Finite Group Databasebeta

← #30677937

Subgroups — #30677937

Order 16768.

394 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#22553680C211yesmaximal, characteristic, complemented
3#22553650C211yesmaximal, characteristic, complemented
4#22553640C211yesmaximal, characteristic, complemented
5#22553612C211yesmaximal, characteristic, complemented
6#22553560C211yesmaximal, characteristic, complemented
7#22553543C211yesmaximal, characteristic, complemented
8#22553514C211yesmaximal, characteristic, complemented
9#22553457C211yesmaximal, characteristic, complemented
10#22553457C211yesmaximal, characteristic, complemented
11#22553422C211yesmaximal, characteristic, complemented
12#22553384C211yesmaximal, characteristic, complemented
13#22553313C211yesmaximal, characteristic, complemented
14#22553308C211yesmaximal, characteristic, complemented
15#547×C131C211yesmaximal, characteristic, complemented
16C2×#14523500C2211yescharacteristic
17C2×#14523500C2211yescharacteristic
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#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
47#16862157C2211yescharacteristic
48C2×#41×C131C2211yescharacteristic
49C2×#41×C131C2211yescharacteristic
50C4⋊Q8×C131C2211yescharacteristic
51#123×C131—21no—
52#123×C131—21no—
53C22⋊Q8×C131C2211yescharacteristic
54C22⋊Q8×C131C2211yescharacteristic
55#121×C131—21no—
56C4×D4×C131C2211yescharacteristic
57C4×D4×C131C2211yescharacteristic
58C2×C4⋊C4×C131—21no—
59C2×D524C2311yescharacteristic
60C2×C22⋊D131—21no—
61C2×C22⋊D131—21no—
62C22×Dic131—21no—
63C2×C4×D131—21no—
64#14523500—41no—
65#14523500—41no—
66#14523498—22no—
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×Q8×C131C2311yescharacteristic
90C2×D4×C131—21no—
91C2×D4×C131—21no—
92C2×D4×C131D411yescharacteristic
93C2×D4×C131C2311yescharacteristic
94C22×C4×C131—21no—
95C22×C4×C131—21no—
96C22×C4×C131—21no—
97C22×C4×C131C2311yescharacteristic
98C22×C4×C131C2311yescharacteristic
99C4⋊C4×C131—21no—
100C4⋊C4×C131—21no—
101C4⋊C4×C131—21no—
102C4⋊C4×C131—21no—
103C4⋊C4×C131D411yescharacteristic
104C4⋊C4×C131C2311yescharacteristic
105C22⋊C4×C131—21no—
106C22⋊C4×C131—21no—
107C22⋊C4×C131—21no—
108C22⋊C4×C131D411yescharacteristic
109C22⋊C4×C131D411yescharacteristic
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—21no—
132D4×C131—21no—
133D4×C131—41no—
134D4×C131—42no—
135D4×C131—42no—
136D4×C131#4112yes—
137C2×C4×C131—21no—
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—42no—
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—22no—
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—21no—
158C22×C131—21no—
159C22×C131—41no—
160C22×C131—41no—
161C22×C131C22×D411yescharacteristic
162C2×D131—41no—
163C2×D131—41no—
164C2×D131—41no—
165C4×C131—21no—
166C4×C131—21no—
167C4×C131—21no—
168C4×C131—21no—
169C4×C131—41no—
170C4×C131—41no—
171C4×C131—41no—
172C4×C131C2×#4111yescharacteristic
173C4×C131H(5,𝔽2)11yescharacteristic
174Dic131—41no—
175Dic131—41no—
176Dic131—41no—
177C2×C131—41no—
178C2×C131—41no—
179C2×C131—42no—
180C2×C131#54611yescharacteristic
181C2×C131#57511yescharacteristic
182C2×C131#57711yescharacteristic
183D131—81no—
184C131#330711yescharacteristic, complemented, Sylow
185#3307—1311nomaximal, Sylow
186#553—1311no—
187#547D13111yescharacteristic, complemented
188#491—1311no—
189#485—1311no—
190#483—1311no—
191#478—1311no—
192#468—1311no—
193#465—1311no—
194#461—1311no—
195#449—1311no—
196#449—1311no—
197#442—1311no—
198#435—1311no—
199#417—1311no—
200#416—1311no—
201C2×#41C2×D13111yescharacteristic
202C2×#41C2×D13111yescharacteristic
203C2×SD16—1311no—
204C2×SD16—1311no—
205C2×D8—1311no—
206C2×M16—1311no—
207C22×C8—1311no—
208C4⋊Q8C2×D13111yescharacteristic
209#126—2621no—
210#125—1311no—
211#123—21no—
212#123—21no—
213#123—2621no—
214C22⋊Q8C2×D13111yescharacteristic
215C22⋊Q8C2×D13111yescharacteristic
216#121—21no—
217#121—2621no—
218#121—1311no—
219#121—1311no—
220C4×D4—2621no—
221C4×D4—1311no—
222C4×D4C2×D13111yescharacteristic
223C4×D4C2×D13111yescharacteristic
224#117—1311no—
225C2×C4⋊C4—21no—
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—
243#97—1311no—
244#41—22no—
245#41—22no—
246#41—41no—
247#41—41no—
248C2×Q8C22×D13111yescharacteristic
249C2×Q8C22×D13111yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—2621no—
253C2×D4—2621no—
254C2×D4—1311no—
255C2×D4C22⋊D13111yescharacteristic
256C2×D4C22×D13111yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2621no—
261C22×C4—2621no—
262C22×C4C22×D13111yescharacteristic
263C22×C4C22×D13111yescharacteristic
264SD16—5241no—
265SD16—5241no—
266D8—2622no—
267M16—5241no—
268C2×C8—5241no—
269C2×C8—1311no—
270C2×C8—1311no—
271C2×C8—1311no—
272C2×C8—1311no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—2621no—
278C4⋊C4—2621no—
279C4⋊C4—2621no—
280C4⋊C4—1311no—
281C4⋊C4—1311no—
282C4⋊C4—1311no—
283C4⋊C4C22⋊D13111yescharacteristic
284C4⋊C4C22×D13111yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—2621no—
289C22⋊C4—2621no—
290C22⋊C4—2621no—
291C22⋊C4—2621no—
292C22⋊C4C22⋊D13111yescharacteristic
293C22⋊C4C22⋊D13111yescharacteristic
294C42—2621no—
295C42C22×D13111yescharacteristic
296C23—21no—
297C23—2621no—
298C23C2×C22⋊D13111yescharacteristic
299C23C2×C22⋊D13111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—21no—
305D4—41no—
306D4—42no—
307D4—42no—
308D4—2621no—
309D4—2621no—
310D4—5242no—
311D4—10481no—
312D4#1452351212yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
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—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—5241no—
347C22—5241no—
348C22—5241no—
349C22C22×C22⋊D13111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5241no—
358C4—5241no—
359C4—5241no—
360C4C2×#1452351211yescharacteristic
361C4#1686229111yescharacteristic
362C2—41no—
363C2—41no—
364C2—42no—
365C2—10481no—
366C2#2255395611yescharacteristic
367C2#2255411011yescharacteristic
368C2#2255412011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#547×C131C2
socleC22×C131C22×D4
hypercenter#547D131
hypocenterC131#3307
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22554008C2semidirect
2. #22553680C2semidirect
3. #22553650C2semidirect
4. #22553640C2semidirect
5. #22553612C2semidirect
6. #22553560C2semidirect
7. #22553543C2semidirect
8. #22553514C2semidirect
9. #22553457C2semidirect
10. #22553457C2semidirect
11. #22553422C2semidirect
12. #22553384C2semidirect
13. #22553313C2semidirect
14. #22553308C2semidirect
15. #547×C131C2semidirect
184. C131#3307semidirect
187. #547D131semidirect

Maximal subgroups

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

subgroupindex
#225540082
#225536802
#225536502
#225536402
#225536122
#225535602
#225535432
#225535142
#225534572
#225534222
#225533842
#225533132
#225533082
#547×C1312
#3307131