Finite Group Databasebeta

← #30677492

Subgroups — #30677492

Order 16768.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862273C211yesmaximal, characteristic, complemented
2C2×#16862267C211yesmaximal, characteristic, complemented
3#22553965C211yesmaximal, characteristic, complemented
4#22553957C211yesmaximal, characteristic, complemented
5#22553704C211yesmaximal, characteristic, complemented
6#22553569C211yesmaximal, characteristic, complemented
7#22553544C211yesmaximal, characteristic, complemented
8#22553528C211yesmaximal, characteristic, complemented
9#22553477C211yesmaximal, characteristic, complemented
10#22553471C211yesmaximal, characteristic, complemented
11#22553461C211yesmaximal, characteristic, complemented
12#22553460C211yesmaximal, characteristic, complemented
13#22553422C211yesmaximal, characteristic, complemented
14#22553382C211yesmaximal, characteristic, complemented
15#461×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×#14523514C2211yescharacteristic, complemented
18C2×#14523512C2211yescharacteristic
19C2×D4×D131C2211yescharacteristic
20#16862273—24no—
21#16862267—24no—
22C2×#14523503C2211yescharacteristic, complemented
23C2×#14523503C2211yescharacteristic, complemented
24C2×#14523502C2211yescharacteristic, complemented
25C2×#14523500C2211yescharacteristic
26C2×#14523500C2211yescharacteristic
27C2×D1048C2211yescharacteristic
28C2×#14523498C2211yescharacteristic
29C2×#14523496C2211yescharacteristic
30C2×C8×D131C2211yescharacteristic, complemented
31#16862234C2211yescharacteristic
32#16862231—21no—
33#16862231—21no—
34#16862230—21no—
35#16862226C2211yescharacteristic
36#16862222C2211yescharacteristic
37#16862222C2211yescharacteristic, complemented
38#16862221—21no—
39#16862221C2211yescharacteristic
40#16862220—21no—
41#16862216—21no—
42C4×D524C2211yescharacteristic
43C4×D524C2211yescharacteristic, complemented
44C4⋊C4×D131—21no—
45C2×(C2×D131)⋊C4—21no—
46#16862177C2211yescharacteristic, complemented
47#16862171C2211yescharacteristic, complemented
48#16862168C2211yescharacteristic
49#16862167C2211yescharacteristic
50#16862166C2211yescharacteristic, complemented
51(C2×D131)⋊C8C2211yescharacteristic
52(C2×D131)⋊C8C2211yescharacteristic, complemented
53#16862157C2211yescharacteristic, complemented
54C2×SD16×C131C2211yescharacteristic
55C2×SD16×C131C2211yescharacteristic
56C4⋊Q8×C131C2211yescharacteristic, complemented
57C4×D4×C131C2211yescharacteristic
58C4⋊C8×C131C2211yescharacteristic, complemented
59#103×C131C2211yescharacteristic, complemented
60#103×C131C2211yescharacteristic, complemented
61C23×D131—21no—
62#14523514—22no—
63#14523514—22no—
64#14523514—41no—
65#14523514—41no—
66#14523512—22no—
67#14523512—41no—
68C2×Dic262C2311yescharacteristic
69C2×D524—21no—
70C2×D524—21no—
71C2×D524—21no—
72C2×D524D411yescharacteristic
73C2×D524C2311yescharacteristic
74C2×D524C2311yescharacteristic
75C2×C22⋊D131—21no—
76C2×C22⋊D131—21no—
77D4×D131—22no—
78D4×D131—41no—
79C22×Dic131—21no—
80C2×C4×D131—21no—
81C2×C4×D131—21no—
82C2×C4×D131—21no—
83C2×C4×D131C2311yescharacteristic
84C2×C4×D131C2311yescharacteristic
85#14523503—22no—
86#14523503—41no—
87#14523502—22no—
88#14523500—22no—
89#14523500—41no—
90D1048—22no—
91#14523498D414yescomplemented
92#14523496—22no—
93C2×C131⋊C8C2311yescharacteristic
94C2×C131⋊C8C2311yescharacteristic
95C8×D131—22no—
96C4⋊Dic131—21no—
97C4⋊Dic131D411yescharacteristic
98C4⋊Dic131C2311yescharacteristic
99Dic131⋊C4—21no—
100Dic131⋊C4—21no—
101(C2×D131)⋊C4—21no—
102(C2×D131)⋊C4—21no—
103(C2×D131)⋊C4—21no—
104(C2×D131)⋊C4—21no—
105(C2×D131)⋊C4—21no—
106(C2×D131)⋊C4—21no—
107(C2×D131)⋊C4—21no—
108(C2×D131)⋊C4D411yescharacteristic
109(C2×D131)⋊C4D411yescharacteristic
110C2×Q8×C131C2311yescharacteristic
111C2×Q8×C131C2311yescharacteristic
112C2×D4×C131C2311yescharacteristic
113C22×C4×C131—21no—
114SD16×C131—22no—
115SD16×C131—22no—
116C2×C8×C131C2311yescharacteristic
117C2×C8×C131C2311yescharacteristic
118C4⋊C4×C131—21no—
119C4⋊C4×C131—21no—
120C4⋊C4×C131C2311yescharacteristic
121C22⋊C4×C131—21no—
122C42×C131C2311yescharacteristic
123C22×D131—21no—
124C22×D131—21no—
125C22×D131—42no—
126C22×D131—42no—
127C22×D131—81no—
128C22×D131C2×D411yescharacteristic
129C22×D131C2×D411yescharacteristic
130Dic262—21no—
131Dic262—21no—
132D524—21no—
133D524—21no—
134D524—21no—
135D524—41no—
136D524—42no—
137D524—42no—
138D524—81no—
139D524C2×D412yes—
140C22⋊D131—42no—
141C22⋊D131—42no—
142C2×Dic131—21no—
143C2×Dic131—21no—
144C2×Dic131—42no—
145C2×Dic131C2×D411yescharacteristic
146C2×Dic131C2×D411yescharacteristic
147C4×D131—21no—
148C4×D131—21no—
149C4×D131—21no—
150C4×D131—21no—
151C4×D131—42no—
152C4×D131—42no—
153C4×D131—42no—
154C131⋊C8—21no—
155C131⋊C8C2×D412yes—
156C23×C131—21no—
157Q8×C131—21no—
158Q8×C131—21no—
159Q8×C131—21no—
160Q8×C131—21no—
161D4×C131—21no—
162D4×C131C2×D412yes—
163C2×C4×C131—21no—
164C2×C4×C131—21no—
165C2×C4×C131—21no—
166C2×C4×C131—81no—
167C2×C4×C131C2×D411yescharacteristic
168C2×C4×C131C2×D411yescharacteristic
169C2×C4×C131C2411yescharacteristic
170C8×C131—21no—
171C8×C131—21no—
172C22×C131—22no—
173C22×C131—41no—
174C22×C131—41no—
175C22×C131C22×D411yescharacteristic
176C2×D131—21no—
177C2×D131—21no—
178C2×D131—22no—
179C2×D131—21no—
180C2×D131—21no—
181C2×D131—21no—
182C2×D131—21no—
183C2×D131—41no—
184C2×D131—41no—
185C2×D131—41no—
186C2×D131—41no—
187C2×D131—41no—
188C2×D131—84no—
189C4×C131—21no—
190C4×C131—21no—
191C4×C131—41no—
192C4×C131—41no—
193C4×C131—41no—
194C4×C131C22×D411yescharacteristic
195C4×C131H(5,𝔽2)11yescharacteristic
196Dic131—21no—
197Dic131—21no—
198Dic131—41no—
199Dic131—41no—
200C2×C131—42no—
201C2×C131D4211yescharacteristic
202C2×C131#57511yescharacteristic
203C2×C131#57711yescharacteristic
204D131—41no—
205D131—41no—
206D131—42no—
207D131—81no—
208C131#326711yescharacteristic, complemented, Sylow
209#3267—1311nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1311no—
211C2×#135—1311no—
212#547—1311no—
213#546—1311no—
214#497—1311no—
215#470—1311no—
216#465—1311no—
217#463—1311no—
218#461D13111yescharacteristic, complemented
219#452—1311no—
220#452—1311no—
221#450—1311no—
222#449—1311no—
223#442—1311no—
224#435—1311no—
225C2×#41—1311no—
226C2×#41—1311no—
227C2×#41—1311no—
228C22×D4—1311no—
229AGL(1,ℤ/8)—2624no—
230#135—2624no—
231C2×Q16—1311no—
232C2×Q16—1311no—
233C2×SD16—1311no—
234C2×SD16—1311no—
235C2×SD16—1311no—
236C2×SD16C2×D13111yescharacteristic
237C2×SD16C2×D13111yescharacteristic
238C2×D8—1311no—
239C2×D8—1311no—
240C2×M16—1311no—
241C22×C8—1311no—
242C4⋊Q8C2×D13111yescharacteristic, complemented
243#124—1311no—
244#123—2621no—
245#123—2621no—
246#123—2621no—
247C22⋊Q8—1311no—
248C22⋊Q8—1311no—
249C22⋊Q8—1311no—
250#121—2621no—
251#121—2621no—
252#121—1311no—
253#120—2621no—
254C4×D4—1311no—
255C4×D4—1311no—
256C4×D4C2×D13111yescharacteristic
257C2×C4⋊C4—2621no—
258C2×C22⋊C4—2621no—
259#106—1311no—
260C4⋊C8C2×D13111yescharacteristic, complemented
261#103—1311no—
262#103C2×D13111yescharacteristic, complemented
263#103C2×D13111yescharacteristic, complemented
264#102—1311no—
265#102—1311no—
266#102—1311no—
267C22⋊C8—1311no—
268C22⋊C8—1311no—
269#97—1311no—
270C24—2621no—
271#41—2622no—
272#41—2622no—
273#41—2622no—
274#41—5241no—
275#41—5241no—
276#41—5241no—
277C2×Q8—1311no—
278C2×Q8C22×D13111yescharacteristic
279C2×Q8C22×D13111yescharacteristic
280C2×D4—2621no—
281C2×D4—2621no—
282C2×D4—2621no—
283C2×D4—2622no—
284C2×D4—2621no—
285C2×D4—2621no—
286C2×D4—5241no—
287C2×D4—1311no—
288C2×D4—1311no—
289C2×D4—1311no—
290C2×D4C22×D13111yescharacteristic
291C22×C4—21no—
292C22×C4—2621no—
293C22×C4—2621no—
294C22×C4—2621no—
295C22×C4—2621no—
296C22×C4—1311no—
297C22×C4—1311no—
298Q16—2622no—
299Q16—5241no—
300SD16—22no—
301SD16—22no—
302SD16—2622no—
303SD16—2622no—
304SD16—5241no—
305D8—2622no—
306D8—1314no—
307M16—2622no—
308C2×C8—2622no—
309C2×C8—1311no—
310C2×C8—1311no—
311C2×C8C22×D13111yescharacteristic
312C2×C8C22×D13111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—2621no—
316C4⋊C4—2621no—
317C4⋊C4—2621no—
318C4⋊C4—1311no—
319C4⋊C4—1311no—
320C4⋊C4C22×D13111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2621no—
323C22⋊C4—2621no—
324C22⋊C4—2621no—
325C22⋊C4—2621no—
326C22⋊C4—2621no—
327C22⋊C4—2621no—
328C22⋊C4—2621no—
329C22⋊C4—1311no—
330C22⋊C4—1311no—
331C42C22×D13111yescharacteristic
332C23—21no—
333C23—2621no—
334C23—2621no—
335C23—5242no—
336C23—5242no—
337C23—10481no—
338C23—1311no—
339C23—1311no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2621no—
345Q8—2621no—
346D4—21no—
347D4—2621no—
348D4—2621no—
349D4—2621no—
350D4—5241no—
351D4—5242no—
352D4—5242no—
353D4—5242no—
354D4—5242no—
355D4—10481no—
356D4—1312no—
357D4D4×D13112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—2621no—
362C2×C4—2621no—
363C2×C4—2621no—
364C2×C4—2621no—
365C2×C4—2621no—
366C2×C4—2621no—
367C2×C4—81no—
368C2×C4—5242no—
369C2×C4—5242no—
370C2×C4—5242no—
371C2×C4—5242no—
372C2×C4—1311no—
373C2×C4—1311no—
374C2×C4D4×D13111yescharacteristic
375C2×C4D4×D13111yescharacteristic
376C2×C4C23×D13111yescharacteristic
377C8—21no—
378C8—21no—
379C8—2621no—
380C8—1312no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—2621no—
385C22—2621no—
386C22—2621no—
387C22—2621no—
388C22—2622no—
389C22—2621no—
390C22—2621no—
391C22—5241no—
392C22—5241no—
393C22—5241no—
394C22—5241no—
395C22—5241no—
396C22—10484no—
397C22C2×D4×D13111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—2621no—
404C4—2621no—
405C4—5241no—
406C4—5241no—
407C4C2×D4×D13111yescharacteristic
408C4#1686229211yescharacteristic
409C2—42no—
410C2—5241no—
411C2—5241no—
412C2—5242no—
413C2—10481no—
414C2#2255394611yescharacteristic
415C2#2255411111yescharacteristic
416C2#2255411811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#461×C131C2
socleC22×C131C22×D4
hypercenter#461D131
hypocenterC131#3267
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#16862273C2semidirect
2. C2×#16862267C2semidirect
3. #22553965C2semidirect
4. #22553957C2semidirect
5. #22553704C2semidirect
6. #22553569C2semidirect
7. #22553544C2semidirect
8. #22553528C2semidirect
9. #22553477C2semidirect
10. #22553471C2semidirect
11. #22553461C2semidirect
12. #22553460C2semidirect
13. #22553422C2semidirect
14. #22553382C2semidirect
15. #461×C131C2semidirect
17. C2×#14523514C22semidirect
22. C2×#14523503C22semidirect
23. C2×#14523503C22semidirect
24. C2×#14523502C22semidirect
30. C2×C8×D131C22semidirect
37. #16862222C22semidirect
43. C4×D524C22semidirect
46. #16862177C22semidirect
47. #16862171C22semidirect
50. #16862166C22semidirect
52. (C2×D131)⋊C8C22semidirect
53. #16862157C22semidirect
56. C4⋊Q8×C131C22semidirect
58. C4⋊C8×C131C22semidirect
59. #103×C131C22semidirect
60. #103×C131C22semidirect
91. #14523498D4semidirect
208. C131#3267semidirect
218. #461D131semidirect
242. C4⋊Q8C2×D131semidirect
260. C4⋊C8C2×D131semidirect
262. #103C2×D131semidirect
263. #103C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622732
C2×#168622672
#225539652
#225539572
#225537042
#225535692
#225535442
#225535282
#225534772
#225534712
#225534612
#225534602
#225534222
#225533822
#461×C1312
#3267131