Finite Group Databasebeta

← #85293

Subgroups — #85293

Order 384.

363 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 46 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×SL(2,ℤ/4)C211yesmaximal, characteristic
2C2×A4⋊Q8C211yesmaximal, characteristic, complemented
3A4×C22⋊C4C211yesmaximal, characteristic
4#5432C211yesmaximal, characteristic
5C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8#2437—31nomaximal, Sylow
9C23×A4C2211yescharacteristic
10C2×C4×A4C2211yescharacteristic
11C2×C4×A4C2211yescharacteristic
12C2×SL(2,ℤ/4)—22no—
13C2×SL(2,ℤ/4)C2211yescharacteristic
14C2×SL(2,ℤ/4)C2211yescharacteristic
15C2×SL(2,ℤ/4)C2211yescharacteristic
16C2×SL(2,ℤ/4)C2211yescharacteristic
17A4⋊Q8—22no—
18#878—41nomaximal
19C2×C22⋊Q8—31no—
20C2×C22⋊Q8—31no—
21C22×C22⋊C4—31no—
22C22×C22⋊C4D311yescharacteristic
23#399—31no—
24#396—31no—
25#396—31no—
26#393—31no—
27#385—31no—
28#385—31no—
29#385—31no—
30#385—31no—
31#380—31no—
32#380—31no—
33#379—31no—
34C22×A4—21no—
35C22×A4—21no—
36C22×A4Q812yescomplemented
37C22×A4C2311yescharacteristic
38C22×Dic3—41no—
39C2×Dic6—41no—
40C4×A4—21no—
41C4×A4—21no—
42SL(2,ℤ/4)—21no—
43SL(2,ℤ/4)—21no—
44SL(2,ℤ/4)—21no—
45SL(2,ℤ/4)D412yes—
46C3×C22⋊C4—41no—
47#217—41no—
48C4⋊Dic3—41no—
49Dic3⋊C4—41no—
50Dic3⋊C4—41no—
51C25C2×D311yescharacteristic
52C22×Q8—31no—
53C23×C4—31no—
54C23×C4—31no—
55C23×C4C2×D311yescharacteristic
56C23×C4C2×D311yescharacteristic
57C22⋊Q8—62no—
58C22⋊Q8—62no—
59C22⋊Q8—61no—
60C22⋊Q8—61no—
61C22⋊Q8—61no—
62C22⋊Q8—61no—
63C2×C4⋊C4—31no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—31no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—31no—
72C2×C22⋊C4—31no—
73C2×C22⋊C4—31no—
74C2×C22⋊C4—31no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—31no—
78C2×C22⋊C4—31no—
79C2×C22⋊C4—31no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—31no—
83C2×C22⋊C4—31no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—62no—
87C2×C22⋊C4—62no—
88#95—31no—
89#95—31no—
90#95—31no—
91#95—31no—
92#95—31no—
93#95—31no—
94C23×C3—41no—
95C2×A4—22no—
96C2×A4C2×D411yescharacteristic
97C2×A4C2×Q811yescharacteristic
98C2×A4#4111yescharacteristic
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C2×Dic3—82no—
106Dic6—82no—
107C24—21no—
108C24—21no—
109C24—31no—
110C24—31no—
111C24—32no—
112C24—61no—
113C24—61no—
114C24Dic612yescomplemented
115C24C22×D311yescharacteristic
116C2×Q8—62no—
117C2×Q8—61no—
118C2×Q8—61no—
119C2×Q8—121no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—62no—
135C22×C4—62no—
136C22×C4—61no—
137C22×C4—61no—
138C22×C4—61no—
139C22×C4—62no—
140C22×C4—61no—
141C22×C4—121no—
142C22×C4—121no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—61no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C4⋊C4—121no—
153C4⋊C4—121no—
154C4⋊C4—121no—
155C22⋊C4—32no—
156C22⋊C4—32no—
157C22⋊C4—31no—
158C22⋊C4—31no—
159C22⋊C4—31no—
160C22⋊C4—61no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—61no—
164C22⋊C4—62no—
165C22⋊C4—61no—
166C22⋊C4—61no—
167C22⋊C4—61no—
168C22⋊C4—61no—
169C22⋊C4—61no—
170C22⋊C4—61no—
171C22⋊C4—61no—
172C22⋊C4—121no—
173C22⋊C4—121no—
174C22⋊C4S411yescharacteristic
175C22×C3—42no—
176C22×C3—41no—
177C22×C3—81no—
178C22×C3—81no—
179A4C22⋊Q811yescharacteristic, complemented
180C3×C4—81no—
181C3×C4—81no—
182Dic3—42no—
183Dic3—81no—
184Dic3—81no—
185Dic3—81no—
186C23—22no—
187C23—31no—
188C23—31no—
189C23—31no—
190C23—31no—
191C23—32no—
192C23—32no—
193C23—32no—
194C23—31no—
195C23—61no—
196C23—61no—
197C23—61no—
198C23—61no—
199C23—61no—
200C23—62no—
201C23—61no—
202C23—61no—
203C23—61no—
204C23—62no—
205C23—62no—
206C23—61no—
207C23—122no—
208C23C2×Dic611yescharacteristic
209C23D3×D411yescharacteristic
210C23#23711yescharacteristic
211C23C2×S411yescharacteristic
212Q8—122no—
213Q8—122no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—122no—
249C2×C4—124no—
250C2×C4—122no—
251C2×C4C2×S411yescharacteristic
252C2×C4C2×S411yescharacteristic
253C2×C3—41no—
254C2×C3—41no—
255C2×C3—41no—
256C2×C3—82no—
257C22—21no—
258C22—21no—
259C22—31no—
260C22—31no—
261C22—31no—
262C22—31no—
263C22—31no—
264C22—31no—
265C22—31no—
266C22—31no—
267C22—31no—
268C22—32no—
269C22—62no—
270C22—62no—
271C22—62no—
272C22—62no—
273C22—61no—
274C22—62no—
275C22—62no—
276C22—61no—
277C22—61no—
278C22—121no—
279C22—121no—
280C22#87811yescharacteristic, complemented
281C22A4⋊Q812yescomplemented
282C22C22×S411yescharacteristic
283C4—21no—
284C4—21no—
285C4—62no—
286C4—61no—
287C4—62no—
288C4—61no—
289C4—121no—
290C4—121no—
291C4—121no—
292C4—121no—
293C4—121no—
294C4—121no—
295C3—41noSylow
296C2—22no—
297C2—31no—
298C2—31no—
299C2—31no—
300C2—31no—
301C2—62no—
302C2C2×A4⋊Q811yescharacteristic
303C2D4×S411yescharacteristic
304C2#591411yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC22×C22⋊C4D3
socleC24C22×D3
prime radicalC23×A4C22
hypercenterC22⋊C4S4
hypocenterA4C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×A4⋊Q8C2semidirect
5. C4⋊SL(2,ℤ/4)C2semidirect
6. SL(2,ℤ/4)⋊C4C2semidirect
7. SL(2,ℤ/4)⋊C4C2semidirect
36. C22×A4Q8semidirect
114. C24Dic6semidirect
179. A4C22⋊Q8semidirect
280. C22#878semidirect
281. C22A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×SL(2,ℤ/4)2
C2×A4⋊Q82
A4×C22⋊C42
#54322
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
#24373
#8784