Finite Group Databasebeta

← #11923083

Subgroups — #11923083

Order 1152.

786 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 145 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
2#10593660C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4#10593654C212yesmaximal, complemented
5C2×C3⋊#859C212yesmaximal, complemented
6#85534—33nomaximal
7#84111—31nomaximal
8C3×2−1+4⋊C3C414yescomplemented
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yes—
11D4×C3⋊Dic3—41nomaximal
12C3⋊#859C2211yescharacteristic
13C3⋊#859C2212yes—
14C3⋊#859C2211yescharacteristic
15C2×C3×2−1+4D311yescharacteristic
16C2×2−1+4⋊C3D313yes—
17#5827—31no—
18C2×#948—31no—
19#5426—33no—
20#5424—33no—
21#5420—36no—
22C2×#859—36no—
23#5233—32no—
24#5233—31no—
25#5232—32no—
26#5232—31no—
27#5225—31no—
28#5225—32no—
29C2×#835—31no—
30C2×#835—32no—
31C2×C32×D4—41no—
32C22×C3⋊Dic3—42no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2×C414yescomplemented
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36#3757—42no—
37C4⋊(C3⋊Dic3)—41no—
38C4×C3⋊Dic3—41no—
39#2898—91noSylow
40C3×2−1+4—34no—
41C3×2−1+4Dic314yescomplemented
42C2×C3×#41—31no—
43C2×C3×#41—32no—
44C2×C3×#41C2×D311yescharacteristic
45C22×C3×Q8—31no—
46C22×C3×Q8C2×D312yes—
472−1+4⋊C3Dic3112yescomplemented
48C2×#41⋊C3C2×D313yes—
49C22×SL(2,𝔽3)C2×D316yes—
50#948—64no—
51Q8×Dic3—61no—
52D4×Dic3—62no—
53D4×Dic3—61no—
54D4×Dic3—123no—
55#926—32no—
56#926—31no—
57C2×C4⋊Dic3—32no—
58C2×C4⋊Dic3—31no—
59C2×#208—32no—
60C2×#208—31no—
61C22×C3⋊C8—32no—
62C22×C3⋊C8—31no—
63#859—33no—
64#859—36no—
65#859—33no—
66#835—32no—
67#835—32no—
68#835—32no—
69#835—32no—
70#835—31no—
71#835—32no—
72#835—31no—
73#832—31no—
74#832—32no—
75#832—31no—
76C23×C32—42no—
77C32×D4—44no—
78C2×C32×C4—41no—
79C2×C3⋊Dic3—41no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3—42no—
82C2×C3⋊Dic3—84no—
83C3×SL(2,𝔽3)C22×C411yescharacteristic
84C2×2−1+4C3⋊D311yescharacteristic
85C2×#131—91no—
86#520—91no—
87#419—91no—
88#419—92no—
89#417—92no—
90#417—91no—
91#416—92no—
92#416—91no—
93C2×#103—91no—
94C2×#103—92no—
95C3×#41—34no—
96C3×#41—34no—
97C3×#41—32no—
98C3×#41—38no—
99C3×#41—64no—
100C3×#41C2×Dic312yescomplemented
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8—34no—
104C2×C3×Q8—34no—
105C2×C3×Q8—62no—
106C2×C3×Q8C2×Dic314yescomplemented
107C2×C3×Q8C22×D311yescharacteristic
108C2×C3×D4—32no—
109C2×C3×D4—31no—
110C2×C3×D4—43no—
111C2×C3×D4S411yescharacteristic
112C22×C3×C4—32no—
113C22×C3×C4—31no—
114C22×C3×C4—61no—
115C22×Dic3—61no—
116C22×Dic3—62no—
117C22×Dic3—126no—
118#41⋊C3C2×Dic316yescomplemented
119C2×SL(2,𝔽3)C2×Dic3112yescomplemented
120C2×SL(2,𝔽3)C22×D313yes—
121#217—61no—
122#217—62no—
123#217—126no—
124C4⋊Dic3—32no—
125C4⋊Dic3—31no—
126C4⋊Dic3—31no—
127C4⋊Dic3—62no—
128C4⋊Dic3—61no—
129C4⋊Dic3—123no—
130C4×Dic3—62no—
131C4×Dic3—61no—
132C4×Dic3—123no—
133#208—64no—
134#208—62no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—32no—
138C2×C3⋊C8—64no—
139C2×C3⋊C8—62no—
140C22×C32—44no—
141C22×C32—41no—
142C22×C32—82no—
143C22×C32—82no—
144C32×C4—42no—
145C3⋊Dic3—42no—
146C3⋊Dic3—81no—
147C3⋊Dic3—82no—
1482−1+4—34no—
1492−1+4C3⋊Dic314yescomplemented
150C2×#41—31no—
151C2×#41—32no—
152C2×#41C2×C3⋊D311yescharacteristic
153C22×Q8—31no—
154C22×Q8C2×C3⋊D312yes—
155#131—184no—
156C2×M16—92no—
157C2×M16—91no—
158C22×C8—92no—
159C22×C8—91no—
160C4×Q8—181no—
161C4×D4—182no—
162C4×D4—181no—
163#117—92no—
164#117—91no—
165C2×C4⋊C4—92no—
166C2×C4⋊C4—91no—
167#103—92no—
168#103—92no—
169#103—92no—
170#103—92no—
171#103—91no—
172#103—92no—
173#103—91no—
174#102—91no—
175#102—92no—
176#102—91no—
177C23×C3—31no—
178C23×C3—46no—
179C23×C3C2×S412yes—
180C3×Q8—34no—
181C3×Q8—31no—
182C3×Q8—34no—
183C3×Q8—62no—
184C3×Q8C22×Dic311yescharacteristic
185C3×D4—32no—
186C3×D4—38no—
187C3×D4—412no—
188C3×D4—61no—
189C3×D4SL(2,ℤ/4)14yes—
190C2×C3×C4—34no—
191C2×C3×C4—34no—
192C2×C3×C4—31no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—43no—
197C2×C3×C4—64no—
198C2×C3×C4C2×S411yescharacteristic
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—62no—
202C2×Dic3—123no—
203C2×Dic3—123no—
204C2×Dic3—126no—
205C2×Dic3—242no—
206C2×Dic3—241no—
207C2×Dic3—2412no—
208SL(2,𝔽3)C22×Dic313yes—
209C3⋊C8—61no—
210C3⋊C8—61no—
211C3⋊C8—62no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—84no—
216#41—34no—
217#41—34no—
218#41—32no—
219#41—38no—
220#41—64no—
221#41C2×C3⋊Dic312yescomplemented
222C2×Q8—31no—
223C2×Q8—32no—
224C2×Q8—34no—
225C2×Q8—34no—
226C2×Q8—62no—
227C2×Q8C2×C3⋊Dic314yescomplemented
228C2×Q8C22×C3⋊D311yescharacteristic
229C2×D4—32no—
230C2×D4—31no—
231C2×D4C3⋊S411yescharacteristic
232C22×C4—32no—
233C22×C4—31no—
234C22×C4—61no—
235C22×C4—181no—
236C22×C4—182no—
237M16—184no—
238M16—182no—
239C2×C8—91no—
240C2×C8—91no—
241C2×C8—92no—
242C2×C8—184no—
243C2×C8—182no—
244C4⋊C4—92no—
245C4⋊C4—91no—
246C4⋊C4—91no—
247C4⋊C4—182no—
248C4⋊C4—181no—
249C22⋊C4—181no—
250C22⋊C4—182no—
251C42—182no—
252C42—181no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—32no—
256C22×C3—412no—
257C22×C3—43no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—86no—
261C22×C3—86no—
262C22×C3C2×SL(2,ℤ/4)14yes—
263C22×C3C22×S411yescharacteristic
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—34no—
267C3×C4—46no—
268C3×C4C2×SL(2,ℤ/4)12yes—
269Dic3—126no—
270Dic3—121no—
271Dic3—122no—
272Dic3—121no—
273Dic3—246no—
274Dic3—243no—
275C32—41noSylow
276C23—31no—
277C23C2×C3⋊S412yes—
278Q8—34no—
279Q8—31no—
280Q8—34no—
281Q8—62no—
282Q8C22×C3⋊Dic311yescharacteristic
283D4—32no—
284D4—38no—
285D4—61no—
286D4C3⋊SL(2,ℤ/4)14yes—
287C2×C4—34no—
288C2×C4—34no—
289C2×C4—32no—
290C2×C4—31no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—64no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—182no—
297C2×C4—722no—
298C2×C4—721no—
299C2×C4C2×C3⋊S411yescharacteristic
300C8—181no—
301C8—181no—
302C8—182no—
303C2×C3—24no—
304C2×C3—43no—
305C2×C3—43no—
306C2×C3—43no—
307C2×C3—62no—
308C2×C3—812no—
309C2×C3#592611yescharacteristic
310C2×C3#592711yescharacteristic
311C2×C3C22×SL(2,ℤ/4)11yescharacteristic
312C22—22no—
313C22—22no—
314C22—32no—
315C22—61no—
316C22—61no—
317C22C2×C3⋊SL(2,ℤ/4)14yes—
318C22C22×C3⋊S411yescharacteristic
319C4—31no—
320C4—31no—
321C4—34no—
322C4—361no—
323C4—362no—
324C4—361no—
325C4C2×C3⋊SL(2,ℤ/4)12yes—
326C3—43no—
327C3#8553411yescharacteristic, complemented
328C2—24no—
329C2—62no—
330C2#1059658911yescharacteristic
331C2#1059659011yescharacteristic
332C2C22×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×2−1+4D3
socleC22×C3C22×S4
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC2×D4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10593660C2semidirect
3. #10593658C2semidirect
4. #10593654C2semidirect
5. C2×C3⋊#859C2semidirect
8. C3×2−1+4⋊C3C4semidirect
33. C3×#41⋊C3C2×C4semidirect
34. C2×C3×SL(2,𝔽3)C2×C4semidirect
41. C3×2−1+4Dic3semidirect
47. 2−1+4⋊C3Dic3semidirect
100. C3×#41C2×Dic3semidirect
106. C2×C3×Q8C2×Dic3semidirect
118. #41⋊C3C2×Dic3semidirect
119. C2×SL(2,𝔽3)C2×Dic3semidirect
149. 2−1+4C3⋊Dic3semidirect
221. #41C2×C3⋊Dic3semidirect
227. C2×Q8C2×C3⋊Dic3semidirect
327. C3#85534semidirect

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
#105936602
#105936582
#105936542
C2×C3⋊#8592
#855343
#841113
D4×C3⋊Dic34