Finite Group Databasebeta

← #30677964

Subgroups — #30677964

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#22554015C211yesmaximal, characteristic, complemented
2#22553984C211yesmaximal, characteristic, complemented
3#22553678C211yesmaximal, characteristic, complemented
4#22553638C211yesmaximal, characteristic, complemented
5#22553627C211yesmaximal, characteristic, complemented
6#22553566C211yesmaximal, characteristic, complemented
7#22553549C211yesmaximal, characteristic, complemented
8#22553508C211yesmaximal, characteristic, complemented
9#22553460C211yesmaximal, characteristic, complemented
10#22553460C211yesmaximal, characteristic, complemented
11#22553410C211yesmaximal, characteristic, complemented
12#22553382C211yesmaximal, characteristic, complemented
13#22553312C211yesmaximal, characteristic, complemented
14#22553307C211yesmaximal, characteristic, complemented
15#479×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×#14523514C2211yescharacteristic
18C2×#14523500C2211yescharacteristic
19C2×D1048C2211yescharacteristic
20C2×D1048C2211yescharacteristic
21C2×#14523496C2211yescharacteristic
22C2×C8×D131C2211yescharacteristic
23#16862243—21no—
24#16862243C2211yescharacteristic
25#16862241—21no—
26#16862231—21no—
27#16862221—21no—
28#16862221—21no—
29#16862221—21no—
30#16862221C2211yescharacteristic
31#16862221C2211yescharacteristic
32#16862221C2211yescharacteristic
33#16862221C2211yescharacteristic
34C4×D524—21no—
35C4×D524C2211yescharacteristic
36C4×D524C2211yescharacteristic
37#16862208—21no—
38#16862204C2211yescharacteristic
39C4⋊C4×D131C2211yescharacteristic
40#16862179C2211yescharacteristic
41#16862170C2211yescharacteristic
42#16862168C2211yescharacteristic
43#16862168C2211yescharacteristic
44#16862166C2211yescharacteristic
45#16862166C2211yescharacteristic
46#16862166C2211yescharacteristic
47#16862166C2211yescharacteristic
48#16862166C2211yescharacteristic
49(C2×D131)⋊C8C2211yescharacteristic
50(C2×D131)⋊C8C2211yescharacteristic
51#16862157C2211yescharacteristic
52#125×C131C2211yescharacteristic
53C4×Q8×C131C2211yescharacteristic
54#107×C131C2211yescharacteristic
55#106×C131C2211yescharacteristic
56C4⋊C8×C131C2211yescharacteristic
57#103×C131C2211yescharacteristic
58#103×C131C2211yescharacteristic
59#14523514—22no—
60#14523514—22no—
61#14523514—41no—
62#14523514—41no—
63C2×D524—21no—
64C2×D524—21no—
65C2×D524—21no—
66C2×D524—21no—
67C2×D524—21no—
68C2×D524—21no—
69C2×D524—21no—
70C2×D524—21no—
71C2×D524D411yescharacteristic
72C2×D524C2311yescharacteristic
73C2×D524C2311yescharacteristic
74C2×D524C2311yescharacteristic
75C2×C4×D131—21no—
76C2×C4×D131—21no—
77C2×C4×D131—21no—
78C2×C4×D131—21no—
79C2×C4×D131C2311yescharacteristic
80C2×C4×D131C2311yescharacteristic
81#14523500—22no—
82D1048—41no—
83D1048—41no—
84#14523496—41no—
85C2×C131⋊C8C2311yescharacteristic
86C2×C131⋊C8C2311yescharacteristic
87C8×D131—41no—
88C4⋊Dic131D411yescharacteristic
89C4⋊Dic131C2311yescharacteristic
90Dic131⋊C4—21no—
91(C2×D131)⋊C4—21no—
92(C2×D131)⋊C4—21no—
93(C2×D131)⋊C4—21no—
94(C2×D131)⋊C4—21no—
95(C2×D131)⋊C4—21no—
96(C2×D131)⋊C4—21no—
97(C2×D131)⋊C4D411yescharacteristic
98(C2×D131)⋊C4D411yescharacteristic
99C4×Dic131—21no—
100C2×Q8×C131C2311yescharacteristic
101C2×C8×C131C2311yescharacteristic
102C2×C8×C131C2311yescharacteristic
103C4⋊C4×C131—21no—
104C4⋊C4×C131—21no—
105C4⋊C4×C131—21no—
106C4⋊C4×C131C2311yescharacteristic
107C4⋊C4×C131C2311yescharacteristic
108C4⋊C4×C131C2311yescharacteristic
109C42×C131—21no—
110C42×C131C2311yescharacteristic
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—21no—
114C22×D131C2×D411yescharacteristic
115C22×D131C2×D411yescharacteristic
116D524—21no—
117D524—21no—
118D524—21no—
119D524—21no—
120D524—21no—
121D524—21no—
122D524—41no—
123D524—42no—
124D524—42no—
125D524—42no—
126D524—42no—
127D524—81no—
128D524—81no—
129D524—81no—
130D524—81no—
131C2×Dic131—21no—
132C2×Dic131C2×D411yescharacteristic
133C2×Dic131C2×D411yescharacteristic
134C4×D131—21no—
135C4×D131—21no—
136C4×D131—21no—
137C4×D131—21no—
138C4×D131—42no—
139C4×D131—42no—
140C4×D131—81no—
141C4×D131—81no—
142C131⋊C8—21no—
143C131⋊C8—21no—
144Q8×C131—21no—
145Q8×C131#4112yes—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—21no—
149C2×C4×C131—21no—
150C2×C4×C131C2×D411yescharacteristic
151C2×C4×C131C2×D411yescharacteristic
152C2×C4×C131C2411yescharacteristic
153C8×C131—21no—
154C8×C131—21no—
155C22×C131C22×D411yescharacteristic
156C2×D131—21no—
157C2×D131—21no—
158C2×D131—21no—
159C2×D131—21no—
160C2×D131—21no—
161C2×D131—21no—
162C2×D131—41no—
163C2×D131—41no—
164C2×D131—41no—
165C2×D131—41no—
166C2×D131—41no—
167C2×D131—41no—
168C2×D131—41no—
169C2×D131—41no—
170C2×D131—41no—
171C4×C131—22no—
172C4×C131—21no—
173C4×C131—21no—
174C4×C131—41no—
175C4×C131—41no—
176C4×C131—41no—
177C4×C131C2×#4111yescharacteristic
178C4×C131H(5,𝔽2)11yescharacteristic
179Dic131—21no—
180Dic131—21no—
181Dic131—41no—
182C2×C131#54611yescharacteristic
183C2×C131#57511yescharacteristic
184C2×C131#57611yescharacteristic
185D131—41no—
186D131—41no—
187D131—81no—
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#479D13111yescharacteristic, complemented
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×#41—1311no—
208C2×#41—1311no—
209C2×SD16—1311no—
210C2×D8—1311no—
211C2×D8—1311no—
212C2×M16—1311no—
213C22×C8—1311no—
214#127—2621no—
215#127—1311no—
216#126—2621no—
217#125C2×D13111yescharacteristic
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×Q8C2×D13111yescharacteristic
227C4×D4—2621no—
228C4×D4—2621no—
229C4×D4—1311no—
230C4×D4—1311no—
231#117—1311no—
232C2×C4⋊C4—1311no—
233#107—1311no—
234#107C2×D13111yescharacteristic
235#106C2×D13111yescharacteristic
236C4⋊C8C2×D13111yescharacteristic
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#102—1311no—
246#102—1311no—
247C22⋊C8—1311no—
248C22⋊C8—1311no—
249#97—1311no—
250#41—2622no—
251#41—2622no—
252#41—5241no—
253#41—5241no—
254C2×Q8C22×D13111yescharacteristic
255C2×D4—2621no—
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×D4—1311no—
267C22×C4—2621no—
268C22×C4—2621no—
269C22×C4—2621no—
270C22×C4—2621no—
271C22×C4—1311no—
272C22×C4—1311no—
273SD16—2622no—
274D8—5241no—
275D8—5241no—
276M16—5241no—
277C2×C8—5241no—
278C2×C8—1311no—
279C2×C8—1311no—
280C2×C8C22×D13111yescharacteristic
281C2×C8C22×D13111yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—2621no—
286C4⋊C4—1311no—
287C4⋊C4—1311no—
288C4⋊C4C22×D13111yescharacteristic
289C4⋊C4C22×D13111yescharacteristic
290C4⋊C4C22×D13111yescharacteristic
291C22⋊C4—2621no—
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—
301C42C22×D13111yescharacteristic
302C23—2621no—
303C23—2621no—
304C23—2621no—
305C23—1311no—
306C23—1311no—
307Q8—21no—
308Q8#1452351412yes—
309D4—2621no—
310D4—2621no—
311D4—2621no—
312D4—2621no—
313D4—2621no—
314D4—2621no—
315D4—5241no—
316D4—5242no—
317D4—5242no—
318D4—5242no—
319D4—5242no—
320D4—10481no—
321D4—10481no—
322D4—10481no—
323D4—10481no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—2621no—
329C2×C4—2621no—
330C2×C4—2621no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—5242no—
334C2×C4—5242no—
335C2×C4—10481no—
336C2×C4—10481no—
337C2×C4—1311no—
338C2×C4—1311no—
339C2×C4D4×D13111yescharacteristic
340C2×C4D4×D13111yescharacteristic
341C2×C4C23×D13111yescharacteristic
342C8—21no—
343C8—21no—
344C8—2621no—
345C8—2621no—
346C22—2621no—
347C22—2621no—
348C22—2621no—
349C22—2621no—
350C22—2621no—
351C22—2621no—
352C22—5241no—
353C22—5241no—
354C22—5241no—
355C22—5241no—
356C22—5241no—
357C22—5241no—
358C22—5241no—
359C22—5241no—
360C22—5241no—
361C22C2×D4×D13111yescharacteristic
362C4—21no—
363C4—22no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—2621no—
369C4—2621no—
370C4—5241no—
371C4C2×#1452351411yescharacteristic
372C4#1686229211yescharacteristic
373C2—5241no—
374C2—5241no—
375C2—10481no—
376C2—10481no—
377C2—10481no—
378C2#2255395311yescharacteristic
379C2#2255411111yescharacteristic
380C2#2255411411yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#479×C131C2
socleC22×C131C22×D4
hypercenter#479D131
hypocenterC131#3309
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #22554015C2semidirect
2. #22553984C2semidirect
3. #22553678C2semidirect
4. #22553638C2semidirect
5. #22553627C2semidirect
6. #22553566C2semidirect
7. #22553549C2semidirect
8. #22553508C2semidirect
9. #22553460C2semidirect
10. #22553460C2semidirect
11. #22553410C2semidirect
12. #22553382C2semidirect
13. #22553312C2semidirect
14. #22553307C2semidirect
15. #479×C131C2semidirect
190. C131#3309semidirect
197. #479D131semidirect

Maximal subgroups

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

subgroupindex
#225540152
#225539842
#225536782
#225536382
#225536272
#225535662
#225535492
#225535082
#225534602
#225534102
#225533822
#225533122
#225533072
#479×C1312
#3309131