Finite Group Databasebeta

← #30675581

Subgroups — #30675581

Order 16768.

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×#16862274C211yesmaximal, characteristic, complemented
2C2×M16×D131C211yesmaximal, characteristic, complemented
3#22553824C211yesmaximal, characteristic, complemented
4C2×#16862221C211yesmaximal, characteristic, complemented
5#22553566C212yesmaximal, complemented
6#22553559C212yesmaximal, complemented
7#22553549C212yesmaximal, complemented
8#22553542C212yesmaximal, complemented
9#22553316C211yesmaximal, characteristic, complemented
10#22553311C211yesmaximal, characteristic, complemented
11#428×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×D524—21no—
14C22×D524C2211yescharacteristic
15C22×C4×D131C2211yescharacteristic
16#16862274—24no—
17C2×#14523502C2212yescomplemented
18C2×D1048C2212yescomplemented
19C2×#14523497C2211yescharacteristic
20C2×#14523496C2212yescomplemented
21M16×D131—24no—
22C2×C8×D131C2212yescomplemented
23#16862230—21no—
24#16862226C2211yescharacteristic, complemented
25#16862226C2211yescharacteristic, complemented
26#16862221—21no—
27#16862221—21no—
28#16862221C2212yes—
29#16862221C2212yes—
30#16862221C2211yescharacteristic, complemented
31#16862221C2211yescharacteristic, complemented
32#16862216—21no—
33C4×D524—21no—
34C4×D524—21no—
35C2×(C2×D131)⋊C4—21no—
36#16862169C2212yescomplemented
37#16862166C2212yescomplemented
38#16862166C2212yescomplemented
39#16862166C2212yescomplemented
40C2×M16×C131C2211yescharacteristic
41#117×C131C2211yescharacteristic
42C2×C4⋊C4×C131C2211yescharacteristic
43#107×C131C2212yescomplemented
44#106×C131C2212yescomplemented
45C23×D131—21no—
46C23×D131D411yescharacteristic
47#14523513—22no—
48#14523513—22no—
49C2×Dic262C2311yescharacteristic
50C2×D524—21no—
51C2×D524—21no—
52C2×D524—22no—
53C2×D524—22no—
54C2×D524—22no—
55C2×D524—21no—
56C2×D524—22no—
57C2×D524—44no—
58C2×D524C2312yes—
59C2×D524C2311yescharacteristic
60C2×C22⋊D131—21no—
61C22×Dic131D411yescharacteristic
62C2×C4×D131—22no—
63C2×C4×D131—21no—
64C2×C4×D131—22no—
65C2×C4×D131D411yescharacteristic
66C2×C4×D131D411yescharacteristic
67C2×C4×D131C2311yescharacteristic
68C2×C4×D131C2311yescharacteristic
69#14523502—24no—
70D1048—24no—
71#14523497—22no—
72#14523496—24no—
73C2×C131⋊C8C2312yes—
74C8×D131—24no—
75C4⋊Dic131—21no—
76C4⋊Dic131—21no—
77(C2×D131)⋊C4—21no—
78(C2×D131)⋊C4—21no—
79(C2×D131)⋊C4—21no—
80(C2×D131)⋊C4—21no—
81(C2×D131)⋊C4—22no—
82(C2×D131)⋊C4—22no—
83C22×C4×C131—21no—
84C22×C4×C131C2311yescharacteristic
85M16×C131D414yescomplemented
86C2×C8×C131C2312yes—
87C4⋊C4×C131—21no—
88C4⋊C4×C131C2312yes—
89C4⋊C4×C131C2311yescharacteristic
90C4⋊C4×C131C2311yescharacteristic
91C22⋊C4×C131—21no—
92C42×C131—21no—
93C22×D131—22no—
94C22×D131—22no—
95C22×D131—21no—
96C22×D131—22no—
97C22×D131—41no—
98C22×D131—42no—
99C22×D131—42no—
100C22×D131—42no—
101C22×D131C2×D411yescharacteristic
102C22×D131C2×D411yescharacteristic
103Dic262—21no—
104Dic262—21no—
105D524—21no—
106D524—22no—
107D524—22no—
108D524—21no—
109D524—42no—
110D524—44no—
111D524—42no—
112D524—44no—
113D524—42no—
114C22⋊D131—42no—
115C2×Dic131—22no—
116C2×Dic131—21no—
117C2×Dic131C2×D411yescharacteristic
118C2×Dic131C2×D411yescharacteristic
119C4×D131—22no—
120C4×D131—22no—
121C4×D131—21no—
122C4×D131—21no—
123C4×D131—21no—
124C4×D131—21no—
125C4×D131—42no—
126C131⋊C8—22no—
127C23×C131C2×D411yescharacteristic
128C2×C4×C131—21no—
129C2×C4×C131—21no—
130C2×C4×C131—22no—
131C2×C4×C131—42no—
132C2×C4×C131C2×D411yescharacteristic
133C2×C4×C131C2×D412yes—
134C2×C4×C131#4112yes—
135C2×C4×C131C2411yescharacteristic
136C8×C131C2×D414yescomplemented
137C22×C131—21no—
138C22×C131—21no—
139C22×C131#12112yes—
140C22×C131C22×D411yescharacteristic
141C2×D131—21no—
142C2×D131—21no—
143C2×D131—21no—
144C2×D131—21no—
145C2×D131—21no—
146C2×D131—44no—
147C2×D131—41no—
148C2×D131—42no—
149C2×D131—42no—
150C2×D131—41no—
151C2×D131—41no—
152C2×D131—42no—
153C2×D131—84no—
154C2×D131AGL(1,ℤ/8)12yescomplemented
155C4×C131—42no—
156C4×C131—41no—
157C4×C131—41no—
158C4×C131#12112yes—
159C4×C131C22×D411yescharacteristic
160C4×C131C2×#4111yescharacteristic
161Dic131—21no—
162Dic131—21no—
163Dic131—41no—
164C2×C131—22no—
165C2×C131C2×#12111yescharacteristic
166C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C131#57511yescharacteristic
168D131—22no—
169D131—41no—
170D131—81no—
171D131—82no—
172C131#305511yescharacteristic, complemented, Sylow
173#3055—1311nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1311no—
175C22×M16—1311no—
176#525—1311no—
177C2×#121—1311no—
178#469—1312no—
179#468—1312no—
180#466—1312no—
181#465—1312no—
182#428D13111yescharacteristic, complemented
183#418—1311no—
184#417—1311no—
185C2×#41—1311no—
186C22×D4—2621no—
187C22×D4—1311no—
188C23×C4—1311no—
189AGL(1,ℤ/8)—2624no—
190C2×SD16—1312no—
191C2×D8—1312no—
192C2×M16—2624no—
193C2×M16—1311no—
194C2×M16—1312no—
195C2×M16C2×D13111yescharacteristic
196C22×C8—1312no—
197#123—2621no—
198C22⋊Q8—1311no—
199C22⋊Q8—1311no—
200#121—2621no—
201#121—2621no—
202#121—1312no—
203#121—1312no—
204#121—1311no—
205#121—1311no—
206#120—2621no—
207C4×D4—2621no—
208C4×D4—2621no—
209#117C2×D13111yescharacteristic
210C2×C4⋊C4C2×D13111yescharacteristic
211C2×C22⋊C4—2621no—
212#107C2×D13112yescomplemented
213#106C2×D13112yescomplemented
214#103—1312no—
215#102—1312no—
216#102—1312no—
217#102—1312no—
218C24—2621no—
219C24—1311no—
220#41—2622no—
221#41—2622no—
222C2×Q8—1311no—
223C2×D4—2621no—
224C2×D4—2621no—
225C2×D4—2622no—
226C2×D4—2622no—
227C2×D4—2621no—
228C2×D4—2621no—
229C2×D4—2622no—
230C2×D4—2622no—
231C2×D4—5244no—
232C2×D4—1311no—
233C2×D4—1312no—
234C22×C4—21no—
235C22×C4—2622no—
236C22×C4—2622no—
237C22×C4—2621no—
238C22×C4—1311no—
239C22×C4—1311no—
240C22×C4—1311no—
241C22×C4—1311no—
242C22×C4—1311no—
243C22×C4C22×D13111yescharacteristic
244SD16—2624no—
245D8—2624no—
246M16—2622no—
247M16—2624no—
248M16D52414yescomplemented
249C2×C8—2624no—
250C2×C8—1312no—
251C2×C8C22×D13112yes—
252C4⋊C4—21no—
253C4⋊C4—2621no—
254C4⋊C4—2621no—
255C4⋊C4C22×D13111yescharacteristic
256C4⋊C4C22×D13111yescharacteristic
257C4⋊C4C22×D13112yes—
258C22⋊C4—21no—
259C22⋊C4—2621no—
260C22⋊C4—2621no—
261C22⋊C4—2622no—
262C22⋊C4—2622no—
263C22⋊C4—2621no—
264C22⋊C4—2621no—
265C42—21no—
266C23—2622no—
267C23—2622no—
268C23—2621no—
269C23—2622no—
270C23—5241no—
271C23—5242no—
272C23—5242no—
273C23—5242no—
274C23—1311no—
275C23—1311no—
276C23D4×D13111yescharacteristic
277Q8—2621no—
278Q8—2621no—
279D4—2621no—
280D4—2622no—
281D4—2622no—
282D4—2621no—
283D4—5244no—
284D4—5242no—
285D4—5244no—
286D4—5242no—
287D4—5242no—
288D4—5242no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—2622no—
294C2×C4—2622no—
295C2×C4—2622no—
296C2×C4—2621no—
297C2×C4—2621no—
298C2×C4—2621no—
299C2×C4—2621no—
300C2×C4—2621no—
301C2×C4—5242no—
302C2×C4—1311no—
303C2×C4—1311no—
304C2×C4D4×D13111yescharacteristic
305C2×C4C2×D52412yes—
306C2×C4#1452351412yes—
307C2×C4C23×D13111yescharacteristic
308C8—2622no—
309C8C2×D52414yescomplemented
310C22—21no—
311C22—21no—
312C22—2621no—
313C22—2621no—
314C22—2621no—
315C22—2621no—
316C22—2621no—
317C22—5244no—
318C22—5241no—
319C22—5242no—
320C22—5242no—
321C22—5241no—
322C22—5241no—
323C22—5242no—
324C22—10484no—
325C22—1312no—
326C22#1686222112yes—
327C22C2×D4×D13111yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—2621no—
332C4—2621no—
333C4—5241no—
334C4#1686222112yes—
335C4C22×D52411yescharacteristic
336C4C2×#1452351411yescharacteristic
337C2—22no—
338C2—2622no—
339C2—5241no—
340C2—10482no—
341C2—10481no—
342C2C2×#1686222111yescharacteristic
343C2AGL(1,ℤ/8)×D13111yescharacteristic
344C2#2255411111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#428×C131C2
socleC22×C131C22×D4
hypercenter#428D131
hypocenterC131#3055
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16862274C2semidirect
2. C2×M16×D131C2semidirect
3. #22553824C2semidirect
4. C2×#16862221C2semidirect
5. #22553566C2semidirect
6. #22553559C2semidirect
7. #22553549C2semidirect
8. #22553542C2semidirect
9. #22553316C2semidirect
10. #22553311C2semidirect
11. #428×C131C2semidirect
17. C2×#14523502C22semidirect
18. C2×D1048C22semidirect
20. C2×#14523496C22semidirect
22. C2×C8×D131C22semidirect
24. #16862226C22semidirect
25. #16862226C22semidirect
30. #16862221C22semidirect
31. #16862221C22semidirect
36. #16862169C22semidirect
37. #16862166C22semidirect
38. #16862166C22semidirect
39. #16862166C22semidirect
43. #107×C131C22semidirect
44. #106×C131C22semidirect
85. M16×C131D4semidirect
136. C8×C131C2×D4semidirect
154. C2×D131AGL(1,ℤ/8)semidirect
172. C131#3055semidirect
182. #428D131semidirect
212. #107C2×D131semidirect
213. #106C2×D131semidirect
248. M16D524semidirect
309. C8C2×D524semidirect

Maximal subgroups

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

subgroupindex
C2×#168622742
C2×M16×D1312
#225538242
C2×#168622212
#225535662
#225535592
#225535492
#225535422
#225533162
#225533112
#428×C1312
#3055131