Finite Group Databasebeta

← #11712654

Subgroups — #11712654

Order 768.

547 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 95 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×#568⋊C3C211yesmaximal, characteristic
2#85519C211yesmaximal, characteristic
3C2×#5426C211yesmaximal, characteristic
4#73126C212yesmaximal, complemented
5#73125C212yesmaximal, complemented
6#20679—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8#568⋊C3C414yescomplemented
9C2×#131⋊C3C2212yes—
10#5426—21no—
11#5426C2212yes—
12#5426C2211yescharacteristic
13#5426C2211yescharacteristic
14#5424—21no—
15#5420—21no—
16C2×#859—21no—
17#5121—41nomaximal
18C2×#568D311yescharacteristic
19C2×#430—31no—
20C2×#428—31no—
21#2896—31no—
22C2×#417—31no—
23#1816—32no—
24#1816—32no—
25#1815—32no—
26#1815—32no—
27#1742—31no—
28C2×#343—31no—
29C2×#41⋊C3C2×C412yes—
30C2×#41⋊C3D411yescharacteristic
31C2×#41⋊C3Q812yes—
32C2×#41⋊C3C2311yescharacteristic
33C22×SL(2,𝔽3)D411yescharacteristic
34C2×C3×M16—41no—
35#926—41no—
36C2×C4⋊Dic3—41no—
37#131⋊C3C2×C414yescomplemented
38#859—22no—
39#859—21no—
40#859—21no—
41#818—42no—
42#817—42no—
43C22×#41C2×D311yescharacteristic
44#568—34no—
45#568Dic314yescomplemented
46C2×#131—32no—
47C2×#131C2×D312yes—
48C22×M16—31no—
49C22×M16—31no—
50C22×M16—31no—
51C22×M16—31no—
52C2×#117—31no—
53C22×C4⋊C4—31no—
54#430—64no—
55C2×#108—32no—
56C2×#108—32no—
57#428—61no—
58#428—62no—
59#428—61no—
60C2×#107—32no—
61C2×#106—32no—
62#419—61no—
63#418—61no—
64#417—31no—
65#417—32no—
66#417—31no—
67#417—31no—
68#417—32no—
69#417—31no—
70#417—61no—
71#417—61no—
72#416—61no—
73#416—61no—
74C2×#103—61no—
75C2×#102—61no—
76#343—32no—
77#343—32no—
78#343—32no—
79#343—32no—
80C22×C3×C4—41no—
81C22×Dic3—81no—
82#41⋊C3C4⋊C412yes—
83#41⋊C3C22×C411yescharacteristic
84#41⋊C3C2×Q811yescharacteristic
85C2×SL(2,𝔽3)C4⋊C412yes—
86C2×SL(2,𝔽3)C2×D411yescharacteristic
87C3×M16—44no—
88C2×C3×C8—42no—
89#217—81no—
90C4⋊Dic3—41no—
91C4⋊Dic3—42no—
92C4⋊Dic3—41no—
93C4⋊Dic3—81no—
94C4×Dic3—81no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—32no—
98C2×#41—31no—
99C2×#41Dic612yes—
100C2×#41D1211yescharacteristic
101C2×#41C2×Dic312yes—
102C2×#41C22×D311yescharacteristic
103C22×Q8D1211yescharacteristic
104C22×D4—31no—
105C23×C4—31no—
106C23×C4—61no—
107#131—34no—
108#131—38no—
109#131—64no—
110#131C2×Dic314yescomplemented
111C2×M16—31no—
112C2×M16—34no—
113C2×M16—34no—
114C2×M16—32no—
115C2×M16—32no—
116C2×M16—34no—
117C2×M16—32no—
118C2×M16—31no—
119C2×M16—31no—
120C2×M16—62no—
121C2×M16—61no—
122C2×M16—61no—
123C2×M16—62no—
124C2×M16—62no—
125C2×M16—62no—
126C2×M16—62no—
127C2×M16S411yescharacteristic
128C22×C8—32no—
129C22×C8—61no—
130C22×C8—61no—
131C22×C8—61no—
132#117—61no—
133#117—61no—
134#117—61no—
135#117—61no—
136C2×C4⋊C4—32no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C42—61no—
146#108—64no—
147#108—64no—
148#107—62no—
149#107—62no—
150#106—62no—
151#106—62no—
152#103—62no—
153#103—61no—
154#103—61no—
155#103—62no—
156#103—61no—
157#103—61no—
158#102—62no—
159#102—62no—
160#102—61no—
161#102—61no—
162#102—61no—
163#102—61no—
164C23×C3—41no—
165C2×C3×C4—42no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—41no—
169C2×Dic3—81no—
170C2×Dic3—81no—
171C2×Dic3—82no—
172C2×Dic3—162no—
173SL(2,𝔽3)C2×C4⋊C411yescharacteristic
174C3×C8—44no—
175C24—31no—
176#41—32no—
177#41—31no—
178#41—32no—
179#41—32no—
180#41—32no—
181#41—32no—
182#41—31no—
183#41—61no—
184#41—61no—
185#41—62no—
186#41C4⋊Dic312yes—
187#41C2×Dic611yescharacteristic
188#41C22×Dic311yescharacteristic
189C2×Q8—31no—
190C2×Q8—32no—
191C2×Q8C4⋊Dic312yes—
192C2×Q8C2×D1211yescharacteristic
193C2×D4—31no—
194C2×D4—31no—
195C2×D4—32no—
196C2×D4—32no—
197C2×D4—61no—
198C2×D4—62no—
199C22×C4—31no—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—32no—
205C22×C4—32no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—62no—
211C22×C4—121no—
212C22×C4—121no—
213C22×C4—242no—
214C22×C4C2×S411yescharacteristic
215M16—34no—
216M16—38no—
217M16—34no—
218M16—62no—
219M16—62no—
220M16—61no—
221M16—61no—
222M16—62no—
223M16—61no—
224M16—61no—
225M16—122no—
226M16—122no—
227M16SL(2,ℤ/4)14yes—
228C2×C8—32no—
229C2×C8—34no—
230C2×C8—34no—
231C2×C8—64no—
232C2×C8—62no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—122no—
236C2×C8—122no—
237C2×C8C2×S412yes—
238C4⋊C4—62no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—62no—
243C4⋊C4—61no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C42—121no—
249C42—121no—
250C22×C3—41no—
251C22×C3—42no—
252C22×C3—81no—
253C22×C3—81no—
254C3×C4—41no—
255C3×C4—42no—
256C3×C4—41no—
257Dic3—161no—
258Dic3—161no—
259Dic3—162no—
260C23—31no—
261C23—31no—
262C23—32no—
263C23—32no—
264C23—121no—
265C23—121no—
266C23C4⋊S411yescharacteristic
267Q8—32no—
268Q8—31no—
269Q8—61no—
270Q8C2×C4⋊Dic311yescharacteristic
271D4—32no—
272D4—31no—
273D4—31no—
274D4—61no—
275D4—61no—
276D4—62no—
277D4—62no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—32no—
285C2×C4—32no—
286C2×C4—32no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—32no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—122no—
295C2×C4—242no—
296C2×C4—242no—
297C2×C4—241no—
298C2×C4—241no—
299C2×C4—242no—
300C2×C4A4⋊Q812yes—
301C2×C4C4⋊S411yescharacteristic
302C2×C4C2×SL(2,ℤ/4)12yes—
303C2×C4C22×S411yescharacteristic
304C8—34no—
305C8—122no—
306C8—121no—
307C8—121no—
308C8C2×SL(2,ℤ/4)14yes—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—82no—
313C22—21no—
314C22—21no—
315C22—31no—
316C22—31no—
317C22—32no—
318C22—61no—
319C22—61no—
320C22—61no—
321C22—61no—
322C22—122no—
323C22—122no—
324C22C4⋊SL(2,ℤ/4)12yes—
325C22C2×C4⋊S411yescharacteristic
326C4—31no—
327C4—31no—
328C4—32no—
329C4—242no—
330C4—241no—
331C4—241no—
332C4C4⋊SL(2,ℤ/4)12yes—
333C4C2×A4⋊Q811yescharacteristic
334C4C22×SL(2,ℤ/4)11yescharacteristic
335C3—41noSylow
336C2—22no—
337C2—61no—
338C2—61no—
339C2—62no—
340C2#8545411yescharacteristic
341C2#8545511yescharacteristic
342C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22C2×C4⋊S4
commutator#41⋊C3C22×C4
FrattiniC2×C4C22×S4
FittingC2×#568D3
prime radicalC22×#41⋊C3C22
cosocleC2×#41⋊C3C23
hypercenterC2×M16S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. #73126C2semidirect
5. #73125C2semidirect
8. #568⋊C3C4semidirect
37. #131⋊C3C2×C4semidirect
45. #568Dic3semidirect
110. #131C2×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×#568⋊C32
#855192
C2×#54262
#731262
#731252
#206793
#51214