Finite Group Databasebeta

← #11922441

Subgroups — #11922441

Order 1152.

681 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 59 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×#986C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4#10593217C214yesmaximal, complemented
5#85540—31nomaximal
6#83948—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8#1010⋊C3—22no—
9C2×D3×SL(2,𝔽3)C2212yescomplemented
10C3×#986—22no—
11C2×C3×GL(2,𝔽3)C2212yes—
12#63914—41nomaximal
13C3⋊#859C2212yescomplemented
14C2×#1010D311yescharacteristic, complemented
15C2×2−1+4⋊C3—31no—
16C2×#986D311yescharacteristic, complemented
17C2×C3×AGL(1,ℤ/8)—31no—
18#5805—31no—
19C22×#214—31no—
20#5424—31no—
21#5424—61no—
22#5421—34no—
23#5233—31no—
24#5170—34no—
25#5158—34no—
26#5129—31no—
27#5120—31no—
28C2×C3×D12—41no—
29C2×C3×D12—41no—
30C3×#41⋊C3D412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C2×#625—44no—
33D3×SL(2,𝔽3)—22no—
34C3×GL(2,𝔽3)—22no—
35C4×C3⋊Dic3—41no—
36#3010—91noSylow
37C2×C3×#41C2×D311yescharacteristic
38C22×C3×D4—31no—
39#1010—22no—
40#1010—62no—
41C2×D3×Q8C2×D312yes—
42C2×#237—32no—
43C2×#235—31no—
44C22×Dic6—31no—
452−1+4⋊C3—62no—
46C2×#41⋊C3—21no—
47C2×#41⋊C3C2×D311yescharacteristic
48C22×SL(2,𝔽3)—32no—
49#986—22no—
50C2×GL(2,𝔽3)C2×D312yes—
51C3×AGL(1,ℤ/8)—62no—
52C3×AGL(1,ℤ/8)—62no—
53C2×C3×SD16—32no—
54C2×C3×D8—32no—
55C2×C3×M16—31no—
56#953—62no—
57#940—61no—
58#940—121no—
59#938—34no—
60#936—61no—
61C2×#214—34no—
62C2×#214—64no—
63#926—31no—
64C22×C3⋊C8—31no—
65#859—32no—
66#859—62no—
67#835—32no—
68#832—32no—
69(C2×D3)⋊C8—34no—
70#816—34no—
71C22×C3×D3—42no—
72C22×C3×D3—42no—
73C2×C32×C4—41no—
74C2×C3⋊Dic3—42no—
75C3×D12—82no—
76C3×D12—82no—
77C3×SL(2,𝔽3)C2×D411yescharacteristic
78#625—88no—
79C2×2−1+4—31no—
80C2×AGL(1,ℤ/8)—31no—
81C22×SD16—91no—
82#535—91no—
83#449—94no—
84#448—94no—
85#419—91no—
86#416—91no—
87#408—91no—
88C24×C3—61no—
89C3×#41—32no—
90C3×#41#6712yescomplemented
91C2×C3×Q8C22×D311yescharacteristic
92C2×C3×D4—31no—
93C2×C3×D4—32no—
94C2×C3×D4—62no—
95C2×C3×D4—62no—
96C2×C3×D4—61no—
97C2×C3×D4—121no—
98C22×C3×C4—31no—
99C2×#67—32no—
100C2×#67—64no—
101C2×#67—124no—
102C22×Dic3—61no—
103D3×Q8—22no—
104D3×Q8—62no—
105#237—64no—
106#237—64no—
107#235—62no—
108#235—62no—
109C2×D12—41no—
110C2×D12S411yescharacteristic
111C2×C4×D3—32no—
112C2×Dic6—32no—
113C2×Dic6—34no—
114C2×Dic6—31no—
115C2×Dic6—62no—
116#41⋊C3—22no—
117#41⋊C3#6712yescomplemented
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)—62no—
120C2×SL(2,𝔽3)C22×D311yescharacteristic
121GL(2,𝔽3)—22no—
122C3×SD16—62no—
123C3×SD16—62no—
124C3×D8—64no—
125C3×M16—62no—
126C2×C3×C8—32no—
127#217—64no—
128#217—61no—
129#214—68no—
130C4⋊Dic3—32no—
131C4×Dic3—61no—
132C4×Dic3—121no—
133C4×Dic3—241no—
134C2×C3⋊C8—32no—
135C2×C3⋊C8—62no—
136C22×C32—41no—
137C2×C3×D3—84no—
138C2×C3×D3—82no—
139C2×C3×D3—84no—
140C2×C3×D3—82no—
141C32×C4—42no—
142C3⋊Dic3—44no—
1432−1+4—62no—
1442−1+4—182no—
145C2×#41—91no—
146C2×#41—92no—
147C2×#41D3211yescharacteristic
148C22×Q8—32no—
149C22×Q8—91no—
150C22×D4—31no—
151AGL(1,ℤ/8)—62no—
152AGL(1,ℤ/8)—62no—
153C2×SD16—32no—
154C2×SD16—94no—
155C2×SD16—184no—
156C2×D8—32no—
157C2×M16—31no—
158C22×C8—91no—
159#127—181no—
160#124—181no—
161#121—94no—
162#120—182no—
163#117—91no—
164#103—94no—
165#103—92no—
166#102—92no—
167C22⋊C8—94no—
168C23×C3—31no—
169C23×C3—62no—
170C23×C3—122no—
171C23×C3—122no—
172C23×C3—242no—
173C22×D3—42no—
174C22×D3C2×S412yes—
175C3×Q8—31no—
176C3×Q8C2×#6711yescharacteristic
177C3×D4—32no—
178C3×D4—61no—
179C3×D4—62no—
180C3×D4—62no—
181C3×D4—122no—
182C3×D4—242no—
183C2×C3×C4—31no—
184C2×C3×C4—32no—
185C2×C3×C4—32no—
186C2×C3×C4—41no—
187C2×C3×C4—81no—
188C2×C3×C4C2×S411yescharacteristic
189#67—64no—
190#67—244no—
191#67—248no—
192C2×Dic3—32no—
193C2×Dic3—64no—
194C2×Dic3—62no—
195C2×Dic3—122no—
196C2×Dic3—242no—
197D12—22no—
198D12—82no—
199C4×D3—62no—
200C4×D3—62no—
201Dic6—34no—
202Dic6—62no—
203Dic6—62no—
204SL(2,𝔽3)—21no—
205SL(2,𝔽3)C2×#6711yescharacteristic
206C3×C8—62no—
207C3⋊C8—62no—
208C2×C32—42no—
209C2×C32—41no—
210C3×D3—162no—
211C3×D3—162no—
212C24—61no—
213#41—32no—
214#41—182no—
215#41—182no—
216#41—184no—
217#41—184no—
218#41#62512yescomplemented
219C2×Q8—62no—
220C2×Q8—92no—
221C2×Q8—94no—
222C2×Q8—91no—
223C2×Q8—182no—
224C2×Q8—182no—
225C2×Q8C2×D3211yescharacteristic
226C2×D4—31no—
227C2×D4—32no—
228C2×D4—31no—
229C2×D4—62no—
230C2×D4—62no—
231C2×D4—61no—
232C2×D4—92no—
233C2×D4—184no—
234C22×C4—31no—
235C22×C4—92no—
236C22×C4—181no—
237SD16—62no—
238SD16—62no—
239SD16—188no—
240D8—64no—
241M16—62no—
242C2×C8—32no—
243C2×C8—92no—
244C2×C8—182no—
245C4⋊C4—92no—
246C22⋊C4—184no—
247C22⋊C4—181no—
248C42—181no—
249C22×C3—32no—
250C22×C3—41no—
251C22×C3—62no—
252C22×C3—81no—
253C22×C3—124no—
254C22×C3—122no—
255C22×C3—244no—
256C22×C3—242no—
257C22×C3—244no—
258C22×C3C22×S411yescharacteristic
259C2×D3—24no—
260C2×D3—22no—
261C2×D3—82no—
262C2×D3—84no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—42no—
266C3×C4—82no—
267C3×C4GL(2,ℤ/4)12yes—
268Dic3—62no—
269Dic3—124no—
270Dic3—122no—
271Dic3—244no—
272C32—41noSylow
273C23—31no—
274C23—32no—
275C23—62no—
276C23—122no—
277C23—242no—
278Q8—31no—
279Q8—94no—
280Q8—182no—
281Q8—182no—
282Q8C2×#62511yescharacteristic
283D4—32no—
284D4—61no—
285D4—62no—
286D4—62no—
287D4—62no—
288D4—122no—
289D4—184no—
290D4—724no—
291C2×C4—31no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—92no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4—184no—
298C2×C4—182no—
299C2×C4D3×S411yescharacteristic
300C8—62no—
301C8—182no—
302C2×C3—42no—
303C2×C3—41no—
304C2×C3—62no—
305C2×C3—82no—
306C2×C3—81no—
307C2×C3—242no—
308C2×C3—482no—
309C2×C3#592612yes—
310C2×C3C2×GL(2,ℤ/4)11yescharacteristic
311D3—42no—
312D3—162no—
313C22—32no—
314C22—62no—
315C22—64no—
316C22—62no—
317C22—124no—
318C22—122no—
319C22—244no—
320C22C2×D3×S411yescharacteristic
321C4—31no—
322C4—31no—
323C4—182no—
324C4—362no—
325C4#6420912yes—
326C3—41no—
327C3—81no—
328C3#8554011yescharacteristic, complemented
329C2—62no—
330C2—122no—
331C2—242no—
332C2#1059646012yes—
333C2C2×#6420911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#1010⋊C3C2semidirect
2. C2×C3×#986C2semidirect
3. #10593658C2semidirect
4. #10593217C2semidirect
9. C2×D3×SL(2,𝔽3)C22semidirect
13. C3⋊#859C22semidirect
14. C2×#1010D3semidirect
16. C2×#986D3semidirect
30. C3×#41⋊C3D4semidirect
90. C3×#41#67semidirect
117. #41⋊C3#67semidirect
218. #41#625semidirect
328. C3#85540semidirect

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×#9862
#105936582
#105932172
#855403
#839483
#639144