Finite Group Databasebeta

← #30677963

Subgroups — #30677963

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#22554006C211yesmaximal, characteristic, complemented
2#22553981C211yesmaximal, characteristic, complemented
3#22553675C211yesmaximal, characteristic, complemented
4#22553636C211yesmaximal, characteristic, complemented
5#22553624C211yesmaximal, characteristic, complemented
6#22553621C211yesmaximal, characteristic, complemented
7#22553564C211yesmaximal, characteristic, complemented
8#22553548C211yesmaximal, characteristic, complemented
9#22553506C211yesmaximal, characteristic, complemented
10#22553460C211yesmaximal, characteristic, complemented
11#22553457C211yesmaximal, characteristic, complemented
12#22553415C211yesmaximal, characteristic, complemented
13#22553383C211yesmaximal, characteristic, complemented
14#22553312C211yesmaximal, characteristic, complemented
15#416×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×#14523500C2211yescharacteristic
18C2×D1048C2211yescharacteristic
19C2×#14523498C2211yescharacteristic
20C2×#14523496C2211yescharacteristic
21#16862242—21no—
22#16862242C2211yescharacteristic
23#16862240—21no—
24#16862237C2211yescharacteristic
25#16862228—21no—
26#16862221C2211yescharacteristic
27#16862220—21no—
28#16862220C2211yescharacteristic
29#16862220C2211yescharacteristic
30#16862219—21no—
31#16862217—21no—
32#16862217C2211yescharacteristic
33Q8×Dic131C2211yescharacteristic
34C4×D524—21no—
35#16862208C2211yescharacteristic
36C4×C22⋊D131—21no—
37C4×C22⋊D131C2211yescharacteristic
38C4⋊C4×D131C2211yescharacteristic
39#16862180C2211yescharacteristic
40#16862179C2211yescharacteristic
41#16862177C2211yescharacteristic
42Dic131⋊C8C2211yescharacteristic
43#16862170C2211yescharacteristic
44#16862168C2211yescharacteristic
45#16862168C2211yescharacteristic
46#16862167C2211yescharacteristic
47#16862166C2211yescharacteristic
48#16862166C2211yescharacteristic
49C22⋊(C131⋊C8)C2211yescharacteristic
50(C2×D131)⋊C8C2211yescharacteristic
51#16862156C2211yescharacteristic
52C2×#41×C131C2211yescharacteristic
53C22×C8×C131C2211yescharacteristic
54#117×C131C2211yescharacteristic
55#103×C131C2211yescharacteristic
56#103×C131C2211yescharacteristic
57#102×C131C2211yescharacteristic
58#102×C131C2211yescharacteristic
59#14523514—22no—
60#14523514—41no—
61C2×D524—21no—
62C2×D524—21no—
63C2×D524C2311yescharacteristic
64C2×D524C2311yescharacteristic
65C2×C22⋊D131—21no—
66C2×C22⋊D131—21no—
67C2×C22⋊D131—21no—
68C2×C22⋊D131—21no—
69C2×C22⋊D131—21no—
70C2×C22⋊D131D411yescharacteristic
71C2×C4×D131—21no—
72C2×C4×D131—21no—
73C2×C4×D131—21no—
74C2×C4×D131C2311yescharacteristic
75#14523500—22no—
76D1048—41no—
77#14523498—41no—
78#14523496—41no—
79C2×C131⋊C8C2311yescharacteristic
80C2×C131⋊C8C2311yescharacteristic
81C4⋊Dic131—21no—
82C4⋊Dic131—21no—
83C4⋊Dic131C2311yescharacteristic
84C4⋊Dic131C2311yescharacteristic
85Dic131⋊C4—21no—
86Dic131⋊C4—21no—
87Dic131⋊C4D411yescharacteristic
88C22⋊Dic131—21no—
89C22⋊Dic131D411yescharacteristic
90(C2×D131)⋊C4—21no—
91(C2×D131)⋊C4—21no—
92(C2×D131)⋊C4—21no—
93(C2×D131)⋊C4—21no—
94(C2×D131)⋊C4D411yescharacteristic
95C4×Dic131—21no—
96C4×Dic131C2311yescharacteristic
97#41×C131—22no—
98#41×C131—41no—
99C2×Q8×C131C2311yescharacteristic
100C2×D4×C131—21no—
101C2×D4×C131C2311yescharacteristic
102C22×C4×C131—21no—
103C22×C4×C131C2311yescharacteristic
104C2×C8×C131—41no—
105C2×C8×C131C2311yescharacteristic
106C2×C8×C131C2311yescharacteristic
107C4⋊C4×C131C2311yescharacteristic
108C4⋊C4×C131C2311yescharacteristic
109C22⋊C4×C131—21no—
110C42×C131—21no—
111C22×D131—21no—
112C22×D131—21no—
113C22×D131C2×D411yescharacteristic
114D524—21no—
115D524—21no—
116D524—21no—
117D524—21no—
118D524—42no—
119D524—42no—
120C22⋊D131—41no—
121C22⋊D131—42no—
122C22⋊D131—81no—
123C22⋊D131—81no—
124C22⋊D131—81no—
125C22⋊D131—81no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131C2×D411yescharacteristic
129C2×Dic131C2×D411yescharacteristic
130C2×Dic131C2×D411yescharacteristic
131C4×D131—21no—
132C4×D131—21no—
133C4×D131—42no—
134C4×D131—81no—
135C4×D131—81no—
136C131⋊C8—21no—
137C131⋊C8—21no—
138C23×C131—21no—
139C23×C131C2×D411yescharacteristic
140Q8×C131—21no—
141Q8×C131#4112yes—
142D4×C131—21no—
143D4×C131—21no—
144D4×C131—42no—
145C2×C4×C131—21no—
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×C131C22×D411yescharacteristic
162C2×D131—21no—
163C2×D131—21no—
164C2×D131—21no—
165C2×D131—41no—
166C2×D131—41no—
167C2×D131—41no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—41no—
171C4×C131—21no—
172C4×C131—22no—
173C4×C131—41no—
174C4×C131—41no—
175C4×C131C2×#4111yescharacteristic
176C4×C131H(5,𝔽2)11yescharacteristic
177Dic131—21no—
178Dic131—21no—
179Dic131—21no—
180Dic131—41no—
181Dic131—41no—
182C2×C131—41no—
183C2×C131—81no—
184C2×C131#54611yescharacteristic
185C2×C131#57511yescharacteristic
186C2×C131#57611yescharacteristic
187D131—41no—
188D131—81no—
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#449—1311no—
203#440—1311no—
204#435—1311no—
205#417—1311no—
206#416D13111yescharacteristic, complemented
207C2×#41—1311no—
208C2×#41C2×D13111yescharacteristic
209C2×SD16—1311no—
210C2×D8—1311no—
211C2×D8—1311no—
212C2×M16—1311no—
213C22×C8C2×D13111yescharacteristic
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#121—1311no—
226C4×Q8—1311no—
227C4×D4—2621no—
228C4×D4—2621no—
229C4×D4—1311no—
230C4×D4—1311no—
231#117C2×D13111yescharacteristic
232C2×C4⋊C4—1311no—
233#107—1311no—
234#107—1311no—
235#106—1311no—
236C4⋊C8—1311no—
237#103—1311no—
238#103C2×D13111yescharacteristic
239#103C2×D13111yescharacteristic
240#102—1311no—
241#102—1311no—
242#102—1311no—
243#102—1311no—
244#102—1311no—
245#102C2×D13111yescharacteristic
246#102C2×D13111yescharacteristic
247C22⋊C8—1311no—
248C22⋊C8—1311no—
249#97—1311no—
250#41—22no—
251#41—41no—
252#41—2622no—
253#41—5241no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—21no—
256C2×D4—2621no—
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×D4—1311no—
266C2×D4C22×D13111yescharacteristic
267C22×C4—21no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4—2621no—
271C22×C4—1311no—
272C22×C4C22×D13111yescharacteristic
273SD16—2622no—
274D8—5241no—
275D8—5241no—
276M16—5241no—
277C2×C8—41no—
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⋊C4C22×D13111yescharacteristic
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—21no—
300C42—2621no—
301C42—1311no—
302C23—21no—
303C23—2621no—
304C23—2621no—
305C23—1311no—
306C23D4×D13111yescharacteristic
307Q8—21no—
308Q8#1452351312yes—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—2621no—
313D4—2621no—
314D4—2621no—
315D4—2621no—
316D4—5242no—
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—21no—
329C2×C4—42no—
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—2621no—
353C22—2621no—
354C22—2621no—
355C22—5241no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22C2×D4×D13111yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—2621no—
367C4—2621no—
368C4—2621no—
369C4—5241no—
370C4—5241no—
371C4C2×#1452351311yescharacteristic
372C4#1686229111yescharacteristic
373C2—41no—
374C2—81no—
375C2—5241no—
376C2—10481no—
377C2—10481no—
378C2#2255395211yescharacteristic
379C2#2255410911yescharacteristic
380C2#2255411411yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#416×C131C2
socleC22×C131C22×D4
hypercenter#416D131
hypocenterC131#3309
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22554006C2semidirect
2. #22553981C2semidirect
3. #22553675C2semidirect
4. #22553636C2semidirect
5. #22553624C2semidirect
6. #22553621C2semidirect
7. #22553564C2semidirect
8. #22553548C2semidirect
9. #22553506C2semidirect
10. #22553460C2semidirect
11. #22553457C2semidirect
12. #22553415C2semidirect
13. #22553383C2semidirect
14. #22553312C2semidirect
15. #416×C131C2semidirect
190. C131#3309semidirect
206. #416D131semidirect

Maximal subgroups

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

subgroupindex
#225540062
#225539812
#225536752
#225536362
#225536242
#225536212
#225535642
#225535482
#225535062
#225534602
#225534572
#225534152
#225533832
#225533122
#416×C1312
#3309131