Finite Group Databasebeta

← #10709172

Subgroups — #10709172

Order 768.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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#79419C211yesmaximal, characteristic
2#70853C211yesmaximal, characteristic
3#69296C211yesmaximal, characteristic, complemented
4#69046C211yesmaximal, characteristic, complemented
5C3×D4⋊C16C211yesmaximal, characteristic
6#67564C211yesmaximal, characteristic, complemented
7#67505C211yesmaximal, characteristic, complemented
8#12397—31nomaximal, Sylow
9#5542—21no—
10C3×C8×D4C2211yescharacteristic
11#5116—21no—
12#5106—21no—
13#5008—21no—
14#4851—21no—
15#4723—22no—
16C4×Dic12C2211yescharacteristic
17#4682C2211yescharacteristic
18#4682C2211yescharacteristic
19C3×C4⋊C16C2211yescharacteristic
20C3×C4×C16C2211yescharacteristic
21#4505—22no—
22#4504—21no—
23#4503—21no—
24#4481—21no—
25#4458C2211yescharacteristic
26#3342—31no—
27#2254—31no—
28#2227—31no—
29#1675—31no—
30#1364—31no—
31#1332—31no—
32D4⋊C16D311yescharacteristic
33C22×C3×C8—21no—
34C3×C4×D4D411yescharacteristic
35#936—41no—
36#926—22no—
37#924—22no—
38C2×Dic12D411yescharacteristic
39#887—41no—
40#879—42no—
41#869—21no—
42#869—21no—
43C4×Dic6—21no—
44C2×C3×C16—21no—
45C2×C3×C16—21no—
46C3×C4⋊C8D411yescharacteristic
47C3×C22⋊C8—21no—
48C3×C4×C8C2311yescharacteristic
49#832—41no—
50#818—21no—
51#818—21no—
52#818—21no—
53#818D412yes—
54#818D411yescharacteristic
55#817—22no—
56#816—22no—
57#816—21no—
58#808—41no—
59C4⋊(C3⋊C8)—21no—
60#556—61no—
61#500—31no—
62#500—31no—
63#485—62no—
64#476—61no—
65#475—61no—
66#467—61no—
67C4×Q16—31no—
68C8×D4C2×D311yescharacteristic
69#427—61no—
70#367—62no—
71#366—61no—
72C4⋊C16C2×D311yescharacteristic
73#358—61no—
74C4×C16C2×D311yescharacteristic
75#335—31no—
76#333—61no—
77C2×C3×D4D811yescharacteristic
78C22×C3×C4—21no—
79C2×Dic6—21no—
80C2×Dic6—41no—
81C2×Dic6—41no—
82C2×C3×C8—21no—
83C2×C3×C8—41no—
84C2×C3×C8—41no—
85C2×C3×C8C2×D411yescharacteristic
86C2×C3×C8C2×D411yescharacteristic
87C3×C4⋊C4D811yescharacteristic
88C3×C22⋊C4—21no—
89C3×C42C2×D411yescharacteristic
90#217—42no—
91#217—42no—
92C4⋊Dic3—21no—
93C4⋊Dic3—22no—
94C4⋊Dic3—41no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97Dic3⋊C4—42no—
98Dic3⋊C4—41no—
99C4×Dic3—42no—
100C4×Dic3—41no—
101C2×C3⋊C8—41no—
102Dic12—21no—
103Dic12—21no—
104C3×C16—22no—
105C3×C16—41no—
106C2×Q16—31no—
107C22×C8—21no—
108C4⋊Q8—61no—
109C4⋊Q8—61no—
110#126—122no—
111#124—121no—
112C22⋊Q8—62no—
113C4×Q8—61no—
114C4×Q8—121no—
115C4×D4D1211yescharacteristic
116#117—62no—
117C2×C16—21no—
118C2×C16—21no—
119#107—31no—
120#107—32no—
121#107—61no—
122#107—61no—
123#107—61no—
124#106—62no—
125#106—121no—
126C4⋊C8—61no—
127C4⋊C8D1211yescharacteristic
128#103—62no—
129#103—61no—
130#102—121no—
131C22⋊C8—21no—
132C4×C8C22×D311yescharacteristic
133C23×C3—21no—
134C3×D4—21no—
135C3×D4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4#12011yescharacteristic
140C2×C3×C4C2×D811yescharacteristic
141C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
142C2×Dic3—41no—
143C2×Dic3—41no—
144C2×Dic3—42no—
145C2×Dic3—81no—
146Dic6—41no—
147Dic6—41no—
148Dic6—81no—
149Dic6—81no—
150Dic6—161no—
151C3×C8—21no—
152C3×C8—41no—
153C3×C8#13511yescharacteristic
154C3×C8#13711yescharacteristic
155C3⋊C8—81no—
156C2×Q8—61no—
157C2×Q8—121no—
158C2×Q8—121no—
159C2×D4D2411yescharacteristic
160C22×C4—21no—
161Q16—61no—
162Q16—61no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—121no—
167C2×C8D3×D411yescharacteristic
168C2×C8D3×D411yescharacteristic
169C4⋊C4—61no—
170C4⋊C4—62no—
171C4⋊C4—122no—
172C4⋊C4—121no—
173C4⋊C4—121no—
174C4⋊C4—121no—
175C4⋊C4—121no—
176C4⋊C4D2411yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—122no—
179C22⋊C4—122no—
180C42—122no—
181C42—121no—
182C42C2×D1211yescharacteristic
183C16—22no—
184C16—41no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3#44711yescharacteristic
189C3×C4—21no—
190C3×C4—41no—
191C3×C4#45211yescharacteristic
192C3×C4#45611yescharacteristic
193C3×C4#50811yescharacteristic
194C3×C4#51011yescharacteristic
195Dic3—81no—
196Dic3—82no—
197Dic3—81no—
198Dic3—161no—
199C23—21no—
200Q8—121no—
201Q8—121no—
202Q8—241no—
203Q8—241no—
204Q8—481no—
205D4—21no—
206D4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—121no—
211C2×C4—122no—
212C2×C4—121no—
213C2×C4—241no—
214C2×C4#88211yescharacteristic
215C2×C4C2×D2411yescharacteristic
216C2×C4#90811yescharacteristic
217C8—21no—
218C8—41no—
219C8—241no—
220C8#91211yescharacteristic
221C8#91511yescharacteristic
222C2×C3—81no—
223C2×C3#164711yescharacteristic
224C2×C3#218911yescharacteristic
225C2×C3#219311yescharacteristic
226C22—41no—
227C22—41no—
228C22—41no—
229C22#473211yescharacteristic
230C4—21no—
231C4—41no—
232C4—241no—
233C4—241no—
234C4—242no—
235C4—481no—
236C4#478311yescharacteristic
237C4#482611yescharacteristic
238C4#490411yescharacteristic
239C4#490911yescharacteristic
240C3#1239711yescharacteristic, complemented, Sylow
241C2—81no—
242C2#6820811yescharacteristic
243C2#6911111yescharacteristic
244C2#6912711yescharacteristic
subgroupG/H
centerC22#4732
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×D4⋊C16C2
socleC22×C3#447
prime radicalC2×C3×D4D8
hypercenterD4⋊C16D3
hypocenterC3#12397
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ D4⋊C16

Complements

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

NH ≅ G/Ntype
3. #69296C2semidirect
4. #69046C2semidirect
6. #67564C2semidirect
7. #67505C2semidirect
240. C3#12397semidirect

Maximal subgroups

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

subgroupindex
#794192
#708532
#692962
#690462
C3×D4⋊C162
#675642
#675052
#123973