Finite Group Databasebeta

← #84110⋊C3 #11921377

Subgroups — #84110⋊C3 #11921377

Order 1152.

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

normal subgroups only

This group has 386 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
1C2×C3×#997C211yesmaximal, characteristic, complemented
2C3⋊(B(4,𝔽2)⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#458⋊C3)C211yesmaximal, characteristic, complemented
4#84110C311yesmaximal, characteristic, complemented
5#3026⋊C3—31nomaximal
6C3×#997C2211yescharacteristic
7A4×#67—41nomaximal
8C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
9C2×#997—21no—
10C2×#997D311yescharacteristic, complemented
11#5804—31no—
12#5244—31no—
13#5244C2×C311yescharacteristic, complemented
14#5243—31no—
15#5243C2×C311yescharacteristic, complemented
16C2×#834—31no—
17#458⋊C3—31no—
18B(4,𝔽2)⋊C3—31no—
19C22×C3×A4—41no—
20C22×C3×A4—81no—
21C2×D3×A4—41no—
22C2×C3×SL(2,𝔽3)—81no—
23Dic3×A4—41no—
24#3026—31noSylow
25C3×H(5,𝔽2)—31no—
26C3×H(5,𝔽2)—61no—
27C3×H(5,𝔽2)—61no—
28C3×H(5,𝔽2)C22×C311yescharacteristic
29C2×C3×#41—61no—
30C22×C3×D4—31no—
31C22×C3×D4—61no—
32C22×#67A411yescharacteristic, complemented
33#997—21no—
34#997C2×D311yescharacteristic
35D4×A4—121no—
36#953—61no—
37C2×#217—31no—
38#939—61no—
39#939—61no—
40#937—31no—
41#937—31no—
42#935—31no—
43#935—31no—
44#929—61no—
45#834—32no—
46#834—31no—
47#834—31no—
48C2×C3×A4—41no—
49C2×C3×A4—81no—
50C2×C3×A4—81no—
51C2×C3×A4—82no—
52D3×A4—81no—
53C3×#67—161no—
54C3×SL(2,𝔽3)—81no—
55C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
56#534—91no—
57#458—31no—
58#458—91no—
59B(4,𝔽2)—31no—
60B(4,𝔽2)—91no—
61C2×#99—91no—
62C24×C3—21no—
63C24×C3—31no—
64C24×C3C2×A411yescharacteristic
65C23×D3C2×A411yescharacteristic, complemented
66C22×A4—41no—
67C22×A4—81no—
68C22×A4—82no—
69C22×A4—81no—
70C22×A4—121no—
71C3×#41—61no—
72C3×#41—62no—
73C3×#41—61no—
74C3×#41—61no—
75C3×#41—62no—
76C3×#41—61no—
77C2×C3×Q8—21no—
78C2×C3×D4—31no—
79C2×C3×D4—31no—
80C2×C3×D4—31no—
81C2×C3×D4—31no—
82C2×C3×D4—61no—
83C2×C3×D4—62no—
84C2×C3×D4—62no—
85C2×C3×D4—62no—
86C2×C3×D4—61no—
87C2×C3×D4—61no—
88C2×C3×D4—62no—
89C2×C3×D4—61no—
90C2×C3×D4—61no—
91C22×C3×C4—31no—
92C22×C3×C4—61no—
93C2×#67—61no—
94C2×#67—61no—
95C2×#67—61no—
96C2×#67—61no—
97C22×Dic3—61no—
98C22×Dic3C2×A411yescharacteristic, complemented
99C2×SL(2,𝔽3)—81no—
100C2×SL(2,𝔽3)—82no—
101C4×A4—121no—
102#217—31no—
103#217—31no—
104#217—61no—
105#217—61no—
106#217—61no—
107(C2×D3)⋊C4—61no—
108(C2×D3)⋊C4—61no—
109Dic3⋊C4—61no—
110Dic3⋊C4—61no—
111C22×C32—161no—
112C2×C3×D3—161no—
113C3×A4—41no—
114C3×A4—81no—
115C3×Dic3—161no—
116H(5,𝔽2)—31no—
117H(5,𝔽2)—61no—
118H(5,𝔽2)—61no—
119H(5,𝔽2)C2×C3×D311yescharacteristic
120C2×#41—61no—
121C22×D4—31no—
122C22×D4—31no—
123C22×D4—61no—
124#123—91no—
125#123—91no—
126#123—181no—
127#121—181no—
128#121—181no—
129#120—91no—
130#120—91no—
131#120—181no—
132C2×C22⋊C4—91no—
133#99—92no—
134#99—91no—
135#99—91no—
136C23×C3—21no—
137C23×C3—31no—
138C23×C3—31no—
139C23×C3—31no—
140C23×C3—31no—
141C23×C3—61no—
142C23×C3—61no—
143C23×C3—61no—
144C23×C3—81no—
145C23×C3—82no—
146C23×C3—241no—
147C23×C3C22×A411yescharacteristic
148C22×D3—61no—
149C22×D3—81no—
150C2×A4—41no—
151C2×A4—81no—
152C2×A4—82no—
153C2×A4—81no—
154C2×A4—82no—
155C2×A4—82no—
156C2×A4—84no—
157C2×A4—81no—
158C2×A4—241no—
159C3×Q8—21no—
160C3×Q8—61no—
161C3×D4—62no—
162C3×D4—62no—
163C3×D4—62no—
164C3×D4—62no—
165C3×D4—61no—
166C3×D4—61no—
167C3×D4—62no—
168C3×D4—481no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—61no—
172C2×C3×C4—62no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176#67—41no—
177#67—121no—
178#67—241no—
179#67—241no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—241no—
184SL(2,𝔽3)—81no—
185SL(2,𝔽3)—82no—
186C2×C32—161no—
187C2×C32—321no—
188C3×D3—321no—
189C24—21no—
190C24—31no—
191C24—31no—
192C24D3×A411yescharacteristic
193#41—61no—
194#41—62no—
195#41—61no—
196#41—61no—
197#41—62no—
198#41—61no—
199C2×Q8—21no—
200C2×D4—31no—
201C2×D4—31no—
202C2×D4—31no—
203C2×D4—31no—
204C2×D4—61no—
205C2×D4—62no—
206C2×D4—62no—
207C2×D4—62no—
208C2×D4—61no—
209C2×D4—61no—
210C2×D4—62no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—181no—
214C2×D4—181no—
215C2×D4—181no—
216C2×D4—181no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—61no—
220C22×C4—181no—
221C4⋊C4—181no—
222C4⋊C4—181no—
223C22⋊C4—91no—
224C22⋊C4—91no—
225C22⋊C4—181no—
226C22⋊C4—181no—
227C22⋊C4—181no—
228C22⋊C4—181no—
229C22⋊C4—181no—
230C22×C3—31no—
231C22×C3—31no—
232C22×C3—41no—
233C22×C3—61no—
234C22×C3—61no—
235C22×C3—81no—
236C22×C3—121no—
237C22×C3—122no—
238C22×C3—121no—
239C22×C3—121no—
240C22×C3—121no—
241C22×C3—161no—
242C22×C3—241no—
243C22×C3—241no—
244C22×C3—321no—
245C22×C3—481no—
246C22×C3#120⋊C311yescharacteristic
247C2×D3—41no—
248C2×D3—121no—
249C2×D3—121no—
250A4—41no—
251A4—81no—
252A4—81no—
253A4—82no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—481no—
257Dic3—41no—
258Dic3—121no—
259Dic3—121no—
260C32—161noSylow
261C23—21no—
262C23—31no—
263C23—31no—
264C23—31no—
265C23—31no—
266C23—61no—
267C23—61no—
268C23—61no—
269C23—81no—
270C23—82no—
271C23—181no—
272C23—241no—
273C23—241no—
274C23C2×D3×A411yescharacteristic
275Q8—21no—
276Q8—61no—
277D4—62no—
278D4—62no—
279D4—62no—
280D4—62no—
281D4—61no—
282D4—61no—
283D4—62no—
284D4—121no—
285D4—361no—
286D4—721no—
287D4—721no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—61no—
291C2×C4—62no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—721no—
299C2×C3—21no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3—121no—
303C2×C3—121no—
304C2×C3—161no—
305C2×C3—321no—
306C2×C3—322no—
307C2×C3—321no—
308C2×C3—961no—
309C2×C3C2×#120⋊C311yescharacteristic
310D3—81no—
311C22—31no—
312C22—31no—
313C22—41no—
314C22—61no—
315C22—61no—
316C22—81no—
317C22—121no—
318C22—122no—
319C22—121no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22—241no—
324C22—241no—
325C22—361no—
326C22—361no—
327C22C3⋊(#120⋊C3)11yescharacteristic
328C4—61no—
329C4—61no—
330C4—121no—
331C4—361no—
332C4—361no—
333C3—161no—
334C3—321no—
335C3#3026⋊C311yescharacteristic, complemented
336C2—21no—
337C2—61no—
338C2—61no—
339C2—121no—
340C2—121no—
341C2—241no—
342C2C2×C3⋊(#120⋊C3)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊(#120⋊C3)
commutator, cosocle, hypocenterC3×H(5,𝔽2)C22×C3
FrattiniC23C2×D3×A4
FittingC2×C3×H(5,𝔽2)C2×C3
socleC2×C3C2×#120⋊C3
hypercenterC22C3⋊(#120⋊C3)
LCSG ▷ C3×H(5,𝔽2)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×#997C2semidirect
2. C3⋊(B(4,𝔽2)⋊C3)C2semidirect
3. C3⋊(#458⋊C3)C2semidirect
4. #84110C3semidirect
8. C2×C3×H(5,𝔽2)C2×C3semidirect
10. C2×#997D3semidirect
13. #5244C2×C3semidirect
15. #5243C2×C3semidirect
32. C22×#67A4semidirect
55. C2×H(5,𝔽2)C3×D3semidirect
65. C23×D3C2×A4semidirect
98. C22×Dic3C2×A4semidirect
335. C3#3026⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9972
C3⋊(B(4,𝔽2)⋊C3)2
C3⋊(#458⋊C3)2
#841103
#3026⋊C33
A4×#674