Finite Group Databasebeta

← #30675212

Subgroups — #30675212

Order 16768.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14523514C211yesmaximal, characteristic, complemented
2C2×#16862273C212yesmaximal, complemented
3#121×D131C211yesmaximal, characteristic, complemented
4#22553460C212yesmaximal, complemented
5#22553456C212yesmaximal, complemented
6#22553451C211yesmaximal, characteristic, complemented
7#22553448C211yesmaximal, characteristic, complemented
8#22553445C211yesmaximal, characteristic, complemented
9#22553312C212yesmaximal, complemented
10#22553274C211yesmaximal, characteristic, complemented
11#448×C131C211yesmaximal, characteristic, complemented
12C2×#14523514—28no—
13C2×#14523514—21no—
14C2×#14523514—21no—
15C2×#14523514C2212yes—
16C2×#14523514C2212yescomplemented
17C22×D524—21no—
18C22×D524C2211yescharacteristic, complemented
19C2×D4×D131—21no—
20C2×D4×D131C2211yescharacteristic, complemented
21C22×C4×D131—21no—
22C22×C4×D131C2211yescharacteristic
23#16862273—28no—
24C2×#14523500C2212yescomplemented
25C2×D1048C2212yescomplemented
26C2×#14523498C2212yescomplemented
27C2×#14523496C2212yescomplemented
28#16862221C2211yescharacteristic, complemented
29#16862220C2211yescharacteristic
30#16862219—21no—
31#16862218—21no—
32#16862217C2211yescharacteristic, complemented
33C4⋊C4×D131C2211yescharacteristic, complemented
34C22⋊C4×D131—21no—
35#16862170C2212yescomplemented
36#16862168C2212yescomplemented
37#16862166C2212yescomplemented
38C22⋊(C131⋊C8)C2211yescharacteristic, complemented
39(C2×D131)⋊C8C2212yescomplemented
40C22×Q8×C131C2211yescharacteristic, complemented
41C2×SD16×C131C2212yescomplemented
42#121×C131C2211yescharacteristic
43#103×C131C2212yescomplemented
44C22⋊C8×C131C2211yescharacteristic, complemented
45C23×D131—21no—
46C23×D131—21no—
47C23×D131D411yescharacteristic
48#14523514—28no—
49#14523514—28no—
50#14523514—24no—
51#14523514—24no—
52#14523514—24no—
53#14523514D418yescomplemented
54C2×D524—28no—
55C2×D524—22no—
56C2×D524—22no—
57C2×D524—24no—
58C2×D524—21no—
59C2×D524—21no—
60C2×D524C2311yescharacteristic
61C2×D524C2312yes—
62C2×C22⋊D131—21no—
63C2×C22⋊D131—21no—
64C2×C22⋊D131—21no—
65D4×D131—24no—
66D4×D131—44no—
67C22×Dic131D411yescharacteristic
68C2×C4×D131—28no—
69C2×C4×D131—22no—
70C2×C4×D131—22no—
71C2×C4×D131—24no—
72C2×C4×D131—21no—
73C2×C4×D131D411yescharacteristic
74C2×C4×D131D411yescharacteristic
75C2×C4×D131C2311yescharacteristic
76C2×C4×D131C2311yescharacteristic
77#14523500—24no—
78D1048—24no—
79#14523498—24no—
80#14523496—24no—
81C2×C131⋊C8C2312yes—
82C4⋊Dic131C2311yescharacteristic
83Dic131⋊C4—21no—
84C22⋊Dic131—21no—
85(C2×D131)⋊C4—21no—
86(C2×D131)⋊C4—21no—
87C2×Q8×C131—24no—
88C2×Q8×C131—21no—
89C2×Q8×C131C2312yes—
90C2×D4×C131—21no—
91C2×D4×C131C2311yescharacteristic
92C22×C4×C131—21no—
93C22×C4×C131C2311yescharacteristic
94SD16×C131—24no—
95C2×C8×C131C2312yes—
96C4⋊C4×C131C2311yescharacteristic
97C22⋊C4×C131—21no—
98C22×D131—24no—
99C22×D131—22no—
100C22×D131—22no—
101C22×D131—22no—
102C22×D131—21no—
103C22×D131—41no—
104C22×D131—42no—
105C22×D131—42no—
106C22×D131—42no—
107C22×D131—42no—
108C22×D131—42no—
109C22×D131C2×D411yescharacteristic
110C22×D131C2×D411yescharacteristic
111D524—28no—
112D524—24no—
113D524—28no—
114D524—22no—
115D524—22no—
116D524—42no—
117D524C2×D414yescomplemented
118C22⋊D131—42no—
119C22⋊D131—42no—
120C22⋊D131—42no—
121C2×Dic131—22no—
122C2×Dic131—21no—
123C2×Dic131C2×D411yescharacteristic
124C2×Dic131C2×D411yescharacteristic
125C4×D131—28no—
126C4×D131—22no—
127C4×D131—22no—
128C4×D131—28no—
129C4×D131—22no—
130C4×D131—42no—
131C4×D131C2×D414yes—
132C131⋊C8—22no—
133C23×C131—21no—
134C23×C131C2×D411yescharacteristic
135Q8×C131—24no—
136Q8×C131—22no—
137Q8×C131C2×D414yescomplemented
138D4×C131—22no—
139D4×C131—42no—
140C2×C4×C131—24no—
141C2×C4×C131—22no—
142C2×C4×C131—22no—
143C2×C4×C131—21no—
144C2×C4×C131C2×D411yescharacteristic
145C2×C4×C131C2411yescharacteristic
146C8×C131—22no—
147C22×C131—21no—
148C22×C131—21no—
149C22×C131—41no—
150C22×C131—41no—
151C22×C131—41no—
152C22×C131AGL(1,ℤ/8)12yescomplemented
153C22×C131C22×D411yescharacteristic
154C2×D131—21no—
155C2×D131—21no—
156C2×D131—21no—
157C2×D131—24no—
158C2×D131—21no—
159C2×D131—21no—
160C2×D131—44no—
161C2×D131—48no—
162C2×D131—42no—
163C2×D131—42no—
164C2×D131—41no—
165C2×D131—41no—
166C2×D131—41no—
167C2×D131—84no—
168C2×D131#12012yes—
169C4×C131—24no—
170C4×C131—21no—
171C4×C131—41no—
172C4×C131C22×D412yes—
173Dic131—21no—
174Dic131—41no—
175Dic131#12012yes—
176C2×C131—22no—
177C2×C131—81no—
178C2×C131C2×#12011yescharacteristic
179C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C131#57511yescharacteristic
181D131—22no—
182D131—41no—
183D131—44no—
184D131—81no—
185C131#300611yescharacteristic, complemented, Sylow
186#3006—1311nomaximal, Sylow
187C22×#41—1311no—
188C2×AGL(1,ℤ/8)—1312no—
189C2×#121—1311no—
190#449—1312no—
191#449—1312no—
192#448—1311no—
193#448D13111yescharacteristic, complemented
194#447—1311no—
195#447—1311no—
196#417—1312no—
197#407—1311no—
198C2×#41—2628no—
199C2×#41—2621no—
200C2×#41—2621no—
201C2×#41—1312no—
202C2×#41—1312no—
203C22×Q8C2×D13111yescharacteristic, complemented
204C22×D4—2621no—
205C22×D4—2621no—
206C22×D4—1311no—
207C22×D4—1311no—
208C23×C4—2621no—
209C23×C4—1311no—
210AGL(1,ℤ/8)—2628no—
211C2×SD16—1312no—
212C2×SD16C2×D13112yescomplemented
213C2×D8—1312no—
214C2×D8—1312no—
215C2×M16—1312no—
216#121—2621no—
217#121—2621no—
218#121—1311no—
219#121—1311no—
220#121—1311no—
221#121C2×D13111yescharacteristic
222C2×C4⋊C4—1311no—
223C2×C22⋊C4—2621no—
224#103—1312no—
225#103C2×D13112yescomplemented
226#102—1312no—
227#102—1312no—
228C22⋊C8—1311no—
229C22⋊C8—1312no—
230C22⋊C8C2×D13111yescharacteristic, complemented
231C24—2621no—
232C24—2621no—
233C24—1311no—
234#41—2628no—
235#41—2628no—
236#41—2624no—
237#41—2624no—
238#41—2624no—
239#41—1318no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D13112yes—
243C2×D4—21no—
244C2×D4—2628no—
245C2×D4—2622no—
246C2×D4—2622no—
247C2×D4—2624no—
248C2×D4—2621no—
249C2×D4—2621no—
250C2×D4—2621no—
251C2×D4—2624no—
252C2×D4—2621no—
253C2×D4—2621no—
254C2×D4—5244no—
255C2×D4—1311no—
256C2×D4—1312no—
257C2×D4C22×D13111yescharacteristic
258C22×C4—21no—
259C22×C4—2628no—
260C22×C4—2622no—
261C22×C4—2622no—
262C22×C4—2624no—
263C22×C4—2621no—
264C22×C4—1311no—
265C22×C4—1311no—
266C22×C4—1311no—
267C22×C4—1311no—
268C22×C4—1311no—
269C22×C4C22×D13111yescharacteristic
270SD16—24no—
271SD16—2624no—
272D8—2624no—
273D8—2624no—
274M16—2624no—
275C2×C8—1312no—
276C2×C8C22×D13112yes—
277C4⋊C4—2621no—
278C4⋊C4—1311no—
279C4⋊C4C22×D13111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—2621no—
282C22⋊C4—2621no—
283C22⋊C4—2621no—
284C23—21no—
285C23—2624no—
286C23—2622no—
287C23—2622no—
288C23—2622no—
289C23—2621no—
290C23—5241no—
291C23—5242no—
292C23—5242no—
293C23—5242no—
294C23—5242no—
295C23—5242no—
296C23—1311no—
297C23—1311no—
298C23D4×D13111yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D13114yescomplemented
302D4—22no—
303D4—42no—
304D4—2628no—
305D4—2624no—
306D4—2628no—
307D4—2622no—
308D4—2622no—
309D4—5242no—
310D4—5242no—
311D4—5242no—
312D4—5242no—
313D4—1314no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—2628no—
319C2×C4—2622no—
320C2×C4—2628no—
321C2×C4—2622no—
322C2×C4—2622no—
323C2×C4—2622no—
324C2×C4—2621no—
325C2×C4—5242no—
326C2×C4—1311no—
327C2×C4—1311no—
328C2×C4—1314no—
329C2×C4D4×D13111yescharacteristic
330C2×C4C23×D13111yescharacteristic
331C8—22no—
332C8—2622no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—2621no—
339C22—2621no—
340C22—2621no—
341C22—2624no—
342C22—2621no—
343C22—2621no—
344C22—5244no—
345C22—5248no—
346C22—5242no—
347C22—5242no—
348C22—5241no—
349C22—5241no—
350C22—5241no—
351C22—10484no—
352C22—1312no—
353C22#1686227312yescomplemented
354C22C2×D4×D13111yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—2621no—
359C4—5241no—
360C4—1312no—
361C4C2×D4×D13112yes—
362C2—22no—
363C2—2622no—
364C2—81no—
365C2—5241no—
366C2—5244no—
367C2—10481no—
368C2#120×D13111yescharacteristic
369C2C2×#1686227311yescharacteristic
370C2#2255411111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#448×C131C2
socleC22×C131C22×D4
hypercenter#448D131
hypocenterC131#3006
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#14523514C2semidirect
2. C2×#16862273C2semidirect
3. #121×D131C2semidirect
4. #22553460C2semidirect
5. #22553456C2semidirect
6. #22553451C2semidirect
7. #22553448C2semidirect
8. #22553445C2semidirect
9. #22553312C2semidirect
10. #22553274C2semidirect
11. #448×C131C2semidirect
16. C2×#14523514C22semidirect
18. C22×D524C22semidirect
20. C2×D4×D131C22semidirect
24. C2×#14523500C22semidirect
25. C2×D1048C22semidirect
26. C2×#14523498C22semidirect
27. C2×#14523496C22semidirect
28. #16862221C22semidirect
32. #16862217C22semidirect
33. C4⋊C4×D131C22semidirect
35. #16862170C22semidirect
36. #16862168C22semidirect
37. #16862166C22semidirect
38. C22⋊(C131⋊C8)C22semidirect
39. (C2×D131)⋊C8C22semidirect
40. C22×Q8×C131C22semidirect
41. C2×SD16×C131C22semidirect
43. #103×C131C22semidirect
44. C22⋊C8×C131C22semidirect
53. #14523514D4semidirect
117. D524C2×D4semidirect
137. Q8×C131C2×D4semidirect
152. C22×C131AGL(1,ℤ/8)semidirect
185. C131#3006semidirect
193. #448D131semidirect
203. C22×Q8C2×D131semidirect
212. C2×SD16C2×D131semidirect
225. #103C2×D131semidirect
230. C22⋊C8C2×D131semidirect
301. Q8D4×D131semidirect
353. C22#16862273semidirect

Maximal subgroups

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

subgroupindex
C22×#145235142
C2×#168622732
#121×D1312
#225534602
#225534562
#225534512
#225534482
#225534452
#225533122
#225532742
#448×C1312
#3006131