Finite Group Databasebeta

← #11713293

Subgroups — #11713293

Order 768.

495 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 61 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
2C4×A4⋊Q8C211yesmaximal, characteristic
3#73052C212yesmaximal
4#73051C212yesmaximal
5C4⋊(A4⋊C8)C211yesmaximal, characteristic
6#24301—31nomaximal, Sylow
7C2×Q8×A4—21no—
8C2×A4⋊Q8C2211yescharacteristic
9A4×C4⋊C4—21no—
10A4×C4⋊C4C2212yes—
11C42×A4C2211yescharacteristic
12C4⋊SL(2,ℤ/4)C2211yescharacteristic
13SL(2,ℤ/4)⋊C4—21no—
14C4×SL(2,ℤ/4)—21no—
15C2×A4⋊C8C2212yes—
16#5094—41nomaximal
17C22×C4⋊Q8D311yescharacteristic
18C2×#475—31no—
19C2×#475—31no—
20C4×C22⋊Q8—31no—
21#2058—32no—
22#2024—32no—
23#1988—31no—
24#1988—31no—
25#1872—32no—
26#1853—32no—
27#1797—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—
35A4⋊Q8—21no—
36A4⋊Q8Q812yescomplemented
37C3×C4⋊Q8—41no—
38C4×Dic6—41no—
39A4⋊C8—22no—
40#810—42no—
41#808—42no—
42C4⋊(C3⋊C8)—41no—
43C23×Q8—21no—
44C2×C4⋊Q8—31no—
45C2×C4⋊Q8—31no—
46C2×C4⋊Q8—32no—
47C2×C4⋊Q8—61no—
48C2×C4⋊Q8—61no—
49C2×C22⋊Q8—31no—
50C2×C4×Q8—31no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×D312yes—
53C22×C42C2×D311yescharacteristic
54#475—62no—
55#475—62no—
56#475—61no—
57#475—62no—
58#475—61no—
59C2×#106—32no—
60C2×#106—32no—
61C2×C4⋊C8—31no—
62C2×C4⋊C8—31no—
63C2×#103—32no—
64C2×#103—32no—
65C2×C22⋊C8—32no—
66#391—31no—
67#389—31no—
68#387—61no—
69#385—61no—
70#380—31no—
71C4×C4⋊C4—31no—
72C4×C4⋊C4—61no—
73C4×C22⋊C4—61no—
74#340—32no—
75#340—32no—
76#340—32no—
77#340—32no—
78#336—32no—
79C22×A4C2×D411yescharacteristic
80C2×C3×Q8—81no—
81C2×Dic6—41no—
82C4×A4—21no—
83C4×A4—42no—
84C4×A4SD1612yes—
85C4×A4C2×Q811yescharacteristic
86C4×A4#4111yescharacteristic
87SL(2,ℤ/4)—22no—
88SL(2,ℤ/4)—41no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—81no—
91C3×C42—41no—
92C4⋊Dic3—41no—
93Dic3⋊C4—81no—
94C4×Dic3—81no—
95C2×C3⋊C8—42no—
96C22×Q8—31no—
97C22×Q8—42no—
98C22×Q8—61no—
99C22×Q8—61no—
100C22×Q8—62no—
101C22×Q8—122no—
102C23×C4—22no—
103C23×C4#6711yescharacteristic
104C23×C4#6711yescharacteristic
105C23×C4C22×D311yescharacteristic
106C22×C8—32no—
107C22×C8—32no—
108C4⋊Q8—32no—
109C4⋊Q8—32no—
110C4⋊Q8—32no—
111C4⋊Q8—31no—
112C4⋊Q8—61no—
113C4⋊Q8—61no—
114C4⋊Q8—61no—
115C4⋊Q8—64no—
116C4⋊Q8S411yescharacteristic
117C22⋊Q8—32no—
118C22⋊Q8—32no—
119C22⋊Q8—61no—
120C22⋊Q8—61no—
121C4×Q8—61no—
122C4×Q8—61no—
123C4×Q8—61no—
124C4×Q8—61no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C42—31no—
140C2×C42—31no—
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—61no—
144C2×C42—61no—
145#106—64no—
146#106—62no—
147#106—62no—
148C4⋊C8—61no—
149C4⋊C8—61no—
150C4⋊C8—61no—
151C4⋊C8—61no—
152#103—64no—
153#103—62no—
154#103—62no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162C2×A4C22⋊Q811yescharacteristic
163C2×A4C2×SD1611yescharacteristic
164C2×A4#13711yescharacteristic
165C3×Q8—162no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—82no—
170C2×Dic3—81no—
171C2×Dic3—81no—
172Dic6—42no—
173Dic6—81no—
174C3⋊C8—82no—
175C24C2×#6711yescharacteristic
176C2×Q8—21no—
177C2×Q8—32no—
178C2×Q8—62no—
179C2×Q8—61no—
180C2×Q8—62no—
181C2×Q8—61no—
182C2×Q8—61no—
183C2×Q8—61no—
184C2×Q8—61no—
185C2×Q8—61no—
186C2×Q8—122no—
187C2×Q8—122no—
188C2×Q8—122no—
189C2×Q8—124no—
190C2×Q8—122no—
191C2×Q8—124no—
192C22×C4—21no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—32no—
201C22×C4—31no—
202C22×C4—42no—
203C22×C4—62no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4—62no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—122no—
211C22×C4#21412yes—
212C22×C4D3×Q811yescharacteristic
213C22×C4#23911yescharacteristic
214C2×C8—62no—
215C2×C8—62no—
216C2×C8—62no—
217C2×C8—62no—
218C2×C8—62no—
219C2×C8—62no—
220C4⋊C4—21no—
221C4⋊C4—34no—
222C4⋊C4—32no—
223C4⋊C4—32no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4—61no—
227C4⋊C4—62no—
228C4⋊C4—62no—
229C4⋊C4—62no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—122no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4C2×S412yes—
239C22⋊C4—62no—
240C22⋊C4—62no—
241C22⋊C4—121no—
242C22⋊C4—121no—
243C42—31no—
244C42—31no—
245C42—31no—
246C42—61no—
247C42—122no—
248C42—121no—
249C42—121no—
250C42C2×S411yescharacteristic
251C22×C3—41no—
252A4#47511yescharacteristic, complemented
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—42no—
256C3×C4—81no—
257C3×C4—162no—
258Dic3—82no—
259Dic3—161no—
260C23—31no—
261C23—31no—
262C23—31no—
263C23—31no—
264C23C2×#21411yescharacteristic
265C23#94211yescharacteristic
266C23#94611yescharacteristic
267Q8—42no—
268Q8—62no—
269Q8—62no—
270Q8—124no—
271Q8—121no—
272Q8—124no—
273Q8—121no—
274Q8—122no—
275C2×C4—22no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—32no—
287C2×C4—32no—
288C2×C4—31no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4GL(2,ℤ/4)11yescharacteristic
312C2×C4GL(2,ℤ/4)11yescharacteristic
313C2×C4C22×S411yescharacteristic
314C8—122no—
315C8—122no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C22—31no—
320C22—31no—
321C22—31no—
322C22—31no—
323C22—31no—
324C22—31no—
325C22—31no—
326C22—31no—
327C22—31no—
328C22—61no—
329C22#509411yescharacteristic, complemented
330C22C2×GL(2,ℤ/4)11yescharacteristic
331C4—21no—
332C4—31no—
333C4—31no—
334C4—32no—
335C4—42no—
336C4—61no—
337C4—122no—
338C4—122no—
339C4—122no—
340C4—241no—
341C4—241no—
342C4#541412yes—
343C4Q8×S411yescharacteristic
344C4#591911yescharacteristic
345C3—41noSylow
346C2—31no—
347C2—31no—
348C2—31no—
349C2—31no—
350C2C2×#541411yescharacteristic
351C2#8549911yescharacteristic
352C2#8550611yescharacteristic
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#475
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
36. A4⋊Q8Q8semidirect
252. A4#475semidirect
329. C22#5094semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
C4×A4⋊Q82
#730522
#730512
C4⋊(A4⋊C8)2
#243013
#50944