Finite Group Databasebeta

← #11921103

Subgroups — #11921103

Order 1152.

502 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 91 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×Q8×A4C211yesmaximal, characteristic
2C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
3C2×C3⋊(A4⋊C8)C211yesmaximal, characteristic
4#73147—33nomaximal
5#72132—31nomaximal
6C3×Q8×A4—21no—
7C3×Q8×A4C412yescomplemented
8C2×C3×C4×A4—21no—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3⋊SL(2,ℤ/4)—21no—
11C3⋊(A4⋊C8)—21no—
12#63662—41nomaximal
13C23×C3×Q8D311yescharacteristic
14C2×Q8×A4D313yes—
15C4⋊SL(2,ℤ/4)—33no—
16C2×A4⋊C8—33no—
17C2×#835—31no—
18C2×#835—31no—
19C2×#835—31no—
20C2×#835—31no—
21#5212—31no—
22C2×#830—31no—
23C22×C3×A4D411yescharacteristic
24C2×C32×Q8—41no—
25C3×C4×A4—22no—
26C3×C4×A4C2×C411yescharacteristic
27C3×C4×A4D411yescharacteristic
28C3⋊SL(2,ℤ/4)—41no—
29C4⋊(C3⋊Dic3)—41no—
30C2×C3⋊(C3⋊C8)—41no—
31#1787—91noSylow
32C22×C3×Q8—21no—
33C22×C3×Q8—31no—
34C22×C3×Q8—31no—
35C22×C3×Q8—32no—
36C22×C3×Q8—61no—
37C22×C3×Q8—61no—
38C22×C3×Q8Dic312yescomplemented
39C23×C3×C4—21no—
40C23×C3×C4C2×D311yescharacteristic
41Q8×A4—23no—
42Q8×A4Dic316yescomplemented
43C2×C4×A4—23no—
44C2×C4×A4C2×D313yes—
45C2×SL(2,ℤ/4)—63no—
46C2×#217—61no—
47C2×C4⋊Dic3—31no—
48C2×C4⋊Dic3—31no—
49C22×C3⋊C8—31no—
50C22×C3⋊C8—31no—
51A4⋊C8—63no—
52#835—62no—
53#835—62no—
54#835—62no—
55#835—61no—
56#835—61no—
57#835—123no—
58C3⋊#95—61no—
59#830—61no—
60#830—61no—
61C2×C3×A4C22⋊C411yescharacteristic
62C2×C3×A4SD1611yescharacteristic
63C2×C3×A4Q1611yescharacteristic
64C32×Q8—42no—
65C32×Q8—81no—
66C2×C32×C4—41no—
67C2×C32×C4—81no—
68C2×C3⋊Dic3—81no—
69C3⋊(C3⋊C8)—81no—
70C23×Q8C3⋊D311yescharacteristic
71C2×#103—91no—
72C2×#103—91no—
73C2×#103—91no—
74C2×#103—91no—
75C2×C22⋊C8—91no—
76#380—91no—
77C24×C3#6711yescharacteristic
78C22×A4#6713yes—
79C2×C3×Q8—32no—
80C2×C3×Q8—32no—
81C2×C3×Q8—32no—
82C2×C3×Q8—32no—
83C2×C3×Q8—32no—
84C2×C3×Q8—31no—
85C2×C3×Q8—43no—
86C2×C3×Q8—61no—
87C2×C3×Q8—61no—
88C2×C3×Q8—61no—
89C2×C3×Q8—62no—
90C2×C3×Q8—61no—
91C2×C3×Q8—61no—
92C2×C3×Q8—62no—
93C2×C3×Q8—64no—
94C2×C3×Q8S411yescharacteristic
95C22×C3×C4—22no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—61no—
101C22×C3×C4—62no—
102C22×C3×C4—61no—
103C22×C3×C4C2×Dic311yescharacteristic
104C22×C3×C4#6711yescharacteristic
105C22×Dic3—61no—
106C22×Dic3—61no—
107C4×A4—26no—
108C4×A4C2×Dic313yes—
109C4×A4#6713yes—
110SL(2,ℤ/4)—123no—
111#217—121no—
112#217—121no—
113C4⋊Dic3—61no—
114C4⋊Dic3—61no—
115C4⋊Dic3—61no—
116C4⋊Dic3—61no—
117C4⋊Dic3—123no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—61no—
120C2×C3⋊C8—61no—
121C2×C3⋊C8—61no—
122C2×C3⋊C8—61no—
123C2×C3⋊C8—61no—
124C2×C3⋊C8—123no—
125C22×C32—41no—
126C3×A4#10311yescharacteristic, complemented
127C32×C4—41no—
128C32×C4—41no—
129C32×C4—82no—
130C3⋊Dic3—161no—
131C22×Q8—21no—
132C22×Q8—31no—
133C22×Q8—31no—
134C22×Q8—32no—
135C22×Q8—61no—
136C22×Q8—61no—
137C22×Q8C3⋊Dic312yescomplemented
138C23×C4—21no—
139C23×C4C2×C3⋊D311yescharacteristic
140C22×C8—91no—
141C22×C8—91no—
142C2×C4⋊C4—91no—
143C2×C4⋊C4—91no—
144C2×C22⋊C4—181no—
145#103—182no—
146#103—182no—
147#103—182no—
148#103—181no—
149#103—181no—
150C22⋊C8—181no—
151C22⋊C8—181no—
152#95—181no—
153C23×C3—31no—
154C23×C3—31no—
155C23×C3—31no—
156C23×C3—31no—
157C23×C3#21511yescharacteristic
158C23×C3#21611yescharacteristic
159C23×C3#21711yescharacteristic
160C2×A4#21513yes—
161C2×A4#21613yes—
162C2×A4#21713yes—
163C3×Q8—21no—
164C3×Q8—34no—
165C3×Q8—32no—
166C3×Q8—46no—
167C3×Q8—62no—
168C3×Q8—62no—
169C3×Q8—62no—
170C3×Q8—61no—
171C3×Q8—83no—
172C3×Q8SL(2,ℤ/4)12yescomplemented
173C2×C3×C4—21no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—43no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—61no—
185C2×C3×C4—62no—
186C2×C3×C4—61no—
187C2×C3×C4—62no—
188C2×C3×C4—83no—
189C2×C3×C4C2×S411yescharacteristic
190C2×Dic3—121no—
191C2×Dic3—121no—
192C2×Dic3—121no—
193C2×Dic3—121no—
194C2×Dic3—121no—
195C2×Dic3—121no—
196C2×Dic3—243no—
197C3⋊C8—121no—
198C3⋊C8—121no—
199C3⋊C8—243no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C24#63811yescharacteristic
204C2×Q8—32no—
205C2×Q8—32no—
206C2×Q8—32no—
207C2×Q8—32no—
208C2×Q8—32no—
209C2×Q8—31no—
210C2×Q8—61no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8—62no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—62no—
217C2×Q8—64no—
218C2×Q8C3⋊S411yescharacteristic
219C22×C4—22no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—61no—
225C22×C4—62no—
226C22×C4—61no—
227C22×C4—181no—
228C22×C4—181no—
229C22×C4C2×C3⋊Dic311yescharacteristic
230C22×C4#63811yescharacteristic
231C2×C8—181no—
232C2×C8—181no—
233C2×C8—181no—
234C2×C8—181no—
235C2×C8—181no—
236C2×C8—181no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4—181no—
241C22⋊C4—361no—
242C22⋊C4—361no—
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—31no—
252C22×C3—43no—
253C22×C3—61no—
254C22×C3#83511yescharacteristic, complemented
255C22×C3GL(2,ℤ/4)11yescharacteristic
256A4#83513yescomplemented
257C3×C4—22no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—43no—
261C3×C4—43no—
262C3×C4—62no—
263C3×C4—86no—
264C3×C4C2×SL(2,ℤ/4)11yescharacteristic
265C3×C4GL(2,ℤ/4)11yescharacteristic
266Dic3—241no—
267Dic3—241no—
268Dic3—483no—
269C32—41noSylow
270C23—31no—
271C23—31no—
272C23—31no—
273C23—31no—
274C23#375511yescharacteristic
275C23#375611yescharacteristic
276C23#375711yescharacteristic
277Q8—21no—
278Q8—34no—
279Q8—32no—
280Q8—62no—
281Q8—62no—
282Q8—62no—
283Q8—61no—
284Q8C3⋊SL(2,ℤ/4)12yescomplemented
285C2×C4—21no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—61no—
296C2×C4—62no—
297C2×C4—61no—
298C2×C4—62no—
299C2×C4—361no—
300C2×C4—361no—
301C2×C4—361no—
302C2×C4—361no—
303C2×C4—361no—
304C2×C4—361no—
305C2×C4C2×C3⋊S411yescharacteristic
306C8—361no—
307C8—361no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—43no—
313C2×C3—43no—
314C2×C3—43no—
315C2×C3#541611yescharacteristic
316C2×C3#541711yescharacteristic
317C2×C3#543211yescharacteristic
318C22—31no—
319C22—31no—
320C22—31no—
321C22—31no—
322C22—31no—
323C22—31no—
324C22—31no—
325C22—31no—
326C22—31no—
327C22—61no—
328C22#6366211yescharacteristic, complemented
329C22#6426911yescharacteristic
330C4—22no—
331C4—31no—
332C4—31no—
333C4—62no—
334C4—721no—
335C4—721no—
336C4C2×C3⋊SL(2,ℤ/4)11yescharacteristic
337C4#6426911yescharacteristic
338C3—43no—
339C3#7314711yescharacteristic, complemented
340C2—31no—
341C2—31no—
342C2—31no—
343C2—31no—
344C2#1059365011yescharacteristic
345C2#1059365111yescharacteristic
346C2#1059366611yescharacteristic
subgroupG/H
centerC22#64269
commutatorC3×C4×A4C2×C4
FrattiniC2×C4C2×C3⋊S4
FittingC23×C3×Q8D3
socleC24×C3#67
prime radicalC22×C3×A4D4
cosocleC2×C3×C4×A4C22
hypercenterC2×Q8C3⋊S4
hypocenterC3×A4#103
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
7. C3×Q8×A4C4semidirect
38. C22×C3×Q8Dic3semidirect
42. Q8×A4Dic3semidirect
126. C3×A4#103semidirect
137. C22×Q8C3⋊Dic3semidirect
172. C3×Q8SL(2,ℤ/4)semidirect
254. C22×C3#835semidirect
256. A4#835semidirect
284. Q8C3⋊SL(2,ℤ/4)semidirect
328. C22#63662semidirect
339. C3#73147semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C4⋊(C3⋊SL(2,ℤ/4))2
C2×C3⋊(A4⋊C8)2
#731473
#721323
#636624