Finite Group Databasebeta

← #11921017

Subgroups — #11921017

Order 1152.

482 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C22×C3⋊#226C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×C3⋊#859C211yesmaximal, characteristic
4#73061—33nomaximal
5#71632—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C3⋊#226—21no—
8C2×C3⋊#226—21no—
9C2×C3⋊#226C412yescomplemented
10C3×C4×SL(2,𝔽3)—21no—
11C3⋊#859—21no—
12#63658—41nomaximal
13C22×#226—33no—
14C2×C3×C4×Q8D311yescharacteristic
15C22×#216—31no—
16C2×C4×SL(2,𝔽3)D313yes—
17C2×#859—33no—
18C2×#835—31no—
19C2×#810—31no—
20#4929—31no—
21#4529—31no—
22#4524—31no—
23C22×C32×C4—41no—
24C22×C3⋊Dic3—41no—
25C22×C3⋊Dic3—41no—
26C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C3⋊#226—22no—
30#1878—91noSylow
31C22×C3×Q8C2×D311yescharacteristic
32C22×Dic6—31no—
33C22×SL(2,𝔽3)C2×D313yes—
34C2×#226—36no—
35C2×#226—63no—
36C2×#226—63no—
37C3×C4×Q8—21no—
38C3×C4×Q8—61no—
39C2×C3×C4⋊C4—31no—
40C2×C3×C42—31no—
41C2×#216—32no—
42C2×#216—32no—
43C2×#216—61no—
44C2×#216—61no—
45C2×#216—61no—
46C2×#216—61no—
47C2×C4⋊Dic3—31no—
48C22×C3⋊C8—31no—
49C22×C3⋊C8—31no—
50C4×SL(2,𝔽3)—23no—
51#859—63no—
52#835—61no—
53#835—61no—
54C3⋊#95—61no—
55C3⋊#95—61no—
56C3⋊#95—123no—
57#810—62no—
58C23×C32—41no—
59C2×C32×C4—42no—
60C2×C32×C4—81no—
61C2×C32×C4—81no—
62C2×C3⋊Dic3—42no—
63C2×C3⋊Dic3—42no—
64C2×C3⋊Dic3—81no—
65C2×C3⋊Dic3—81no—
66C2×C3⋊Dic3—81no—
67C2×C3⋊Dic3—81no—
68C3×SL(2,𝔽3)C22⋊C411yescharacteristic
69C22×Q16—91no—
70C2×C4×Q8C3⋊D311yescharacteristic
71C2×#103—91no—
72C2×#103—91no—
73#391—91no—
74#340—91no—
75#336—91no—
76C2×C3×Q8—31no—
77C2×C3×Q8—31no—
78C2×C3×Q8—31no—
79C2×C3×Q8C4×D311yescharacteristic
80C2×C3×Q8D1211yescharacteristic
81C2×C3×Q8#6711yescharacteristic
82C22×C3×C4—31no—
83C22×C3×C4—31no—
84C22×C3×C4—43no—
85C22×C3×C4S411yescharacteristic
86C22×Dic3—61no—
87C22×Dic3—61no—
88C22×Dic3—123no—
89C22×Dic3—123no—
90C2×Dic6—32no—
91C2×Dic6—61no—
92C2×Dic6—61no—
93C2×Dic6—61no—
94C2×Dic6—61no—
95C2×Dic6—61no—
96C2×SL(2,𝔽3)C4×D313yes—
97C2×SL(2,𝔽3)D1213yes—
98C2×SL(2,𝔽3)#6713yes—
99#226—66no—
100C3×C4⋊C4—32no—
101C3×C4⋊C4—61no—
102C3×C42—61no—
103C3×C42—61no—
104#216—62no—
105#216—62no—
106#216—62no—
107#216—62no—
108C4⋊Dic3—61no—
109C4⋊Dic3—61no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—61no—
112C2×C3⋊C8—61no—
113C2×C3⋊C8—61no—
114C2×C3⋊C8—61no—
115C2×C3⋊C8—61no—
116C22×C32—41no—
117C22×C32—41no—
118C22×C32—41no—
119C22×C32—41no—
120C22×C32—41no—
121C22×C32—41no—
122C22×C32—41no—
123C32×C4—82no—
124C3⋊Dic3—82no—
125C3⋊Dic3—82no—
126C22×Q8—91no—
127C22×Q8C2×C3⋊D311yescharacteristic
128C2×Q16—92no—
129C2×Q16—92no—
130C2×Q16—181no—
131C2×Q16—181no—
132C2×Q16—181no—
133C2×Q16—181no—
134C22×C8—91no—
135C22×C8—91no—
136C4×Q8—21no—
137C4×Q8—61no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—91no—
140C2×C42—31no—
141#103—182no—
142#103—181no—
143#103—181no—
144#95—181no—
145#95—181no—
146C23×C3—43no—
147C23×C3C2×S411yescharacteristic
148C3×Q8—31no—
149C3×Q8—31no—
150C3×Q8—31no—
151C3×Q8—61no—
152C3×Q8(C2×D3)⋊C411yescharacteristic
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—46no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—83no—
166C2×C3×C4—83no—
167C2×C3×C4#22612yescomplemented
168C2×Dic3—62no—
169C2×Dic3—126no—
170C2×Dic3—121no—
171C2×Dic3—121no—
172C2×Dic3—126no—
173C2×Dic3—121no—
174C2×Dic3—121no—
175C2×Dic3—121no—
176C2×Dic3—243no—
177C2×Dic3—243no—
178C2×Dic3—243no—
179C2×Dic3—243no—
180Dic6—62no—
181Dic6—62no—
182Dic6—121no—
183Dic6—121no—
184SL(2,𝔽3)(C2×D3)⋊C413yes—
185C3⋊C8—62no—
186C3⋊C8—121no—
187C2×C32—41no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C2×Q8—31no—
195C2×Q8—31no—
196C2×Q8—31no—
197C2×Q8—92no—
198C2×Q8—181no—
199C2×Q8—181no—
200C2×Q8—181no—
201C2×Q8—181no—
202C2×Q8—181no—
203C2×Q8C4×C3⋊D311yescharacteristic
204C2×Q8C4⋊(C3⋊D3)11yescharacteristic
205C2×Q8#63811yescharacteristic
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—181no—
209C22×C4—181no—
210C22×C4C3⋊S411yescharacteristic
211Q16—182no—
212Q16—182no—
213Q16—182no—
214Q16—182no—
215C2×C8—92no—
216C2×C8—181no—
217C2×C8—181no—
218C2×C8—181no—
219C2×C8—181no—
220C2×C8—181no—
221C4⋊C4—32no—
222C4⋊C4—61no—
223C4⋊C4—181no—
224C4⋊C4—181no—
225C42—61no—
226C42—61no—
227C22×C3—43no—
228C22×C3—43no—
229C22×C3—43no—
230C22×C3—43no—
231C22×C3—43no—
232C22×C3—43no—
233C22×C3—43no—
234C22×C3C4×S411yescharacteristic
235C22×C3C4⋊S411yescharacteristic
236C22×C3C2×#22611yescharacteristic
237C22×C3#98311yescharacteristic
238C22×C3#98411yescharacteristic
239C22×C3#98511yescharacteristic
240C22×C3GL(2,ℤ/4)11yescharacteristic
241C3×C4—22no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—86no—
247C3×C4—121no—
248Dic3—122no—
249Dic3—246no—
250Dic3—246no—
251Dic3—241no—
252C32—41noSylow
253C23C2×C3⋊S411yescharacteristic
254Q8—31no—
255Q8—31no—
256Q8—31no—
257Q8—61no—
258Q8—182no—
259Q8—182no—
260Q8—361no—
261Q8—361no—
262Q8#375211yescharacteristic
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—182no—
275C2×C4—361no—
276C2×C4—361no—
277C2×C4—361no—
278C2×C4—361no—
279C2×C4—361no—
280C2×C4C3⋊#22612yescomplemented
281C8—182no—
282C8—361no—
283C2×C3—43no—
284C2×C3—43no—
285C2×C3—43no—
286C2×C3—43no—
287C2×C3—43no—
288C2×C3—43no—
289C2×C3—43no—
290C2×C3C4×#22611yescharacteristic
291C2×C3#538811yescharacteristic
292C2×C3C4⋊#22611yescharacteristic
293C2×C3#539511yescharacteristic
294C2×C3(C2×S4)⋊C411yescharacteristic
295C2×C3#541911yescharacteristic
296C2×C3#542511yescharacteristic
297C22C4×C3⋊S411yescharacteristic
298C22C4⋊(C3⋊S4)11yescharacteristic
299C22C2×C3⋊#22611yescharacteristic
300C22#6426411yescharacteristic
301C22#6426511yescharacteristic
302C22#6426611yescharacteristic
303C22#6426911yescharacteristic
304C4—22no—
305C4—31no—
306C4—31no—
307C4—31no—
308C4—31no—
309C4—121no—
310C4—362no—
311C4—721no—
312C3—43no—
313C3#7306111yescharacteristic, complemented
314C2C4×C3⋊#22611yescharacteristic
315C2#1059362211yescharacteristic
316C2C4⋊(C3⋊#226)11yescharacteristic
317C2#1059362911yescharacteristic
318C2#1059364711yescharacteristic
319C2#1059365311yescharacteristic
320C2#1059365911yescharacteristic
subgroupG/H
center, FrattiniC23C2×C3⋊S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radical, cosocleC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
9. C2×C3⋊#226C4semidirect
167. C2×C3×C4#226semidirect
280. C2×C4C3⋊#226semidirect
313. C3#73061semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#2262
C2×C3×C4×SL(2,𝔽3)2
C2×C3⋊#8592
#730613
#716323
#636584