Finite Group Databasebeta

← #11713981

Subgroups — #11713981

Order 768.

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

normal subgroups only

This group has 576 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×#5425C212yesmaximal
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4#73064C211yesmaximal, characteristic, complemented
5#73061C212yesmaximal, complemented
6#32916—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×#226C2212yes—
9#520⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12#5425—24no—
13C2×#859C2212yes—
14C4⋊#226—22no—
15#4981—41nomaximal
16C2×#520D311yescharacteristic
17C2×#462—31no—
18C2×#461—31no—
19C2×#452—32no—
20#2621—31no—
21#1878—32no—
22#1877—32no—
23#1872—32no—
24#1805—31no—
25#1804—31no—
26#1798—31no—
27C2×#41⋊C3—21no—
28C2×#41⋊C3—21no—
29C2×#41⋊C3D412yes—
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×#226—22no—
32C2×#226—22no—
33C2×#226—22no—
34C2×C3×C4⋊C4—41no—
35C2×C4⋊Dic3—41no—
36C2×Dic3⋊C4—42no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)D412yes—
39#859—22no—
40C3⋊#95—42no—
41C3⋊#95—41no—
42C22×#41C2×D311yescharacteristic
43C22×Q16—32no—
44C22×SD16—32no—
45C2×C4⋊Q8—31no—
46C2×C22⋊Q8—32no—
47#520—22no—
48#520—62no—
49C2×C4×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×D4—31no—
52C2×C4×D4—31no—
53C2×#117—31no—
54C22×C4⋊C4—31no—
55#462—62no—
56#462—62no—
57#461—64no—
58#452—64no—
59#452—64no—
60C2×C4⋊C8—31no—
61C2×C4⋊C8—31no—
62C2×#103—32no—
63C2×#103—32no—
64C2×#103—32no—
65C2×#102—32no—
66C2×C22⋊C8—32no—
67#398—61no—
68#398—61no—
69#397—61no—
70#397—61no—
71#391—32no—
72#382—31no—
73#340—32no—
74#340—32no—
75#336—32no—
76C22×C3×C4—41no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×Dic3—42no—
80C22×Dic3—42no—
81#41⋊C3—22no—
82C2×SL(2,𝔽3)C2×D411yescharacteristic
83C2×SL(2,𝔽3)C2×D411yescharacteristic
84C2×SL(2,𝔽3)#4111yescharacteristic
85#226—42no—
86C3×C4⋊C4—82no—
87C4⋊Dic3—82no—
88Dic3⋊C4—84no—
89C2×#41—21no—
90C2×#41—21no—
91C2×#41—32no—
92C2×#41—61no—
93C2×#41—61no—
94C2×#41D1212yes—
95C22×Q8—32no—
96C22×Q8C22×D311yescharacteristic
97C22×D4—31no—
98C23×C4—31no—
99C23×C4—61no—
100C2×Q16—62no—
101C2×Q16—62no—
102C2×Q16—62no—
103C2×Q16—62no—
104C2×Q16—62no—
105C2×Q16—62no—
106C2×SD16—64no—
107C2×SD16—64no—
108C2×SD16—64no—
109C22×C8—32no—
110C22×C8—32no—
111C4⋊Q8—62no—
112C4⋊Q8—62no—
113C22⋊Q8—64no—
114C22⋊Q8—64no—
115C4×Q8—21no—
116C4×Q8—32no—
117C4×Q8—61no—
118C4×Q8#6712yes—
119C4×D4—34no—
120C4×D4—62no—
121C4×D4—62no—
122C4×D4—62no—
123#117—62no—
124#117—62no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—32no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4S411yescharacteristic
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—62no—
138C2×C42—31no—
139C2×C42—31no—
140C4⋊C8—62no—
141C4⋊C8—62no—
142#103—64no—
143#103—64no—
144#103—62no—
145#103—62no—
146#102—64no—
147C22⋊C8—64no—
148#95—61no—
149#95—62no—
150#95—61no—
151#95—62no—
152C23×C3—41no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—81no—
156C2×C3×C4—81no—
157C2×C3×C4—82no—
158C2×C3×C4—81no—
159C2×C3×C4—82no—
160C2×Dic3—44no—
161C2×Dic3—82no—
162C2×Dic3—82no—
163C2×Dic3—82no—
164C2×Dic3—82no—
165C2×Dic3—82no—
166SL(2,𝔽3)#12111yescharacteristic
167C24—31no—
168#41—22no—
169#41—64no—
170#41—62no—
171#41—62no—
172#41—62no—
173C2×Q8—31no—
174C2×Q8—31no—
175C2×Q8—31no—
176C2×Q8—62no—
177C2×Q8—62no—
178C2×Q8—62no—
179C2×Q8—62no—
180C2×Q8—62no—
181C2×Q8—62no—
182C2×Q8C2×D1211yescharacteristic
183C2×Q8#23511yescharacteristic
184C2×Q8C2×#6711yescharacteristic
185C2×D4—32no—
186C2×D4—32no—
187C2×D4—32no—
188C2×D4—61no—
189C2×D4—61no—
190C2×D4—61no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—32no—
194C22×C4—32no—
195C22×C4—31no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4—64no—
201C22×C4—62no—
202C22×C4—62no—
203C22×C4—242no—
204C22×C4C2×S411yescharacteristic
205C22×C4C2×S411yescharacteristic
206C22×C4C2×S411yescharacteristic
207Q16—122no—
208Q16—122no—
209Q16—122no—
210Q16—122no—
211SD16—128no—
212C2×C8—62no—
213C2×C8—62no—
214C2×C8—62no—
215C2×C8—62no—
216C2×C8—62no—
217C2×C8—62no—
218C4⋊C4—22no—
219C4⋊C4—32no—
220C4⋊C4—32no—
221C4⋊C4—62no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4—122no—
227C4⋊C4—124no—
228C4⋊C4—122no—
229C22⋊C4—62no—
230C22⋊C4—62no—
231C22⋊C4—124no—
232C42—32no—
233C42—32no—
234C42—61no—
235C42—61no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—41no—
241C3×C4—82no—
242C3×C4—82no—
243C3×C4—161no—
244Dic3—84no—
245Dic3—162no—
246C23—32no—
247C23—61no—
248C23—62no—
249C23—62no—
250C23—61no—
251C23C22×S411yescharacteristic
252Q8—31no—
253Q8—31no—
254Q8—31no—
255Q8—61no—
256Q8—122no—
257Q8—122no—
258Q8—122no—
259Q8—122no—
260Q8#93011yescharacteristic
261D4—34no—
262D4—62no—
263D4—62no—
264D4—62no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—61no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—244no—
291C2×C4C4⋊S412yes—
292C2×C4GL(2,ℤ/4)12yes—
293C8—122no—
294C8—122no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—42no—
298C2×C3—42no—
299C2×C3—41no—
300C22—64no—
301C22—62no—
302C22—121no—
303C22—121no—
304C22—122no—
305C22C2×C4⋊S411yescharacteristic
306C22#591211yescharacteristic
307C22#592412yes—
308C22#592712yes—
309C22C2×GL(2,ℤ/4)11yescharacteristic
310C4—22no—
311C4—22no—
312C4—31no—
313C4—31no—
314C4—31no—
315C4—31no—
316C4—41no—
317C4—62no—
318C4—121no—
319C4—242no—
320C4—242no—
321C3—41noSylow
322C2—122no—
323C2#8536212yes—
324C2#8537012yes—
325C2#8548611yescharacteristic
326C2#8552711yescharacteristic
327C2#8553511yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#121
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
3. C2×C4⋊#226C2semidirect
4. #73064C2semidirect
5. #73061C2semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#54252
C2×C4⋊#2262
#730642
#730612
#329163
#49814