Finite Group Databasebeta

← #10713694

Subgroups — #10713694

Order 768.

566 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#5151C211yesmaximal, characteristic
2C3×#2140C211yesmaximal, characteristic, complemented
3C2×#4563C211yesmaximal, characteristic, complemented
4C2×#4516C211yesmaximal, characteristic, complemented
5C2×#4511C211yesmaximal, characteristic, complemented
6#70659C211yesmaximal, characteristic, complemented
7#70652C211yesmaximal, characteristic, complemented
8#13261—31nomaximal, Sylow
9C22×C3×D8C2211yescharacteristic
10C2×#940—21no—
11C22×#213—21no—
12C22×D24C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#427C2211yescharacteristic
15#5151—22no—
16#5151C414yescomplemented
17C2×#819C2211yescharacteristic
18C2×C8×Dic3C2211yescharacteristic
19C2×#812C2211yescharacteristic
20#4997—21no—
21C3×#357—22no—
22#4563—22no—
23#4532—21no—
24#4516—22no—
25#4511—22no—
26#4491—22no—
27C2×#493—31no—
28C2×#364—31no—
29C2×#361—31no—
30C2×#361—31no—
31#2140—31no—
32#2140D311yescharacteristic, complemented
33#1772—31no—
34C22×C3×D4—21no—
35C22×#67—42no—
36C22×D12—21no—
37C2×C3×D8—21no—
38C2×C3×D8—21no—
39C2×C3×D8—21no—
40C2×C3×D8—21no—
41C2×C3×D8—21no—
42C2×C3×D8C2×C412yes—
43C22×C3×C8C2311yescharacteristic
44C3×#117—21no—
45#940—24no—
46#940—42no—
47C2×#213—24no—
48C2×#213—41no—
49C2×#213—41no—
50C2×#213—42no—
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53C22×C3⋊C8D411yescharacteristic
54C2×D24—21no—
55C2×D24—21no—
56C2×D24—21no—
57C2×D24—21no—
58C2×D24—21no—
59C2×D24C2×C412yes—
60C3×M32—21no—
61C3×M32—21no—
62C2×C3×C16—21no—
63C3×#107D412yescomplemented
64C3×#106—21no—
65#819—21no—
66#819D412yescomplemented
67C8×Dic3C2×C412yes—
68C8×Dic3D412yes—
69#812—21no—
70#812—21no—
71C2×C3⋊C16—21no—
72#803—41no—
73C4×C3⋊C8—41no—
74C22×D8—31no—
75C22×D8—61no—
76C22×D8C2×D311yescharacteristic
77C2×#127—61no—
78C2×M32—31no—
79C2×M32C2×D311yescharacteristic
80#493—34no—
81#493—62no—
82C2×#108—31no—
83#427C2×D311yescharacteristic
84#405—61no—
85C2×C4×C8—31no—
86#364—62no—
87#361—62no—
88#361—62no—
89#357—22no—
90#357—62no—
91#339—61no—
92C24×C3—41no—
93C23×D3—41no—
94C2×C3×D4—22no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C22×C3×C4C2×D411yescharacteristic
101C2×#67—48no—
102C2×#67—88no—
103C22×Dic3—21no—
104C2×D12—22no—
105C2×D12—41no—
106C2×D12—41no—
107C2×D12—41no—
108C2×D12—41no—
109C2×D12—41no—
110C3×D8—22no—
111C3×D8—22no—
112C3×D8—41no—
113C3×D8—41no—
114C2×C3×C8C22×C411yescharacteristic
115C2×C3×C8C2×D411yescharacteristic
116C2×C3×C8C2×D411yescharacteristic
117C2×C3×C8C2×D411yescharacteristic
118C2×C3×C8#4111yescharacteristic
119C2×C3×C8#4111yescharacteristic
120C3×C4⋊C4—22no—
121C3×C22⋊C4—41no—
122C3×C42—41no—
123#213—44no—
124#213—44no—
125C4×Dic3—21no—
126C4×Dic3C22⋊C412yes—
127#208—22no—
128#208—41no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8C22⋊C412yes—
131C2×C3⋊C8D811yescharacteristic
132C2×C3⋊C8D811yescharacteristic
133C2×C3⋊C8SD1612yes—
134D24—22no—
135D24—22no—
136D24—41no—
137D24—41no—
138C3×C16—41no—
139C3⋊C16—41no—
140C22×D4—21no—
141C22×D4—61no—
142C22×D4—122no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—32no—
149C2×D8—64no—
150C2×D8—61no—
151C2×D8—61no—
152C2×D8—61no—
153C2×D8—61no—
154C2×D8—61no—
155C2×D8—121no—
156C2×D8—121no—
157C2×D8—122no—
158C2×D8C4×D312yes—
159C2×M16—61no—
160C22×C8—31no—
161C22×C8C22×D311yescharacteristic
162#127—64no—
163#127—122no—
164#117—21no—
165C2×C42—31no—
166M32—21no—
167M32—21no—
168M32—61no—
169M32—61no—
170C2×C16—21no—
171C2×C16—61no—
172#108—32no—
173#108—61no—
174#107#6712yescomplemented
175#106—21no—
176#97—121no—
177C4×C8—32no—
178C4×C8—32no—
179C4×C8—121no—
180C23×C3—42no—
181C23×C3—84no—
182C23×C3—81no—
183C23×C3—81no—
184C23×C3AGL(1,ℤ/8)11yescharacteristic
185C22×D3—42no—
186C22×D3—84no—
187C22×D3—81no—
188C22×D3—81no—
189C3×D4—42no—
190C3×D4—42no—
191C3×D4—81no—
192C3×D4—81no—
193C2×C3×C4—42no—
194C2×C3×C4C2×C22⋊C411yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4C2×D811yescharacteristic
198C2×C3×C4C2×SD1611yescharacteristic
199C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
200#67—816no—
201C2×Dic3—24no—
202C2×Dic3—22no—
203D12—42no—
204D12—42no—
205D12—81no—
206D12—81no—
207C3×C8C4×D411yescharacteristic
208C3×C8C4×D411yescharacteristic
209C3×C8#12411yescharacteristic
210C3×C8#12711yescharacteristic
211C3⋊C8—42no—
212C3⋊C8#10212yes—
213C3⋊C8#10212yes—
214C24—41no—
215C24—121no—
216C2×D4—22no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—62no—
223C2×D4—128no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—248no—
230C22×C4—61no—
231C22×C4D3×D411yescharacteristic
232D8—22no—
233D8—22no—
234D8—41no—
235D8—41no—
236D8—62no—
237D8—62no—
238D8—124no—
239D8—124no—
240D8—121no—
241D8—121no—
242M16—62no—
243M16—121no—
244C2×C8—32no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—32no—
248C2×C8—121no—
249C2×C8C2×C4×D311yescharacteristic
250C2×C8#23511yescharacteristic
251C2×C8D3×D411yescharacteristic
252C2×C8D3×D411yescharacteristic
253C2×C8#23711yescharacteristic
254C2×C8C2×#6711yescharacteristic
255C4⋊C4—22no—
256C22⋊C4—41no—
257C42—32no—
258C42—41no—
259C42—61no—
260C16—41no—
261C16—121no—
262C22×C3—22no—
263C22×C3—84no—
264C22×C3—82no—
265C22×C3—162no—
266C22×C3—162no—
267C22×C3#41811yescharacteristic
268C22×C3#45911yescharacteristic
269C22×C3#46011yescharacteristic
270C2×D3—84no—
271C2×D3—82no—
272C2×D3—162no—
273C2×D3—162no—
274C3×C4—82no—
275C3×C4#39011yescharacteristic
276C3×C4C2×#10211yescharacteristic
277C3×C4#45911yescharacteristic
278C3×C4#46011yescharacteristic
279Dic3—24no—
280C23—42no—
281C23—84no—
282C23—81no—
283C23—81no—
284C23—122no—
285C23—244no—
286C23—241no—
287C23—241no—
288C23#91111yescharacteristic
289D4—42no—
290D4—42no—
291D4—81no—
292D4—81no—
293D4—122no—
294D4—122no—
295D4—2416no—
296D4—241no—
297D4—241no—
298C2×C4—42no—
299C2×C4—64no—
300C2×C4—62no—
301C2×C4D3×C22⋊C411yescharacteristic
302C2×C4#88411yescharacteristic
303C2×C4D3×D811yescharacteristic
304C2×C4D3×SD1611yescharacteristic
305C2×C4#91411yescharacteristic
306C2×C4#93911yescharacteristic
307C8—32no—
308C8—32no—
309C8—122no—
310C8#67⋊C411yescharacteristic
311C8#88511yescharacteristic
312C8C4×#6711yescharacteristic
313C8#94011yescharacteristic
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—162no—
317C2×C3#198211yescharacteristic
318C2×C3#221312yes—
319D3—162no—
320C22—22no—
321C22—84no—
322C22—82no—
323C22—162no—
324C22—162no—
325C22—244no—
326C22—242no—
327C22—482no—
328C22—482no—
329C22#476211yescharacteristic
330C22#477011yescharacteristic
331C22#515011yescharacteristic
332C4—64no—
333C4—82no—
334C4D3×#10211yescharacteristic
335C4#478611yescharacteristic
336C4#495711yescharacteristic
337C4#516311yescharacteristic
338C3#1326111yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—162no—
342C2—482no—
343C2#7016111yescharacteristic
344C2#7151412yes—
subgroupG/H
centerC22#5150
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2140C2
socleC22×C3#459
prime radicalC2×#5151C2
cosocleC22×C3×C8C23
hypercenter#2140D3
hypocenterC3#13261
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2140

Complements

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

NH ≅ G/Ntype
2. C3×#2140C2semidirect
3. C2×#4563C2semidirect
4. C2×#4516C2semidirect
5. C2×#4511C2semidirect
6. #70659C2semidirect
7. #70652C2semidirect
16. #5151C4semidirect
32. #2140D3semidirect
63. C3×#107D4semidirect
66. #819D4semidirect
174. #107#67semidirect
338. C3#13261semidirect

Maximal subgroups

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

subgroupindex
C2×#51512
C3×#21402
C2×#45632
C2×#45162
C2×#45112
#706592
#706522
#132613