Finite Group Databasebeta

← #10709170

Subgroups — #10709170

Order 768.

278 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C3×#3342C211yesmaximal, characteristic
2#71385C211yesmaximal, characteristic, complemented
3#70911C211yesmaximal, characteristic, complemented
4#70853C211yesmaximal, characteristic
5#67801C211yesmaximal, characteristic, complemented
6#67765C211yesmaximal, characteristic, complemented
7D4⋊(C3⋊C16)C211yesmaximal, characteristic
8#12397—31nomaximal, Sylow
9C3×#556—21no—
10C3×#500C2211yescharacteristic
11C3×#485—22no—
12C3×#475—21no—
13C3×#467—21no—
14C3×C4×Q16C2211yescharacteristic
15C3×C8×D4C2211yescharacteristic
16C3×#427—21no—
17#5008—21no—
18#4682C2211yescharacteristic
19#4564—21no—
20#4490—22no—
21#4489—21no—
22#4481—21no—
23C4⋊(C3⋊C16)C2211yescharacteristic
24C4×C3⋊C16C2211yescharacteristic
25#4458C2211yescharacteristic
26#3342D311yescharacteristic
27#2254—31no—
28#2227—31no—
29#1675—31no—
30#1364—31no—
31#1332—31no—
32D4⋊C16—31no—
33C2×C3×Q16D411yescharacteristic
34C22×C3×C8—21no—
35C3×C4⋊Q8—21no—
36C3×#126—42no—
37C3×#124—41no—
38C3×C22⋊Q8—22no—
39C3×C4×Q8—21no—
40C3×C4×Q8—41no—
41C3×C4×D4D411yescharacteristic
42C3×#117—22no—
43#869—21no—
44C3×#107—21no—
45C3×#107D411yescharacteristic
46C3×#107D412yes—
47C3×#106—22no—
48C3×C4⋊C8D411yescharacteristic
49C3×#103—22no—
50C3×#103—21no—
51C3×C22⋊C8—21no—
52C3×C4×C8C2311yescharacteristic
53#832—41no—
54#818—21no—
55#818—21no—
56C2×C3⋊C16—21no—
57C2×C3⋊C16—21no—
58#808—41no—
59C4⋊(C3⋊C8)—21no—
60#556—21no—
61#500—31no—
62#500C2×D311yescharacteristic
63#485—22no—
64#476—61no—
65#475—21no—
66#467—21no—
67C4×Q16C2×D311yescharacteristic
68C8×D4C2×D311yescharacteristic
69#427—21no—
70#367—62no—
71#366—61no—
72C4⋊C16—31no—
73#358—61no—
74C4×C16—31no—
75#335—31no—
76#333—61no—
77C2×C3×Q8—21no—
78C2×C3×Q8—41no—
79C2×C3×D4D811yescharacteristic
80C22×C3×C4—21no—
81C2×Dic6—41no—
82C3×Q16—21no—
83C3×Q16—21no—
84C2×C3×C8—21no—
85C2×C3×C8—41no—
86C2×C3×C8—41no—
87C2×C3×C8C2×D411yescharacteristic
88C2×C3×C8C2×D411yescharacteristic
89C3×C4⋊C4—21no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—41no—
94C3×C4⋊C4D811yescharacteristic
95C3×C22⋊C4—21no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—42no—
98C3×C42—42no—
99C3×C42—41no—
100C3×C42C2×D411yescharacteristic
101C4⋊Dic3—41no—
102C4⋊Dic3—41no—
103C2×C3⋊C8—41no—
104C3⋊C16—22no—
105C3⋊C16—41no—
106C2×Q16#6711yescharacteristic
107C22×C8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—61no—
110#126—42no—
111#124—41no—
112C22⋊Q8—22no—
113C4×Q8—21no—
114C4×Q8—41no—
115C4×D4#6711yescharacteristic
116#117—22no—
117C2×C16—61no—
118C2×C16—61no—
119#107—21no—
120#107—61no—
121#107—61no—
122#107#6711yescharacteristic
123#107#6712yes—
124#106—22no—
125#106—121no—
126C4⋊C8—61no—
127C4⋊C8#6711yescharacteristic
128#103—22no—
129#103—21no—
130#102—121no—
131C22⋊C8—21no—
132C4×C8C22×D311yescharacteristic
133C23×C3—21no—
134C3×Q8—41no—
135C3×Q8—41no—
136C3×Q8—81no—
137C3×Q8—81no—
138C3×D4—21no—
139C3×D4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—42no—
145C2×C3×C4—41no—
146C2×C3×C4#12011yescharacteristic
147C2×C3×C4C2×D811yescharacteristic
148C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
149C2×Dic3—81no—
150Dic6—161no—
151C3×C8—21no—
152C3×C8—41no—
153C3×C8#13511yescharacteristic
154C3×C8#13711yescharacteristic
155C3⋊C8—81no—
156C2×Q8—21no—
157C2×Q8—41no—
158C2×Q8—121no—
159C2×D4#21311yescharacteristic
160C22×C4—21no—
161Q16—21no—
162Q16—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—121no—
167C2×C8C2×#6711yescharacteristic
168C2×C8C2×#6711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—121no—
175C4⋊C4—121no—
176C4⋊C4#21311yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C42—42no—
181C42—41no—
182C42C2×#6711yescharacteristic
183C16—62no—
184C16—121no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3#44711yescharacteristic
189C3×C4—21no—
190C3×C4—41no—
191C3×C4—81no—
192C3×C4—82no—
193C3×C4—81no—
194C3×C4#45211yescharacteristic
195C3×C4#45611yescharacteristic
196C3×C4#50811yescharacteristic
197C3×C4#51011yescharacteristic
198Dic3—161no—
199C23—21no—
200Q8—41no—
201Q8—41no—
202Q8—81no—
203Q8—81no—
204Q8—481no—
205D4—21no—
206D4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—241no—
214C2×C4C2×#21311yescharacteristic
215C2×C4#93211yescharacteristic
216C2×C4#95311yescharacteristic
217C8—21no—
218C8—41no—
219C8—241no—
220C8#95011yescharacteristic
221C8#95111yescharacteristic
222C2×C3—81no—
223C2×C3#164711yescharacteristic
224C2×C3#218911yescharacteristic
225C2×C3#219311yescharacteristic
226C22—41no—
227C22—41no—
228C22—41no—
229C22#521711yescharacteristic
230C4—21no—
231C4—41no—
232C4—82no—
233C4—81no—
234C4—81no—
235C4—481no—
236C4#519411yescharacteristic
237C4#519511yescharacteristic
238C4#523911yescharacteristic
239C4#524611yescharacteristic
240C3#1239711yescharacteristic, complemented, Sylow
241C2—81no—
242C2#7191211yescharacteristic
243C2#7208311yescharacteristic
244C2#7212411yescharacteristic
subgroupG/H
centerC22#5217
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#3342C2
socleC22×C3#447
prime radicalC2×C3×D4D8
hypercenter#3342D3
hypocenterC3#12397
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #3342

Complements

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

NH ≅ G/Ntype
2. #71385C2semidirect
3. #70911C2semidirect
5. #67801C2semidirect
6. #67765C2semidirect
240. C3#12397semidirect

Maximal subgroups

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

subgroupindex
C3×#33422
#713852
#709112
#708532
#678012
#677652
D4⋊(C3⋊C16)2
#123973