Finite Group Databasebeta

← #11712972

Subgroups — #11712972

Order 768.

486 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 85 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
2C4⋊(A4⋊C8)C212yesmaximal
3C4⋊(A4⋊C8)C211yesmaximal, characteristic
4#73044C212yesmaximal
5C4×A4⋊C8C211yesmaximal, characteristic
6#22510—31nomaximal, Sylow
7C2×Q8×A4C411yescharacteristic
8A4×C4⋊C4C412yes—
9A4×C4⋊C4C411yescharacteristic
10C42×A4C2211yescharacteristic
11C42×A4C2212yes—
12C2×A4⋊C8C2211yescharacteristic
13C2×A4⋊C8C2212yes—
14C2×A4⋊C8C2211yescharacteristic
15#5024—41nomaximal
16C22×C4×Q8D311yescharacteristic
17C2×#446—31no—
18C2×#446—31no—
19#1985—32no—
20#1985—32no—
21#1984—31no—
22#1984—31no—
23#1797—32no—
24#1797—31no—
25#1749—32no—
26C4×C22⋊C8—31no—
27Q8×A4—22no—
28C2×C4×A4C2×C412yes—
29C2×C4×A4C2×C411yescharacteristic
30C2×C4×A4C2×C412yes—
31C2×C4×A4C2×C411yescharacteristic
32C2×C4×A4C2311yescharacteristic
33C3×C4×Q8—41no—
34A4⋊C8—21no—
35A4⋊C8—22no—
36A4⋊C8Q812yescomplemented
37C4⋊(C3⋊C8)—42no—
38C4⋊(C3⋊C8)—41no—
39#803—42no—
40C4×C3⋊C8—41no—
41C23×Q8Dic311yescharacteristic
42C2×C4×Q8—31no—
43C2×C4×Q8—31no—
44C2×C4×Q8—32no—
45C2×C4×Q8—31no—
46C2×C4×Q8—31no—
47C2×C4×Q8—32no—
48C22×C4⋊C4Dic312yes—
49C22×C4⋊C4Dic311yescharacteristic
50C22×C42C2×D311yescharacteristic
51C22×C42C2×D312yes—
52#446—61no—
53#446—62no—
54#446—61no—
55#446—62no—
56#446—61no—
57#446—61no—
58C2×C4⋊C8—31no—
59C2×C4⋊C8—31no—
60C2×C4⋊C8—32no—
61C2×C4⋊C8—32no—
62C2×C22⋊C8—31no—
63C2×C22⋊C8—31no—
64C2×C22⋊C8—32no—
65C2×#97—32no—
66C2×#97—32no—
67C2×C4×C8—31no—
68C2×C4×C8—31no—
69#336—32no—
70#336—32no—
71#336—32no—
72#336—32no—
73#336—32no—
74#336—31no—
75#336—31no—
76C22×A4C22×C411yescharacteristic
77C2×C3×Q8—41no—
78C4×A4—22no—
79C4×A4—21no—
80C4×A4—22no—
81C4×A4M1612yes—
82C4×A4C2×Q811yescharacteristic
83C4×A4#4111yescharacteristic
84C3×C4⋊C4—41no—
85C3×C4⋊C4—42no—
86C3×C42—41no—
87C3×C42—42no—
88C2×C3⋊C8—41no—
89C2×C3⋊C8—41no—
90C2×C3⋊C8—42no—
91C22×Q8—22no—
92C22×Q8—31no—
93C22×Q8—31no—
94C22×Q8—32no—
95C22×Q8—62no—
96C23×C4C2×Dic311yescharacteristic
97C23×C4C2×Dic312yes—
98C23×C4C2×Dic312yes—
99C23×C4C2×Dic311yescharacteristic
100C23×C4C22×D311yescharacteristic
101C22×C8—31no—
102C22×C8—31no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8—32no—
106C22×C8—32no—
107C4×Q8—31no—
108C4×Q8—32no—
109C4×Q8—31no—
110C4×Q8—32no—
111C4×Q8—31no—
112C4×Q8—61no—
113C4×Q8—61no—
114C4×Q8—62no—
115C4×Q8—61no—
116C4×Q8—62no—
117C4×Q8S411yescharacteristic
118C2×C4⋊C4—31no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—32no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—32no—
125C2×C42—31no—
126C2×C42—32no—
127C2×C42—31no—
128C2×C42—32no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—32no—
132C2×C42—32no—
133C4⋊C8—61no—
134C4⋊C8—61no—
135C4⋊C8—62no—
136C4⋊C8—62no—
137C4⋊C8—61no—
138C4⋊C8—62no—
139C4⋊C8—61no—
140C4⋊C8—62no—
141C22⋊C8—32no—
142C22⋊C8—32no—
143C22⋊C8—61no—
144C22⋊C8—62no—
145C22⋊C8—61no—
146C22⋊C8—62no—
147#97—62no—
148#97—62no—
149#97—62no—
150#97—62no—
151C4×C8—61no—
152C4×C8—61no—
153C4×C8—61no—
154C4×C8—61no—
155C2×A4C4×Q811yescharacteristic
156C2×A4C2×M1611yescharacteristic
157C2×A4#13111yescharacteristic
158C3×Q8—82no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—41no—
164C3⋊C8—42no—
165C3⋊C8—81no—
166C3⋊C8—82no—
167C24C22×Dic311yescharacteristic
168C2×Q8—34no—
169C2×Q8—32no—
170C2×Q8—31no—
171C2×Q8—61no—
172C2×Q8—62no—
173C2×Q8—62no—
174C2×Q8—62no—
175C2×Q8—64no—
176C2×Q8—64no—
177C2×Q8SL(2,ℤ/4)11yescharacteristic
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—32no—
188C22×C4—32no—
189C22×C4—31no—
190C22×C4—32no—
191C22×C4—32no—
192C22×C4—31no—
193C22×C4—32no—
194C22×C4—62no—
195C22×C4—61no—
196C22×C4—62no—
197C22×C4#20812yes—
198C22×C4D3×Q811yescharacteristic
199C22×C4#23911yescharacteristic
200C2×C8—32no—
201C2×C8—32no—
202C2×C8—61no—
203C2×C8—61no—
204C2×C8—61no—
205C2×C8—61no—
206C2×C8—61no—
207C2×C8—62no—
208C2×C8—61no—
209C2×C8—62no—
210C2×C8—61no—
211C2×C8—62no—
212C2×C8—61no—
213C2×C8—62no—
214C2×C8—61no—
215C2×C8—62no—
216C2×C8—61no—
217C2×C8—62no—
218C4⋊C4—32no—
219C4⋊C4—32no—
220C4⋊C4—31no—
221C4⋊C4—32no—
222C4⋊C4—32no—
223C4⋊C4—61no—
224C4⋊C4—62no—
225C4⋊C4SL(2,ℤ/4)12yes—
226C4⋊C4SL(2,ℤ/4)11yescharacteristic
227C42—31no—
228C42—31no—
229C42—32no—
230C42—32no—
231C42—31no—
232C42—32no—
233C42—61no—
234C42—62no—
235C42C2×S411yescharacteristic
236C42C2×S412yes—
237C22×C3—41no—
238A4#44611yescharacteristic, complemented
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—42no—
242C3×C4—82no—
243C3×C4—81no—
244C3×C4—82no—
245C23—31no—
246C23—31no—
247C23—31no—
248C23—31no—
249C23C2×#20811yescharacteristic
250C23Q8×Dic311yescharacteristic
251C23#94811yescharacteristic
252Q8—22no—
253Q8—64no—
254Q8—64no—
255Q8—62no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—32no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—34no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—61no—
283C2×C4—62no—
284C2×C4C2×SL(2,ℤ/4)11yescharacteristic
285C2×C4C2×SL(2,ℤ/4)12yes—
286C2×C4C2×SL(2,ℤ/4)12yes—
287C2×C4C2×SL(2,ℤ/4)11yescharacteristic
288C2×C4C22×S411yescharacteristic
289C8—62no—
290C8—62no—
291C8—121no—
292C8—122no—
293C8—121no—
294C8—122no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—31no—
305C22—31no—
306C22—31no—
307C22—61no—
308C22#502411yescharacteristic, complemented
309C22C22×SL(2,ℤ/4)11yescharacteristic
310C4—21no—
311C4—22no—
312C4—22no—
313C4—31no—
314C4—31no—
315C4—32no—
316C4—61no—
317C4—62no—
318C4—62no—
319C4#540912yes—
320C4Q8×S411yescharacteristic
321C4#591911yescharacteristic
322C3—41noSylow
323C2—31no—
324C2—31no—
325C2—31no—
326C2—31no—
327C2C2×#540911yescharacteristic
328C2Q8×SL(2,ℤ/4)11yescharacteristic
329C2#8551111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutator, prime radicalC22×A4C22×C4
FittingC22×C4×Q8D3
socleC24C22×Dic3
cosocleC2×C4×A4C23
hypercenterC4×Q8S4
hypocenterA4#446
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
36. A4⋊C8Q8semidirect
238. A4#446semidirect
308. C22#5024semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×A42
C4⋊(A4⋊C8)2
#730442
C4×A4⋊C82
#225103
#50244