Finite Group Databasebeta

← #62509⋊C3 #11710539

Subgroups — #62509⋊C3 #11710539

Order 768.

493 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 69 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#3518⋊C3C211yesmaximal, characteristic, complemented
2#3499⋊C3C211yesmaximal, characteristic, complemented
3SD16×SL(2,𝔽3)C212yesmaximal, complemented
4D8×SL(2,𝔽3)C212yesmaximal, complemented
5#2971⋊C3C211yesmaximal, characteristic, complemented
6#62509C311yesmaximal, characteristic, complemented, Sylow
7C2×2−1+4⋊C3—21no—
8Q8×SL(2,𝔽3)C2211yescharacteristic
9D4×SL(2,𝔽3)—21no—
10D4×SL(2,𝔽3)C2211yescharacteristic
11D4×SL(2,𝔽3)C2212yes—
12#548⋊C3—21no—
13#519⋊C3—21no—
14C4×#41⋊C3C2211yescharacteristic
15C3×#496—41nomaximal
16C8×SL(2,𝔽3)C2212yes—
17#3518C2×C311yescharacteristic, complemented
18#3499C2×C311yescharacteristic, complemented
19Q8×SD16C2×C312yescomplemented
20Q8×D8C2×C312yescomplemented
21#3354—32no—
22#3353—32no—
23#3239—31no—
24#3235—31no—
25#3171—31no—
26#3165—31no—
27#2971C2×C311yescharacteristic, complemented
282−1+4⋊C3—42no—
29C2×#41⋊C3D411yescharacteristic
30C22×SL(2,𝔽3)—22no—
31C22×SL(2,𝔽3)—21no—
32C2×C3×SD16—42no—
33C2×C3×D8—42no—
34C3×#127—41no—
35C3×#124—41no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)D411yescharacteristic
38C4×SL(2,𝔽3)C2311yescharacteristic
39C3×#97—41no—
40C2×2−1+4—21no—
41Q82C22×C311yescharacteristic
42#556—31no—
43#556—61no—
44#555—31no—
45D4×Q8—21no—
46D4×Q8C22×C311yescharacteristic
47D4×Q8C22×C312yes—
48#548—21no—
49#547—32no—
50#547—61no—
51#542—61no—
52#536—61no—
53#533—31no—
54#532—31no—
55C2×C4⋊Q8—31no—
56#529—31no—
57#519—21no—
58C4×#41C22×C311yescharacteristic
59#501—32no—
60#501—31no—
61#500—32no—
62#498—32no—
63#496A411yescharacteristic
64#490—31no—
65#489—31no—
66#484—32no—
67#482—34no—
68#481—32no—
69#476—32no—
70#475—32no—
71#474—32no—
72#474—32no—
73#469—32no—
74#468—32no—
75C8×Q8C22×C312yes—
76C4×SD16—32no—
77C4×D8—32no—
78#436—32no—
79#432—31no—
80#428—34no—
81C4×M16—31no—
82C2×C3×Q8—41no—
83C2×C3×D4—41no—
84C2×C3×D4—42no—
85C2×C3×D4—81no—
86C2×C3×D4—81no—
87#41⋊C3—21no—
88C2×SL(2,𝔽3)—41no—
89C2×SL(2,𝔽3)—42no—
90C2×SL(2,𝔽3)C2×D411yescharacteristic
91C3×SD16—84no—
92C3×D8—84no—
93C2×C3×C8—42no—
94C3×C22⋊C4—81no—
95C3×C22⋊C4—81no—
96C3×C42—41no—
972−1+4—42no—
982−1+4—122no—
99C2×#41—61no—
100C2×#41—62no—
101C2×#41C3×D411yescharacteristic
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—61no—
105C2×SD16C2×A412yes—
106C2×D8C2×A412yes—
107C2×M16—32no—
108C4⋊Q8—32no—
109C4⋊Q8—31no—
110C4⋊Q8—31no—
111C4⋊Q8—31no—
112C4⋊Q8—61no—
113C4⋊Q8—61no—
114C4⋊Q8—61no—
115#127C2×A411yescharacteristic
116#126—61no—
117#126—62no—
118#125—31no—
119#124—61no—
120#124—62no—
121#124—61no—
122#124C2×A411yescharacteristic
123#123—61no—
124#123—64no—
125C22⋊Q8—31no—
126C22⋊Q8—61no—
127C22⋊Q8—61no—
128C22⋊Q8—62no—
129C22⋊Q8—62no—
130C22⋊Q8—61no—
131#121—31no—
132#121—32no—
133#121—61no—
134#121—61no—
135C4×Q8—21no—
136C4×Q8—31no—
137C4×Q8—61no—
138C4×Q8—61no—
139C4×Q8—61no—
140C4×Q8C3×D411yescharacteristic
141C4×Q8C23×C311yescharacteristic
142C4×D4—31no—
143C4×D4—31no—
144C4×D4—31no—
145C4×D4—32no—
146C4×D4—61no—
147C4×D4—61no—
148#117—31no—
149#117—32no—
150#117—61no—
151#117—61no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—61no—
154C2×C42—31no—
155#107—34no—
156#107—32no—
157#106—34no—
158#106—32no—
159C4⋊C8—32no—
160#103—32no—
161#102—32no—
162#102—32no—
163#102—32no—
164C22⋊C8—32no—
165#97—31no—
166#97C2×A411yescharacteristic
167C4×C8—32no—
168C23×C3—81no—
169C23×C3—82no—
170C3×Q8—82no—
171C3×D4—82no—
172C3×D4—82no—
173C3×D4—82no—
174C3×D4—162no—
175C3×D4—162no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—81no—
180SL(2,𝔽3)AGL(1,ℤ/8)11yescharacteristic
181C3×C8—44no—
182#41—21no—
183#41—61no—
184#41—122no—
185#41—122no—
186#41—124no—
187#41—124no—
188C2×Q8—42no—
189C2×Q8—41no—
190C2×Q8—61no—
191C2×Q8—62no—
192C2×Q8—61no—
193C2×Q8—61no—
194C2×Q8—122no—
195C2×Q8—122no—
196C2×Q8—121no—
197C2×Q8—122no—
198C2×Q8C2×C3×D411yescharacteristic
199C2×Q8C22×A411yescharacteristic
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—31no—
203C2×D4—62no—
204C2×D4—61no—
205C2×D4C22×A411yescharacteristic
206C2×D4C22×A412yes—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—61no—
211C22×C4—62no—
212C22×C4—62no—
213SD16—24no—
214D8—24no—
215M16—64no—
216C2×C8—32no—
217C2×C8C22×A412yes—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—32no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—62no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—62no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C22⋊C4—21no—
233C22⋊C4—21no—
234C22⋊C4—32no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—62no—
238C22⋊C4—61no—
239C22⋊C4—62no—
240C22⋊C4—62no—
241C42—31no—
242C42—31no—
243C42—31no—
244C42—61no—
245C42—62no—
246C42C22×A411yescharacteristic
247C22×C3—41no—
248C22×C3—161no—
249C22×C3—162no—
250C22×C3—162no—
251C22×C3—162no—
252C22×C3—161no—
253C22×C3—161no—
254C3×C4—42no—
255C3×C4—81no—
256C3×C4—81no—
257C3×C4—161no—
258C23—21no—
259C23—22no—
260C23—31no—
261Q8—22no—
262Q8—31no—
263Q8—122no—
264Q8—121no—
265Q8—122no—
266Q8—121no—
267Q8—122no—
268Q8—241no—
269Q8C3×AGL(1,ℤ/8)11yescharacteristic
270D4—22no—
271D4—22no—
272D4—22no—
273D4—42no—
274D4—42no—
275D4—61no—
276D4—61no—
277D4—124no—
278D4—241no—
279C2×C4—21no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—62no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—62no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—122no—
292C2×C4—121no—
293C2×C4—122no—
294C2×C4—124no—
295C2×C4D4×A411yescharacteristic
296C2×C4D4×A411yescharacteristic
297C2×C4C23×A411yescharacteristic
298C8—62no—
299C82−1+4⋊C314yescomplemented
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—322no—
304C2×C3—321no—
305C22—41no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—42no—
310C22—41no—
311C22—61no—
312C22—61no—
313C22—61no—
314C22C2×D4×A411yescharacteristic
315C4—21no—
316C4—21no—
317C4—31no—
318C4—31no—
319C4—41no—
320C4—61no—
321C4—61no—
322C4—122no—
323C4—121no—
324C4—121no—
325C4C2×2−1+4⋊C312yes—
326C3—41noSylow
327C2—82no—
328C2—81no—
329C2—121no—
330C2#3499⋊C311yescharacteristic
331C2A4×AGL(1,ℤ/8)11yescharacteristic
332C2#3586⋊C311yescharacteristic
subgroupG/H
center, socleC22C2×D4×A4
commutator, cosocleC4×Q8C23×C3
FrattiniC2×C4C23×A4
Fitting#62509C3
hypercenter#496A4
hypocenterQ8C3×AGL(1,ℤ/8)
LCSG ▷ C4×Q8 ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #3518⋊C3C2semidirect
2. #3499⋊C3C2semidirect
3. SD16×SL(2,𝔽3)C2semidirect
4. D8×SL(2,𝔽3)C2semidirect
5. #2971⋊C3C2semidirect
6. #62509C3semidirect
17. #3518C2×C3semidirect
18. #3499C2×C3semidirect
19. Q8×SD16C2×C3semidirect
20. Q8×D8C2×C3semidirect
27. #2971C2×C3semidirect
299. C82−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
#3518⋊C32
#3499⋊C32
SD16×SL(2,𝔽3)2
D8×SL(2,𝔽3)2
#2971⋊C32
#625093
C3×#4964