Finite Group Databasebeta

← #11712862

Subgroups — #11712862

Order 768.

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

normal subgroups only

This group has 483 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
1#85440C211yesmaximal, characteristic
2#85437C212yesmaximal, complemented
3A4×#97C211yesmaximal, characteristic, complemented
4#73105C211yesmaximal, characteristic, complemented
5#73046C212yesmaximal, complemented
6#21546—31nomaximal, Sylow
7#5912C2212yes—
8C2×C8×A4C2211yescharacteristic
9C2×C8×A4C2211yescharacteristic
10C42×A4C2211yescharacteristic
11#5409C2212yes—
12#5402C411yescharacteristic, complemented
13#5401C412yescomplemented
14#5400—22no—
15C8×S4—22no—
16#5398C411yescharacteristic, complemented
17#4717—41nomaximal
18#3050—31no—
19C2×#444—31no—
20#2929—32no—
21C22×#97D311yescharacteristic, complemented
22#1975—31no—
23#1946—31no—
24#1938—32no—
25#1896—32no—
26#1810—31no—
27#1773—31no—
28#1765—32no—
29C2×C4×A4—21no—
30C2×C4×A4C2311yescharacteristic
31GL(2,ℤ/4)—22no—
32C4⋊S4C2×C412yes—
33C4×S4—22no—
34A4⋊Q8C2×C412yes—
35#904—41no—
36#901—42no—
37C8×A4C2×C411yescharacteristic
38C8×A4C2×C411yescharacteristic
39C8×A4D412yes—
40A4⋊C8—22no—
41C3×#97—41no—
42#819—41no—
43#805—42no—
44C2×#135—31no—
45C2×#131—32no—
46C23×C8C2×D311yescharacteristic
47C23×C8C2×D311yescharacteristic
48#525—32no—
49C22×C42C2×D311yescharacteristic
50#467—31no—
51#467—31no—
52#466—31no—
53#466—31no—
54#465—32no—
55#465—32no—
56#444—61no—
57#444—61no—
58#444—61no—
59#444—61no—
60#435—62no—
61#435—62no—
62C8×D4—62no—
63C8×D4—62no—
64C2×#108—31no—
65C2×#108—31no—
66#427—31no—
67#424—32no—
68C2×#104—32no—
69C2×#104—32no—
70#416—32no—
71#408—32no—
72#407—32no—
73#405—32no—
74C2×#97—31no—
75C2×#97—31no—
76C2×#97—31no—
77C2×#97—31no—
78#341—32no—
79#339—32no—
80#339—32no—
81C22×A4#4111yescharacteristic
82C2×S4—22no—
83#235—42no—
84C4×A4—41no—
85C4×A4C22×C411yescharacteristic
86C4×A4C2×D411yescharacteristic
87SL(2,ℤ/4)—22no—
88C2×C3×C8—41no—
89C2×C3×C8—41no—
90C3×C42—41no—
91#208—42no—
92Dic12—41no—
93D24—41no—
94#204—42no—
95ΓL(1,𝔽25)—82no—
96D3×C8—82no—
97C2×#41—32no—
98C23×C4—21no—
99C23×C4C22×D311yescharacteristic
100#135—61no—
101#135—61no—
102#135—61no—
103#135—61no—
104C2×Q16—31no—
105C2×SD16—32no—
106C2×D8—31no—
107#131—62no—
108#131—62no—
109#131—62no—
110#131—62no—
111C2×M16—32no—
112C2×M16—32no—
113C2×M16—62no—
114C22×C8—31no—
115C22×C8—31no—
116C22×C8—31no—
117C22×C8—31no—
118C22×C8—32no—
119C22×C8—31no—
120C22×C8—31no—
121C22×C8—62no—
122C22×C8C4×D311yescharacteristic
123C22×C8C4×D311yescharacteristic
124C22×C8D1212yes—
125#123—62no—
126C22⋊Q8—32no—
127C22⋊Q8—32no—
128#121—32no—
129#121—32no—
130#120—62no—
131C4×D4—62no—
132C4×D4—62no—
133#117—32no—
134C2×C42—31no—
135C2×C42—31no—
136C2×C42—61no—
137#108—61no—
138#108—61no—
139#108—61no—
140#108—61no—
141#107—31no—
142#107—31no—
143#106—32no—
144C4⋊C8—62no—
145C4⋊C8—62no—
146#104—64no—
147#104—62no—
148#104—62no—
149#103—32no—
150#103—32no—
151#102—32no—
152#102—32no—
153C22⋊C8—62no—
154C22⋊C8—62no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157#97—31no—
158#97—31no—
159#97—31no—
160#97—61no—
161#97—62no—
162#97S411yescharacteristic, complemented
163C4×C8—62no—
164C2×A4—21no—
165C2×A4C4×D411yescharacteristic
166S4—42no—
167C2×C3×C4—41no—
168C2×C3×C4—81no—
169#67—82no—
170D12—42no—
171C4×D3—82no—
172Dic6—42no—
173C3×C8—41no—
174C3×C8—41no—
175C3×C8—42no—
176C3⋊C8—82no—
177C24#23511yescharacteristic
178#41—62no—
179#41—62no—
180#41—62no—
181#41—62no—
182C2×Q8—32no—
183C2×D4—32no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—62no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—31no—
191C22×C4—41no—
192C22×C4—61no—
193C22×C4—62no—
194C22×C4—61no—
195C22×C4—121no—
196C22×C4C2×C4×D311yescharacteristic
197C22×C4C2×D1211yescharacteristic
198Q16—61no—
199Q16—61no—
200SD16—62no—
201SD16—62no—
202D8—61no—
203D8—61no—
204M16—62no—
205M16—62no—
206M16—62no—
207M16—62no—
208M16—122no—
209M16—122no—
210C2×C8—31no—
211C2×C8—31no—
212C2×C8—31no—
213C2×C8—31no—
214C2×C8—31no—
215C2×C8—31no—
216C2×C8—32no—
217C2×C8—32no—
218C2×C8—32no—
219C2×C8—31no—
220C2×C8—31no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—62no—
224C2×C8—62no—
225C2×C8—122no—
226C2×C8—122no—
227C2×C8C2×S411yescharacteristic
228C2×C8C2×S411yescharacteristic
229C4⋊C4—32no—
230C4⋊C4—32no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C22⋊C4—62no—
234C22⋊C4—62no—
235C22⋊C4—62no—
236C22⋊C4—62no—
237C22⋊C4—62no—
238C42—32no—
239C42—31no—
240C42—61no—
241C42—62no—
242C42C2×S411yescharacteristic
243C22×C3—41no—
244C2×D3—82no—
245A4#44411yescharacteristic, complemented
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—161no—
249Dic3—82no—
250C23—21no—
251C23—31no—
252C23—31no—
253C23—61no—
254C23—62no—
255C23C4×D1211yescharacteristic
256Q8—62no—
257Q8—62no—
258D4—62no—
259D4—62no—
260D4—122no—
261D4—124no—
262D4—122no—
263D4—122no—
264D4—122no—
265C2×C4—21no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—61no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—61no—
278C2×C4—122no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—122no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4C22×S411yescharacteristic
285C8—31no—
286C8—31no—
287C8—32no—
288C8—122no—
289C8—122no—
290C8C4×S411yescharacteristic
291C8C4×S411yescharacteristic
292C8C4⋊S412yes—
293C2×C3—41no—
294C2×C3—81no—
295D3—162no—
296C22—31no—
297C22—31no—
298C22—32no—
299C22—31no—
300C22—61no—
301C22—61no—
302C22—61no—
303C22—122no—
304C22—122no—
305C22—122no—
306C22#471711yescharacteristic, complemented
307C22#591211yescharacteristic
308C4—31no—
309C4—31no—
310C4—41no—
311C4—121no—
312C4—122no—
313C4—122no—
314C4—122no—
315C4C2×C4×S411yescharacteristic
316C4C2×C4⋊S411yescharacteristic
317C3—41noSylow
318C2—21no—
319C2—31no—
320C2—31no—
321C2—61no—
322C2—242no—
323C2C4×C4⋊S411yescharacteristic
subgroupG/H
centerC4C2×C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#97D3
socleC23C4×D12
prime radical#85440C2
cosocleC2×C4×A4C23
hypercenter#97S4
hypocenterA4#444
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C4 ◁ C2×C8 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #85437C2semidirect
3. A4×#97C2semidirect
4. #73105C2semidirect
5. #73046C2semidirect
12. #5402C4semidirect
13. #5401C4semidirect
16. #5398C4semidirect
21. C22×#97D3semidirect
162. #97S4semidirect
245. A4#444semidirect
306. C22#4717semidirect

Maximal subgroups

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

subgroupindex
#854402
#854372
A4×#972
#731052
#730462
#215463
#47174