Finite Group Databasebeta

← #11714023

Subgroups — #11714023

Order 768.

520 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 61 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×#520⋊C3C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic, complemented
3#73151C212yesmaximal
4#73064C211yesmaximal, characteristic, complemented
5#73059C212yesmaximal, complemented
6#32965—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8#520⋊C3—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859C2212yes—
12C2×#859C2212yes—
13#5386—22no—
14#4979—41nomaximal
15C2×#520D311yescharacteristic
16C2×#475—31no—
17C2×#476—31no—
18#2628—31no—
19#1865—32no—
20#1864—32no—
21#1853—32no—
22#1805—31no—
23#1804—31no—
24#1798—31no—
25#1788—32no—
26C22×Dic6—41no—
27C2×#41⋊C3—21no—
28C2×#41⋊C3—21no—
29C2×#41⋊C3Q812yescomplemented
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×C3×C4⋊C4—41no—
32C2×Dic3⋊C4—42no—
33C4×SL(2,𝔽3)—21no—
34C4×SL(2,𝔽3)D412yes—
35#859—22no—
36#859—22no—
37C3⋊#95—41no—
38C3⋊#95—42no—
39C22×#41C2×D311yescharacteristic
40C2×C4⋊Q8—31no—
41#520—22no—
42#520—62no—
43C2×C4×Q8C2×D311yescharacteristic
44C2×C4×Q8C2×D311yescharacteristic
45C2×C4×D4—31no—
46C2×C4×D4—31no—
47C2×#117—31no—
48C22×C4⋊C4—31no—
49#476—64no—
50#475—62no—
51#475—62no—
52C2×#106—32no—
53C2×#106—32no—
54C2×C4⋊C8—31no—
55C2×C4⋊C8—31no—
56C2×#103—32no—
57C2×#103—32no—
58C2×#102—32no—
59C2×#102—32no—
60C2×C22⋊C8—32no—
61#400—61no—
62#399—61no—
63#395—61no—
64#393—61no—
65#389—32no—
66#382—31no—
67#380—32no—
68#340—32no—
69#340—32no—
70#336—32no—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×Dic3—42no—
75C22×Dic3—42no—
76C2×Dic6—82no—
77C2×Dic6—84no—
78#41⋊C3—22no—
79C2×SL(2,𝔽3)C2×D411yescharacteristic
80C2×SL(2,𝔽3)C2×Q811yescharacteristic
81C2×SL(2,𝔽3)#4111yescharacteristic
82C3×C4⋊C4—82no—
83Dic3⋊C4—84no—
84C2×#41—21no—
85C2×#41—21no—
86C2×#41—32no—
87C2×#41—61no—
88C2×#41—61no—
89C2×#41Dic612yescomplemented
90C22×Q8—61no—
91C22×Q8C22×D311yescharacteristic
92C22×D4—31no—
93C23×C4—31no—
94C23×C4—61no—
95C22×C8—32no—
96C22×C8—32no—
97C4⋊Q8—62no—
98C4⋊Q8—62no—
99C4×Q8—21no—
100C4×Q8—32no—
101C4×Q8—61no—
102C4×Q8#6712yes—
103C4×D4—34no—
104C4×D4—62no—
105C4×D4—62no—
106C4×D4—62no—
107#117—62no—
108#117—62no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4S411yescharacteristic
119C2×C22⋊C4—31no—
120C2×C22⋊C4—31no—
121C2×C22⋊C4—62no—
122C2×C42—31no—
123C2×C42—31no—
124#106—64no—
125#106—64no—
126C4⋊C8—62no—
127C4⋊C8—62no—
128#103—62no—
129#103—62no—
130#103—62no—
131#103—62no—
132#102—64no—
133#102—64no—
134C22⋊C8—64no—
135#95—61no—
136#95—62no—
137#95—62no—
138#95—61no—
139C23×C3—41no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—81no—
143C2×C3×C4—81no—
144C2×C3×C4—82no—
145C2×C3×C4—81no—
146C2×C3×C4—82no—
147C2×Dic3—44no—
148C2×Dic3—44no—
149C2×Dic3—82no—
150C2×Dic3—82no—
151C2×Dic3—82no—
152C2×Dic3—82no—
153Dic6—88no—
154SL(2,𝔽3)C22⋊Q811yescharacteristic
155C24—31no—
156#41—22no—
157#41—64no—
158#41—62no—
159#41—62no—
160#41—62no—
161C2×Q8—31no—
162C2×Q8—31no—
163C2×Q8—31no—
164C2×Q8—122no—
165C2×Q8—242no—
166C2×Q8C2×Dic611yescharacteristic
167C2×Q8#23511yescharacteristic
168C2×Q8C2×#6711yescharacteristic
169C2×D4—32no—
170C2×D4—32no—
171C2×D4—32no—
172C2×D4—61no—
173C2×D4—61no—
174C2×D4—61no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—31no—
180C22×C4—61no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4—64no—
185C22×C4—62no—
186C22×C4—62no—
187C22×C4—242no—
188C22×C4C2×S411yescharacteristic
189C22×C4C2×S411yescharacteristic
190C22×C4C2×S411yescharacteristic
191C2×C8—62no—
192C2×C8—62no—
193C2×C8—62no—
194C2×C8—62no—
195C2×C8—62no—
196C2×C8—62no—
197C4⋊C4—22no—
198C4⋊C4—32no—
199C4⋊C4—32no—
200C4⋊C4—62no—
201C4⋊C4—61no—
202C4⋊C4—61no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C4⋊C4—122no—
208C4⋊C4—242no—
209C22⋊C4—62no—
210C22⋊C4—62no—
211C22⋊C4—242no—
212C42—32no—
213C42—32no—
214C42—61no—
215C42—61no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—41no—
221C3×C4—82no—
222C3×C4—82no—
223C3×C4—161no—
224Dic3—84no—
225Dic3—84no—
226C23—32no—
227C23—61no—
228C23—62no—
229C23—62no—
230C23—61no—
231C23C22×S411yescharacteristic
232Q8—31no—
233Q8—31no—
234Q8—31no—
235Q8—61no—
236Q8—244no—
237Q8#92411yescharacteristic
238D4—34no—
239D4—62no—
240D4—62no—
241D4—62no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—244no—
268C2×C4A4⋊Q812yes—
269C2×C4GL(2,ℤ/4)12yes—
270C8—122no—
271C8—122no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—42no—
275C2×C3—42no—
276C2×C3—41no—
277C22—64no—
278C22—62no—
279C22—121no—
280C22—121no—
281C22—122no—
282C22C2×A4⋊Q811yescharacteristic
283C22#591211yescharacteristic
284C22#592412yes—
285C22#592612yes—
286C22C2×GL(2,ℤ/4)11yescharacteristic
287C4—22no—
288C4—22no—
289C4—31no—
290C4—31no—
291C4—31no—
292C4—31no—
293C4—41no—
294C4—62no—
295C4—121no—
296C4—242no—
297C4—242no—
298C3—41noSylow
299C2—122no—
300C2#8536112yes—
301C2#8536912yes—
302C2#8548011yescharacteristic
303C2#8552711yescharacteristic
304C2#8553711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#5386C2semidirect
4. #73064C2semidirect
5. #73059C2semidirect
29. C2×#41⋊C3Q8semidirect
89. C2×#41Dic6semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#53862
#731512
#730642
#730592
#329653
#49794