Finite Group Databasebeta

← #11921905

Subgroups — #11921905

Order 1152.

672 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 83 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×#64259C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic
3C4×Dic3×A4C211yesmaximal, characteristic
4Dic3⋊SL(2,ℤ/4)C214yesmaximal
5#85283—31nomaximal
6#82811—31nomaximal
7C2×C3×C4×A4C2211yescharacteristic
8C2×Dic3×A4C2212yes—
9C2×C3⋊SL(2,ℤ/4)C2212yes—
10#64259—22no—
11C2×C3×SL(2,ℤ/4)C2212yes—
12C3×A4⋊Q8—22no—
13#63866—41nomaximal
14C2×A4⋊Q8—31no—
15C2×A4⋊Q8—61no—
16C2×A4⋊Q8D311yescharacteristic
17C2×C3×C22⋊Q8—31no—
18C2×#944—31no—
19C2×#924—31no—
20C22×C4×Dic3D311yescharacteristic
21C42×A4—31no—
22C4⋊SL(2,ℤ/4)—34no—
23#5229—31no—
24#4976—34no—
25#4975—31no—
26#4973—31no—
27#4942—34no—
28C22×C3×A4C2311yescharacteristic
29C2×#634—41no—
30C2×C3×Dic6—41no—
31C3×C4×A4D412yes—
32Dic3×A4Q814yescomplemented
33C3⋊SL(2,ℤ/4)—22no—
34C3×SL(2,ℤ/4)—22no—
35C3×C4×Dic3—41no—
36Dic3⋊Dic3—44no—
37#2857—91noSylow
38C22×C3×Q8—31no—
39C23×C3×C4C2×D311yescharacteristic
40C23×Dic3C2×D312yes—
41C22×Dic6—31no—
42C2×C4×A4—21no—
43C2×C4×A4—32no—
44C2×C4×A4C2×D311yescharacteristic
45C2×SL(2,ℤ/4)—32no—
46C2×SL(2,ℤ/4)—62no—
47C2×SL(2,ℤ/4)C2×D312yes—
48A4⋊Q8—22no—
49A4⋊Q8—64no—
50A4⋊Q8—62no—
51C3×C22⋊Q8—62no—
52C3×C22⋊Q8—62no—
53C2×C3×C4⋊C4—31no—
54C2×C3×C4⋊C4—32no—
55C2×C3×C22⋊C4—32no—
56C2×#217—32no—
57#944—61no—
58#944—62no—
59#944—61no—
60C2×C4⋊Dic3—31no—
61#924—62no—
62#924—62no—
63C2×Dic3⋊C4—34no—
64C2×Dic3⋊C4—34no—
65C2×Dic3⋊C4—32no—
66C2×C4×Dic3—31no—
67C2×C4×Dic3—31no—
68C2×C4×Dic3—32no—
69#869—121no—
70C3⋊#95—34no—
71C3⋊#95—34no—
72C2×C3×A4C2×D411yescharacteristic
73C2×C3×A4C2×Q812yes—
74C2×C32×C4—41no—
75C2×C3⋊Dic3—42no—
76#634—82no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—42no—
79C3×Dic6—82no—
80C2×C4⋊Q8—91no—
81C2×C22⋊Q8—31no—
82C2×C22⋊Q8—91no—
83C22×C42—31no—
84#391—91no—
85#391—91no—
86#389—94no—
87#384—91no—
88#380—94no—
89C24×C3C22×D311yescharacteristic
90C22×A4—21no—
91C22×A4C22×D311yescharacteristic
92C2×C3×Q8—61no—
93C2×C3×Q8—62no—
94C2×C3×Q8—62no—
95C2×C3×Q8—61no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—32no—
99C22×C3×C4—32no—
100C22×C3×C4—32no—
101C22×C3×C4D1212yes—
102C22×Dic3—32no—
103C22×Dic3—32no—
104C22×Dic3—34no—
105C22×Dic3—32no—
106C22×Dic3—32no—
107C22×Dic3Dic614yescomplemented
108C2×Dic6—41no—
109C2×Dic6—62no—
110C2×Dic6—61no—
111C2×Dic6—62no—
112C2×Dic6—61no—
113C2×Dic6—121no—
114C2×Dic6—241no—
115C4×A4—22no—
116C4×A4—34no—
117C4×A4#6712yes—
118SL(2,ℤ/4)—22no—
119SL(2,ℤ/4)—64no—
120SL(2,ℤ/4)—62no—
121C3×C4⋊C4—62no—
122C3×C4⋊C4—64no—
123C3×C22⋊C4—62no—
124C3×C22⋊C4—62no—
125C3×C42—121no—
126#217—62no—
127#217—62no—
128C4⋊Dic3—62no—
129C4⋊Dic3—124no—
130Dic3⋊C4—68no—
131Dic3⋊C4—64no—
132Dic3⋊C4—64no—
133Dic3⋊C4—64no—
134C4×Dic3—31no—
135C4×Dic3—32no—
136C4×Dic3—61no—
137C4×Dic3S411yescharacteristic
138C22×C32—41no—
139C3×A4C4⋊Q811yescharacteristic, complemented
140C32×C4—42no—
141C3⋊Dic3—82no—
142C3×Dic3—44no—
143C3×Dic3—82no—
144C22×Q8—31no—
145C22×Q8—91no—
146C23×C4—32no—
147C23×C4D3211yescharacteristic
148C4⋊Q8—181no—
149C4⋊Q8—182no—
150C4⋊Q8—181no—
151C22⋊Q8—62no—
152C22⋊Q8—62no—
153C22⋊Q8—182no—
154C22⋊Q8—182no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—32no—
157C2×C4⋊C4—91no—
158C2×C4⋊C4—92no—
159C2×C4⋊C4—94no—
160C2×C4⋊C4—94no—
161C2×C22⋊C4—32no—
162C2×C22⋊C4—92no—
163C2×C42—91no—
164C2×C42—92no—
165C2×C42—91no—
166#95—94no—
167#95—94no—
168C23×C3—31no—
169C23×C3—31no—
170C23×C3—32no—
171C23×C3C2×Dic612yes—
172C23×C3C2×D1211yescharacteristic
173C2×A4—22no—
174C2×A4—21no—
175C2×A4D3×Q812yes—
176C2×A4C2×#6711yescharacteristic
177C3×Q8—122no—
178C3×Q8—122no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—41no—
184C2×C3×C4—61no—
185C2×C3×C4—64no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—81no—
191C2×C3×C4—122no—
192C2×C3×C4C2×S411yescharacteristic
193C2×Dic3—32no—
194C2×Dic3—34no—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—42no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—64no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—122no—
205C2×Dic3—242no—
206C2×Dic3C2×S412yes—
207Dic6—82no—
208Dic6—122no—
209Dic6—122no—
210Dic6—242no—
211Dic6—244no—
212C2×C32—42no—
213C2×C32—41no—
214C24C2×D3211yescharacteristic
215C2×Q8—62no—
216C2×Q8—61no—
217C2×Q8—62no—
218C2×Q8—61no—
219C2×Q8—182no—
220C2×Q8—181no—
221C2×Q8—182no—
222C2×Q8—181no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—32no—
226C22×C4—32no—
227C22×C4—32no—
228C22×C4—34no—
229C22×C4—92no—
230C22×C4—92no—
231C22×C4—94no—
232C22×C4—92no—
233C22×C4—92no—
234C22×C4#62612yes—
235C4⋊C4—62no—
236C4⋊C4—64no—
237C4⋊C4—184no—
238C4⋊C4—184no—
239C4⋊C4—182no—
240C4⋊C4—184no—
241C4⋊C4—188no—
242C22⋊C4—62no—
243C22⋊C4—62no—
244C22⋊C4—182no—
245C22⋊C4—182no—
246C42—31no—
247C42—92no—
248C42—91no—
249C42—181no—
250C22×C3—32no—
251C22×C3—31no—
252C22×C3—32no—
253C22×C3—32no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—41no—
257C22×C3—61no—
258C22×C3—81no—
259C22×C3#86911yescharacteristic, complemented
260C22×C3C22×S411yescharacteristic
261A4—21no—
262A4#94411yescharacteristic, complemented
263C3×C4—32no—
264C3×C4—42no—
265C3×C4—82no—
266C3×C4—124no—
267C3×C4—122no—
268C3×C4—122no—
269C3×C4C4⋊S412yes—
270Dic3—34no—
271Dic3—82no—
272Dic3—122no—
273Dic3—122no—
274Dic3—242no—
275Dic3—244no—
276Dic3A4⋊Q814yescomplemented
277C32—41noSylow
278C23—31no—
279C23—31no—
280C23—32no—
281C23D3×Dic612yes—
282C23C2×#62611yescharacteristic
283Q8—122no—
284Q8—122no—
285Q8—362no—
286Q8—362no—
287C2×C4—32no—
288C2×C4—31no—
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—61no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—64no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—94no—
299C2×C4—94no—
300C2×C4—92no—
301C2×C4—94no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4—182no—
305C2×C4—184no—
306C2×C4—182no—
307C2×C4—182no—
308C2×C4D3×S411yescharacteristic
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—32no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—82no—
315C2×C3—81no—
316C2×C3C2×A4⋊Q812yes—
317C2×C3C2×C4⋊S411yescharacteristic
318C22—32no—
319C22—31no—
320C22—32no—
321C22—32no—
322C22—31no—
323C22—31no—
324C22—61no—
325C22#6386611yescharacteristic, complemented
326C22C2×D3×S411yescharacteristic
327C4—34no—
328C4—32no—
329C4—94no—
330C4—122no—
331C4—122no—
332C4—362no—
333C4—362no—
334C4#6420712yes—
335C3—41no—
336C3—81no—
337C3#8528311yescharacteristic, complemented
338C2—31no—
339C2—31no—
340C2—32no—
341C2D3×A4⋊Q812yes—
342C2C2×#6420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, prime radical, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
hypercenterC2×C4D3×S4
hypocenterC3×A4C4⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
32. Dic3×A4Q8semidirect
107. C22×Dic3Dic6semidirect
139. C3×A4C4⋊Q8semidirect
259. C22×C3#869semidirect
262. A4#944semidirect
276. Dic3A4⋊Q8semidirect
325. C22#63866semidirect
337. C3#85283semidirect

Maximal subgroups

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

subgroupindex
C2×#642592
C2×C3×A4⋊Q82
C4×Dic3×A42
Dic3⋊SL(2,ℤ/4)2
#852833
#828113
#638664