Finite Group Databasebeta

← #11713227

Subgroups — #11713227

Order 768.

507 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 49 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#3489⋊C3C211yesmaximal, characteristic
2C2×#5426C211yesmaximal, characteristic, complemented
3#73155C212yesmaximal, complemented
4#73149C211yesmaximal, characteristic, complemented
5#73076C212yesmaximal, complemented
6#24030—31nomaximal, Sylow
7C2×2−1+4⋊C3C2211yescharacteristic
8C22×#41⋊C3C2211yescharacteristic
9D4×SL(2,𝔽3)—21no—
10#548⋊C3—21no—
11#520⋊C3C2211yescharacteristic
12#5426—21no—
13#5426C2212yes—
14C2×GU(2,𝔽3)C2212yes—
15C2×#859—21no—
16#5034—41nomaximal
17#3489D311yescharacteristic
18#3092—31no—
19C2×#417—31no—
20#2502—31no—
21#2039—32no—
22#2038—32no—
23#1901—31no—
24#1871—31no—
25#1859—32no—
26#1793—32no—
27#1789—31no—
282−1+4⋊C3—22no—
29C2×#41⋊C3—22no—
30C2×#41⋊C3D411yescharacteristic
31C2×#41⋊C3C2311yescharacteristic
32C22×SL(2,𝔽3)—21no—
33C22×SL(2,𝔽3)D411yescharacteristic
34C3×#121—41no—
35C2×C4⋊Dic3—41no—
36C4×SL(2,𝔽3)—21no—
37GU(2,𝔽3)—22no—
38#859—22no—
39#832—42no—
40#830—41no—
41#808—42no—
42C2×2−1+4C2×D311yescharacteristic
43C22×#41C2×D311yescharacteristic
44C22×M16—31no—
45D4×Q8—21no—
46#548—21no—
47#547—61no—
48#546—61no—
49#537—31no—
50#536—31no—
51#525—31no—
52C2×C22⋊Q8—31no—
53#520C2×D311yescharacteristic
54C22×C4⋊C4—31no—
55#484—61no—
56#483—61no—
57#481—61no—
58#480—61no—
59#428—32no—
60C2×#104—32no—
61C2×#104—32no—
62C2×#104—32no—
63#417—32no—
64#417—32no—
65#417—32no—
66#417—61no—
67#417—61no—
68C2×#103—61no—
69C2×#102—61no—
70C2×#101—32no—
71#407—31no—
72#398—62no—
73#391—32no—
74#385—62no—
75#384—32no—
76C4×C22⋊C4—31no—
77#343—32no—
78#339—32no—
79C2×C3×D4—41no—
80C2×C3×D4—81no—
81C22×C3×C4—41no—
82C22×Dic3—81no—
83#41⋊C3—21no—
84#41⋊C3#4112yes—
85C2×SL(2,𝔽3)—21no—
86C2×SL(2,𝔽3)—41no—
87C2×SL(2,𝔽3)C2×D411yescharacteristic
88C3×C4⋊C4—41no—
89C3×C22⋊C4—81no—
90C4⋊Dic3—42no—
91C4⋊Dic3—81no—
92C2×C3⋊C8—42no—
932−1+4—22no—
942−1+4—62no—
95C2×#41—22no—
96C2×#41—31no—
97C2×#41—31no—
98C2×#41—31no—
99C2×#41—61no—
100C2×#41—62no—
101C2×#41#6711yescharacteristic
102C2×#41C22×D311yescharacteristic
103C22×Q8—21no—
104C22×Q8—31no—
105C22×Q8#6711yescharacteristic
106C22×D4—31no—
107C23×C4—31no—
108C23×C4—61no—
109C2×M16—32no—
110C2×M16—32no—
111C2×M16—61no—
112C2×M16—122no—
113C22×C8—61no—
114C4⋊Q8—61no—
115#124—61no—
116#123—61no—
117#123—62no—
118C22⋊Q8—31no—
119C22⋊Q8—31no—
120C22⋊Q8—61no—
121C22⋊Q8—61no—
122C22⋊Q8—61no—
123C22⋊Q8—61no—
124#121—31no—
125#121—62no—
126#121—61no—
127#121S411yescharacteristic
128#120—61no—
129C4×Q8—21no—
130C4×D4—61no—
131C4×D4—61no—
132C4×D4—61no—
133#117—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—32no—
136C2×C4⋊C4—32no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—64no—
143C2×C22⋊C4—32no—
144C2×C22⋊C4—61no—
145C2×C42—32no—
146#107—62no—
147#106—62no—
148#104—62no—
149#104—62no—
150#104—122no—
151#104—122no—
152#103—62no—
153#103—62no—
154#103—62no—
155#102—62no—
156#102—62no—
157#102—62no—
158#101—122no—
159C22⋊C8—61no—
160C22⋊C8—61no—
161#95—61no—
162#95—64no—
163C23×C3—41no—
164C23×C3—81no—
165C3×D4—82no—
166C3×D4—162no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—82no—
170C2×C3×C4—81no—
171C2×Dic3—82no—
172C2×Dic3—162no—
173SL(2,𝔽3)#12311yescharacteristic
174C3⋊C8—82no—
175C24—31no—
176#41—21no—
177#41—32no—
178#41—62no—
179#41—62no—
180#41—62no—
181#41—62no—
182#41—62no—
183#41—61no—
184#41—121no—
185#41—122no—
186#41—122no—
187#41#23712yes—
188C2×Q8—21no—
189C2×Q8—31no—
190C2×Q8—31no—
191C2×Q8—41no—
192C2×Q8—62no—
193C2×Q8—62no—
194C2×Q8—61no—
195C2×Q8—61no—
196C2×Q8—121no—
197C2×Q8C2×#6711yescharacteristic
198C2×D4—21no—
199C2×D4—31no—
200C2×D4—31no—
201C2×D4—61no—
202C2×D4—62no—
203C2×D4—61no—
204C2×D4—61no—
205C2×D4—121no—
206C2×D4C2×S411yescharacteristic
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—32no—
213C22×C4—32no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4—61no—
217C22×C4—62no—
218C22×C4—62no—
219C22×C4—121no—
220C22×C4—121no—
221C22×C4—242no—
222C22×C4C2×S411yescharacteristic
223M16—62no—
224M16—62no—
225M16—62no—
226M16—62no—
227M16—122no—
228C2×C8—62no—
229C2×C8—62no—
230C2×C8—122no—
231C4⋊C4—31no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—244no—
241C4⋊C4C2×S411yescharacteristic
242C22⋊C4—21no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—62no—
247C22⋊C4—64no—
248C22⋊C4—61no—
249C42—61no—
250C42—62no—
251C42—62no—
252C22×C3—41no—
253C22×C3—42no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3—161no—
257C22×C3—161no—
258C22×C3—161no—
259C3×C4—42no—
260C3×C4—81no—
261C3×C4—161no—
262Dic3—162no—
263C23—21no—
264C23—31no—
265C23—31no—
266C23—62no—
267C23—62no—
268C23—121no—
269C23GL(2,ℤ/4)11yescharacteristic
270Q8—31no—
271Q8—61no—
272Q8—61no—
273Q8—62no—
274Q8—121no—
275Q8—121no—
276Q8#93511yescharacteristic
277D4—22no—
278D4—32no—
279D4—42no—
280D4—61no—
281D4—61no—
282D4—61no—
283D4—122no—
284D4—122no—
285D4—241no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—31no—
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—31no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—62no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—62no—
298C2×C4—64no—
299C2×C4—64no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4—124no—
308C2×C4—122no—
309C2×C4—242no—
310C2×C4—242no—
311C2×C4GL(2,ℤ/4)11yescharacteristic
312C2×C4C22×S411yescharacteristic
313C8—122no—
314C8—122no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—82no—
319C2×C3—321no—
320C22—21no—
321C22—21no—
322C22—32no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—61no—
327C22—61no—
328C22—61no—
329C22—61no—
330C22—61no—
331C22—124no—
332C22#543012yes—
333C22C2×GL(2,ℤ/4)11yescharacteristic
334C4—21no—
335C4—31no—
336C4—31no—
337C4—41no—
338C4—61no—
339C4—124no—
340C4—121no—
341C4—121no—
342C4—242no—
343C4#591412yes—
344C3—41noSylow
345C2—22no—
346C2—62no—
347C2—81no—
348C2—121no—
349C2#8549111yescharacteristic
350C2C2×#543011yescharacteristic
351C2#8554311yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3489D3
prime radical#3489⋊C3C2
hypercenter#121S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C2×#5426C2semidirect
3. #73155C2semidirect
4. #73149C2semidirect
5. #73076C2semidirect

Maximal subgroups

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

subgroupindex
#3489⋊C32
C2×#54262
#731552
#731492
#730762
#240303
#50344