Finite Group Databasebeta

← #10708554

Subgroups — #10708554

Order 768.

274 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 53 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#80859C211yesmaximal, characteristic
2C3×#2238C211yesmaximal, characteristic
3#70726C211yesmaximal, characteristic, complemented
4#70724C211yesmaximal, characteristic, complemented
5#67779C211yesmaximal, characteristic
6#67537C211yesmaximal, characteristic, complemented
7#67533C211yesmaximal, characteristic, complemented
8#12306—31nomaximal, Sylow
9#5603—21no—
10C3×#469C2211yescharacteristic
11C3×#430C2211yescharacteristic
12#5153—21no—
13#5137C2211yescharacteristic
14#5119C2211yescharacteristic
15#5041—21no—
16#4854—21no—
17#4767—21no—
18#4761—21no—
19C3×#365—21no—
20C3×#361—21no—
21C3×#349C2211yescharacteristic
22C3⋊#338C2211yescharacteristic
23#4558—21no—
24#4556—21no—
25#4553C2211yescharacteristic
26#4496—21no—
27#4494—21no—
28#3263—31no—
29#2239—31no—
30#2238D311yescharacteristic
31#2103—31no—
32#1342—31no—
33#1341—31no—
34#1328—31no—
35C2×C3×D8D411yescharacteristic
36C2×C3×M16—21no—
37C2×C3×M16C2311yescharacteristic
38C3×#121—21no—
39#936—21no—
40#936—41no—
41#935—41no—
42D4×Dic3—21no—
43C2×#214—21no—
44#926D411yescharacteristic
45#924—21no—
46C2×#208—21no—
47C22×C3⋊C8D411yescharacteristic
48C2×Dic12D411yescharacteristic
49#888—21no—
50#887—21no—
51#879—41no—
52#878—41no—
53C3×M32—21no—
54C3×#108—21no—
55C3×#108—21no—
56C3×#106D411yescharacteristic
57C3×#102—21no—
58#832—21no—
59#817D411yescharacteristic
60#816—21no—
61Dic3⋊C8—21no—
62Dic3⋊C8—21no—
63#812—21no—
64#810—21no—
65#808—21no—
66#542—61no—
67#476—61no—
68#475—61no—
69#470—31no—
70#469C2×D311yescharacteristic
71#465—61no—
72#464—61no—
73#463—61no—
74#430C2×D311yescharacteristic
75#424—31no—
76#365—21no—
77#365—61no—
78#362—61no—
79#361—21no—
80#349—31no—
81#349C2×D311yescharacteristic
82#344—61no—
83#341—61no—
84#338—31no—
85C2×C3×D4—21no—
86C2×C3×D4—41no—
87C22×C3×C4C2×D411yescharacteristic
88C22×Dic3—41no—
89C2×Dic6—21no—
90C2×Dic6—41no—
91C3×D8—41no—
92C3×M16—41no—
93C3×M16—41no—
94C3×M16—41no—
95C2×C3×C8—41no—
96C2×C3×C8C2×D411yescharacteristic
97C2×C3×C8C2×D411yescharacteristic
98C3×C4⋊C4—21no—
99C3×C22⋊C4—41no—
100#217—41no—
101#217—41no—
102#217—41no—
103#217—41no—
104#217SD1611yescharacteristic
105#217SD1611yescharacteristic
106#214—81no—
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109Dic3⋊C4—41no—
110Dic3⋊C4—41no—
111Dic3⋊C4—41no—
112C4×Dic3—21no—
113C4×Dic3—41no—
114#208—41no—
115#208—41no—
116C2×C3⋊C8—21no—
117C2×C3⋊C8—41no—
118C2×C3⋊C8—41no—
119Dic12—41no—
120C3×C16—41no—
121C3⋊C16—41no—
122C2×Q16—31no—
123C2×SD16—61no—
124C2×D8#6711yescharacteristic
125C2×M16—21no—
126C2×M16—61no—
127C2×M16C22×D311yescharacteristic
128C22×C8—31no—
129C4⋊Q8—61no—
130#126—121no—
131#124—61no—
132#124—121no—
133#123—121no—
134C22⋊Q8—61no—
135C22⋊Q8—121no—
136#121—21no—
137C4×Q8—61no—
138C4×D4—61no—
139#117—31no—
140M32—21no—
141M32—61no—
142#108—21no—
143#108—21no—
144#106—31no—
145#106—61no—
146#106#6711yescharacteristic
147C4⋊C8—61no—
148C4⋊C8—61no—
149#103—61no—
150#103—61no—
151#102—21no—
152#102—61no—
153C23×C3—41no—
154C23×C3C2×SD1611yescharacteristic
155C3×D4—41no—
156C3×D4—41no—
157C3×D4—161no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—41no—
161C2×C3×C4#12011yescharacteristic
162C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
163C2×Dic3—21no—
164C2×Dic3—21no—
165C2×Dic3—41no—
166C2×Dic3—41no—
167C2×Dic3—161no—
168Dic6—41no—
169Dic6—41no—
170Dic6—161no—
171C3×C8—21no—
172C3×C8—21no—
173C3×C8—81no—
174C3⋊C8—41no—
175C3⋊C8—81no—
176C2×Q8—61no—
177C2×Q8—121no—
178C2×D4—21no—
179C2×D4—41no—
180C22×C4—121no—
181C22×C4D3×D411yescharacteristic
182Q16—121no—
183SD16—241no—
184D8—41no—
185M16—41no—
186M16—41no—
187M16—41no—
188M16—121no—
189M16—121no—
190C2×C8—41no—
191C2×C8—61no—
192C2×C8—121no—
193C2×C8—121no—
194C2×C8D3×D411yescharacteristic
195C2×C8C2×#6711yescharacteristic
196C4⋊C4—21no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—121no—
200C4⋊C4—121no—
201C4⋊C4—121no—
202C22⋊C4—31no—
203C22⋊C4—31no—
204C22⋊C4—41no—
205C22⋊C4—121no—
206C22⋊C4—121no—
207C22⋊C4—121no—
208C22⋊C4—121no—
209C42—61no—
210C42—121no—
211C16—41no—
212C16—121no—
213C22×C3—21no—
214C22×C3—41no—
215C22×C3—81no—
216C22×C3—81no—
217C22×C3—81no—
218C22×C3#45011yescharacteristic
219C3×C4—21no—
220C3×C4—81no—
221C3×C4#45411yescharacteristic
222C3×C4#45811yescharacteristic
223Dic3—41no—
224Dic3—41no—
225Dic3—81no—
226Dic3—81no—
227C23—41no—
228C23D3×SD1611yescharacteristic
229Q8—121no—
230Q8—121no—
231Q8—481no—
232D4—41no—
233D4—41no—
234D4—161no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—121no—
241C2×C4—121no—
242C2×C4—481no—
243C2×C4#91411yescharacteristic
244C2×C4#93711yescharacteristic
245C8—21no—
246C8—21no—
247C8—81no—
248C8—121no—
249C8—241no—
250C2×C3—21no—
251C2×C3—41no—
252C2×C3—161no—
253C2×C3#161611yescharacteristic
254C22—21no—
255C22—41no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22#516911yescharacteristic
260C4—21no—
261C4—81no—
262C4—121no—
263C4—121no—
264C4—241no—
265C4—241no—
266C4#505311yescharacteristic
267C4#520111yescharacteristic
268C3#1230611yescharacteristic, complemented, Sylow
269C2—21no—
270C2—41no—
271C2—161no—
272C2#7117111yescharacteristic
subgroupG/H
centerC2#71171
commutator, cosocleC2×C3×M16C23
FrattiniC2×M16C22×D3
FittingC3×#2238C2
socleC2×C3#1616
prime radicalC3×#469C22
hypercenter#2238D3
hypocenterC3#12306
LCSG ▷ C2×C3×M16 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2238

Complements

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

NH ≅ G/Ntype
3. #70726C2semidirect
4. #70724C2semidirect
6. #67537C2semidirect
7. #67533C2semidirect
268. C3#12306semidirect

Maximal subgroups

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

subgroupindex
#808592
C3×#22382
#707262
#707242
#677792
#675372
#675332
#123063