Finite Group Databasebeta

← #10690690

Subgroups — #10690690

Order 768.

630 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 135 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#82329C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic, complemented
3C3×#1460C211yesmaximal, characteristic, complemented
4#70756C212yesmaximal, complemented
5#67737C212yesmaximal, complemented
6#9101—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#940C411yescharacteristic
9C2×C4×D12C2211yescharacteristic
10C3×#431C2212yes—
11#5221C411yescharacteristic
12#5213—22no—
13C2×#830—22no—
14D4×C3⋊C8—24no—
15#4988C411yescharacteristic
16#4977C411yescharacteristic
17#4956—22no—
18#4932C2211yescharacteristic
19#4925C2212yes—
20C3×#323—22no—
21#4483—24no—
22#2972—31no—
23#2328—31no—
24#1478—32no—
25#1460D311yescharacteristic, complemented
26#1274—32no—
27C22×C3×D4C2×C411yescharacteristic
28C23×C3×C4D412yes—
29C22×#67—22no—
30C22×D12C2×C411yescharacteristic
31C22×C4×D3—21no—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C2×C3×C4⋊C4C2×C411yescharacteristic
35C2×C3×C22⋊C4D412yes—
36C2×C3×C42C2311yescharacteristic
37C2×#217—22no—
38#940—22no—
39#940C814yescomplemented
40C2×(C2×D3)⋊C4—21no—
41C2×(C2×D3)⋊C4—22no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×C4×Dic3C2×C411yescharacteristic
44C2×C4×Dic3C2×C411yescharacteristic
45C22×C3⋊C8—24no—
46C4×D12—22no—
47C4×D12—22no—
48C3×C4⋊C8—22no—
49C3×C22⋊C8—22no—
50C3×C4×C8—22no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95—21no—
54#830—24no—
55#830—22no—
56C4⋊(C3⋊C8)—22no—
57C4×C3⋊C8D414yescomplemented
58C2×#127—31no—
59C2×C4×D4—31no—
60C2×C4×D4C2×D311yescharacteristic
61C8×D4—64no—
62#431—32no—
63#431C2×D312yes—
64C2×C22⋊C8—62no—
65#390—31no—
66#390—31no—
67#386—62no—
68#386—62no—
69#382—31no—
70#376—31no—
71D4⋊C8—64no—
72#323—22no—
73C24×C3C22⋊C412yes—
74C23×D3—21no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4C2×C812yes—
78C2×C3×D4M1612yes—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—82no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22×C411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C2×#67—28no—
87C2×#67—82no—
88C2×#67—82no—
89C22×Dic3—21no—
90C22×Dic3—21no—
91C22×Dic3—21no—
92C2×D12—22no—
93C2×D12—22no—
94C2×D12—21no—
95C2×D12C2×C812yescomplemented
96C2×C4×D3—42no—
97C2×C4×D3—82no—
98C2×C3×C8—42no—
99C3×C4⋊C4—22no—
100C3×C22⋊C4—24no—
101C3×C42C2×D412yes—
102C3×C42#4112yes—
103#217—82no—
104(C2×D3)⋊C4—41no—
105(C2×D3)⋊C4—41no—
106(C2×D3)⋊C4—82no—
107C4⋊Dic3—22no—
108C4×Dic3—21no—
109C4×Dic3—22no—
110C4×Dic3C2×C812yescomplemented
111C2×C3⋊C8—24no—
112C2×C3⋊C8—84no—
113C22×D4—31no—
114C22×D4—62no—
115C22×D4C4×D311yescharacteristic
116C23×C4—61no—
117C23×C4D1212yes—
118C22×C8—64no—
119#127—34no—
120#127—62no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—62no—
124C4×D4—62no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4C4×D311yescharacteristic
127C2×C22⋊C4—61no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—62no—
130C2×C22⋊C4#6712yes—
131C2×C42—31no—
132C2×C42—31no—
133C2×C42C22×D311yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—62no—
136C22⋊C8—22no—
137C22⋊C8—64no—
138C22⋊C8—62no—
139C4×C8—22no—
140C4×C8—34no—
141#95—62no—
142#95—61no—
143#95—61no—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3C22⋊C814yes—
149C23×C3C2×C22⋊C411yescharacteristic
150C22×D3—22no—
151C22×D3—42no—
152C22×D3—42no—
153C22×D3—41no—
154C22×D3—41no—
155C3×D4—24no—
156C3×D4—24no—
157C2×C3×C4—22no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—44no—
161C2×C3×C4—42no—
162C2×C3×C4—82no—
163C2×C3×C4C4×D412yes—
164C2×C3×C4#12011yescharacteristic
165C2×C3×C4#12112yes—
166C2×C3×C4#12311yescharacteristic
167C2×C3×C4C22×C811yescharacteristic
168C2×C3×C4C2×M1611yescharacteristic
169C2×C3×C4#13112yes—
170#67—88no—
171C2×Dic3—22no—
172C2×Dic3—41no—
173C2×Dic3—41no—
174C2×Dic3—41no—
175C2×Dic3—41no—
176C2×Dic3—42no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179D12—24no—
180D12—42no—
181C4×D3—84no—
182C3×C8—82no—
183C3⋊C8—44no—
184C24—61no—
185C24(C2×D3)⋊C412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—32no—
189C2×D4—68no—
190C2×D4—62no—
191C2×D4—61no—
192C2×D4—62no—
193C2×D4—242no—
194C2×D4—242no—
195C2×D4D3×C812yes—
196C2×D4ΓL(1,𝔽25)12yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4—61no—
202C22×C4—82no—
203C22×C4—122no—
204C22×C4—242no—
205C22×C4(C2×D3)⋊C412yes—
206C22×C4C2×C4×D311yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4C2×#6711yescharacteristic
209C2×C8—42no—
210C2×C8—64no—
211C2×C8—244no—
212C4⋊C4—22no—
213C4⋊C4—62no—
214C22⋊C4—24no—
215C22⋊C4—121no—
216C22⋊C4—121no—
217C22⋊C4—242no—
218C22⋊C4—242no—
219C42—32no—
220C42—61no—
221C42—62no—
222C42D3×D412yes—
223C42#23712yes—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—24no—
227C22×C3—44no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—82no—
231C22×C3#38611yescharacteristic
232C22×C3C2×C22⋊C811yescharacteristic
233C22×C3#40811yescharacteristic
234C2×D3—44no—
235C2×D3—42no—
236C2×D3—82no—
237C2×D3—81no—
238C2×D3—81no—
239C3×C4—22no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—42no—
243C3×C4C8×D412yes—
244C3×C4#43512yes—
245Dic3—42no—
246Dic3—81no—
247Dic3—81no—
248C23—22no—
249C23—22no—
250C23—42no—
251C23—42no—
252C23—62no—
253C23—122no—
254C23—122no—
255C23—121no—
256C23—121no—
257C23(C2×D3)⋊C814yes—
258C23C2×(C2×D3)⋊C411yescharacteristic
259D4—24no—
260D4—24no—
261D4—64no—
262D4—122no—
263D4—248no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—62no—
270C2×C4—82no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—122no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—244no—
279C2×C4#67⋊C412yes—
280C2×C4#88211yescharacteristic
281C2×C4#88611yescharacteristic
282C2×C4C2×D3×C811yescharacteristic
283C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
284C2×C4#90712yes—
285C2×C4#93912yes—
286C8—82no—
287C8—124no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—44no—
291C2×C3#186911yescharacteristic
292C2×C3#191011yescharacteristic
293C2×C3#191111yescharacteristic
294D3—82no—
295C22—21no—
296C22—21no—
297C22—24no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—124no—
303C22—122no—
304C22—242no—
305C22—241no—
306C22—241no—
307C22#495811yescharacteristic
308C22C2×(C2×D3)⋊C811yescharacteristic
309C22#512911yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—122no—
315C4—241no—
316C4—241no—
317C4#67⋊C812yes—
318C4#473012yes—
319C3#910111yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—44no—
323C2—242no—
324C2#6995211yescharacteristic
325C2#7004711yescharacteristic
326C2#7007111yescharacteristic
subgroupG/H
centerC22#4958
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1460C2
socleC22×C3#386
prime radicalC2×#940C4
cosocleC2×C3×C42C23
hypercenter#1460D3
hypocenterC3#9101
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #82301C2semidirect
3. C3×#1460C2semidirect
4. #70756C2semidirect
5. #67737C2semidirect
25. #1460D3semidirect
39. #940C8semidirect
57. C4×C3⋊C8D4semidirect
95. C2×D12C2×C8semidirect
110. C4×Dic3C2×C8semidirect
319. C3#9101semidirect

Maximal subgroups

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

subgroupindex
#823292
#823012
C3×#14602
#707562
#677372
#91013