacteristic, complemented
11#5828—31no—
12#5244—31no—
13#5244C2×C311yescharacteristic
14#5243—31no—
15#5243C2×C311yescharacteristic, complemented
16#5240—31no—
17#458⋊C3—31no—
18B(4,𝔽2)⋊C3—31no—
19C2×D3×A4—41no—
20C3×#41⋊C3—81no—
21C3×C4×A4—41no—
22C3×C4×A4—81no—
23Dic3×A4—41no—
24#3025—31noSylow
25C3×2−1+4—61no—
26C3×H(5,𝔽2)—31no—
27C3×H(5,𝔽2)—61no—
28C3×H(5,𝔽2)C22×C311yescharacteristic
29C2×C3×#41—31no—
30C2×C3×#41—61no—
31C2×C3×#41—61no—
32C22×C4×D3A411yescharacteristic, complemented
33#997—21no—
34#997C2×D311yescharacteristic
35C2×C4×A4—121no—
36#946—61no—
37#938—61no—
38#937—31no—
39#937—31no—
40#935—31no—
41#935—31no—
42C4×#67—61no—
43C4×#67—61no—
44#926—31no—
45#834—32no—
46#834—31no—
47#834—31no—
48C2×C3×A4—41no—
49C2×C3×A4—81no—
50D3×A4—42no—
51C3×C4×D3—161no—
52C3×SL(2,𝔽3)—81no—
53#585C3×D311yescharacteristic, complemented
54#525—91no—
55#458—31no—
56#458—91no—
57B(4,𝔽2)—31no—
58B(4,𝔽2)—91no—
59#410—91no—
60C23×D3C2×A411yescharacteristic
61C22×A4—121no—
62C3×#41—21no—
63C3×#41—62no—
64C3×#41—62no—
65C3×#41—61no—
66C3×#41—62no—
67C3×#41—61no—
68C3×#41—62no—
69C3×#41—62no—
70C3×#41—61no—
71C2×C3×Q8—31no—
72C2×C3×Q8—61no—
73C2×C3×Q8—61no—
74C2×C3×D4—31no—
75C2×C3×D4—31no—
76C2×C3×D4—31no—
77C2×C3×D4—61no—
78C2×C3×D4—62no—
79C2×C3×D4—61no—
80C2×C3×D4—61no—
81C2×C3×D4—61no—
82C22×C3×C4—21no—
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—61no—
86C22×C3×C4C2×A411yescharacteristic
87C2×#67—61no—
88C2×#67—61no—
89C22×Dic3C2×A411yescharacteristic
90C2×C4×D3—61no—
91C2×C4×D3—61no—
92#41⋊C3—81no—
93#41⋊C3—82no—
94C4×A4—41no—
95C4×A4—81no—
96C4×A4—82no—
97C4×A4—81no—
98C4×A4—121no—
99#217—31no—
100#217—31no—
101#217—61no—
102#217—61no—
103(C2×D3)⋊C4—61no—
104(C2×D3)⋊C4—61no—
105C4⋊Dic3—61no—
106Dic3⋊C4—61no—
107Dic3⋊C4—61no—
108C4×Dic3—61no—
109C2×C3×D3—161no—
110C3×A4—41no—
111C3×A4—81no—
112C32×C4—161no—
113C3×Dic3—161no—
1142−1+4—61no—
115H(5,𝔽2)—31no—
116H(5,𝔽2)—61no—
117H(5,𝔽2)C2×C3×D311yescharacteristic
118C2×#41—31no—
119C2×#41—61no—
120C2×#41—61no—
121C23×C4—31no—
122#123—91no—
123#123—91no—
124C22⋊Q8—181no—
125#121—181no—
126#120—91no—
127#120—91no—
128C4×D4—181no—
129C4×D4—181no—
130#117—91no—
131#99—92no—
132#99—91no—
133#99—91no—
134C23×C3—21no—
135C23×C3—31no—
136C23×C3—31no—
137C23×C3—61no—
138C23×C3C22×A411yescharacteristic
139C22×D3—42no—
140C22×D3—61no—
141C2×A4—41no—
142C2×A4—81no—
143C2×A4—82no—
144C2×A4—81no—
145C2×A4—122no—
146C3×Q8—21no—
147C3×Q8—62no—
148C3×Q8—61no—
149C3×D4—62no—
150C3×D4—62no—
151C3×D4—62no—
152C3×D4—61no—
153C3×D4—61no—
154C3×D4—62no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—61no—
160C2×C3×C4—61no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—61no—
166C2×C3×C4—481no—
167#67—122no—
168#67—122no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C4×D3—41no—
173C4×D3—121no—
174SL(2,𝔽3)—81no—
175SL(2,𝔽3)—82no—
176C2×C32—161no—
177C3×D3—162no—
178C24—31no—
179#41—21no—
180#41—62no—
181#41—62no—
182#41—61no—
183#41—62no—
184#41—61no—
185#41—62no—
186#41—62no—
187#41—61no—
188C2×Q8—31no—
189C2×Q8—61no—
190C2×Q8—61no—
191C2×D4—31no—
192C2×D4—31no—
193C2×D4—31no—
194C2×D4—61no—
195C2×D4—62no—
196C2×D4—61no—
197C2×D4—61no—
198C2×D4—61no—
199C2×D4—181no—
200C2×D4—181no—
201C22×C4—21no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—61no—
206C22×C4—181no—
207C22×C4—181no—
208C22×C4D3×A411yescharacteristic
209C4⋊C4—181no—
210C4⋊C4—181no—
211C4⋊C4—181no—
212C22⋊C4—91no—
213C22⋊C4—91no—
214C22⋊C4—181no—
215C22⋊C4—181no—
216C22⋊C4—181no—
217C22⋊C4—181no—
218C42—181no—
219C22×C3—31no—
220C22×C3—41no—
221C22×C3—61no—
222C22×C3—61no—
223C22×C3—81no—
224C22×C3—121no—
225C22×C3—121no—
226C22×C3—241no—
227C22×C3—481no—
228C2×D3—41no—
229C2×D3—62no—
230C2×D3—121no—
231A4—41no—
232A4—81no—
233A4—81no—
234A4—82no—
235C3×C4—31no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4—161no—
239C3×C4—321no—
240C3×C4—481no—
241C3×C4#120⋊C311yescharacteristic
242Dic3—41no—
243Dic3—121no—
244Dic3—121no—
245C32—161noSylow
246C23—21no—
247C23—31no—
248C23—31no—
249C23—61no—
250C23—122no—
251C23—181no—
252C23C2×D3×A411yescharacteristic
253Q8—21no—
254Q8—62no—
255Q8—61no—
256D4—62no—
257D4—62no—
258D4—62no—
259D4—61no—
260D4—61no—
261D4—62no—
262D4—362no—
263D4—362no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—121no—
276C2×C4—181no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—361no—
280C2×C3—61no—
281C2×C3—121no—
282C2×C3—121no—
283C2×C3—161no—
284C2×C3—321no—
285C2×C3—482no—
286C2×C3C2×#120⋊C311yescharacteristic
287D3—42no—
288C22—31no—
289C22—41no—
290C22—61no—
291C22—61no—
292C22—81no—
293C22—121no—
294C22—121no—
295C22—121no—
296C22—182no—
297C22—241no—
298C22—361no—
299C4—31no—
300C4—61no—
301C4—61no—
302C4—121no—
303C4—361no—
304C4—361no—
305C4C3⋊(#120⋊C3)11yescharacteristic
306C3—161no—
307C3—321no—
308C3#3025⋊C311yescharacteristic, complemented
309C2—61no—
310C2—121no—
311C2—121no—
312C2—122no—
313C2C2×C3⋊(#120⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC4C3⋊(#120⋊C3)
commutator, cosocle, hypocenterC3×H(5,𝔽2)C22×C3
FrattiniC23C2×D3×A4
FittingC3×#585C2×C3
socleC2×C3C2×#120⋊C3
LCSG ▷ C3×H(5,𝔽2)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×#5950C2semidirect
2. C3⋊(B(4,𝔽2)⋊C3)C2semidirect
3. C3⋊(#458⋊C3)C2semidirect
4. #84125C3semidirect
8. C3×#585C2×C3semidirect
10. #5950D3semidirect
15. #5243C2×C3semidirect
32. C22×C4×D3A4semidirect
53. #585C3×D3semidirect
308. C3#3025⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#59502
C3⋊(B(4,𝔽2)⋊C3)2
C3⋊(#458⋊C3)2
#841253
#3025⋊C33
C4×D3×A44