Finite Group Databasebeta

← #30677811

Subgroups — #30677811

Order 16768.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22554008C211yesmaximal, characteristic, complemented
2#22553666C211yesmaximal, characteristic, complemented
3#22553645C211yesmaximal, characteristic, complemented
4#22553633C211yesmaximal, characteristic, complemented
5#22553612C211yesmaximal, characteristic, complemented
6#22553546C211yesmaximal, characteristic, complemented
7#22553543C211yesmaximal, characteristic, complemented
8#22553524C211yesmaximal, characteristic, complemented
9#22553457C211yesmaximal, characteristic, complemented
10#22553446C211yesmaximal, characteristic, complemented
11C4×#14523498C211yesmaximal, characteristic, complemented
12D4×C131⋊C8C211yesmaximal, characteristic, complemented
13#22553308C211yesmaximal, characteristic, complemented
14C2×#16862167C211yesmaximal, characteristic, complemented
15#546×C131C211yesmaximal, characteristic, complemented
16C2×#14523500C2211yescharacteristic, complemented
17C2×#14523498C2211yescharacteristic
18C2×#14523498C2211yescharacteristic
19C22×C131⋊C8C2211yescharacteristic
20C22×C131⋊C8C2211yescharacteristic, complemented
21#16862241—21no—
22#16862238C2211yescharacteristic, complemented
23#16862228—21no—
24#16862220C2211yescharacteristic
25#16862220C2211yescharacteristic, complemented
26#16862218—21no—
27C4×D524C2211yescharacteristic, complemented
28(C22⋊D131)⋊C4—21no—
29#16862205C2211yescharacteristic, complemented
30C2×C4⋊Dic131C2211yescharacteristic
31#16862179C2211yescharacteristic, complemented
32#16862179C2211yescharacteristic, complemented
33#16862177C2211yescharacteristic, complemented
34C4⋊(C131⋊C8)C2211yescharacteristic, complemented
35#16862170C2211yescharacteristic, complemented
36#16862170C2211yescharacteristic, complemented
37#16862170C2211yescharacteristic, complemented
38#16862167C2211yescharacteristic
39#16862167C2211yescharacteristic
40#16862167C2211yescharacteristic
41#16862167C2211yescharacteristic
42#16862167C2211yescharacteristic
43#16862167C2211yescharacteristic
44#16862166C2211yescharacteristic, complemented
45C22⋊(C131⋊C8)C2211yescharacteristic
46C22⋊(C131⋊C8)C2211yescharacteristic, complemented
47C4×C131⋊C8C2211yescharacteristic, complemented
48C2×#41×C131C2211yescharacteristic
49C22×D4×C131C2211yescharacteristic
50#124×C131C2211yescharacteristic
51#123×C131—21no—
52C22⋊Q8×C131C2211yescharacteristic
53#121×C131—21no—
54#121×C131C2211yescharacteristic
55#120×C131—21no—
56C4×D4×C131C2211yescharacteristic
57C4×D4×C131C2211yescharacteristic
58C2×C22⋊C4×C131—21no—
59C2×D524C2311yescharacteristic
60C2×C22⋊D131—21no—
61C2×C22⋊D131—21no—
62C22×Dic131—21no—
63C2×C4×D131—21no—
64#14523500—41no—
65#14523498—22no—
66#14523498—41no—
67C2×C131⋊C8—41no—
68C2×C131⋊C8—41no—
69C2×C131⋊C8C2311yescharacteristic
70C2×C131⋊C8C2311yescharacteristic
71C2×C131⋊C8C2311yescharacteristic
72C2×C131⋊C8C2311yescharacteristic
73C4⋊Dic131—21no—
74C4⋊Dic131C2311yescharacteristic
75C4⋊Dic131C2311yescharacteristic
76C4⋊Dic131C2311yescharacteristic
77Dic131⋊C4—21no—
78Dic131⋊C4—21no—
79C22⋊Dic131—21no—
80(C2×D131)⋊C4—21no—
81(C2×D131)⋊C4—21no—
82(C2×D131)⋊C4—21no—
83C4×Dic131—21no—
84C24×C131—21no—
85#41×C131—22no—
86#41×C131—41no—
87C2×Q8×C131C2311yescharacteristic
88C2×D4×C131—22no—
89C2×D4×C131—21no—
90C2×D4×C131—21no—
91C2×D4×C131—21no—
92C2×D4×C131—41no—
93C2×D4×C131D411yescharacteristic
94C2×D4×C131C2311yescharacteristic
95C2×D4×C131C2311yescharacteristic
96C22×C4×C131—21no—
97C22×C4×C131—21no—
98C22×C4×C131C2311yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C4⋊C4×C131—21no—
101C4⋊C4×C131D411yescharacteristic
102C4⋊C4×C131C2311yescharacteristic
103C22⋊C4×C131—21no—
104C22⋊C4×C131—21no—
105C22⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C22⋊C4×C131—21no—
108C22⋊C4×C131D411yescharacteristic
109C22⋊C4×C131D411yescharacteristic
110C42×C131C2311yescharacteristic
111C22×D131—21no—
112D524—21no—
113D524—21no—
114C22⋊D131—42no—
115C22⋊D131—81no—
116C2×Dic131—21no—
117C2×Dic131—21no—
118C2×Dic131—21no—
119C2×Dic131—81no—
120C4×D131—81no—
121C131⋊C8—21no—
122C131⋊C8—21no—
123C131⋊C8—21no—
124C131⋊C8—21no—
125C23×C131—21no—
126C23×C131—21no—
127C23×C131—42no—
128C23×C131—42no—
129C23×C131—81no—
130C23×C131C2×D411yescharacteristic
131C23×C131C2×D411yescharacteristic
132Q8×C131—21no—
133Q8×C131—21no—
134D4×C131—22no—
135D4×C131—21no—
136D4×C131—21no—
137D4×C131—21no—
138D4×C131—42no—
139D4×C131—42no—
140D4×C131—81no—
141D4×C131#4112yes—
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—81no—
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—22no—
159C22×C131—21no—
160C22×C131—21no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131—41no—
164C22×C131—41no—
165C22×C131—41no—
166C22×C131—84no—
167C22×C131#13512yes—
168C22×C131C22×D411yescharacteristic
169C2×D131—41no—
170C2×D131—41no—
171C2×D131—41no—
172C4×C131—21no—
173C4×C131—21no—
174C4×C131—21no—
175C4×C131—21no—
176C4×C131—41no—
177C4×C131—41no—
178C4×C131C2×#4111yescharacteristic
179C4×C131H(5,𝔽2)11yescharacteristic
180Dic131—41no—
181Dic131—41no—
182Dic131—41no—
183C2×C131—22no—
184C2×C131—41no—
185C2×C131—42no—
186C2×C131—81no—
187C2×C131#54611yescharacteristic
188C2×C131C2×#13511yescharacteristic
189C2×C131#57611yescharacteristic
190D131—81no—
191C131#329711yescharacteristic, complemented, Sylow
192#3297—1311nomaximal, Sylow
193#553—1311no—
194#546D13111yescharacteristic, complemented
195#488—1311no—
196#484—1311no—
197#482—1311no—
198#478—1311no—
199#466—1311no—
200#465—1311no—
201#463—1311no—
202#449—1311no—
203#447—1311no—
204C4×D8—1311no—
205C8×D4—1311no—
206#416—1311no—
207C2×#102—1311no—
208C2×#41C2×D13111yescharacteristic
209C22×D4C2×D13111yescharacteristic
210C2×SD16—1311no—
211C2×D8—1311no—
212C2×D8—1311no—
213C22×C8—1311no—
214C22×C8—1311no—
215#126—2621no—
216#125—1311no—
217#124C2×D13111yescharacteristic
218#123—21no—
219#123—2621no—
220C22⋊Q8C2×D13111yescharacteristic
221#121—21no—
222#121—2621no—
223#121—1311no—
224#121—1311no—
225#121C2×D13111yescharacteristic
226#120—21no—
227C4×D4—2621no—
228C4×D4—1311no—
229C4×D4C2×D13111yescharacteristic
230C4×D4C2×D13111yescharacteristic
231#117—1311no—
232C2×C4⋊C4—1311no—
233C2×C22⋊C4—21no—
234#107—1311no—
235#107—1311no—
236#106—1311no—
237C4⋊C8—1311no—
238#103—1311no—
239#103—1311no—
240#103—1311no—
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102—1311no—
246#102—1311no—
247#102—1311no—
248C22⋊C8—1311no—
249C22⋊C8—1311no—
250C4×C8—1311no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—2621no—
261C2×D4—2621no—
262C2×D4—1311no—
263C2×D4C22⋊D13111yescharacteristic
264C2×D4C22×D13111yescharacteristic
265C2×D4C22×D13111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4C22×D13111yescharacteristic
271C22×C4C22×D13111yescharacteristic
272SD16—5241no—
273D8—2622no—
274D8—5241no—
275C2×C8—5241no—
276C2×C8—5241no—
277C2×C8—1311no—
278C2×C8—1311no—
279C2×C8—1311no—
280C2×C8—1311no—
281C4⋊C4—21no—
282C4⋊C4—2621no—
283C4⋊C4—2621no—
284C4⋊C4—2621no—
285C4⋊C4—1311no—
286C4⋊C4—1311no—
287C4⋊C4—1311no—
288C4⋊C4C22⋊D13111yescharacteristic
289C4⋊C4C22×D13111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—2621no—
296C22⋊C4—2621no—
297C22⋊C4—2621no—
298C22⋊C4—2621no—
299C22⋊C4C22⋊D13111yescharacteristic
300C22⋊C4C22⋊D13111yescharacteristic
301C42—2621no—
302C42C22×D13111yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—2621no—
308C23—81no—
309C23C2×C22⋊D13111yescharacteristic
310C23C2×C22⋊D13111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—2621no—
320D4—2621no—
321D4—81no—
322D4—5242no—
323D4—10481no—
324D4#1452351212yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—2621no—
335C2×C4—81no—
336C2×C4—10481no—
337C2×C4—10481no—
338C2×C4C2×C22⋊D13111yescharacteristic
339C2×C4C2×C22⋊D13111yescharacteristic
340C2×C4C2×C22⋊D13111yescharacteristic
341C2×C4C2×C22⋊D13111yescharacteristic
342C2×C4C23×D13111yescharacteristic
343C8—2621no—
344C8—2621no—
345C8—2621no—
346C8—2621no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—5241no—
360C22—5241no—
361C22—5241no—
362C22#1686226612yes—
363C22C22×C22⋊D13111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—5241no—
371C4—5241no—
372C4—5241no—
373C4C2×#1452351211yescharacteristic
374C4#1686229111yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—10481no—
380C2#2255395611yescharacteristic
381C2C2×#1686226611yescharacteristic
382C2#2255411511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#546×C131C2
socleC22×C131C22×D4
hypercenter#546D131
hypocenterC131#3297
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #22554008C2semidirect
2. #22553666C2semidirect
3. #22553645C2semidirect
4. #22553633C2semidirect
5. #22553612C2semidirect
6. #22553546C2semidirect
7. #22553543C2semidirect
8. #22553524C2semidirect
9. #22553457C2semidirect
10. #22553446C2semidirect
11. C4×#14523498C2semidirect
12. D4×C131⋊C8C2semidirect
13. #22553308C2semidirect
14. C2×#16862167C2semidirect
15. #546×C131C2semidirect
16. C2×#14523500C22semidirect
20. C22×C131⋊C8C22semidirect
22. #16862238C22semidirect
25. #16862220C22semidirect
27. C4×D524C22semidirect
29. #16862205C22semidirect
31. #16862179C22semidirect
32. #16862179C22semidirect
33. #16862177C22semidirect
34. C4⋊(C131⋊C8)C22semidirect
35. #16862170C22semidirect
36. #16862170C22semidirect
37. #16862170C22semidirect
44. #16862166C22semidirect
46. C22⋊(C131⋊C8)C22semidirect
47. C4×C131⋊C8C22semidirect
191. C131#3297semidirect
194. #546D131semidirect

Maximal subgroups

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

subgroupindex
#225540082
#225536662
#225536452
#225536332
#225536122
#225535462
#225535432
#225535242
#225534572
#225534462
C4×#145234982
D4×C131⋊C82
#225533082
C2×#168621672
#546×C1312
#3297131