Finite Group Databasebeta

← #30677952

Subgroups — #30677952

Order 16768.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22553966C211yesmaximal, characteristic, complemented
2#22553671C211yesmaximal, characteristic, complemented
3#22553645C211yesmaximal, characteristic, complemented
4#22553623C211yesmaximal, characteristic, complemented
5#22553607C211yesmaximal, characteristic, complemented
6#22553560C211yesmaximal, characteristic, complemented
7#22553546C211yesmaximal, characteristic, complemented
8#22553524C211yesmaximal, characteristic, complemented
9#22553457C211yesmaximal, characteristic, complemented
10#22553451C211yesmaximal, characteristic, complemented
11#22553410C211yesmaximal, characteristic, complemented
12#22553384C211yesmaximal, characteristic, complemented
13#22553313C211yesmaximal, characteristic, complemented
14C2×#16862170C211yesmaximal, characteristic, complemented
15#548×C131C211yesmaximal, characteristic, complemented
16C2×#14523512—21no—
17C2×#14523500C2211yescharacteristic
18C2×#14523500C2211yescharacteristic
19C2×#14523498C2211yescharacteristic
20C2×#14523497C2211yescharacteristic
21C22×C131⋊C8C2211yescharacteristic
22#16862246C2211yescharacteristic
23#16862230—21no—
24#16862230—21no—
25#16862226—21no—
26#16862220C2211yescharacteristic
27#16862220C2211yescharacteristic
28C4×D524C2211yescharacteristic
29C2×C4⋊Dic131C2211yescharacteristic
30C2×C4⋊Dic131C2211yescharacteristic
31#16862179C2211yescharacteristic
32#16862179C2211yescharacteristic
33#16862179C2211yescharacteristic
34C4⋊(C131⋊C8)C2211yescharacteristic
35#16862170C2211yescharacteristic
36#16862170C2211yescharacteristic
37#16862170C2211yescharacteristic
38#16862170C2211yescharacteristic
39#16862170C2211yescharacteristic
40#16862170C2211yescharacteristic
41#16862167C2211yescharacteristic
42#16862167C2211yescharacteristic
43#16862167C2211yescharacteristic
44#16862166C2211yescharacteristic
45C22⋊(C131⋊C8)C2211yescharacteristic
46C22⋊(C131⋊C8)C2211yescharacteristic
47#16862157C2211yescharacteristic
48C2×#41×C131C2211yescharacteristic
49C22×Q8×C131C2211yescharacteristic
50#124×C131—21no—
51#124×C131C2211yescharacteristic
52C22⋊Q8×C131—21no—
53C22⋊Q8×C131C2211yescharacteristic
54#121×C131—21no—
55#121×C131C2211yescharacteristic
56C4×Q8×C131C2211yescharacteristic
57C4×D4×C131—21no—
58C4×D4×C131C2211yescharacteristic
59#14523512—44no—
60C2×Dic262—21no—
61C2×D524C2311yescharacteristic
62C2×C22⋊D131—21no—
63C2×C22⋊D131—21no—
64C22×Dic131—21no—
65C22×Dic131—21no—
66C2×C4×D131—21no—
67#14523500—22no—
68#14523500—41no—
69#14523498—41no—
70#14523497—41no—
71C2×C131⋊C8—41no—
72C2×C131⋊C8C2311yescharacteristic
73C2×C131⋊C8C2311yescharacteristic
74C2×C131⋊C8C2311yescharacteristic
75C2×C131⋊C8C2311yescharacteristic
76C4⋊Dic131—21no—
77C4⋊Dic131—21no—
78C4⋊Dic131—21no—
79C4⋊Dic131C2311yescharacteristic
80C4⋊Dic131C2311yescharacteristic
81C4⋊Dic131C2311yescharacteristic
82(C2×D131)⋊C4—21no—
83(C2×D131)⋊C4—21no—
84(C2×D131)⋊C4—21no—
85#41×C131—22no—
86#41×C131—41no—
87C2×Q8×C131—21no—
88C2×Q8×C131—22no—
89C2×Q8×C131—41no—
90C2×Q8×C131C2311yescharacteristic
91C2×Q8×C131C2311yescharacteristic
92C2×D4×C131—21no—
93C2×D4×C131—21no—
94C2×D4×C131D411yescharacteristic
95C2×D4×C131C2311yescharacteristic
96C22×C4×C131—21no—
97C22×C4×C131—21no—
98C22×C4×C131C2311yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C4⋊C4×C131—21no—
101C4⋊C4×C131—21no—
102C4⋊C4×C131D411yescharacteristic
103C4⋊C4×C131C2311yescharacteristic
104C22⋊C4×C131—21no—
105C22⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C22⋊C4×C131—21no—
108C22⋊C4×C131D411yescharacteristic
109C22⋊C4×C131D411yescharacteristic
110C42×C131—21no—
111C42×C131C2311yescharacteristic
112C22×D131—21no—
113Dic262—42no—
114D524—21no—
115D524—21no—
116C22⋊D131—42no—
117C22⋊D131—42no—
118C2×Dic131—21no—
119C2×Dic131—21no—
120C2×Dic131—21no—
121C2×Dic131—42no—
122C2×Dic131—42no—
123C4×D131—42no—
124C131⋊C8—21no—
125C131⋊C8—21no—
126C131⋊C8—21no—
127C131⋊C8—21no—
128C23×C131—21no—
129C23×C131C2×D411yescharacteristic
130C23×C131C2×D411yescharacteristic
131Q8×C131—21no—
132Q8×C131—21no—
133Q8×C131—21no—
134Q8×C131—42no—
135Q8×C131#4112yescomplemented
136D4×C131—22no—
137D4×C131—21no—
138D4×C131—21no—
139D4×C131—42no—
140D4×C131—42no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—42no—
149C2×C4×C131—42no—
150C2×C4×C131C2×D411yescharacteristic
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2×D411yescharacteristic
153C2×C4×C131C2×D411yescharacteristic
154C2×C4×C131C2411yescharacteristic
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—21no—
158C22×C131—21no—
159C22×C131—21no—
160C22×C131—41no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131AGL(1,ℤ/8)12yes—
164C22×C131C22×D411yescharacteristic
165C2×D131—41no—
166C2×D131—41no—
167C2×D131—41no—
168C4×C131—21no—
169C4×C131—21no—
170C4×C131—22no—
171C4×C131—21no—
172C4×C131—21no—
173C4×C131—41no—
174C4×C131—41no—
175C4×C131C2×#4111yescharacteristic
176C4×C131H(5,𝔽2)11yescharacteristic
177Dic131—41no—
178Dic131—41no—
179Dic131—41no—
180C2×C131—22no—
181C2×C131—41no—
182C2×C131—81no—
183C2×C131#54611yescharacteristic
184C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C131#57811yescharacteristic
186D131—81no—
187C131#330811yescharacteristic, complemented, Sylow
188#3308—1311nomaximal, Sylow
189#548D13111yescharacteristic, complemented
190#547—1311no—
191#489—1311no—
192#484—1311no—
193#480—1311no—
194#477—1311no—
195#468—1311no—
196#466—1311no—
197#463—1311no—
198#449—1311no—
199#448—1311no—
200#440—1311no—
201#435—1311no—
202#417—1311no—
203C2×#103—1311no—
204C2×#41—2621no—
205C2×#41C2×D13111yescharacteristic
206C22×Q8C2×D13111yescharacteristic
207C2×SD16—1311no—
208C2×SD16—1311no—
209C2×D8—1311no—
210C2×M16—1311no—
211C22×C8—1311no—
212C4⋊Q8—1311no—
213#124—21no—
214#124C2×D13111yescharacteristic
215#123—2621no—
216#123—2621no—
217C22⋊Q8—21no—
218C22⋊Q8—2621no—
219C22⋊Q8C2×D13111yescharacteristic
220#121—21no—
221#121—1311no—
222#121—1311no—
223#121C2×D13111yescharacteristic
224C4×Q8C2×D13111yescharacteristic
225C4×D4—21no—
226C4×D4—1311no—
227C4×D4C2×D13111yescharacteristic
228C2×C4⋊C4—1311no—
229C2×C4⋊C4—1311no—
230#107—1311no—
231#107—1311no—
232#107—1311no—
233C4⋊C8—1311no—
234#103—1311no—
235#103—1311no—
236#103—1311no—
237#103—1311no—
238#103—1311no—
239#103—1311no—
240#102—1311no—
241#102—1311no—
242#102—1311no—
243#102—1311no—
244C22⋊C8—1311no—
245C22⋊C8—1311no—
246#97—1311no—
247#41—22no—
248#41—41no—
249#41—5244no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—2621no—
254C2×Q8C22×D13111yescharacteristic
255C2×Q8C22×D13111yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2621no—
259C2×D4—2621no—
260C2×D4—1311no—
261C2×D4C22⋊D13111yescharacteristic
262C2×D4C22×D13111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—2621no—
266C22×C4—2621no—
267C22×C4—2621no—
268C22×C4C22×D13111yescharacteristic
269C22×C4C22×D13111yescharacteristic
270SD16—2622no—
271SD16—5241no—
272D8—5241no—
273M16—5241no—
274C2×C8—5241no—
275C2×C8—1311no—
276C2×C8—1311no—
277C2×C8—1311no—
278C2×C8—1311no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—2621no—
282C4⋊C4—2621no—
283C4⋊C4—2621no—
284C4⋊C4—1311no—
285C4⋊C4—1311no—
286C4⋊C4—1311no—
287C4⋊C4C22⋊D13111yescharacteristic
288C4⋊C4C22×D13111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—2621no—
294C22⋊C4—2621no—
295C22⋊C4—2621no—
296C22⋊C4C22⋊D13111yescharacteristic
297C22⋊C4C22⋊D13111yescharacteristic
298C42—21no—
299C42C22×D13111yescharacteristic
300C23—21no—
301C23—2621no—
302C23C2×C22⋊D13111yescharacteristic
303C23C2×C22⋊D13111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—5242no—
309Q8#1452351212yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—2621no—
316D4—2621no—
317D4—5242no—
318D4—5242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—2621no—
329C2×C4—2621no—
330C2×C4—2621no—
331C2×C4—5242no—
332C2×C4—5242no—
333C2×C4—5242no—
334C2×C4C2×C22⋊D13111yescharacteristic
335C2×C4C2×C22⋊D13111yescharacteristic
336C2×C4C2×C22⋊D13111yescharacteristic
337C2×C4C2×C22⋊D13111yescharacteristic
338C2×C4C23×D13111yescharacteristic
339C8—2621no—
340C8—2621no—
341C8—2621no—
342C8—2621no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—5241no—
352C22—5241no—
353C22—5241no—
354C22#1686227112yes—
355C22C22×C22⋊D13111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—5241no—
364C4—5241no—
365C4—5241no—
366C4C2×#1452351211yescharacteristic
367C4#1686229111yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—10481no—
372C2#2255395611yescharacteristic
373C2C2×#1686227111yescharacteristic
374C2#2255412711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#548×C131C2
socleC22×C131C22×D4
hypercenter#548D131
hypocenterC131#3308
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22553966C2semidirect
2. #22553671C2semidirect
3. #22553645C2semidirect
4. #22553623C2semidirect
5. #22553607C2semidirect
6. #22553560C2semidirect
7. #22553546C2semidirect
8. #22553524C2semidirect
9. #22553457C2semidirect
10. #22553451C2semidirect
11. #22553410C2semidirect
12. #22553384C2semidirect
13. #22553313C2semidirect
14. C2×#16862170C2semidirect
15. #548×C131C2semidirect
135. Q8×C131#41semidirect
187. C131#3308semidirect
189. #548D131semidirect
309. Q8#14523512semidirect

Maximal subgroups

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

subgroupindex
#225539662
#225536712
#225536452
#225536232
#225536072
#225535602
#225535462
#225535242
#225534572
#225534512
#225534102
#225533842
#225533132
C2×#168621702
#548×C1312
#3308131