Finite Group Databasebeta

← #18010633

Subgroups — #18010633

Order 4736.

332 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 66 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#41×#3866C211yesmaximal, characteristic, complemented
2#14558530C211yesmaximal, characteristic, complemented
3C2×#11924600C211yesmaximal, characteristic, complemented
4#14558513C211yesmaximal, characteristic, complemented
5#14558416C211yesmaximal, characteristic, complemented
6#14558405C211yesmaximal, characteristic, complemented
7#14558329C211yesmaximal, characteristic
8C2×#10596953C2211yescharacteristic
9Q8×#3866—21no—
10D4×#3866—21no—
11D4×#3866—22no—
12#11924615—21no—
13#11924615C2211yescharacteristic
14#11924615C2211yescharacteristic
15C2×C4×#3866—21no—
16C2×C4×#3866C2211yescharacteristic
17#11924601—21no—
18#11924600—21no—
19#11924600—21no—
20#11924600—21no—
21#11924594—22no—
22C2×#10596964C2211yescharacteristic
23#41×D37C2211yescharacteristic
24#11924569C411yescharacteristic, complemented
25#11924566C411yescharacteristic, complemented
26#11924565—21no—
27#11924562C411yescharacteristic, complemented
28#11924560—21no—
29#11924557—21no—
30#11924555—21no—
31M16×D37C2211yescharacteristic
32AGL(1,ℤ/8)×C37C411yescharacteristic, complemented
33C22×#3866—22no—
34C22×#3866—21no—
35#41×C37C2×C411yescharacteristic
36C2×D4×C37C2×C411yescharacteristic, complemented
37C2×C22⋊D37—21no—
38C22×Dic37—21no—
39#10596966D411yescharacteristic
40Q8×D37D411yescharacteristic
41#10596964—21no—
42#10596964—21no—
43#10596964—21no—
44#10596964—21no—
45#10596964—21no—
46#10596964D411yescharacteristic
47D4×D37—21no—
48D4×D37D411yescharacteristic
49#10596962—21no—
50#10596962C2×C411yescharacteristic
51C2×C4×D37—21no—
52C2×C4×D37C2311yescharacteristic
53C2×Dic74C2×C411yescharacteristic, complemented
54C22⋊#3866—22no—
55C22⋊#3866—21no—
56C22⋊#3866D412yes—
57C2×#64438—21no—
58C4⋊#3866—22no—
59C4⋊#3866—21no—
60C4⋊#3866—21no—
61C4×#3866—22no—
62C4×#3866—21no—
63C4×#3866—21no—
64C4×#3866—21no—
65C4×#3866D411yescharacteristic
66C4×#3866D411yescharacteristic
67#10596953—21no—
68#10596953—21no—
69SD16×C37—21no—
70D8×C37—21no—
71M16×C37C2×C411yescharacteristic, complemented
72#10596942—21no—
73#10596940—21no—
74#10596940—21no—
75#10596939—21no—
76#10596934C2×C411yescharacteristic, complemented
77Dic148—21no—
78#10596930—21no—
79#10596929—21no—
80C8×D37—21no—
81C23×C37—21no—
82C22×D37—21no—
83C22×D37C2×D411yescharacteristic
84C2×#3866—21no—
85C2×#3866—22no—
86C2×#3866—21no—
87C2×#3866—42no—
88C2×#3866—44no—
89C2×#3866—41no—
90Q8×C37C22⋊C411yescharacteristic
91D4×C37—21no—
92D4×C37—21no—
93D4×C37—21no—
94D4×C37C22⋊C411yescharacteristic
95C2×C4×C37—21no—
96C2×C4×C37C22×C411yescharacteristic
97C22⋊D37—21no—
98C22⋊D37—21no—
99C22⋊D37—21no—
100C22⋊D37—41no—
101C22⋊D37—41no—
102C2×Dic37—21no—
103C2×Dic37—21no—
104C2×Dic37—41no—
105C2×Dic37—41no—
106C2×Dic37#4111yescharacteristic
107D148—21no—
108D148C22⋊C411yescharacteristic
109C4×D37—21no—
110C4×D37—21no—
111C4×D37—21no—
112C4×D37C2×D411yescharacteristic
113C4×D37C2×D411yescharacteristic
114C4×D37C2×D411yescharacteristic
115C4×D37#4111yescharacteristic
116Dic74—21no—
117Dic74—21no—
118Dic74—21no—
119Dic74C22⋊C411yescharacteristic
120#64438—41no—
121C8×C37—21no—
122#64436—21no—
123C22×C37—21no—
124C22×C37—41no—
125C22×C37—81no—
126C22×C37C4×D411yescharacteristic
127C2×D37—21no—
128C2×D37—41no—
129C2×D37—42no—
130C2×D37#12011yescharacteristic
131C2×D37#12111yescharacteristic
132#3866—22no—
133#3866—42no—
134#3866—41no—
135#3866—81no—
136C4×C37—21no—
137C4×C37C2×C22⋊C411yescharacteristic
138C4×C37C4×D411yescharacteristic
139Dic37—21no—
140Dic37—41no—
141Dic37#12111yescharacteristic
142Dic37#12311yescharacteristic
143#1906—371nomaximal, Sylow
144C2×C37—21no—
145C2×C37—41no—
146C2×C37—81no—
147C2×C37#38611yescharacteristic
148D37—21no—
149D37—21no—
150D37—41no—
151#575—371no—
152C4×#41—371no—
153#421—371no—
154C2×#104—371no—
155#410—371no—
156#344—371no—
157#339—371no—
158C37#190611yescharacteristic, complemented, Sylow
159C2×#41—371no—
160C2×#41—371no—
161#137—371no—
162#137—371no—
163#137—741no—
164AGL(1,ℤ/8)—371no—
165AGL(1,ℤ/8)—741no—
166AGL(1,ℤ/8)#386611yescharacteristic, complemented
167#135—741no—
168#135—741no—
169C2×M16—371no—
170C2×M16—371no—
171C4×Q8—741no—
172C4×D4—742no—
173C4×D4—741no—
174#117—371no—
175#117—371no—
176#117—741no—
177C2×C42—371no—
178C2×C42—741no—
179#104—741no—
180#104—741no—
181#104—741no—
182#104—741no—
183#99—742no—
184#41—371no—
185#41—371no—
186#41—371no—
187#41—741no—
188#41—741no—
189#41—741no—
190#41—741no—
191#41—741no—
192#41—741no—
193#41C2×#386611yescharacteristic
194C2×Q8—371no—
195C2×Q8—371no—
196C2×D4—371no—
197C2×D4—741no—
198C2×D4—741no—
199C2×D4C2×#386611yescharacteristic, complemented
200C22×C4—371no—
201C22×C4—741no—
202C22×C4—741no—
203C22×C4—741no—
204C22×C4—742no—
205Q16—741no—
206Q16—741no—
207SD16—21no—
208SD16—741no—
209SD16—741no—
210SD16—741no—
211D8—21no—
212D8—741no—
213M16—371no—
214M16—741no—
215M16—741no—
216M16—741no—
217M16C2×#386611yescharacteristic, complemented
218C2×C8—741no—
219C2×C8—741no—
220C4⋊C4—741no—
221C4⋊C4—741no—
222C4⋊C4—742no—
223C22⋊C4—372no—
224C22⋊C4—742no—
225C22⋊C4—741no—
226C42—371no—
227C42—371no—
228C42—741no—
229C42—741no—
230C42—741no—
231C42—742no—
232C23—21no—
233C23—371no—
234C23—741no—
235Q8—371no—
236Q8—741no—
237Q8—741no—
238Q8—741no—
239Q8C22⋊#386611yescharacteristic
240D4—21no—
241D4—21no—
242D4—21no—
243D4—371no—
244D4—741no—
245D4—741no—
246D4—741no—
247D4—741no—
248D4—1481no—
249D4—1481no—
250D4C22⋊#386611yescharacteristic
251C2×C4—21no—
252C2×C4—371no—
253C2×C4—371no—
254C2×C4—371no—
255C2×C4—371no—
256C2×C4—371no—
257C2×C4—741no—
258C2×C4—741no—
259C2×C4—741no—
260C2×C4—741no—
261C2×C4—741no—
262C2×C4—741no—
263C2×C4—742no—
264C2×C4—741no—
265C2×C4—1481no—
266C2×C4—1481no—
267C2×C4—1481no—
268C2×C4—1482no—
269C2×C4—1484no—
270C2×C4C22×#386611yescharacteristic
271C8—21no—
272C8—741no—
273C8—1481no—
274C22—21no—
275C22—41no—
276C22—81no—
277C22—371no—
278C22—371no—
279C22—741no—
280C22—1481no—
281C22—1482no—
282C22D4×#386611yescharacteristic
283C4—21no—
284C4—371no—
285C4—371no—
286C4—2961no—
287C4—742no—
288C4—741no—
289C4—1481no—
290C4—1482no—
291C4—1481no—
292C4C2×C22⋊#386611yescharacteristic
293C4D4×#386611yescharacteristic
294C2—21no—
295C2—41no—
296C2—81no—
297C2—741no—
298C2—741no—
299C2—1481no—
300C2#1455848111yescharacteristic
subgroupG/H
centerC2#14558481
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
FittingAGL(1,ℤ/8)×C37C4
socleC2×C37#386
prime radical#14558329C2
cosocleC2×C4×D37C23
hypercenterAGL(1,ℤ/8)#3866
hypocenterC37#1906
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×#3866C2semidirect
2. #14558530C2semidirect
3. C2×#11924600C2semidirect
4. #14558513C2semidirect
5. #14558416C2semidirect
6. #14558405C2semidirect
24. #11924569C4semidirect
25. #11924566C4semidirect
27. #11924562C4semidirect
32. AGL(1,ℤ/8)×C37C4semidirect
36. C2×D4×C37C2×C4semidirect
53. C2×Dic74C2×C4semidirect
71. M16×C37C2×C4semidirect
76. #10596934C2×C4semidirect
158. C37#1906semidirect
166. AGL(1,ℤ/8)#3866semidirect
199. C2×D4C2×#3866semidirect
217. M16C2×#3866semidirect

Maximal subgroups

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

subgroupindex
#41×#38662
#145585302
C2×#119246002
#145585132
#145584162
#145584052
#145583292
#190637