Finite Group Databasebeta

← #11714009

Subgroups — #11714009

Order 768.

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

normal subgroups only

This group has 508 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
1C4×Q8×A4C211yesmaximal, characteristic
2#85283C211yesmaximal, characteristic
3#73147C212yesmaximal
4#73051C212yesmaximal
5C4⋊(A4⋊C8)C211yesmaximal, characteristic
6#32954—31nomaximal, Sylow
7C2×Q8×A4C2211yescharacteristic
8C2×A4⋊Q8—21no—
9A4×C4⋊C4—21no—
10A4×C4⋊C4C2211yescharacteristic
11C42×A4—21no—
12C42×A4C2211yescharacteristic
13C4⋊SL(2,ℤ/4)—21no—
14C4⋊SL(2,ℤ/4)C2212yes—
15C2×A4⋊C8C2212yes—
16#5020—41nomaximal
17C22×C4×Q8D311yescharacteristic
18C2×#475—31no—
19C2×#475—31no—
20#2857—31no—
21#1865—32no—
22#1865—32no—
23#1853—32no—
24#1803—31no—
25#1803—31no—
26#1797—31no—
27#1787—32no—
28Q8×A4—21no—
29Q8×A4Q812yescomplemented
30C2×C4×A4—21no—
31C2×C4×A4—21no—
32C2×C4×A4D411yescharacteristic
33C2×C4×A4D411yescharacteristic
34C2×C4×A4C2311yescharacteristic
35C2×SL(2,ℤ/4)—22no—
36A4⋊Q8—42no—
37C3×C4×Q8—41no—
38#869—41no—
39A4⋊C8—22no—
40#835—42no—
41#808—42no—
42C4⋊(C3⋊C8)—41no—
43C23×Q8C2×D311yescharacteristic
44C2×C4⋊Q8—31no—
45C2×C22⋊Q8—61no—
46C2×C4×Q8—31no—
47C2×C4×Q8—31no—
48C2×C4×Q8—31no—
49C2×C4×Q8—31no—
50C2×C4×Q8—61no—
51C2×C4×Q8—61no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×D311yescharacteristic
54C22×C42—21no—
55C22×C42C2×D311yescharacteristic
56#475—62no—
57#475—62no—
58#475—61no—
59#475—61no—
60#475—61no—
61#475—61no—
62C2×#106—32no—
63C2×#106—32no—
64C2×C4⋊C8—31no—
65C2×C4⋊C8—31no—
66C2×#103—32no—
67C2×#103—32no—
68C2×#103—32no—
69C2×#103—32no—
70C2×C22⋊C8—32no—
71#391—61no—
72#389—32no—
73#389—61no—
74#384—31no—
75#380—32no—
76#380—61no—
77#340—32no—
78#340—32no—
79#336—32no—
80C22×A4C2×D411yescharacteristic
81C2×C3×Q8—41no—
82C2×Dic6—81no—
83C4×A4—22no—
84C4×A4—21no—
85C4×A4—41no—
86C4×A4SD1612yes—
87C4×A4C2×Q811yescharacteristic
88C4×A4#4111yescharacteristic
89SL(2,ℤ/4)—42no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—81no—
92C3×C42—41no—
93C3×C42—81no—
94C4⋊Dic3—42no—
95C4⋊Dic3—81no—
96C2×C3⋊C8—42no—
97C22×Q8—21no—
98C22×Q8—31no—
99C22×Q8—31no—
100C22×Q8—32no—
101C22×Q8—61no—
102C22×Q8—61no—
103C22×Q8—61no—
104C22×Q8Dic612yescomplemented
105C23×C4—21no—
106C23×C4—21no—
107C23×C4#6711yescharacteristic
108C23×C4#6711yescharacteristic
109C23×C4C22×D311yescharacteristic
110C22×C8—32no—
111C22×C8—32no—
112C4⋊Q8—61no—
113C4⋊Q8—61no—
114C4⋊Q8—61no—
115C4⋊Q8—61no—
116C22⋊Q8—122no—
117C22⋊Q8—122no—
118C4×Q8—32no—
119C4×Q8—32no—
120C4×Q8—31no—
121C4×Q8—31no—
122C4×Q8—31no—
123C4×Q8—61no—
124C4×Q8—61no—
125C4×Q8—61no—
126C4×Q8—62no—
127C4×Q8—62no—
128C4×Q8S411yescharacteristic
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—32no—
132C2×C4⋊C4—32no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C22⋊C4—62no—
143C2×C42—31no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147C2×C42—61no—
148C2×C42—61no—
149C2×C42—61no—
150C2×C42—61no—
151#106—64no—
152#106—62no—
153#106—62no—
154C4⋊C8—61no—
155C4⋊C8—61no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158#103—64no—
159#103—64no—
160#103—64no—
161#103—62no—
162#103—62no—
163C22⋊C8—62no—
164C22⋊C8—62no—
165#95—62no—
166#95—62no—
167C2×A4C22⋊Q811yescharacteristic
168C2×A4C2×SD1611yescharacteristic
169C2×A4#13711yescharacteristic
170C3×Q8—42no—
171C3×Q8—81no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×Dic3—82no—
178Dic6—162no—
179C3⋊C8—82no—
180C24C2×#6711yescharacteristic
181C2×Q8—32no—
182C2×Q8—32no—
183C2×Q8—32no—
184C2×Q8—32no—
185C2×Q8—32no—
186C2×Q8—31no—
187C2×Q8—61no—
188C2×Q8—61no—
189C2×Q8—61no—
190C2×Q8—62no—
191C2×Q8—61no—
192C2×Q8—61no—
193C2×Q8—62no—
194C2×Q8—64no—
195C2×Q8—122no—
196C2×Q8—121no—
197C2×Q8—122no—
198C2×Q8—121no—
199C2×Q8C2×S411yescharacteristic
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—32no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—41no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—62no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4—121no—
221C22×C4#21512yes—
222C22×C4C2×Dic611yescharacteristic
223C22×C4#23511yescharacteristic
224C2×C8—62no—
225C2×C8—62no—
226C2×C8—62no—
227C2×C8—62no—
228C2×C8—62no—
229C2×C8—62no—
230C4⋊C4—21no—
231C4⋊C4—32no—
232C4⋊C4—31no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—62no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C4⋊C4—62no—
242C4⋊C4—122no—
243C4⋊C4—124no—
244C4⋊C4—121no—
245C4⋊C4—122no—
246C4⋊C4—121no—
247C4⋊C4C2×S411yescharacteristic
248C22⋊C4—122no—
249C22⋊C4—122no—
250C42—21no—
251C42—31no—
252C42—31no—
253C42—31no—
254C42—61no—
255C42—61no—
256C42—61no—
257C42—62no—
258C42—61no—
259C42C2×S411yescharacteristic
260C22×C3—41no—
261A4#47511yescharacteristic, complemented
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—42no—
265C3×C4—81no—
266C3×C4—82no—
267C3×C4—161no—
268Dic3—162no—
269C23—31no—
270C23—31no—
271C23—31no—
272C23—31no—
273C23#92411yescharacteristic
274C23C2×#21511yescharacteristic
275C23#95111yescharacteristic
276Q8—21no—
277Q8—34no—
278Q8—32no—
279Q8—62no—
280Q8—62no—
281Q8—62no—
282Q8—61no—
283Q8—242no—
284Q8—242no—
285Q8A4⋊Q812yescomplemented
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—62no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4GL(2,ℤ/4)11yescharacteristic
323C2×C4GL(2,ℤ/4)11yescharacteristic
324C2×C4C22×S411yescharacteristic
325C8—122no—
326C8—122no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—41no—
330C22—31no—
331C22—31no—
332C22—31no—
333C22—31no—
334C22—31no—
335C22—31no—
336C22—31no—
337C22—31no—
338C22—31no—
339C22—61no—
340C22#502011yescharacteristic, complemented
341C22C2×GL(2,ℤ/4)11yescharacteristic
342C4—21no—
343C4—22no—
344C4—31no—
345C4—31no—
346C4—32no—
347C4—41no—
348C4—61no—
349C4—62no—
350C4—121no—
351C4—242no—
352C4—242no—
353C4#541712yes—
354C4C2×A4⋊Q811yescharacteristic
355C4#591211yescharacteristic
356C3—41noSylow
357C2—31no—
358C2—31no—
359C2—31no—
360C2—31no—
361C2#8548011yescharacteristic
362C2C2×#541711yescharacteristic
363C2#8551211yescharacteristic
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
29. Q8×A4Q8semidirect
104. C22×Q8Dic6semidirect
261. A4#475semidirect
285. Q8A4⋊Q8semidirect
340. C22#5020semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×A42
#852832
#731472
#730512
C4⋊(A4⋊C8)2
#329543
#50204