Finite Group Databasebeta

← #10694870

Subgroups — #10694870

Order 768.

402 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 117 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×#5078C211yesmaximal, characteristic
2C3×#1997C211yesmaximal, characteristic
3#71775C211yesmaximal, characteristic
4#70508C211yesmaximal, characteristic
5#70490C211yesmaximal, characteristic
6#69879C211yesmaximal, characteristic
7#69847C211yesmaximal, characteristic
8#9810—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5078—21no—
14#5078—21no—
15#5078—22no—
16C2×#807C414yes—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20C3×#343—22no—
21#4554—22no—
22#4547—22no—
23#4486—22no—
24#4480—22no—
25#4454—21no—
26#4454—21no—
27C2×#500—31no—
28#1997—31no—
29#1997D311yescharacteristic
30#1566—31no—
31#1549—31no—
32#1549—31no—
33#1451—31no—
34C22×C3×Q8—21no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C2×C3×M16D412yes—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C412yes—
44C2×C3×C42C2311yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3C2×C412yes—
47C2×#208—21no—
48C22×C3⋊C8C2×C412yes—
49#869—21no—
50#869—21no—
51#869—21no—
52#869—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8—21no—
55C3×#97—21no—
56C3×C4×C8—21no—
57#830—21no—
58#807—24no—
59#807—24no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)D412yes—
63C4×C3⋊C8—21no—
64C4×C3⋊C8D411yescharacteristic
65C4×C3⋊C8D411yescharacteristic
66C4×C3⋊C8D412yes—
67C2×C4⋊Q8—31no—
68C2×C4⋊Q8C2×D311yescharacteristic
69#500—61no—
70#500—61no—
71#500—62no—
72#431—31no—
73C2×#107—34no—
74#423—31no—
75#423C2×D311yescharacteristic
76C4×M16C2×D311yescharacteristic
77C2×C4×C8—31no—
78#343—22no—
79#343—62no—
80#334—62no—
81#332—62no—
82#332—62no—
83#322—61no—
84#322—61no—
85C2×C3×Q8—41no—
86C2×C3×Q8—41no—
87C2×C3×Q8—82no—
88C22×C3×C4—22no—
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—22no—
93C2×Dic6—41no—
94C2×Dic6—41no—
95C2×Dic6—82no—
96C3×M16—22no—
97C3×M16—22no—
98C3×M16—42no—
99C2×C3×C8—22no—
100C2×C3×C8—41no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C42C2×D411yescharacteristic
106C3×C42C2×D411yescharacteristic
107C3×C42C2×D411yescharacteristic
108C3×C42#4111yescharacteristic
109C4⋊Dic3—22no—
110C4⋊Dic3—22no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113#208—42no—
114C2×C3⋊C8—24no—
115C2×C3⋊C8—22no—
116C2×C3⋊C8—41no—
117C2×C3⋊C8C22⋊C412yes—
118C2×C3⋊C8C22⋊C412yes—
119C22×Q8—21no—
120C22×Q8—61no—
121C2×M16—21no—
122C2×M16—61no—
123C2×M16#6712yes—
124C22×C8—32no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—61no—
130C4⋊Q8—61no—
131C4⋊Q8—61no—
132C4⋊Q8—61no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4C4×D312yes—
137C2×C42C22×D311yescharacteristic
138#107—64no—
139#107—64no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—32no—
143C4⋊C8—61no—
144C4⋊C8—61no—
145C22⋊C8—61no—
146#97—21no—
147C4×C8—21no—
148C4×C8—31no—
149C4×C8—31no—
150C4×C8—32no—
151C4×C8—61no—
152C23×C3C4×D411yescharacteristic
153C3×Q8—82no—
154C3×Q8—82no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—42no—
158C2×C3×C4—82no—
159C2×C3×C4C2×C22⋊C411yescharacteristic
160C2×C3×C4C4×D411yescharacteristic
161C2×C3×C4#12111yescharacteristic
162C2×C3×C4#12111yescharacteristic
163C2×C3×C4#12411yescharacteristic
164C2×C3×C4#12711yescharacteristic
165C2×C3×C4#13712yes—
166C2×C3×C4#13712yes—
167C2×Dic3—42no—
168C2×Dic3—82no—
169Dic6—82no—
170Dic6—82no—
171C3×C8—42no—
172C3×C8—81no—
173C3⋊C8—22no—
174C3⋊C8—42no—
175C3⋊C8—81no—
176C3⋊C8#10114yes—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—82no—
180C2×Q8—121no—
181C2×Q8—121no—
182C2×Q8—242no—
183C22×C4—22no—
184C22×C4—62no—
185C22×C4C2×C4×D311yescharacteristic
186C22×C4#23511yescharacteristic
187C22×C4C2×#6711yescharacteristic
188M16—22no—
189M16—22no—
190M16—42no—
191M16—122no—
192C2×C8—22no—
193C2×C8—32no—
194C2×C8—32no—
195C2×C8—41no—
196C2×C8—64no—
197C2×C8—62no—
198C2×C8—121no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C4⋊C4—121no—
206C4⋊C4—121no—
207C42D3×D411yescharacteristic
208C42D3×D411yescharacteristic
209C42D3×D411yescharacteristic
210C42#23711yescharacteristic
211C22×C3—21no—
212C22×C3—21no—
213C22×C3#39011yescharacteristic
214C22×C3#44112yes—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—82no—
218C3×C4C2×#10112yes—
219C3×C4#47012yes—
220C3×C4#47212yes—
221C3×C4#49711yescharacteristic
222C3×C4#49711yescharacteristic
223Dic3—82no—
224C23C4×#6711yescharacteristic
225Q8—82no—
226Q8—82no—
227Q8—242no—
228Q8—242no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—122no—
234C2×C4—242no—
235C2×C4D3×C22⋊C411yescharacteristic
236C2×C4#67⋊C411yescharacteristic
237C2×C4#88411yescharacteristic
238C2×C4#88511yescharacteristic
239C2×C4#93911yescharacteristic
240C2×C4#94011yescharacteristic
241C2×C4#94212yes—
242C2×C4#95112yes—
243C8—34no—
244C8—42no—
245C8—62no—
246C8—81no—
247C8—122no—
248C8—241no—
249C2×C3—22no—
250C2×C3#196611yescharacteristic
251C2×C3#197111yescharacteristic
252C2×C3#197811yescharacteristic
253C22—21no—
254C22—21no—
255C22#495711yescharacteristic
256C22#503012yes—
257C4—21no—
258C4—21no—
259C4—82no—
260C4—242no—
261C4#474912yes—
262C4D3×#10112yes—
263C4#505212yes—
264C4#506011yescharacteristic
265C4#508611yescharacteristic
266C3#981011yescharacteristic, complemented, Sylow
267C2—22no—
268C2#6967811yescharacteristic
269C2#7007411yescharacteristic
270C2#7009811yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1997D3
hypocenterC3#9810
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
266. C3#9810semidirect

Maximal subgroups

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

subgroupindex
C2×#50782
C3×#19972
#717752
#705082
#704902
#698792
#698472
#98103