Finite Group Databasebeta

← #30677960

Subgroups — #30677960

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#22554014C211yesmaximal, characteristic, complemented
2#22553981C211yesmaximal, characteristic, complemented
3#22553675C211yesmaximal, characteristic, complemented
4#22553634C211yesmaximal, characteristic, complemented
5#22553626C211yesmaximal, characteristic, complemented
6#22553621C211yesmaximal, characteristic, complemented
7#22553563C211yesmaximal, characteristic, complemented
8#22553545C211yesmaximal, characteristic, complemented
9#22553505C211yesmaximal, characteristic, complemented
10#22553456C211yesmaximal, characteristic, complemented
11#22553413C211yesmaximal, characteristic, complemented
12#22553386C211yesmaximal, characteristic, complemented
13#22553313C211yesmaximal, characteristic, complemented
14#22553307C211yesmaximal, characteristic, complemented
15#449×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×#14523498C2211yescharacteristic
18C2×#14523497C2211yescharacteristic
19C2×C8×D131C2211yescharacteristic
20#16862242—21no—
21#16862242C2211yescharacteristic
22#16862239—21no—
23#16862237C2211yescharacteristic
24#16862230—21no—
25#16862220—21no—
26#16862220C2211yescharacteristic
27#16862218—21no—
28#16862217—21no—
29#16862217C2211yescharacteristic
30#16862217C2211yescharacteristic
31Q8×Dic131C2211yescharacteristic
32D4×Dic131—21no—
33D4×Dic131C2211yescharacteristic
34C4×C22⋊D131—21no—
35C4×C22⋊D131C2211yescharacteristic
36#16862204C2211yescharacteristic
37C2×C4⋊Dic131C2211yescharacteristic
38#16862180C2211yescharacteristic
39#16862180C2211yescharacteristic
40#16862177C2211yescharacteristic
41Dic131⋊C8C2211yescharacteristic
42#16862170C2211yescharacteristic
43#16862170C2211yescharacteristic
44#16862168C2211yescharacteristic
45#16862168C2211yescharacteristic
46#16862167C2211yescharacteristic
47#16862167C2211yescharacteristic
48#16862167C2211yescharacteristic
49#16862166C2211yescharacteristic
50(C2×D131)⋊C8C2211yescharacteristic
51#16862156C2211yescharacteristic
52C2×#41×C131C2211yescharacteristic
53C2×SD16×C131C2211yescharacteristic
54C2×D8×C131C2211yescharacteristic
55#121×C131C2211yescharacteristic
56#103×C131C2211yescharacteristic
57#102×C131C2211yescharacteristic
58C22⋊C8×C131C2211yescharacteristic
59#14523514—22no—
60#14523514—41no—
61C2×D524—21no—
62C2×D524C2311yescharacteristic
63C2×C22⋊D131—21no—
64C2×C22⋊D131—21no—
65C2×C22⋊D131—21no—
66C2×C22⋊D131—21no—
67C2×C22⋊D131—21no—
68C2×C22⋊D131D411yescharacteristic
69C22×Dic131—21no—
70C22×Dic131—21no—
71C2×C4×D131—21no—
72C2×C4×D131C2311yescharacteristic
73#14523498—41no—
74#14523497—41no—
75C2×C131⋊C8C2311yescharacteristic
76C2×C131⋊C8C2311yescharacteristic
77C8×D131—41no—
78C4⋊Dic131—21no—
79C4⋊Dic131—21no—
80C4⋊Dic131—21no—
81C4⋊Dic131C2311yescharacteristic
82C4⋊Dic131C2311yescharacteristic
83C4⋊Dic131C2311yescharacteristic
84Dic131⋊C4—21no—
85Dic131⋊C4D411yescharacteristic
86C22⋊Dic131—21no—
87C22⋊Dic131—21no—
88C22⋊Dic131—21no—
89C22⋊Dic131D411yescharacteristic
90(C2×D131)⋊C4—21no—
91(C2×D131)⋊C4—21no—
92(C2×D131)⋊C4D411yescharacteristic
93C4×Dic131—21no—
94C4×Dic131—21no—
95C4×Dic131C2311yescharacteristic
96#41×C131—22no—
97#41×C131—41no—
98C2×Q8×C131C2311yescharacteristic
99C2×D4×C131—21no—
100C2×D4×C131—21no—
101C2×D4×C131C2311yescharacteristic
102C2×D4×C131C2311yescharacteristic
103C22×C4×C131—21no—
104C22×C4×C131C2311yescharacteristic
105SD16×C131—22no—
106D8×C131—41no—
107C2×C8×C131C2311yescharacteristic
108C2×C8×C131C2311yescharacteristic
109C4⋊C4×C131C2311yescharacteristic
110C22⋊C4×C131—21no—
111C22×D131—21no—
112C22×D131C2×D411yescharacteristic
113D524—21no—
114D524—21no—
115D524—42no—
116C22⋊D131—41no—
117C22⋊D131—42no—
118C22⋊D131—81no—
119C22⋊D131—81no—
120C22⋊D131—81no—
121C22⋊D131—81no—
122C2×Dic131—21no—
123C2×Dic131—21no—
124C2×Dic131—21no—
125C2×Dic131—81no—
126C2×Dic131—81no—
127C2×Dic131C2×D411yescharacteristic
128C2×Dic131C2×D411yescharacteristic
129C2×Dic131C2×D411yescharacteristic
130C4×D131—21no—
131C4×D131—21no—
132C4×D131—42no—
133C131⋊C8—21no—
134C131⋊C8—21no—
135C23×C131—21no—
136C23×C131—21no—
137C23×C131C2×D411yescharacteristic
138Q8×C131—21no—
139Q8×C131#4112yes—
140D4×C131—21no—
141D4×C131—21no—
142D4×C131—21no—
143D4×C131—21no—
144D4×C131—42no—
145D4×C131—42no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131—42no—
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2411yescharacteristic
153C8×C131—21no—
154C8×C131—21no—
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—21no—
158C22×C131—41no—
159C22×C131—41no—
160C22×C131—41no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131—41no—
164C22×C131C22×D411yescharacteristic
165C2×D131—21no—
166C2×D131—21no—
167C2×D131—21no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—41no—
171C4×C131—21no—
172C4×C131—22no—
173C4×C131—41no—
174C4×C131C2×#4111yescharacteristic
175C4×C131H(5,𝔽2)11yescharacteristic
176Dic131—21no—
177Dic131—21no—
178Dic131—21no—
179Dic131—41no—
180Dic131—41no—
181Dic131—41no—
182C2×C131—41no—
183C2×C131—81no—
184C2×C131—81no—
185C2×C131#54611yescharacteristic
186C2×C131#57511yescharacteristic
187C2×C131#57611yescharacteristic
188D131—41no—
189D131—81no—
190C131#330911yescharacteristic, complemented, Sylow
191#3309—1311nomaximal, Sylow
192#553—1311no—
193#550—1311no—
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#449D13111yescharacteristic, complemented
203#440—1311no—
204#435—1311no—
205#417—1311no—
206#416—1311no—
207C2×#41—1311no—
208C2×#41C2×D13111yescharacteristic
209C2×SD16C2×D13111yescharacteristic
210C2×D8—1311no—
211C2×D8C2×D13111yescharacteristic
212C2×M16—1311no—
213C22×C8—1311no—
214#127—2621no—
215#127—1311no—
216#126—2621no—
217#125—1311no—
218#123—2621no—
219#121—2621no—
220#121—2621no—
221#121—2621no—
222#121—1311no—
223#121—1311no—
224#121—1311no—
225#121C2×D13111yescharacteristic
226C4×Q8—1311no—
227C4×D4—2621no—
228C4×D4—2621no—
229C4×D4—1311no—
230C4×D4—1311no—
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#103C2×D13111yescharacteristic
240#102—1311no—
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102—1311no—
246#102C2×D13111yescharacteristic
247C22⋊C8—1311no—
248C22⋊C8C2×D13111yescharacteristic
249#97—1311no—
250#41—22no—
251#41—41no—
252#41—2622no—
253#41—5241no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2621no—
258C2×D4—2621no—
259C2×D4—2621no—
260C2×D4—2621no—
261C2×D4—2621no—
262C2×D4—2621no—
263C2×D4—1311no—
264C2×D4—1311no—
265C2×D4C22×D13111yescharacteristic
266C2×D4C22×D13111yescharacteristic
267C22×C4—21no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4—2621no—
271C22×C4—1311no—
272C22×C4C22×D13111yescharacteristic
273SD16—22no—
274D8—41no—
275D8—5241no—
276M16—5241no—
277C2×C8—5241no—
278C2×C8—1311no—
279C2×C8—1311no—
280C2×C8C22×D13111yescharacteristic
281C2×C8C22×D13111yescharacteristic
282C4⋊C4—2621no—
283C4⋊C4—2621no—
284C4⋊C4—2621no—
285C4⋊C4—2621no—
286C4⋊C4—1311no—
287C4⋊C4—1311no—
288C4⋊C4—1311no—
289C4⋊C4—1311no—
290C4⋊C4C22×D13111yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—2621no—
293C22⋊C4—2621no—
294C22⋊C4—2621no—
295C22⋊C4—2621no—
296C22⋊C4—2621no—
297C22⋊C4—1311no—
298C22⋊C4—1311no—
299C42—2621no—
300C42—2621no—
301C42—1311no—
302C23—21no—
303C23—21no—
304C23—2621no—
305C23—1311no—
306C23D4×D13111yescharacteristic
307Q8—21no—
308Q8#1452351212yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—2621no—
316D4—2621no—
317D4—5242no—
318D4—5241no—
319D4—5242no—
320D4—10481no—
321D4—10481no—
322D4—10481no—
323D4—10481no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—2621no—
330C2×C4—2621no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—5242no—
335C2×C4—10481no—
336C2×C4—10481no—
337C2×C4—1311no—
338C2×C4—1311no—
339C2×C4—1311no—
340C2×C4D4×D13111yescharacteristic
341C2×C4C23×D13111yescharacteristic
342C8—21no—
343C8—21no—
344C8—2621no—
345C8—2621no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—2621no—
356C22—2621no—
357C22—2621no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22C2×D4×D13111yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—2621no—
366C4—2621no—
367C4—2621no—
368C4—5241no—
369C4—5241no—
370C4—5241no—
371C4C2×#1452351211yescharacteristic
372C4#1686229111yescharacteristic
373C2—41no—
374C2—81no—
375C2—81no—
376C2—5241no—
377C2—10481no—
378C2#2255394911yescharacteristic
379C2#2255410811yescharacteristic
380C2#2255411411yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#449×C131C2
socleC22×C131C22×D4
hypercenter#449D131
hypocenterC131#3309
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22554014C2semidirect
2. #22553981C2semidirect
3. #22553675C2semidirect
4. #22553634C2semidirect
5. #22553626C2semidirect
6. #22553621C2semidirect
7. #22553563C2semidirect
8. #22553545C2semidirect
9. #22553505C2semidirect
10. #22553456C2semidirect
11. #22553413C2semidirect
12. #22553386C2semidirect
13. #22553313C2semidirect
14. #22553307C2semidirect
15. #449×C131C2semidirect
190. C131#3309semidirect
202. #449D131semidirect

Maximal subgroups

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

subgroupindex
#225540142
#225539812
#225536752
#225536342
#225536262
#225536212
#225535632
#225535452
#225535052
#225534562
#225534132
#225533862
#225533132
#225533072
#449×C1312
#3309131