Finite Group Databasebeta

← #11713668

Subgroups — #11713668

Order 768.

449 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 57 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
1A4×#125C211yesmaximal, characteristic
2#85283C211yesmaximal, characteristic
3#73051C212yesmaximal
4#73050C212yesmaximal
5#73044C211yesmaximal, characteristic
6#25819—31nomaximal, Sylow
7C2×A4⋊Q8—21no—
8A4×C4⋊C4—21no—
9A4×C4⋊C4—21no—
10A4×C4⋊C4C2211yescharacteristic
11A4×C4⋊C4C2211yescharacteristic
12C42×A4C2211yescharacteristic
13C4⋊SL(2,ℤ/4)—21no—
14C4⋊SL(2,ℤ/4)C2212yes—
15C2×A4⋊C8C2212yes—
16#5064—41nomaximal
17C22×#125D311yescharacteristic
18C2×#501—31no—
19C2×#501—31no—
20#2857—31no—
21#2059—32no—
22#2059—32no—
23#1992—31no—
24#1992—31no—
25#1853—32no—
26#1852—32no—
27#1749—31no—
28C2×C4×A4—21no—
29C2×C4×A4—21no—
30C2×C4×A4D411yescharacteristic
31C2×C4×A4D411yescharacteristic
32C2×C4×A4C2311yescharacteristic
33C2×SL(2,ℤ/4)—22no—
34A4⋊Q8—42no—
35C3×#125—41no—
36#869—41no—
37A4⋊C8Q814yescomplemented
38#808—42no—
39#807—42no—
40#803—41no—
41C2×C4⋊Q8—31no—
42C2×#125—31no—
43C2×#125—31no—
44C2×#125—31no—
45C2×#125—31no—
46C2×#125—61no—
47C2×#125—61no—
48C2×C22⋊Q8—61no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×D311yescharacteristic
52C22×C4⋊C4C2×D311yescharacteristic
53C22×C42C2×D311yescharacteristic
54#501—62no—
55#501—62no—
56#501—61no—
57#501—61no—
58#501—61no—
59#501—61no—
60C2×#107—32no—
61C2×#107—32no—
62C2×#106—32no—
63C2×#106—32no—
64C2×C22⋊C8—32no—
65C2×#97—31no—
66C2×#97—31no—
67#391—61no—
68#389—32no—
69#389—61no—
70#384—31no—
71#380—32no—
72#380—61no—
73#340—32no—
74#340—32no—
75#340—32no—
76#340—32no—
77#336—32no—
78C22×A4C2×D411yescharacteristic
79C2×Dic6—81no—
80C4×A4—21no—
81C4×A4—21no—
82C4×A4—41no—
83C4×A4—41no—
84C4×A4C2×Q812yes—
85SL(2,ℤ/4)—42no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—81no—
89C3×C4⋊C4—81no—
90C3×C42—41no—
91C4⋊Dic3—42no—
92C4⋊Dic3—81no—
93C2×C3⋊C8—42no—
94C22×Q8—61no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4#6711yescharacteristic
98C23×C4#6711yescharacteristic
99C23×C4C22×D311yescharacteristic
100C22×C8—32no—
101C22×C8—32no—
102C4⋊Q8—61no—
103C4⋊Q8—61no—
104C4⋊Q8—61no—
105C4⋊Q8—61no—
106#125—32no—
107#125—32no—
108#125—31no—
109#125—31no—
110#125—31no—
111#125—61no—
112#125—61no—
113#125—61no—
114#125—62no—
115#125—62no—
116#125S411yescharacteristic
117C22⋊Q8—122no—
118C22⋊Q8—122no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—32no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—62no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C22⋊C4—62no—
140C2×C42—31no—
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—31no—
144#107—64no—
145#107—62no—
146#107—62no—
147#106—64no—
148#106—62no—
149#106—62no—
150C22⋊C8—34no—
151C22⋊C8—34no—
152#97—61no—
153#97—61no—
154#97—61no—
155#97—61no—
156#95—62no—
157#95—62no—
158C2×A4C4⋊Q811yescharacteristic
159C2×A4AGL(1,ℤ/8)11yescharacteristic
160C2×A4#13711yescharacteristic
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×Dic3—82no—
167Dic6—162no—
168C3⋊C8—44no—
169C24C2×#6711yescharacteristic
170C2×Q8—121no—
171C2×Q8—122no—
172C2×Q8—122no—
173C2×Q8—121no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—61no—
186C22×C4—61no—
187C22×C4—61no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—61no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—121no—
194C22×C4—121no—
195C22×C4D3×Q812yes—
196C2×C8—34no—
197C2×C8—34no—
198C2×C8—62no—
199C2×C8—62no—
200C2×C8—62no—
201C2×C8—62no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—32no—
205C4⋊C4—32no—
206C4⋊C4—31no—
207C4⋊C4—31no—
208C4⋊C4—61no—
209C4⋊C4—61no—
210C4⋊C4—61no—
211C4⋊C4—61no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C4⋊C4—64no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—62no—
221C4⋊C4—121no—
222C4⋊C4—122no—
223C4⋊C4—122no—
224C4⋊C4—124no—
225C4⋊C4—121no—
226C4⋊C4C2×S411yescharacteristic
227C4⋊C4C2×S411yescharacteristic
228C22⋊C4—122no—
229C22⋊C4—122no—
230C42—31no—
231C42—31no—
232C42—31no—
233C42—61no—
234C42C2×S411yescharacteristic
235C22×C3—41no—
236A4#50111yescharacteristic, complemented
237C3×C4—42no—
238C3×C4—81no—
239C3×C4—81no—
240C3×C4—161no—
241C3×C4—161no—
242Dic3—162no—
243C23—31no—
244C23—31no—
245C23—31no—
246C23—31no—
247C23#94411yescharacteristic
248C23#94911yescharacteristic
249C23#95111yescharacteristic
250Q8—242no—
251Q8—242no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—31no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4GL(2,ℤ/4)11yescharacteristic
286C2×C4GL(2,ℤ/4)11yescharacteristic
287C2×C4C22×S411yescharacteristic
288C8—64no—
289C8—64no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C22—31no—
294C22—31no—
295C22—31no—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—61no—
303C22#506411yescharacteristic, complemented
304C22C2×GL(2,ℤ/4)11yescharacteristic
305C4—21no—
306C4—21no—
307C4—32no—
308C4—41no—
309C4—41no—
310C4—61no—
311C4—61no—
312C4—121no—
313C4—121no—
314C4—242no—
315C4—242no—
316C4Q8×S412yes—
317C3—41noSylow
318C2—31no—
319C2—31no—
320C2—31no—
321C2—31no—
322C2#8550211yescharacteristic
323C2#8551211yescharacteristic
324C2#8551311yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#125D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenter#125S4
hypocenterA4#501
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
37. A4⋊C8Q8semidirect
236. A4#501semidirect
303. C22#5064semidirect

Maximal subgroups

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

subgroupindex
A4×#1252
#852832
#730512
#730502
#730442
#258193
#50644