Finite Group Databasebeta

← #11923076

Subgroups — #11923076

Order 1152.

720 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 81 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×#994C211yesmaximal, characteristic
2C2×#64265C211yesmaximal, characteristic, complemented
3#10593659C214yesmaximal, complemented
4#10593658C211yesmaximal, characteristic, complemented
5#85527—33nomaximal
6#84105—31nomaximal
7C3×#994—22no—
8C2×C3×#41⋊C3C2212yes—
9C2×C3×#41⋊C3C2211yescharacteristic
10#64265—22no—
11C2×C3⋊#226C2212yescomplemented
12#64153—41nomaximal
13C3⋊#859C2212yescomplemented
14C2×C3×H(5,𝔽2)D311yescharacteristic
15C2×#994D313yes—
16C2×#984—33no—
17#5825—31no—
18C2×#951—31no—
19C22×#214—31no—
20#5425—312no—
21#5424—33no—
22#5239—34no—
23#5234—31no—
24#5233—31no—
25#5218—34no—
26#5215—31no—
27C2×C32×Q8—41no—
28C2×#634—41no—
29C3×#41⋊C3—22no—
30C3×#41⋊C3D412yes—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊#226—22no—
33#3750—44no—
34C4×C3⋊Dic3—41no—
35#3009—91noSylow
36C3×H(5,𝔽2)—22no—
37C3×H(5,𝔽2)—62no—
38C2×C3×#41—31no—
39C2×C3×#41C2×D312yes—
40C2×C3×#41C2×D311yescharacteristic
41C22×C3×D4—31no—
42C22×C3×D4—32no—
43C22×Dic6—31no—
44#994—26no—
45C2×#41⋊C3C2×D316yes—
46C2×#41⋊C3C2×D313yes—
47#984—66no—
48C2×#226—36no—
49#951—62no—
50#951—62no—
51#944—61no—
52#944—123no—
53C2×#216—32no—
54#936—61no—
55#935—62no—
56C2×#214—32no—
57C2×#214—34no—
58C2×#214—64no—
59#926—31no—
60#924—34no—
61C2×#208—31no—
62C22×C3⋊C8—31no—
63#859—36no—
64#835—32no—
65#832—34no—
66#832—32no—
67#830—34no—
68C32×Q8—82no—
69C2×C32×C4—41no—
70C2×C32×C4—42no—
71C2×C3⋊Dic3—42no—
72C2×C3⋊Dic3—42no—
73#634—82no—
74C3×SL(2,𝔽3)C2×D411yescharacteristic
75C2×H(5,𝔽2)C3⋊D311yescharacteristic
76C2×#137—91no—
77C22×SD16—91no—
78#536—91no—
79#452—94no—
80#450—94no—
81#418—91no—
82#416—91no—
83#408—91no—
84C24×C3—61no—
85C3×#41—22no—
86C3×#41—32no—
87C3×#41—62no—
88C3×#41—62no—
89C3×#41—62no—
90C3×#41#6712yes—
91C2×C3×Q8—43no—
92C2×C3×Q8S411yescharacteristic
93C2×C3×Q8C22×D311yescharacteristic
94C2×C3×D4—31no—
95C2×C3×D4—32no—
96C2×C3×D4—34no—
97C2×C3×D4—32no—
98C2×C3×D4—31no—
99C2×C3×D4—62no—
100C2×C3×D4—64no—
101C2×C3×D4—64no—
102C22×C3×C4—31no—
103C22×C3×C4—32no—
104C22×Dic3—61no—
105C2×Dic6—32no—
106C2×Dic6—62no—
107C2×Dic6—62no—
108C2×Dic6—61no—
109C2×Dic6—123no—
110#41⋊C3—26no—
111#41⋊C3#6716yes—
112C2×SL(2,𝔽3)C22×D313yes—
113#226—66no—
114#217—64no—
115#217—61no—
116#216—62no—
117#216—62no—
118#214—68no—
119#214—64no—
120C4⋊Dic3—32no—
121Dic3⋊C4—64no—
122Dic3⋊C4—1212no—
123C4×Dic3—61no—
124C4×Dic3—123no—
125#208—62no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—32no—
128C2×C3⋊C8—62no—
129C22×C32—41no—
130C32×C4—42no—
131C32×C4—82no—
132C3⋊Dic3—44no—
133C3⋊Dic3—82no—
134H(5,𝔽2)—22no—
135H(5,𝔽2)—62no—
136C2×#41—31no—
137C2×#41C2×C3⋊D312yes—
138C2×#41C2×C3⋊D311yescharacteristic
139C22×Q8—91no—
140C22×D4—31no—
141C22×D4—32no—
142#137—182no—
143#137—182no—
144C2×Q16—92no—
145C2×SD16—94no—
146C2×SD16—92no—
147C2×SD16—184no—
148C2×M16—91no—
149C22×C8—91no—
150C4⋊Q8—181no—
151#124—181no—
152#123—182no—
153C22⋊Q8—94no—
154#117—91no—
155#103—92no—
156#102—92no—
157#102—94no—
158C22⋊C8—94no—
159C23×C3—31no—
160C23×C3—32no—
161C23×C3—64no—
162C23×C3—242no—
163C3×Q8—22no—
164C3×Q8—31no—
165C3×Q8—86no—
166C3×Q8C2×#6711yescharacteristic
167C3×D4—32no—
168C3×D4—34no—
169C3×D4—64no—
170C3×D4—61no—
171C3×D4—62no—
172C3×D4—62no—
173C2×C3×C4—31no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—43no—
177C2×C3×C4—46no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4C2×S412yes—
181C2×C3×C4C2×S411yescharacteristic
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—126no—
185C2×Dic3—126no—
186C2×Dic3—122no—
187Dic6—62no—
188Dic6—62no—
189Dic6—122no—
190Dic6—246no—
191SL(2,𝔽3)C2×#6713yes—
192C3⋊C8—62no—
193C3⋊C8—62no—
194C2×C32—42no—
195C2×C32—41no—
196C24—61no—
197#41—22no—
198#41—32no—
199#41—62no—
200#41—62no—
201#41—62no—
202#41#63812yes—
203C2×Q8—92no—
204C2×Q8—182no—
205C2×Q8—182no—
206C2×Q8—181no—
207C2×Q8C3⋊S411yescharacteristic
208C2×Q8C22×C3⋊D311yescharacteristic
209C2×D4—31no—
210C2×D4—32no—
211C2×D4—34no—
212C2×D4—32no—
213C2×D4—31no—
214C2×D4—62no—
215C2×D4—64no—
216C2×D4—64no—
217C22×C4—31no—
218C22×C4—32no—
219C22×C4—181no—
220Q16—182no—
221Q16—182no—
222SD16—188no—
223SD16—184no—
224M16—182no—
225C2×C8—92no—
226C2×C8—92no—
227C2×C8—182no—
228C4⋊C4—92no—
229C4⋊C4—184no—
230C22⋊C4—181no—
231C22⋊C4—184no—
232C42—181no—
233C22×C3—32no—
234C22×C3—43no—
235C22×C3—62no—
236C22×C3—64no—
237C22×C3—62no—
238C22×C3—244no—
239C22×C3C22×S411yescharacteristic
240C3×C4—22no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—46no—
244C3×C4—86no—
245C3×C4GL(2,ℤ/4)12yes—
246Dic3—1212no—
247Dic3—122no—
248Dic3—122no—
249Dic3—246no—
250C32—41noSylow
251C23—31no—
252C23—32no—
253C23—64no—
254C23—242no—
255Q8—22no—
256Q8—31no—
257Q8—182no—
258Q8—182no—
259Q8—362no—
260Q8C2×#63811yescharacteristic
261D4—32no—
262D4—34no—
263D4—64no—
264D4—61no—
265D4—62no—
266D4—62no—
267C2×C4—31no—
268C2×C4—32no—
269C2×C4—32no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—182no—
273C2×C4—182no—
274C2×C4—362no—
275C2×C4C2×C3⋊S412yes—
276C2×C4C2×C3⋊S411yescharacteristic
277C8—182no—
278C8—182no—
279C2×C3—46no—
280C2×C3—43no—
281C2×C3—62no—
282C2×C3—122no—
283C2×C3#592412yes—
284C2×C3C2×GL(2,ℤ/4)11yescharacteristic
285C22—32no—
286C22—62no—
287C22—64no—
288C22—62no—
289C22—244no—
290C22C22×C3⋊S411yescharacteristic
291C4—22no—
292C4—31no—
293C4—31no—
294C4—362no—
295C4—362no—
296C4#6426912yes—
297C3—43no—
298C3#8552711yescharacteristic, complemented
299C2—62no—
300C2—122no—
301C2#1059658712yes—
302C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×H(5,𝔽2)D3
socleC22×C3C22×S4
prime radicalC2×C3×#994C2
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#64265C2semidirect
3. #10593659C2semidirect
4. #10593658C2semidirect
11. C2×C3⋊#226C22semidirect
13. C3⋊#859C22semidirect
298. C3#85527semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C2×#642652
#105936592
#105936582
#855273
#841053
#641534