Finite Group Databasebeta

← #11713667

Subgroups — #11713667

Order 768.

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

normal subgroups only

This group has 467 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×C4⋊Q8C211yesmaximal, characteristic
2#85284C211yesmaximal, characteristic
3#73051C212yesmaximal
4#73050C212yesmaximal
5#73044C211yesmaximal, characteristic
6#25816—31nomaximal, Sylow
7C2×Q8×A4—21no—
8A4×C4⋊C4—21no—
9A4×C4⋊C4C2212yes—
10C42×A4C2211yescharacteristic
11C4⋊SL(2,ℤ/4)C2211yescharacteristic
12C4⋊SL(2,ℤ/4)C2211yescharacteristic
13SL(2,ℤ/4)⋊C4—21no—
14SL(2,ℤ/4)⋊C4—21no—
15C2×A4⋊C8C2212yes—
16#5081—41nomaximal
17C22×C4⋊Q8D311yescharacteristic
18C2×#501—31no—
19C2×#501—31no—
20#2851—31no—
21#2058—32no—
22#2057—32no—
23#1993—31no—
24#1993—31no—
25#1853—32no—
26#1852—32no—
27#1749—31no—
28Q8×A4—42no—
29C2×C4×A4—22no—
30C2×C4×A4D411yescharacteristic
31C2×C4×A4D411yescharacteristic
32C2×C4×A4C2311yescharacteristic
33C2×SL(2,ℤ/4)—21no—
34C2×SL(2,ℤ/4)—21no—
35C3×C4⋊Q8—41no—
36#870—41no—
37A4⋊C8Q814yescomplemented
38#808—42no—
39#807—42no—
40#803—41no—
41C23×Q8—21no—
42C2×C4⋊Q8—31no—
43C2×C4⋊Q8—31no—
44C2×C4⋊Q8—32no—
45C2×C4⋊Q8—61no—
46C2×C4⋊Q8—61no—
47C2×#125—31no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×D312yes—
50C22×C42C2×D311yescharacteristic
51#501—62no—
52#501—62no—
53#501—62no—
54#501—61no—
55#501—61no—
56C2×#107—32no—
57C2×#107—32no—
58C2×#106—32no—
59C2×#106—32no—
60C2×C22⋊C8—32no—
61C2×#97—31no—
62C2×#97—31no—
63#389—31no—
64#389—31no—
65#387—61no—
66#387—61no—
67#385—61no—
68#385—61no—
69#382—31no—
70#380—31no—
71#380—31no—
72#340—32no—
73#340—32no—
74#340—32no—
75#340—32no—
76#336—32no—
77C22×A4C2×D411yescharacteristic
78C2×C3×Q8—81no—
79C4×A4—21no—
80C4×A4—21no—
81C4×A4—42no—
82C4×A4C2×Q812yes—
83SL(2,ℤ/4)—41no—
84SL(2,ℤ/4)—41no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—81no—
87C3×C42—41no—
88C4⋊Dic3—41no—
89C4⋊Dic3—41no—
90Dic3⋊C4—81no—
91Dic3⋊C4—81no—
92C2×C3⋊C8—42no—
93C22×Q8—42no—
94C22×Q8—61no—
95C22×Q8—62no—
96C22×Q8—61no—
97C22×Q8—122no—
98C23×C4—22no—
99C23×C4#6711yescharacteristic
100C23×C4#6711yescharacteristic
101C23×C4C22×D311yescharacteristic
102C22×C8—32no—
103C22×C8—32no—
104C4⋊Q8—32no—
105C4⋊Q8—32no—
106C4⋊Q8—32no—
107C4⋊Q8—31no—
108C4⋊Q8—61no—
109C4⋊Q8—61no—
110C4⋊Q8—61no—
111C4⋊Q8—64no—
112C4⋊Q8S411yescharacteristic
113#125—61no—
114#125—61no—
115#125—61no—
116#125—61no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—61no—
131C2×C22⋊C4—61no—
132C2×C22⋊C4—61no—
133C2×C42—31no—
134C2×C42—31no—
135C2×C42—31no—
136C2×C42—31no—
137#107—64no—
138#107—62no—
139#107—62no—
140#106—64no—
141#106—62no—
142#106—62no—
143C22⋊C8—34no—
144C22⋊C8—34no—
145#97—61no—
146#97—61no—
147#97—61no—
148#97—61no—
149#95—61no—
150#95—61no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155C2×A4C4⋊Q811yescharacteristic
156C2×A4AGL(1,ℤ/8)11yescharacteristic
157C2×A4#13711yescharacteristic
158C3×Q8—162no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—82no—
163C2×Dic3—81no—
164C2×Dic3—81no—
165C3⋊C8—44no—
166C24C2×#6711yescharacteristic
167C2×Q8—21no—
168C2×Q8—62no—
169C2×Q8—62no—
170C2×Q8—61no—
171C2×Q8—124no—
172C2×Q8—122no—
173C2×Q8—122no—
174C2×Q8—122no—
175C2×Q8—124no—
176C2×Q8—122no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—32no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—42no—
187C22×C4—62no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—62no—
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—61no—
194C22×C4—61no—
195C22×C4—122no—
196C22×C4D3×Q812yes—
197C2×C8—34no—
198C2×C8—34no—
199C2×C8—62no—
200C2×C8—62no—
201C2×C8—62no—
202C2×C8—62no—
203C4⋊C4—21no—
204C4⋊C4—34no—
205C4⋊C4—32no—
206C4⋊C4—62no—
207C4⋊C4—62no—
208C4⋊C4—62no—
209C4⋊C4—61no—
210C4⋊C4—61no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C4⋊C4—122no—
217C4⋊C4—122no—
218C4⋊C4—121no—
219C4⋊C4—121no—
220C4⋊C4—121no—
221C4⋊C4—121no—
222C4⋊C4C2×S412yes—
223C22⋊C4—121no—
224C22⋊C4—121no—
225C22⋊C4—121no—
226C22⋊C4—121no—
227C42—31no—
228C42—31no—
229C42—31no—
230C42—61no—
231C42C2×S411yescharacteristic
232C22×C3—41no—
233A4#50111yescharacteristic, complemented
234C3×C4—42no—
235C3×C4—81no—
236C3×C4—81no—
237C3×C4—162no—
238Dic3—161no—
239Dic3—161no—
240C23—31no—
241C23—31no—
242C23—31no—
243C23—31no—
244C23#93211yescharacteristic
245C23#94211yescharacteristic
246C23#94411yescharacteristic
247Q8—42no—
248Q8—122no—
249Q8—124no—
250Q8—124no—
251C2×C4—22no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—32no—
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—62no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4GL(2,ℤ/4)11yescharacteristic
285C2×C4GL(2,ℤ/4)11yescharacteristic
286C2×C4C22×S411yescharacteristic
287C8—64no—
288C8—64no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C22—31no—
293C22—31no—
294C22—31no—
295C22—31no—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—61no—
302C22#508111yescharacteristic, complemented
303C22C2×GL(2,ℤ/4)11yescharacteristic
304C4—21no—
305C4—21no—
306C4—32no—
307C4—42no—
308C4—61no—
309C4—61no—
310C4—122no—
311C4—241no—
312C4—241no—
313C4—241no—
314C4—241no—
315C4Q8×S412yes—
316C3—41noSylow
317C2—31no—
318C2—31no—
319C2—31no—
320C2—31no—
321C2#8548911yescharacteristic
322C2#8549911yescharacteristic
323C2#8550211yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4⋊Q8D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenterC4⋊Q8S4
hypocenterA4#501
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
37. A4⋊C8Q8semidirect
233. A4#501semidirect
302. C22#5081semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#852842
#730512
#730502
#730442
#258163
#50814