Finite Group Databasebeta

← #10709243

Subgroups — #10709243

Order 768.

290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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#79574C211yesmaximal, characteristic
2#70886C211yesmaximal, characteristic
3#69046C211yesmaximal, characteristic, complemented
4#68588C211yesmaximal, characteristic, complemented
5C3×#1331C211yesmaximal, characteristic
6#67506C211yesmaximal, characteristic, complemented
7#67476C211yesmaximal, characteristic, complemented
8#12407—31nomaximal, Sylow
9#5543—21no—
10C3×C4×D8C2211yescharacteristic
11Dic3×D8—21no—
12#5008—21no—
13#4878—21no—
14#4849—21no—
15#4766—22no—
16#4682C2211yescharacteristic
17#4682C2211yescharacteristic
18C8×Dic6C2211yescharacteristic
19C3×C4×C16C2211yescharacteristic
20C3×#335C2211yescharacteristic
21#4562—21no—
22#4505—22no—
23#4489—21no—
24#4482—21no—
25C4⋊(C3⋊C16)C2211yescharacteristic
26#3380—31no—
27#2254—31no—
28#2224—31no—
29#1674—31no—
30#1365—31no—
31Q8⋊C16—31no—
32#1331D311yescharacteristic
33C2×C3×D8D411yescharacteristic
34C3×C4×D4—21no—
35D4×Dic3—22no—
36C2×C4⋊Dic3—41no—
37#924—41no—
38C2×Dic12—41no—
39#890—22no—
40#878—42no—
41#869—21no—
42#869—21no—
43C4×Dic6D411yescharacteristic
44C2×C3×C16—21no—
45C3×#107D411yescharacteristic
46C3×C4⋊C8—21no—
47C3×#102—21no—
48C3×C4×C8C2311yescharacteristic
49#832—22no—
50#818—21no—
51#818—21no—
52#818—21no—
53#818D412yes—
54#816—41no—
55Dic3⋊C8—21no—
56C8×Dic3—21no—
57C2×C3⋊C16—21no—
58#808—22no—
59C4⋊(C3⋊C8)D411yescharacteristic
60#554—61no—
61#500—31no—
62#500—31no—
63#499—61no—
64#478—62no—
65#476—61no—
66#462—61no—
67C8×Q8—31no—
68C4×D8—61no—
69C4×D8C2×D311yescharacteristic
70#367—62no—
71#366—61no—
72C4⋊C16—31no—
73#357—61no—
74C4×C16C2×D311yescharacteristic
75#335C2×D311yescharacteristic
76#332—61no—
77C2×C3×D4—21no—
78C22×C3×C4—41no—
79C22×Dic3—42no—
80C2×Dic6—41no—
81C2×Dic6—41no—
82C2×Dic6D811yescharacteristic
83C3×D8—21no—
84C3×D8Q1612yescomplemented
85C2×C3×C8—41no—
86C2×C3×C8C2×D411yescharacteristic
87C2×C3×C8C2×D411yescharacteristic
88C3×C4⋊C4—21no—
89C3×C22⋊C4—41no—
90C3×C42C2×D411yescharacteristic
91#217—42no—
92C4⋊Dic3—22no—
93C4⋊Dic3—42no—
94C4⋊Dic3—41no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97C4⋊Dic3D811yescharacteristic
98Dic3⋊C4—21no—
99Dic3⋊C4—42no—
100C4×Dic3—21no—
101C2×C3⋊C8—21no—
102Dic12—82no—
103C3×C16—22no—
104C3⋊C16—41no—
105C2×Q16—121no—
106C2×D8D1211yescharacteristic
107C4⋊Q8—61no—
108C4⋊Q8—61no—
109#125—62no—
110C22⋊Q8—122no—
111C22⋊Q8—121no—
112C4×Q8—31no—
113C4×D4—21no—
114C4×D4—62no—
115C2×C4⋊C4—121no—
116C2×C16—21no—
117C2×C16—61no—
118#107—32no—
119#107—61no—
120#107—61no—
121#107—61no—
122#107D1211yescharacteristic
123#106—62no—
124C4⋊C8—21no—
125C4⋊C8—31no—
126C4⋊C8—61no—
127#103—121no—
128#102—21no—
129#102—62no—
130C4×C8—61no—
131C4×C8C22×D311yescharacteristic
132C23×C3—41no—
133C3×D4—22no—
134C3×D4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—81no—
137C2×C3×C4—81no—
138C2×C3×C4#12011yescharacteristic
139C2×C3×C4C2×D811yescharacteristic
140C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—42no—
144C2×Dic3—81no—
145C2×Dic3—162no—
146Dic6—21no—
147Dic6—21no—
148Dic6—81no—
149Dic6—81no—
150Dic6—161no—
151C3×C8—21no—
152C3×C8—81no—
153C3×C8C2×Q1611yescharacteristic
154C3×C8#13711yescharacteristic
155C3⋊C8—41no—
156C2×Q8—31no—
157C2×Q8—121no—
158C2×Q8—121no—
159C2×D4—21no—
160C22×C4—41no—
161C22×C4—122no—
162Q16—242no—
163D8—21no—
164D8Dic1212yescomplemented
165C2×C8—41no—
166C2×C8—61no—
167C2×C8C2×D1211yescharacteristic
168C2×C8D3×D411yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—31no—
171C4⋊C4—61no—
172C4⋊C4—62no—
173C4⋊C4—122no—
174C4⋊C4—122no—
175C4⋊C4—121no—
176C4⋊C4—121no—
177C4⋊C4—121no—
178C22⋊C4—41no—
179C22⋊C4—122no—
180C42—61no—
181C42D3×D411yescharacteristic
182C16—22no—
183C16—121no—
184C22×C3—42no—
185C22×C3—81no—
186C22×C3—81no—
187C22×C3#44711yescharacteristic
188C3×C4—21no—
189C3×C4—81no—
190C3×C4#45111yescharacteristic
191C3×C4#45611yescharacteristic
192C3×C4#50811yescharacteristic
193C3×C4#50911yescharacteristic
194Dic3—41no—
195Dic3—41no—
196Dic3—82no—
197Dic3—161no—
198C23—41no—
199Q8—61no—
200Q8—61no—
201Q8—241no—
202Q8—241no—
203Q8—481no—
204D4—22no—
205D4—41no—
206C2×C4—41no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—122no—
212C2×C4—241no—
213C2×C4—482no—
214C2×C4#88211yescharacteristic
215C2×C4D3×D811yescharacteristic
216C2×C4#91411yescharacteristic
217C8—21no—
218C8—81no—
219C8—121no—
220C8C2×Dic1211yescharacteristic
221C8#90911yescharacteristic
222C2×C3—82no—
223C2×C3#164911yescharacteristic
224C2×C3#218611yescharacteristic
225C2×C3#219411yescharacteristic
226C22—42no—
227C22—81no—
228C22—81no—
229C22#477311yescharacteristic
230C4—21no—
231C4—81no—
232C4—121no—
233C4—121no—
234C4—242no—
235C4—481no—
236C4#473911yescharacteristic
237C4#482611yescharacteristic
238C4#491111yescharacteristic
239C4#491611yescharacteristic
240C3#1240711yescharacteristic, complemented, Sylow
241C2—82no—
242C2#6819211yescharacteristic
243C2#6919211yescharacteristic
244C2#6924411yescharacteristic
subgroupG/H
centerC22#4773
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#1331C2
socleC22×C3#447
prime radicalC2×C3×D8D4
hypercenter#1331D3
hypocenterC3#12407
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
3. #69046C2semidirect
4. #68588C2semidirect
6. #67506C2semidirect
7. #67476C2semidirect
84. C3×D8Q16semidirect
164. D8Dic12semidirect
240. C3#12407semidirect

Maximal subgroups

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

subgroupindex
#795742
#708862
#690462
#685882
C3×#13312
#675062
#674762
#124073