Finite Group Databasebeta

← #30670891

Subgroups — #30670891

Order 16768.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16862229C211yesmaximal, characteristic, complemented
2C2×#16862218C211yesmaximal, characteristic, complemented
3C2×#16862216C211yesmaximal, characteristic, complemented
4C2×#16862214C211yesmaximal, characteristic, complemented
5C22×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
6#22553247C212yesmaximal, complemented
7#22553221C212yesmaximal, complemented
8#22553213C212yesmaximal, complemented
9#22553212C212yesmaximal, complemented
10C22⋊C4×Dic131C211yesmaximal, characteristic, complemented
11C2×#123×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×D524C2212yes—
14C22×C22⋊D131C2211yescharacteristic
15C22×C22⋊D131C2212yes—
16C22×C22⋊D131C2211yescharacteristic, complemented
17C23×Dic131C2211yescharacteristic
18#16862229—24no—
19#16862218—24no—
20#16862216—24no—
21#16862214—24no—
22C2×Dic131⋊C4C2212yescomplemented
23C2×Dic131⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic131C2211yescharacteristic
25C2×C22⋊Dic131C2211yescharacteristic, complemented
26C2×(C2×D131)⋊C4—24no—
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2212yes—
29C2×(C2×D131)⋊C4C2212yes—
30C2×(C2×D131)⋊C4C2212yes—
31C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
32C2×(C2×D131)⋊C4C2212yescomplemented
33C2×C4×Dic131C2212yescomplemented
34C131⋊#95C2212yescomplemented
35C22×D4×C131C2211yescharacteristic
36C23×C4×C131C2211yescharacteristic
37#123×C131—24no—
38C2×C4⋊C4×C131C2212yescomplemented
39C2×C22⋊C4×C131C2211yescharacteristic
40C2×C22⋊C4×C131C2212yescomplemented
41C23×D131—24no—
42C23×D131—22no—
43C23×D131—22no—
44C23×D131—22no—
45C23×D131—42no—
46C23×D131C2311yescharacteristic
47C23×D131C2312yes—
48C2×D524—24no—
49C2×D524—24no—
50C2×D524—42no—
51C2×C22⋊D131—24no—
52C2×C22⋊D131—24no—
53C2×C22⋊D131—22no—
54C2×C22⋊D131—22no—
55C2×C22⋊D131—22no—
56C2×C22⋊D131—22no—
57C2×C22⋊D131—42no—
58C2×C22⋊D131—42no—
59C22×Dic131—42no—
60C22×Dic131D414yescomplemented
61C22×Dic131C2311yescharacteristic
62C22×Dic131C2311yescharacteristic
63C22×Dic131C2311yescharacteristic
64C22×Dic131C2312yes—
65Dic131⋊C4—22no—
66Dic131⋊C4—24no—
67C22⋊Dic131—22no—
68C22⋊Dic131D414yescomplemented
69(C2×D131)⋊C4—24no—
70(C2×D131)⋊C4—22no—
71(C2×D131)⋊C4—24no—
72(C2×D131)⋊C4—24no—
73(C2×D131)⋊C4—24no—
74(C2×D131)⋊C4—24no—
75C4×Dic131—42no—
76C24×C131C2311yescharacteristic
77C24×C131C2311yescharacteristic
78C2×D4×C131—22no—
79C2×D4×C131—22no—
80C2×D4×C131—22no—
81C22×C4×C131—22no—
82C22×C4×C131—22no—
83C22×C4×C131—22no—
84C22×C4×C131C2312yes—
85C22×C4×C131C2311yescharacteristic
86C22×C4×C131C2312yes—
87C4⋊C4×C131—24no—
88C22⋊C4×C131—24no—
89C22⋊C4×C131D414yescomplemented
90C22×D131—28no—
91C22×D131—22no—
92C22×D131—22no—
93C22×D131—22no—
94C22×D131—22no—
95C22×D131—22no—
96C22×D131—22no—
97C22×D131—21no—
98C22×D131—21no—
99C22×D131—21no—
100C22×D131—22no—
101C22×D131—22no—
102C22×D131—48no—
103C22×D131—48no—
104C22×D131—44no—
105C22×D131—44no—
106C22×D131#4114yes—
107D524—48no—
108C22⋊D131—48no—
109C22⋊D131—44no—
110C22⋊D131—44no—
111C2×Dic131—22no—
112C2×Dic131—22no—
113C2×Dic131—22no—
114C2×Dic131—22no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131—21no—
118C2×Dic131—44no—
119C2×Dic131C2×D412yes—
120C2×Dic131C2×D414yes—
121C2×Dic131C2×D412yes—
122C23×C131—22no—
123C23×C131—22no—
124C23×C131—21no—
125C23×C131—21no—
126C23×C131—21no—
127C23×C131—22no—
128C23×C131—22no—
129C23×C131C2×D412yes—
130C23×C131C2×D412yes—
131C23×C131C2×D412yes—
132C23×C131C2411yescharacteristic
133D4×C131—44no—
134C2×C4×C131—22no—
135C2×C4×C131—22no—
136C2×C4×C131—22no—
137C2×C4×C131—21no—
138C2×C4×C131—22no—
139C2×C4×C131—22no—
140C2×C4×C131—21no—
141C2×C4×C131—21no—
142C2×C4×C131—44no—
143C2×C4×C131C2×D414yes—
144C22×C131—22no—
145C22×C131—22no—
146C22×C131—24no—
147C22×C131—24no—
148C22×C131—41no—
149C22×C131—41no—
150C22×C131—41no—
151C22×C131—42no—
152C22×C131—42no—
153C22×C131#12714yes—
154C22×C131C22×D411yescharacteristic
155C22×C131C22×D411yescharacteristic
156C22×C131C22×D411yescharacteristic
157C22×C131C2×#4112yes—
158C22×C131H(5,𝔽2)12yes—
159C2×D131—24no—
160C2×D131—24no—
161C2×D131—24no—
162C2×D131—416no—
163C2×D131—42no—
164C2×D131—42no—
165C2×D131—42no—
166C2×D131—42no—
167C2×D131—41no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—42no—
171C2×D131—42no—
172C4×C131—24no—
173C4×C131—42no—
174C4×C131—41no—
175Dic131—22no—
176Dic131—22no—
177Dic131—24no—
178Dic131—41no—
179C2×C131—24no—
180C2×C131—81no—
181C2×C131#52511yescharacteristic
182C2×C131C2×#12711yescharacteristic
183C2×C131#53411yescharacteristic
184C2×C131#54612yes—
185C2×C131#54612yes—
186D131—44no—
187D131—81no—
188C131#253911yescharacteristic, complemented, Sylow
189#2539—1311nomaximal, Sylow
190C2×#123—1311no—
191C2×#123D13111yescharacteristic, complemented
192C2×#121—1311no—
193C2×#120—1311no—
194C2×#120—1311no—
195C22×C22⋊C4—1311no—
196#399—1312no—
197#392—1312no—
198#390—1312no—
199#390—1312no—
200C4×C22⋊C4—1311no—
201C25—1311no—
202C22×D4—1311no—
203C22×D4—1312no—
204C22×D4—1311no—
205C22×D4—1312no—
206C22×D4C2×D13111yescharacteristic
207C23×C4—1311no—
208C23×C4C2×D13111yescharacteristic
209#123—24no—
210#123—2624no—
211#121—2624no—
212#120—2624no—
213#120—2624no—
214C2×C4⋊C4—1311no—
215C2×C4⋊C4—1312no—
216C2×C4⋊C4C2×D13112yescomplemented
217C2×C22⋊C4—2624no—
218C2×C22⋊C4—1311no—
219C2×C22⋊C4—1312no—
220C2×C22⋊C4—1312no—
221C2×C22⋊C4—1312no—
222C2×C22⋊C4—1312no—
223C2×C22⋊C4—1311no—
224C2×C22⋊C4—1312no—
225C2×C22⋊C4—1311no—
226C2×C22⋊C4C2×D13111yescharacteristic
227C2×C22⋊C4C2×D13112yescomplemented
228C2×C42—1312no—
229#95—1312no—
230C24—2624no—
231C24—2622no—
232C24—2622no—
233C24—2622no—
234C24—5242no—
235C24—1311no—
236C24—1312no—
237C24C22×D13111yescharacteristic
238C24C22×D13111yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—2624no—
243C2×D4—2624no—
244C2×D4—2622no—
245C2×D4—2622no—
246C2×D4—2622no—
247C2×D4—2624no—
248C2×D4—2622no—
249C2×D4—2624no—
250C2×D4—5242no—
251C2×D4—5242no—
252C2×D4—5242no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—5242no—
257C22×C4—1311no—
258C22×C4—1311no—
259C22×C4—1312no—
260C22×C4—1311no—
261C22×C4—1314no—
262C22×C4C22×D13112yes—
263C22×C4C22×D13111yescharacteristic
264C22×C4C22×D13112yes—
265C4⋊C4—24no—
266C4⋊C4—2622no—
267C4⋊C4—2624no—
268C22⋊C4—24no—
269C22⋊C4—2624no—
270C22⋊C4—2624no—
271C22⋊C4—2624no—
272C22⋊C4—2622no—
273C22⋊C4—2624no—
274C22⋊C4—2622no—
275C22⋊C4—2624no—
276C22⋊C4—1314no—
277C22⋊C4C22⋊D13114yescomplemented
278C42—5242no—
279C23—22no—
280C23—22no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—2628no—
287C23—2622no—
288C23—2622no—
289C23—2622no—
290C23—2622no—
291C23—2622no—
292C23—2622no—
293C23—2621no—
294C23—2621no—
295C23—2621no—
296C23—2622no—
297C23—2622no—
298C23—5248no—
299C23—5248no—
300C23—5244no—
301C23—5244no—
302C23—1314no—
303C23D4×D13112yes—
304C23D4×D13112yes—
305C23C2×C22⋊D13112yes—
306C23C23×D13111yescharacteristic
307D4—44no—
308D4—5248no—
309D4—5248no—
310D4—5244no—
311D4—5244no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—2622no—
322C2×C4—2622no—
323C2×C4—2622no—
324C2×C4—2622no—
325C2×C4—2621no—
326C2×C4—2621no—
327C2×C4—2621no—
328C2×C4—5244no—
329C2×C4—1312no—
330C2×C4—1314no—
331C2×C4—1312no—
332C2×C4C2×C22⋊D13114yes—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—22no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—2624no—
343C22—2624no—
344C22—2624no—
345C22—52416no—
346C22—5242no—
347C22—5242no—
348C22—5242no—
349C22—5242no—
350C22—5241no—
351C22—5242no—
352C22—5242no—
353C22—5241no—
354C22—5241no—
355C22#1686224214yes—
356C22C2×D4×D13111yescharacteristic
357C22C2×D4×D13111yescharacteristic
358C22C22×C22⋊D13111yescharacteristic
359C22#41×D13112yes—
360C22#1686229212yes—
361C4—24no—
362C4—42no—
363C4—41no—
364C4—2622no—
365C4—2622no—
366C4—2624no—
367C4—5241no—
368C2—24no—
369C2—81no—
370C2—5244no—
371C2—10481no—
372C2#2255382011yescharacteristic
373C2C2×#1686224211yescharacteristic
374C2#2255387311yescharacteristic
375C2#2255395012yes—
376C2#2255395412yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#123×C131C2
hypercenterC2×#123D131
hypocenterC131#2539
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16862229C2semidirect
2. C2×#16862218C2semidirect
3. C2×#16862216C2semidirect
4. C2×#16862214C2semidirect
5. C22×(C2×D131)⋊C4C2semidirect
6. #22553247C2semidirect
7. #22553221C2semidirect
8. #22553213C2semidirect
9. #22553212C2semidirect
10. C22⋊C4×Dic131C2semidirect
11. C2×#123×C131C2semidirect
16. C22×C22⋊D131C22semidirect
22. C2×Dic131⋊C4C22semidirect
23. C2×Dic131⋊C4C22semidirect
25. C2×C22⋊Dic131C22semidirect
31. C2×(C2×D131)⋊C4C22semidirect
32. C2×(C2×D131)⋊C4C22semidirect
33. C2×C4×Dic131C22semidirect
34. C131⋊#95C22semidirect
38. C2×C4⋊C4×C131C22semidirect
40. C2×C22⋊C4×C131C22semidirect
60. C22×Dic131D4semidirect
68. C22⋊Dic131D4semidirect
89. C22⋊C4×C131D4semidirect
188. C131#2539semidirect
191. C2×#123D131semidirect
216. C2×C4⋊C4C2×D131semidirect
227. C2×C22⋊C4C2×D131semidirect
277. C22⋊C4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622292
C2×#168622182
C2×#168622162
C2×#168622142
C22×(C2×D131)⋊C42
#225532472
#225532212
#225532132
#225532122
C22⋊C4×Dic1312
C2×#123×C1312
#2539131