Finite Group Databasebeta

← #49168246

Subgroups — #49168246

Order 37504.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#22627650C211yesmaximal, characteristic, complemented
2#34036675C211yesmaximal, characteristic, complemented
3C2×D4×Dic293C211yesmaximal, characteristic, complemented
4#34036346C212yesmaximal, complemented
5#34036342C212yesmaximal, complemented
6#34036332C212yesmaximal, complemented
7#34036322C212yesmaximal, complemented
8#34036152C211yesmaximal, characteristic, complemented
9#34036131C211yesmaximal, characteristic, complemented
10#34036130C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C23×Dic293—21no—
14#22627650—24no—
15C2×#17989790C2212yes—
16C2×#17989787C2212yes—
17C2×#17989786C2211yescharacteristic
18#22627622C2211yescharacteristic, complemented
19#22627620C2211yescharacteristic, complemented
20#22627613C2211yescharacteristic, complemented
21#22627611C2211yescharacteristic, complemented
22#22627596—21no—
23#22627596—21no—
24D4×Dic293—21no—
25D4×Dic293—21no—
26D4×Dic293C2212yes—
27D4×Dic293C2212yes—
28C4×C22⋊D293—21no—
29C4×C22⋊D293—21no—
30C2×C4⋊Dic293C2211yescharacteristic
31C2×C22⋊Dic293—21no—
32C2×C4×Dic293C2211yescharacteristic
33#22627553C2212yescomplemented
34#22627553C2212yescomplemented
35#22627549C2212yescomplemented
36#22627548C2212yescomplemented
37#22627546C2212yescomplemented
38#22627545C2212yescomplemented
39C2×#41×C293C2211yescharacteristic
40C22×D4×C293C2211yescharacteristic
41AGL(1,ℤ/8)×C293—24no—
42C2×SD16×C293C2212yes—
43C2×D8×C293C2212yes—
44C2×M16×C293C2211yescharacteristic
45#17989802—22no—
46#17989802—22no—
47C2×Dic586C2311yescharacteristic
48C2×D1172C2311yescharacteristic
49C2×C22⋊D293—21no—
50C2×C22⋊D293—21no—
51C2×C22⋊D293—21no—
52C22×Dic293—22no—
53C22×Dic293—22no—
54C22×Dic293—21no—
55C22×Dic293—44no—
56C22×Dic293D411yescharacteristic
57C22×Dic293D411yescharacteristic
58C2×C4×D293—21no—
59#17989790—24no—
60#17989787—24no—
61#17989786—22no—
62C2×#14556873C2312yes—
63C4⋊Dic293—21no—
64C4⋊Dic293C2312yes—
65#17989781—21no—
66#17989781—21no—
67C22⋊Dic293—21no—
68C22⋊Dic293—21no—
69C22⋊Dic293—22no—
70C22⋊Dic293—22no—
71#17989779—21no—
72#17989779—21no—
73C4×Dic293D411yescharacteristic
74C4×Dic293D411yescharacteristic
75C4×Dic293C2311yescharacteristic
76C4×Dic293C2311yescharacteristic
77C24×C293—21no—
78#41×C293—22no—
79#41×C293—22no—
80C2×Q8×C293C2311yescharacteristic
81C2×D4×C293—21no—
82C2×D4×C293—21no—
83C2×D4×C293—22no—
84C2×D4×C293D414yescomplemented
85C2×D4×C293C2311yescharacteristic
86C2×D4×C293C2312yes—
87C22×C4×C293—21no—
88C22×C4×C293C2311yescharacteristic
89SD16×C293—24no—
90D8×C293—24no—
91M16×C293—22no—
92C2×C8×C293C2312yes—
93C22×D293—21no—
94Dic586—21no—
95Dic586—21no—
96D1172—21no—
97D1172—21no—
98C22⋊D293—42no—
99C22⋊D293—42no—
100C22⋊D293—42no—
101C2×Dic293—22no—
102C2×Dic293—21no—
103C2×Dic293—22no—
104C2×Dic293—41no—
105C2×Dic293—44no—
106C2×Dic293—42no—
107C2×Dic293—44no—
108C2×Dic293C2×D411yescharacteristic
109C2×Dic293C2×D411yescharacteristic
110C2×Dic293C2×D411yescharacteristic
111C2×Dic293C2×D411yescharacteristic
112C4×D293—42no—
113#14556873—22no—
114C23×C293—24no—
115C23×C293—22no—
116C23×C293—21no—
117C23×C293—42no—
118C23×C293—42no—
119C23×C293C2×D411yescharacteristic
120Q8×C293—21no—
121Q8×C293—21no—
122D4×C293—24no—
123D4×C293—21no—
124D4×C293—22no—
125D4×C293—21no—
126D4×C293—42no—
127D4×C293C2×D414yes—
128C2×C4×C293—21no—
129C2×C4×C293—42no—
130C2×C4×C293C2×D411yescharacteristic
131C2×C4×C293C2×D412yes—
132C2×C4×C293#4112yes—
133C2×C4×C293C2411yescharacteristic
134C8×C293—22no—
135C22×C293—21no—
136C22×C293—21no—
137C22×C293—24no—
138C22×C293—48no—
139C22×C293—42no—
140C22×C293—42no—
141C22×C293—41no—
142C22×C293—41no—
143C22×C293—41no—
144C22×C293#12112yes—
145C22×C293C22×D411yescharacteristic
146C2×D293—41no—
147C2×D293—41no—
148C2×D293—41no—
149C4×C293—41no—
150C4×C293#12112yes—
151C4×C293C22×D411yescharacteristic
152C4×C293C2×#4111yescharacteristic
153Dic293—21no—
154Dic293—21no—
155Dic293—21no—
156Dic293—41no—
157Dic293—42no—
158Dic293AGL(1,ℤ/8)12yescomplemented
159C2×C293—22no—
160C2×C293—44no—
161C2×C293—81no—
162C2×C293C2×#12111yescharacteristic
163C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C293#57511yescharacteristic
165D293—81no—
166C293#303611yescharacteristic, complemented, Sylow
167#3036—2931nomaximal, Sylow
168C2×AGL(1,ℤ/8)—2931no—
169C2×AGL(1,ℤ/8)D29311yescharacteristic, complemented
170#532—2931no—
171C2×C4×D4—2931no—
172#463—2932no—
173#463—2932no—
174#461—2932no—
175#459—2932no—
176#423—2931no—
177#417—2931no—
178#417—2931no—
179C2×#41—2931no—
180C2×#41C2×D29311yescharacteristic
181C22×D4C2×D29311yescharacteristic
182C23×C4—5861no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—5864no—
185C2×SD16—2932no—
186C2×SD16C2×D29312yes—
187C2×D8—2932no—
188C2×D8C2×D29312yes—
189C2×M16—2931no—
190C2×M16C2×D29311yescharacteristic
191C4⋊Q8—2931no—
192#127—2931no—
193#124—2931no—
194#124—2931no—
195#121—5861no—
196#121—5861no—
197C4×D4—2932no—
198C4×D4—2932no—
199C4×D4—5861no—
200C4×D4—5861no—
201C4×D4—5861no—
202C4×D4—5861no—
203C2×C4⋊C4—2931no—
204C2×C22⋊C4—5861no—
205C2×C42—2931no—
206C4⋊C8—2932no—
207C4⋊C8—2932no—
208#103—2932no—
209#103—2932no—
210#102—2932no—
211#102—2932no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—5862no—
216#41—5862no—
217C2×Q8—2931no—
218C2×Q8C22×D29311yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—2931no—
223C2×D4—5861no—
224C2×D4—5861no—
225C2×D4—5861no—
226C2×D4C22⋊D29314yescomplemented
227C2×D4C22×D29311yescharacteristic
228C2×D4C22×D29312yes—
229C22×C4—21no—
230C22×C4—2931no—
231C22×C4—2931no—
232C22×C4—5862no—
233C22×C4—5862no—
234C22×C4—5861no—
235C22×C4—5861no—
236C22×C4—11724no—
237C22×C4C22×D29311yescharacteristic
238SD16—24no—
239SD16—5864no—
240D8—24no—
241D8—5864no—
242M16—22no—
243M16—5862no—
244C2×C8—2932no—
245C2×C8C22×D29312yes—
246C4⋊C4—2932no—
247C4⋊C4—5861no—
248C4⋊C4—5861no—
249C4⋊C4—5861no—
250C22⋊C4—5861no—
251C22⋊C4—5861no—
252C22⋊C4—5862no—
253C22⋊C4—5862no—
254C22⋊C4—5861no—
255C22⋊C4—5861no—
256C42—2931no—
257C42—2931no—
258C42—2931no—
259C42—2931no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—5861no—
266C23D4×D29311yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—5861no—
270Q8—5861no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—5861no—
277D4—5861no—
278D4—11722no—
279D4—11722no—
280D4—11722no—
281D4C2×C22⋊D29314yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—2931no—
285C2×C4—2931no—
286C2×C4—2931no—
287C2×C4—2931no—
288C2×C4—5862no—
289C2×C4—5862no—
290C2×C4—5861no—
291C2×C4—11724no—
292C2×C4—11721no—
293C2×C4—11724no—
294C2×C4—11722no—
295C2×C4—11722no—
296C2×C4D4×D29311yescharacteristic
297C2×C4C2×C22⋊D29312yes—
298C2×C4#1798980112yes—
299C2×C4C23×D29311yescharacteristic
300C8—22no—
301C8—5862no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—11721no—
312C22—11721no—
313C22—11721no—
314C22#2262759612yes—
315C22C2×D4×D29311yescharacteristic
316C4—41no—
317C4—2932no—
318C4—5861no—
319C4—5861no—
320C4—5861no—
321C4—11721no—
322C4—11722no—
323C4#2262759612yes—
324C4C22×C22⋊D29311yescharacteristic
325C4C2×#1798980111yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—23441no—
330C2C2×#2262759611yescharacteristic
331C2AGL(1,ℤ/8)×D29311yescharacteristic
332C2#3403692511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×AGL(1,ℤ/8)×C293C2
socleC22×C293C22×D4
hypercenterC2×AGL(1,ℤ/8)D293
hypocenterC293#3036
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22627650C2semidirect
2. #34036675C2semidirect
3. C2×D4×Dic293C2semidirect
4. #34036346C2semidirect
5. #34036342C2semidirect
6. #34036332C2semidirect
7. #34036322C2semidirect
8. #34036152C2semidirect
9. #34036131C2semidirect
10. #34036130C2semidirect
11. C2×AGL(1,ℤ/8)×C293C2semidirect
18. #22627622C22semidirect
19. #22627620C22semidirect
20. #22627613C22semidirect
21. #22627611C22semidirect
33. #22627553C22semidirect
34. #22627553C22semidirect
35. #22627549C22semidirect
36. #22627548C22semidirect
37. #22627546C22semidirect
38. #22627545C22semidirect
84. C2×D4×C293D4semidirect
158. Dic293AGL(1,ℤ/8)semidirect
166. C293#3036semidirect
169. C2×AGL(1,ℤ/8)D293semidirect
226. C2×D4C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276502
#340366752
C2×D4×Dic2932
#340363462
#340363422
#340363322
#340363222
#340361522
#340361312
#340361302
C2×AGL(1,ℤ/8)×C2932
#3036293