Finite Group Databasebeta

← #30677863

Subgroups — #30677863

Order 16768.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 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#22553650C211yesmaximal, characteristic, complemented
4#22553628C211yesmaximal, characteristic, complemented
5#22553619C211yesmaximal, characteristic, complemented
6#22553546C211yesmaximal, characteristic, complemented
7#22553543C211yesmaximal, characteristic, complemented
8#22553524C211yesmaximal, characteristic, complemented
9#22553457C211yesmaximal, characteristic, complemented
10#22553451C211yesmaximal, characteristic, complemented
11C4×#14523500C211yesmaximal, characteristic, complemented
12D4×C131⋊C8C211yesmaximal, characteristic, complemented
13#22553308C211yesmaximal, characteristic, complemented
14C2×#16862170C211yesmaximal, characteristic, complemented
15#548×C131C211yesmaximal, characteristic, complemented
16C2×#14523500C2211yescharacteristic
17C2×#14523500C2211yescharacteristic
18C2×#14523498C2211yescharacteristic
19C22×C131⋊C8C2211yescharacteristic
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#16862177C2211yescharacteristic
33#16862177C2211yescharacteristic
34C4⋊(C131⋊C8)C2211yescharacteristic
35#16862170C2211yescharacteristic
36#16862170C2211yescharacteristic
37#16862170C2211yescharacteristic
38#16862170C2211yescharacteristic
39#16862170C2211yescharacteristic
40#16862170C2211yescharacteristic
41#16862167C2211yescharacteristic
42#16862167C2211yescharacteristic
43#16862167C2211yescharacteristic
44#16862166C2211yescharacteristic
45C22⋊(C131⋊C8)C2211yescharacteristic
46C22⋊(C131⋊C8)C2211yescharacteristic
47C4×C131⋊C8C2211yescharacteristic
48C2×#41×C131C2211yescharacteristic
49C22×Q8×C131C2211yescharacteristic
50#124×C131—21no—
51#124×C131C2211yescharacteristic
52C22⋊Q8×C131—21no—
53C22⋊Q8×C131C2211yescharacteristic
54#121×C131—21no—
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#14523500—41no—
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—
84#41×C131—22no—
85#41×C131—41no—
86C2×Q8×C131—21no—
87C2×Q8×C131—22no—
88C2×Q8×C131—41no—
89C2×Q8×C131C2311yescharacteristic
90C2×Q8×C131C2311yescharacteristic
91C2×D4×C131—21no—
92C2×D4×C131—21no—
93C2×D4×C131D411yescharacteristic
94C2×D4×C131C2311yescharacteristic
95C22×C4×C131—21no—
96C22×C4×C131—21no—
97C22×C4×C131C2311yescharacteristic
98C22×C4×C131C2311yescharacteristic
99C4⋊C4×C131—21no—
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×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×C131C2×D411yescharacteristic
127C23×C131C2×D411yescharacteristic
128Q8×C131—21no—
129Q8×C131—21no—
130Q8×C131—21no—
131Q8×C131—42no—
132Q8×C131#4112yes—
133D4×C131—22no—
134D4×C131—21no—
135D4×C131—21no—
136D4×C131—42no—
137D4×C131—42no—
138C2×C4×C131—21no—
139C2×C4×C131—21no—
140C2×C4×C131—21no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—42no—
146C2×C4×C131—42no—
147C2×C4×C131C2×D411yescharacteristic
148C2×C4×C131C2×D411yescharacteristic
149C2×C4×C131C2×D411yescharacteristic
150C2×C4×C131C2×D411yescharacteristic
151C2×C4×C131C2411yescharacteristic
152C22×C131—21no—
153C22×C131—21no—
154C22×C131—21no—
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—41no—
158C22×C131—41no—
159C22×C131—41no—
160C22×C131#13512yes—
161C22×C131C22×D411yescharacteristic
162C2×D131—41no—
163C2×D131—41no—
164C2×D131—41no—
165C4×C131—21no—
166C4×C131—21no—
167C4×C131—22no—
168C4×C131—21no—
169C4×C131—21no—
170C4×C131—41no—
171C4×C131—41no—
172C4×C131C2×#4111yescharacteristic
173C4×C131H(5,𝔽2)11yescharacteristic
174Dic131—41no—
175Dic131—41no—
176Dic131—41no—
177C2×C131—22no—
178C2×C131—41no—
179C2×C131—81no—
180C2×C131#54611yescharacteristic
181C2×C131C2×#13511yescharacteristic
182C2×C131#57711yescharacteristic
183D131—81no—
184C131#330111yescharacteristic, complemented, Sylow
185#3301—1311nomaximal, Sylow
186#553—1311no—
187#548D13111yescharacteristic, complemented
188#488—1311no—
189#485—1311no—
190#481—1311no—
191#479—1311no—
192#466—1311no—
193#465—1311no—
194#463—1311no—
195#449—1311no—
196#448—1311no—
197C4×SD16—1311no—
198C8×D4—1311no—
199#416—1311no—
200C2×#103—1311no—
201C2×#41C2×D13111yescharacteristic
202C22×Q8C2×D13111yescharacteristic
203C2×SD16—1311no—
204C2×SD16—1311no—
205C2×D8—1311no—
206C22×C8—1311no—
207C22×C8—1311no—
208#126—2621no—
209#125—1311no—
210#124—21no—
211#124C2×D13111yescharacteristic
212#123—2621no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D13111yescharacteristic
215#121—21no—
216#121—2621no—
217#121—1311no—
218#121—1311no—
219#121C2×D13111yescharacteristic
220C4×Q8C2×D13111yescharacteristic
221C4×D4—21no—
222C4×D4—2621no—
223C4×D4—1311no—
224C4×D4C2×D13111yescharacteristic
225#117—1311no—
226C2×C4⋊C4—1311no—
227#107—1311no—
228#106—1311no—
229#106—1311no—
230C4⋊C8—1311no—
231#103—1311no—
232#103—1311no—
233#103—1311no—
234#103—1311no—
235#103—1311no—
236#103—1311no—
237#102—1311no—
238#102—1311no—
239#102—1311no—
240#102—1311no—
241C22⋊C8—1311no—
242C22⋊C8—1311no—
243C4×C8—1311no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D13111yescharacteristic
250C2×Q8C22×D13111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2621no—
254C2×D4—2621no—
255C2×D4—1311no—
256C2×D4C22⋊D13111yescharacteristic
257C2×D4C22×D13111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2621no—
261C22×C4—2621no—
262C22×C4C22×D13111yescharacteristic
263C22×C4C22×D13111yescharacteristic
264SD16—2622no—
265SD16—5241no—
266D8—5241no—
267C2×C8—5241no—
268C2×C8—5241no—
269C2×C8—1311no—
270C2×C8—1311no—
271C2×C8—1311no—
272C2×C8—1311no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—2621no—
276C4⋊C4—2621no—
277C4⋊C4—2621no—
278C4⋊C4—1311no—
279C4⋊C4—1311no—
280C4⋊C4—1311no—
281C4⋊C4C22⋊D13111yescharacteristic
282C4⋊C4C22×D13111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2621no—
288C22⋊C4—2621no—
289C22⋊C4—2621no—
290C22⋊C4—2621no—
291C22⋊C4C22⋊D13111yescharacteristic
292C22⋊C4C22⋊D13111yescharacteristic
293C42—21no—
294C42—2621no—
295C42C22×D13111yescharacteristic
296C23—21no—
297C23—2621no—
298C23C2×C22⋊D13111yescharacteristic
299C23C2×C22⋊D13111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1452351212yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—2621no—
311D4—2621no—
312D4—5242no—
313D4—10481no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—2621no—
324C2×C4—2621no—
325C2×C4—2621no—
326C2×C4—10481no—
327C2×C4—10481no—
328C2×C4C2×C22⋊D13111yescharacteristic
329C2×C4C2×C22⋊D13111yescharacteristic
330C2×C4C2×C22⋊D13111yescharacteristic
331C2×C4C2×C22⋊D13111yescharacteristic
332C2×C4C23×D13111yescharacteristic
333C8—2621no—
334C8—2621no—
335C8—2621no—
336C8—2621no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—5241no—
346C22—5241no—
347C22—5241no—
348C22#1686226612yes—
349C22C22×C22⋊D13111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—5241no—
358C4—5241no—
359C4—5241no—
360C4C2×#1452351211yescharacteristic
361C4#1686229111yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—10481no—
366C2#2255395611yescharacteristic
367C2C2×#1686226611yescharacteristic
368C2#2255412011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#548×C131C2
socleC22×C131C22×D4
hypercenter#548D131
hypocenterC131#3301
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22554008C2semidirect
2. #22553666C2semidirect
3. #22553650C2semidirect
4. #22553628C2semidirect
5. #22553619C2semidirect
6. #22553546C2semidirect
7. #22553543C2semidirect
8. #22553524C2semidirect
9. #22553457C2semidirect
10. #22553451C2semidirect
11. C4×#14523500C2semidirect
12. D4×C131⋊C8C2semidirect
13. #22553308C2semidirect
14. C2×#16862170C2semidirect
15. #548×C131C2semidirect
184. C131#3301semidirect
187. #548D131semidirect

Maximal subgroups

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

subgroupindex
#225540082
#225536662
#225536502
#225536282
#225536192
#225535462
#225535432
#225535242
#225534572
#225534512
C4×#145235002
D4×C131⋊C82
#225533082
C2×#168621702
#548×C1312
#3301131