Finite Group Databasebeta

← #10709075

Subgroups — #10709075

Order 768.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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#79579C211yesmaximal, characteristic
2#70886C211yesmaximal, characteristic
3#69065C211yesmaximal, characteristic, complemented
4#68588C211yesmaximal, characteristic, complemented
5C3×#1331C211yesmaximal, characteristic
6#67506C211yesmaximal, characteristic, complemented
7#67474C211yesmaximal, characteristic, complemented
8#12383—31nomaximal, Sylow
9#5542—21no—
10C3×C4×D8C2211yescharacteristic
11#5155—21no—
12#5019—21no—
13#4873—21no—
14#4849—21no—
15#4767—22no—
16#4699C2211yescharacteristic
17#4682C2211yescharacteristic
18C8×Dic6C2211yescharacteristic
19C3×C4×C16C2211yescharacteristic
20C3×#335C2211yescharacteristic
21#4562—21no—
22#4503—22no—
23#4489—21no—
24#4482—21no—
25C4⋊(C3⋊C16)C2211yescharacteristic
26#3393—31no—
27#2241—31no—
28#2224—31no—
29#1674—31no—
30#1362—31no—
31Q8⋊C16—31no—
32#1331D311yescharacteristic
33C2×C3×D8D411yescharacteristic
34C3×C4×D4—21no—
35#936—22no—
36C2×#214—22no—
37#926—41no—
38#924—41no—
39C2×Dic12—21no—
40C2×Dic12—41no—
41C2×Dic12D412yes—
42#887—22no—
43#879—42no—
44#869—21no—
45#869—21no—
46C4×Dic6D411yescharacteristic
47C2×C3×C16—21no—
48C3×#107D411yescharacteristic
49C3×C4⋊C8—21no—
50C3×#102—21no—
51C3×C4×C8C2311yescharacteristic
52#818—21no—
53#818—21no—
54#816—41no—
55Dic3⋊C8—21no—
56C8×Dic3—21no—
57C2×C3⋊C16—21no—
58#810—22no—
59C4⋊(C3⋊C8)D411yescharacteristic
60#556—61no—
61#500—31no—
62#495—61no—
63#494—31no—
64#464—62no—
65#462—61no—
66#461—61no—
67C8×Q8—31no—
68C4×Q16—61no—
69C4×D8C2×D311yescharacteristic
70#366—61no—
71C4⋊C16—31no—
72#358—62no—
73#357—61no—
74C4×C16C2×D311yescharacteristic
75#335C2×D311yescharacteristic
76#332—61no—
77C2×C3×D4—21no—
78C22×C3×C4—41no—
79C2×Dic6—22no—
80C2×Dic6—41no—
81C2×Dic6D811yescharacteristic
82C3×D8—21no—
83C3×D8—21no—
84C2×C3×C8—41no—
85C2×C3×C8C2×D411yescharacteristic
86C2×C3×C8C2×D411yescharacteristic
87C3×C4⋊C4—21no—
88C3×C22⋊C4—41no—
89C3×C42C2×D411yescharacteristic
90#217—42no—
91#217—42no—
92#214—82no—
93C4⋊Dic3—41no—
94C4⋊Dic3—41no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97C4⋊Dic3D811yescharacteristic
98Dic3⋊C4—21no—
99Dic3⋊C4—42no—
100C4×Dic3—21no—
101C4×Dic3—42no—
102C2×C3⋊C8—21no—
103Dic12—42no—
104Dic12—41no—
105Dic12—41no—
106Dic12—82no—
107C3×C16—41no—
108C3⋊C16—41no—
109C2×Q16—32no—
110C2×Q16—61no—
111C2×Q16—121no—
112C2×SD16—62no—
113C2×D8D1211yescharacteristic
114C4⋊Q8—61no—
115C4⋊Q8—61no—
116#126—122no—
117#124—62no—
118C22⋊Q8—121no—
119C4×Q8—31no—
120C4×Q8—62no—
121C4×D4—21no—
122#117—121no—
123C2×C16—21no—
124C2×C16—61no—
125#107—61no—
126#107—61no—
127#107D1211yescharacteristic
128C4⋊C8—21no—
129C4⋊C8—31no—
130C4⋊C8—61no—
131#103—62no—
132#103—121no—
133#102—21no—
134C4×C8—61no—
135C4×C8C22×D311yescharacteristic
136C23×C3—41no—
137C3×D4—41no—
138C3×D4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—81no—
141C2×C3×C4—81no—
142C2×C3×C4#12011yescharacteristic
143C2×C3×C4C2×D811yescharacteristic
144C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—42no—
148C2×Dic3—81no—
149Dic6—21no—
150Dic6—21no—
151Dic6—42no—
152Dic6—42no—
153Dic6—81no—
154Dic6—81no—
155C3×C8—21no—
156C3×C8—81no—
157C3×C8#13511yescharacteristic
158C3×C8#13711yescharacteristic
159C3⋊C8—41no—
160C2×Q8—31no—
161C2×Q8—62no—
162C2×Q8—121no—
163C2×D4—21no—
164C22×C4—41no—
165Q16—122no—
166Q16—121no—
167Q16—121no—
168Q16—242no—
169SD16—242no—
170D8—21no—
171D8—21no—
172C2×C8—41no—
173C2×C8—61no—
174C2×C8C2×D1211yescharacteristic
175C2×C8D3×D411yescharacteristic
176C4⋊C4—21no—
177C4⋊C4—31no—
178C4⋊C4—61no—
179C4⋊C4—122no—
180C4⋊C4—121no—
181C4⋊C4—121no—
182C4⋊C4—121no—
183C4⋊C4—121no—
184C22⋊C4—41no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C42—61no—
188C42—122no—
189C42D3×D411yescharacteristic
190C16—41no—
191C16—121no—
192C22×C3—81no—
193C22×C3—81no—
194C22×C3—81no—
195C22×C3#44711yescharacteristic
196C3×C4—21no—
197C3×C4—81no—
198C3×C4#45211yescharacteristic
199C3×C4#45511yescharacteristic
200C3×C4#50811yescharacteristic
201C3×C4#50911yescharacteristic
202Dic3—41no—
203Dic3—41no—
204Dic3—82no—
205Dic3—161no—
206C23—41no—
207Q8—61no—
208Q8—61no—
209Q8—122no—
210Q8—122no—
211Q8—241no—
212Q8—241no—
213D4—41no—
214D4—41no—
215C2×C4—41no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—122no—
221C2×C4—241no—
222C2×C4#88211yescharacteristic
223C2×C4D3×D811yescharacteristic
224C2×C4#91411yescharacteristic
225C8—21no—
226C8—81no—
227C8—121no—
228C8#90411yescharacteristic
229C8#90911yescharacteristic
230C2×C3—161no—
231C2×C3#164111yescharacteristic
232C2×C3#218611yescharacteristic
233C2×C3#219511yescharacteristic
234C22—81no—
235C22—81no—
236C22—81no—
237C22#477311yescharacteristic
238C4—21no—
239C4—81no—
240C4—121no—
241C4—121no—
242C4—242no—
243C4—481no—
244C4#473311yescharacteristic
245C4#482511yescharacteristic
246C4#491111yescharacteristic
247C4#491611yescharacteristic
248C3#1238311yescharacteristic, complemented, Sylow
249C2—161no—
250C2#6819011yescharacteristic
251C2#6919211yescharacteristic
252C2#6924511yescharacteristic
subgroupG/H
centerC22#4773
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#1331C2
socleC22×C3#447
prime radicalC2×C3×D8D4
hypercenter#1331D3
hypocenterC3#12383
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. #69065C2semidirect
4. #68588C2semidirect
6. #67506C2semidirect
7. #67474C2semidirect
248. C3#12383semidirect

Maximal subgroups

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

subgroupindex
#795792
#708862
#690652
#685882
C3×#13312
#675062
#674742
#123833