Finite Group Databasebeta

← #11713131

Subgroups — #11713131

Order 768.

513 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 65 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×#5398C211yesmaximal, characteristic
2#85302C212yesmaximal
3A4×#107C211yesmaximal, characteristic
4C8×SL(2,ℤ/4)C211yesmaximal, characteristic
5#73052C212yesmaximal
6#23024—31nomaximal, Sylow
7C2×A4⋊Q8C2212yes—
8C2×C8×A4C2211yescharacteristic
9A4×C4⋊C4C2212yes—
10SL(2,ℤ/4)⋊C4—22no—
11C4×SL(2,ℤ/4)—22no—
12C4×SL(2,ℤ/4)C2211yescharacteristic
13C2×A4⋊C8C2211yescharacteristic
14#5398C414yescomplemented
15#4873—41nomaximal
16C2×#467—31no—
17C2×C4×Q16—31no—
18C22×#107D311yescharacteristic
19#2360—32no—
20#1969—31no—
21#1942—31no—
22#1928—32no—
23#1872—32no—
24#1863—32no—
25C4×#107—31no—
26C8×C22⋊C4—31no—
27C2×C4×A4—22no—
28C2×C4×A4C2311yescharacteristic
29C2×SL(2,ℤ/4)—22no—
30C2×SL(2,ℤ/4)D411yescharacteristic
31C2×SL(2,ℤ/4)D411yescharacteristic
32A4⋊Q8—22no—
33A4⋊Q8C2×C414yes—
34C2×Dic12—41no—
35#887—42no—
36C8×A4C2×C412yes—
37A4⋊C8—21no—
38C3×#107—41no—
39C8×Dic3—41no—
40#810—42no—
41C22×Q16—31no—
42C23×C8C2×D311yescharacteristic
43C2×C22⋊Q8—32no—
44C2×C4×Q8—32no—
45C22×C4⋊C4C2×D312yes—
46#467—34no—
47#467—34no—
48C4×Q16—62no—
49C4×Q16—61no—
50C4×Q16—61no—
51C2×#107—31no—
52C2×#107—31no—
53C2×#107—32no—
54C2×#107—31no—
55C2×#103—32no—
56C2×#103—32no—
57C2×#103—32no—
58C2×C22⋊C8—31no—
59C2×C4×C8—31no—
60C2×C4×C8—31no—
61#391—32no—
62#389—32no—
63#389—62no—
64#387—62no—
65#385—62no—
66C4×C4⋊C4—32no—
67C4×C22⋊C4—31no—
68C4×C22⋊C4—62no—
69#340—32no—
70#340—32no—
71#336—31no—
72#336—31no—
73C22×A4C2×D411yescharacteristic
74C2×Dic6—42no—
75C4×A4—42no—
76C4×A4C22×C411yescharacteristic
77C4×A4#4111yescharacteristic
78SL(2,ℤ/4)—24no—
79SL(2,ℤ/4)—21no—
80SL(2,ℤ/4)Q1612yescomplemented
81C2×C3×C8—41no—
82C3×C4⋊C4—42no—
83Dic3⋊C4—82no—
84C4×Dic3—41no—
85C4×Dic3—82no—
86C2×C3⋊C8—41no—
87Dic12—44no—
88C22×Q8—32no—
89C23×C4—22no—
90C23×C4C22×D311yescharacteristic
91C2×Q16—34no—
92C2×Q16—62no—
93C2×Q16—61no—
94C2×Q16—61no—
95C22×C8—31no—
96C22×C8—31no—
97C22×C8—32no—
98C22×C8—31no—
99C22×C8—31no—
100C22×C8C4×D312yes—
101C22⋊Q8—34no—
102C22⋊Q8—34no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C4×Q8—62no—
106C4×Q8—62no—
107C4×Q8—122no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4—62no—
114C2×C4⋊C4—62no—
115C2×C22⋊C4—31no—
116C2×C22⋊C4—31no—
117C2×C22⋊C4—62no—
118C2×C42—31no—
119C2×C42—31no—
120C2×C42—62no—
121C2×C42—62no—
122#107—34no—
123#107—31no—
124#107—32no—
125#107—61no—
126#107S411yescharacteristic
127#103—38no—
128#103—62no—
129#103—62no—
130#103—122no—
131C22⋊C8—61no—
132C22⋊C8—61no—
133C4×C8—62no—
134C4×C8—61no—
135C4×C8—61no—
136#95—31no—
137#95—31no—
138#95—62no—
139#95—62no—
140#95—62no—
141#95—62no—
142C2×A4C4×D411yescharacteristic
143C2×A4C2×Q1611yescharacteristic
144C2×A4#13511yescharacteristic
145C2×C3×C4—41no—
146C2×C3×C4—82no—
147C2×Dic3—41no—
148C2×Dic3—41no—
149C2×Dic3—82no—
150Dic6—44no—
151Dic6—82no—
152C3×C8—42no—
153C3⋊C8—81no—
154C24D3×D411yescharacteristic
155C2×Q8—34no—
156C2×Q8—62no—
157C2×Q8—62no—
158C2×Q8—62no—
159C2×Q8—62no—
160C2×Q8—62no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—42no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4—62no—
174C22×C4—122no—
175C22×C4C2×C4×D311yescharacteristic
176C22×C4#23911yescharacteristic
177Q16—64no—
178Q16—64no—
179C2×C8—31no—
180C2×C8—32no—
181C2×C8—32no—
182C2×C8—32no—
183C2×C8—61no—
184C2×C8—61no—
185C2×C8—61no—
186C2×C8—61no—
187C2×C8—61no—
188C2×C8—61no—
189C2×C8—61no—
190C2×C8C2×S411yescharacteristic
191C4⋊C4—34no—
192C4⋊C4—32no—
193C4⋊C4—62no—
194C4⋊C4—62no—
195C4⋊C4—62no—
196C4⋊C4—64no—
197C4⋊C4—64no—
198C4⋊C4—122no—
199C4⋊C4—122no—
200C4⋊C4—242no—
201C4⋊C4C2×S412yes—
202C22⋊C4—32no—
203C22⋊C4—32no—
204C22⋊C4—61no—
205C22⋊C4—64no—
206C22⋊C4—64no—
207C22⋊C4—61no—
208C42—61no—
209C42—61no—
210C42—61no—
211C42—61no—
212C42—122no—
213C42—122no—
214C42—242no—
215C22×C3—41no—
216A4C4×Q1611yescharacteristic, complemented
217C3×C4—41no—
218C3×C4—41no—
219C3×C4—162no—
220Dic3—42no—
221Dic3—84no—
222Dic3—81no—
223C23—31no—
224C23—31no—
225C23—31no—
226C23—31no—
227C23#89311yescharacteristic
228C23#91211yescharacteristic
229C23D3×Q1611yescharacteristic
230Q8—64no—
231Q8—64no—
232Q8—122no—
233Q8—122no—
234C2×C4—22no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—32no—
243C2×C4—32no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—64no—
251C2×C4—64no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—62no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4C22×S411yescharacteristic
267C8—32no—
268C8—121no—
269C8—121no—
270C8C4×S412yes—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C22—31no—
275C22—31no—
276C22—31no—
277C22—31no—
278C22—31no—
279C22—31no—
280C22—31no—
281C22—31no—
282C22—31no—
283C22—61no—
284C22#487311yescharacteristic, complemented
285C22D4×S411yescharacteristic
286C4—31no—
287C4—31no—
288C4—42no—
289C4—62no—
290C4—62no—
291C4—121no—
292C4—124no—
293C4—124no—
294C4—121no—
295C4—122no—
296C4C2×C4×S411yescharacteristic
297C4#591911yescharacteristic
298C3—41noSylow
299C2—31no—
300C2—31no—
301C2—31no—
302C2—31no—
303C2#8530811yescharacteristic
304C2#8545811yescharacteristic
305C2Q16×S411yescharacteristic
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#107D3
socleC24D3×D4
prime radicalC22×A4C2×D4
cosocleC2×C4×A4C23
hypercenter#107S4
hypocenterA4C4×Q16
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
14. #5398C4semidirect
80. SL(2,ℤ/4)Q16semidirect
216. A4C4×Q16semidirect
284. C22#4873semidirect

Maximal subgroups

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

subgroupindex
C2×#53982
#853022
A4×#1072
C8×SL(2,ℤ/4)2
#730522
#230243
#48734