Finite Group Databasebeta

← #11713701

Subgroups — #11713701

Order 768.

512 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 55 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic, complemented
3#73151C212yesmaximal, complemented
4#73151C212yesmaximal, complemented
5#73063C211yesmaximal, characteristic, complemented
6#25893—31nomaximal, Sylow
7C22×#41⋊C3C2212yes—
8C4⋊(#41⋊C3)—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2211yescharacteristic
11C2×#859C2211yescharacteristic
12C2×#859C2212yes—
13#5389—22no—
14#4985—41nomaximal
15C2×#550D311yescharacteristic
16C2×#490—31no—
17C2×#479—31no—
18#2562—31no—
19#2034—32no—
20#2033—32no—
21#1982—31no—
22#1955—31no—
23#1788—32no—
24#1788—32no—
25#1763—31no—
26C2×#41⋊C3—22no—
27C2×#41⋊C3—22no—
28C2×#41⋊C3—22no—
29C22×SL(2,𝔽3)C2311yescharacteristic
30C2×C3×C4⋊C4—41no—
31C2×Dic3⋊C4—41no—
32C4×SL(2,𝔽3)D412yes—
33#859—21no—
34#859—21no—
35#859—22no—
36C3⋊#95—41no—
37C3⋊#95—42no—
38C3⋊#95—42no—
39C22×#41C2×D312yes—
40#550—22no—
41#550—62no—
42C2×#127—31no—
43C2×#125—31no—
44C2×#121—32no—
45C2×C4×Q8C2×D311yescharacteristic
46C2×C4×D4—31no—
47#490—64no—
48#479—62no—
49#479—62no—
50C2×#107—31no—
51C2×#106—31no—
52C2×C4⋊C8—31no—
53C2×#103—31no—
54C2×#103—31no—
55C2×#103—32no—
56C2×#102—32no—
57C2×#102—32no—
58C2×#102—32no—
59C2×#102—32no—
60C2×C22⋊C8—32no—
61C2×C22⋊C8—32no—
62C2×#97—31no—
63#400—61no—
64#394—61no—
65#388—62no—
66#380—32no—
67#380—32no—
68C4×C4⋊C4—31no—
69#340—32no—
70C22×C3×C4—41no—
71C22×C3×C4—42no—
72C22×Dic3—41no—
73C22×Dic3—42no—
74C22×Dic3—41no—
75#41⋊C3—42no—
76C2×SL(2,𝔽3)C2×D411yescharacteristic
77C2×SL(2,𝔽3)#4112yes—
78C3×C4⋊C4—82no—
79Dic3⋊C4—82no—
80C2×#41—22no—
81C2×#41—22no—
82C2×#41—22no—
83C2×#41—62no—
84C2×#41—62no—
85C2×#41—62no—
86C22×Q8C22×D311yescharacteristic
87C22×D4—32no—
88C22×D4—31no—
89C22×D4—61no—
90C23×C4—32no—
91C22×C8—32no—
92C22×C8—31no—
93C22×C8—31no—
94#127—34no—
95#127—62no—
96#125—62no—
97#125—62no—
98#121—64no—
99#121—64no—
100C4×Q8—32no—
101C4×Q8#6712yes—
102C4×D4—62no—
103C4×D4—62no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—61no—
110C2×C4⋊C4S411yescharacteristic
111C2×C22⋊C4—32no—
112C2×C22⋊C4—62no—
113C2×C22⋊C4—62no—
114C2×C42—31no—
115C2×C42—61no—
116#107—62no—
117#106—62no—
118C4⋊C8—62no—
119#103—62no—
120#103—61no—
121#103—61no—
122#103—62no—
123#103—61no—
124#103—61no—
125#102—64no—
126#102—64no—
127#102—122no—
128#102—122no—
129C22⋊C8—122no—
130C22⋊C8—122no—
131#97—62no—
132#95—61no—
133#95—62no—
134#95—62no—
135C23×C3—41no—
136C2×C3×C4—42no—
137C2×C3×C4—82no—
138C2×C3×C4—82no—
139C2×C3×C4—82no—
140C2×C3×C4—82no—
141C2×Dic3—42no—
142C2×Dic3—81no—
143C2×Dic3—81no—
144C2×Dic3—81no—
145C2×Dic3—82no—
146C2×Dic3—82no—
147C2×Dic3—82no—
148C2×Dic3—82no—
149SL(2,𝔽3)#12411yescharacteristic
150C24—32no—
151#41—42no—
152#41—124no—
153#41—122no—
154#41—122no—
155#41—122no—
156C2×Q8—31no—
157C2×Q8—32no—
158C2×Q8#23712yes—
159C2×Q8C2×#6711yescharacteristic
160C2×D4—34no—
161C2×D4—62no—
162C2×D4—62no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—62no—
166C2×D4—64no—
167C2×D4—62no—
168C2×D4—122no—
169C2×D4—241no—
170C2×D4—241no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—61no—
181C22×C4—61no—
182C22×C4C2×S412yes—
183C22×C4C2×S411yescharacteristic
184C2×C8—62no—
185C2×C8—62no—
186C2×C8—62no—
187C2×C8—62no—
188C2×C8—61no—
189C2×C8—61no—
190C2×C8—62no—
191C4⋊C4—22no—
192C4⋊C4—32no—
193C4⋊C4—61no—
194C4⋊C4—61no—
195C4⋊C4—61no—
196C4⋊C4—62no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—62no—
200C4⋊C4—122no—
201C4⋊C4—122no—
202C22⋊C4—64no—
203C22⋊C4—124no—
204C22⋊C4—242no—
205C42—32no—
206C42—32no—
207C42—241no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—41no—
213C3×C4—82no—
214C3×C4—162no—
215Dic3—82no—
216Dic3—161no—
217Dic3—162no—
218C23—62no—
219C23—62no—
220C23—62no—
221C23—62no—
222C23—62no—
223C23—62no—
224C23—62no—
225C23C22×S411yescharacteristic
226Q8—32no—
227Q8—31no—
228Q8—61no—
229Q8#93611yescharacteristic
230D4—64no—
231D4—64no—
232D4—124no—
233D4—122no—
234D4—122no—
235D4—244no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—31no—
243C2×C4—31no—
244C2×C4—32no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—121no—
255C2×C4—122no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—121no—
259C2×C4GL(2,ℤ/4)12yes—
260C8—121no—
261C8—121no—
262C8—122no—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—42no—
266C2×C3—42no—
267C2×C3—41no—
268C22—122no—
269C22—122no—
270C22—122no—
271C22—122no—
272C22—122no—
273C22—122no—
274C22—122no—
275C22#591412yes—
276C22#592312yes—
277C22#592411yescharacteristic
278C22#592511yescharacteristic
279C22C2×GL(2,ℤ/4)11yescharacteristic
280C4—22no—
281C4—31no—
282C4—32no—
283C4—31no—
284C4—42no—
285C4—62no—
286C4—241no—
287C4—241no—
288C4—242no—
289C3—41noSylow
290C2—242no—
291C2#8540512yes—
292C2#8541312yes—
293C2#8549211yescharacteristic
294C2#8552211yescharacteristic
295C2#8552911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
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×#5389C2semidirect
3. #73151C2semidirect
4. #73151C2semidirect
5. #73063C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#41⋊C3)2
C2×#53892
#731512
#730632
#258933
#49854