Finite Group Databasebeta

← #30677967

Subgroups — #30677967

Order 16768.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22554008C211yesmaximal, characteristic, complemented
2#22553676C211yesmaximal, characteristic, complemented
3#22553633C211yesmaximal, characteristic, complemented
4#22553623C211yesmaximal, characteristic, complemented
5#22553619C211yesmaximal, characteristic, complemented
6#22553563C211yesmaximal, characteristic, complemented
7#22553546C211yesmaximal, characteristic, complemented
8#22553504C211yesmaximal, characteristic, complemented
9#22553457C211yesmaximal, characteristic, complemented
10#22553457C211yesmaximal, characteristic, complemented
11#22553410C211yesmaximal, characteristic, complemented
12#22553384C211yesmaximal, characteristic, complemented
13#22553313C211yesmaximal, characteristic, complemented
14#22553308C211yesmaximal, characteristic, complemented
15#550×C131C211yesmaximal, characteristic, complemented
16C2×#14523500C2211yescharacteristic
17C2×#14523498C2211yescharacteristic
18C2×#14523498C2211yescharacteristic
19C2×#14523497C2211yescharacteristic
20C22×C131⋊C8C2211yescharacteristic
21#16862241—21no—
22#16862238C2211yescharacteristic
23#16862228—21no—
24#16862220C2211yescharacteristic
25#16862220C2211yescharacteristic
26#16862218—21no—
27C4×D524C2211yescharacteristic
28(C22⋊D131)⋊C4—21no—
29#16862205C2211yescharacteristic
30C2×C4⋊Dic131C2211yescharacteristic
31#16862179C2211yescharacteristic
32#16862179C2211yescharacteristic
33#16862177C2211yescharacteristic
34C4⋊(C131⋊C8)C2211yescharacteristic
35#16862170C2211yescharacteristic
36#16862170C2211yescharacteristic
37#16862170C2211yescharacteristic
38#16862167C2211yescharacteristic
39#16862167C2211yescharacteristic
40#16862167C2211yescharacteristic
41#16862167C2211yescharacteristic
42#16862167C2211yescharacteristic
43#16862167C2211yescharacteristic
44#16862166C2211yescharacteristic
45C22⋊(C131⋊C8)C2211yescharacteristic
46C22⋊(C131⋊C8)C2211yescharacteristic
47#16862157C2211yescharacteristic
48C2×#41×C131C2211yescharacteristic
49C2×#41×C131C2211yescharacteristic
50#127×C131—21no—
51#127×C131C2211yescharacteristic
52#121×C131—21no—
53#121×C131—21no—
54#121×C131C2211yescharacteristic
55#121×C131C2211yescharacteristic
56C4×Q8×C131C2211yescharacteristic
57C4×D4×C131—21no—
58C4×D4×C131C2211yescharacteristic
59C2×D524C2311yescharacteristic
60C2×C22⋊D131—21no—
61C2×C22⋊D131—21no—
62C22×Dic131—21no—
63C2×C4×D131—21no—
64#14523500—22no—
65#14523498—41no—
66#14523498—41no—
67#14523497—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—
84#41×C131—22no—
85#41×C131—22no—
86#41×C131—41no—
87#41×C131—41no—
88C2×Q8×C131C2311yescharacteristic
89C2×D4×C131—21no—
90C2×D4×C131—21no—
91C2×D4×C131—21no—
92C2×D4×C131—21no—
93C2×D4×C131—21no—
94C2×D4×C131—21no—
95C2×D4×C131D411yescharacteristic
96C2×D4×C131C2311yescharacteristic
97C2×D4×C131C2311yescharacteristic
98C22×C4×C131—21no—
99C22×C4×C131—21no—
100C22×C4×C131C2311yescharacteristic
101C22×C4×C131C2311yescharacteristic
102C4⋊C4×C131—21no—
103C4⋊C4×C131D411yescharacteristic
104C4⋊C4×C131C2311yescharacteristic
105C22⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C22⋊C4×C131D411yescharacteristic
108C22⋊C4×C131D411yescharacteristic
109C42×C131—21no—
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×C131C2×D411yescharacteristic
128C23×C131C2×D411yescharacteristic
129Q8×C131—21no—
130Q8×C131#4112yes—
131D4×C131—21no—
132D4×C131—21no—
133D4×C131—21no—
134D4×C131—21no—
135D4×C131—41no—
136D4×C131—42no—
137D4×C131—42no—
138D4×C131—42no—
139D4×C131—81no—
140D4×C131—81no—
141D4×C131—81no—
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—21no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131—41no—
164C22×C131—41no—
165C22×C131—41no—
166C22×C131—41no—
167C22×C131C22×D411yescharacteristic
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—41no—
171C4×C131—21no—
172C4×C131—21no—
173C4×C131—22no—
174C4×C131—21no—
175C4×C131—21no—
176C4×C131—41no—
177C4×C131C2×#4111yescharacteristic
178C4×C131H(5,𝔽2)11yescharacteristic
179Dic131—41no—
180Dic131—41no—
181Dic131—41no—
182C2×C131—41no—
183C2×C131—41no—
184C2×C131—81no—
185C2×C131—81no—
186C2×C131#54611yescharacteristic
187C2×C131#57511yescharacteristic
188C2×C131#57611yescharacteristic
189D131—81no—
190C131#330911yescharacteristic, complemented, Sylow
191#3309—1311nomaximal, Sylow
192#553—1311no—
193#550D13111yescharacteristic, complemented
194#490—1311no—
195#482—1311no—
196#480—1311no—
197#479—1311no—
198#469—1311no—
199#466—1311no—
200#459—1311no—
201#449—1311no—
202#449—1311no—
203#440—1311no—
204#435—1311no—
205#417—1311no—
206#416—1311no—
207C2×#41C2×D13111yescharacteristic
208C2×#41C2×D13111yescharacteristic
209C2×SD16—1311no—
210C2×D8—1311no—
211C2×D8—1311no—
212C2×M16—1311no—
213C22×C8—1311no—
214#127—21no—
215#127C2×D13111yescharacteristic
216#126—2621no—
217#125—1311no—
218#123—2621no—
219#121—21no—
220#121—21no—
221#121—2621no—
222#121—1311no—
223#121—1311no—
224#121C2×D13111yescharacteristic
225#121C2×D13111yescharacteristic
226C4×Q8C2×D13111yescharacteristic
227C4×D4—21no—
228C4×D4—2621no—
229C4×D4—1311no—
230C4×D4C2×D13111yescharacteristic
231#117—1311no—
232C2×C4⋊C4—1311no—
233#107—1311no—
234#107—1311no—
235#106—1311no—
236C4⋊C8—1311no—
237#103—1311no—
238#103—1311no—
239#103—1311no—
240#102—1311no—
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102—1311no—
246#102—1311no—
247C22⋊C8—1311no—
248C22⋊C8—1311no—
249#97—1311no—
250#41—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—2621no—
262C2×D4—2621no—
263C2×D4—1311no—
264C2×D4C22⋊D13111yescharacteristic
265C2×D4C22×D13111yescharacteristic
266C2×D4C22×D13111yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2621no—
270C22×C4—2621no—
271C22×C4C22×D13111yescharacteristic
272C22×C4C22×D13111yescharacteristic
273SD16—2622no—
274D8—5241no—
275D8—5241no—
276M16—5241no—
277C2×C8—5241no—
278C2×C8—1311no—
279C2×C8—1311no—
280C2×C8—1311no—
281C2×C8—1311no—
282C4⋊C4—21no—
283C4⋊C4—2621no—
284C4⋊C4—2621no—
285C4⋊C4—2621no—
286C4⋊C4—1311no—
287C4⋊C4—1311no—
288C4⋊C4—1311no—
289C4⋊C4C22⋊D13111yescharacteristic
290C4⋊C4C22×D13111yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—2621no—
294C22⋊C4—2621no—
295C22⋊C4—2621no—
296C22⋊C4—2621no—
297C22⋊C4C22⋊D13111yescharacteristic
298C22⋊C4C22⋊D13111yescharacteristic
299C42—21no—
300C42—2621no—
301C42C22×D13111yescharacteristic
302C23—21no—
303C23—21no—
304C23—2621no—
305C23C2×C22⋊D13111yescharacteristic
306C23C2×C22⋊D13111yescharacteristic
307Q8—21no—
308Q8#1452351212yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—2621no—
318D4—2621no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—5242no—
323D4—10481no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—2621no—
335C2×C4—10481no—
336C2×C4—10481no—
337C2×C4C2×C22⋊D13111yescharacteristic
338C2×C4C2×C22⋊D13111yescharacteristic
339C2×C4C2×C22⋊D13111yescharacteristic
340C2×C4C2×C22⋊D13111yescharacteristic
341C2×C4C23×D13111yescharacteristic
342C8—2621no—
343C8—2621no—
344C8—2621no—
345C8—2621no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22C22×C22⋊D13111yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—22no—
366C4—21no—
367C4—41no—
368C4—5241no—
369C4—5241no—
370C4—5241no—
371C4C2×#1452351211yescharacteristic
372C4#1686229111yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—10481no—
378C2#2255395611yescharacteristic
379C2#2255411011yescharacteristic
380C2#2255411511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#550×C131C2
socleC22×C131C22×D4
hypercenter#550D131
hypocenterC131#3309
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #22554008C2semidirect
2. #22553676C2semidirect
3. #22553633C2semidirect
4. #22553623C2semidirect
5. #22553619C2semidirect
6. #22553563C2semidirect
7. #22553546C2semidirect
8. #22553504C2semidirect
9. #22553457C2semidirect
10. #22553457C2semidirect
11. #22553410C2semidirect
12. #22553384C2semidirect
13. #22553313C2semidirect
14. #22553308C2semidirect
15. #550×C131C2semidirect
190. C131#3309semidirect
193. #550D131semidirect

Maximal subgroups

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

subgroupindex
#225540082
#225536762
#225536332
#225536232
#225536192
#225535632
#225535462
#225535042
#225534572
#225534102
#225533842
#225533132
#225533082
#550×C1312
#3309131