Finite Group Databasebeta

← #30677495

Subgroups — #30677495

Order 16768.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862267C211yesmaximal, characteristic, complemented
2#22553981C211yesmaximal, characteristic, complemented
3#22553956C211yesmaximal, characteristic, complemented
4#22553700C211yesmaximal, characteristic, complemented
5#22553558C211yesmaximal, characteristic, complemented
6#22553545C211yesmaximal, characteristic, complemented
7#22553532C211yesmaximal, characteristic, complemented
8#22553505C211yesmaximal, characteristic, complemented
9#22553474C211yesmaximal, characteristic, complemented
10#22553456C211yesmaximal, characteristic, complemented
11#22553454C211yesmaximal, characteristic, complemented
12#22553446C211yesmaximal, characteristic, complemented
13#22553413C211yesmaximal, characteristic, complemented
14#22553381C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic, complemented
17C2×#14523512C2211yescharacteristic
18#16862267—24no—
19C2×#14523503C2211yescharacteristic, complemented
20C2×#14523502C2211yescharacteristic, complemented
21C2×#14523501C2211yescharacteristic, complemented
22C2×#14523498C2211yescharacteristic
23C2×#14523498C2211yescharacteristic
24C2×C8×D131C2211yescharacteristic, complemented
25#16862242—21no—
26#16862242C2211yescharacteristic, complemented
27#16862233C2211yescharacteristic
28#16862227—21no—
29#16862225C2211yescharacteristic
30#16862220—21no—
31#16862220C2211yescharacteristic
32#16862218—21no—
33#16862217—21no—
34#16862217C2211yescharacteristic
35#16862217C2211yescharacteristic, complemented
36#16862215—21no—
37Q8×Dic131C2211yescharacteristic, complemented
38D4×Dic131C2211yescharacteristic
39C4×C22⋊D131—21no—
40C4×C22⋊D131C2211yescharacteristic
41C2×C22⋊Dic131—21no—
42#16862180C2211yescharacteristic, complemented
43Dic131⋊C8C2211yescharacteristic, complemented
44#16862171C2211yescharacteristic, complemented
45#16862170C2211yescharacteristic, complemented
46#16862168C2211yescharacteristic, complemented
47#16862167C2211yescharacteristic
48#16862167C2211yescharacteristic
49#16862167C2211yescharacteristic, complemented
50C22⋊(C131⋊C8)C2211yescharacteristic
51(C2×D131)⋊C8C2211yescharacteristic, complemented
52#16862156C2211yescharacteristic, complemented
53C2×#41×C131C2211yescharacteristic
54C22×D4×C131C2211yescharacteristic
55AGL(1,ℤ/8)×C131—24no—
56C2×SD16×C131C2211yescharacteristic
57C2×SD16×C131C2211yescharacteristic
58C2×D8×C131C2211yescharacteristic
59C2×D8×C131C2211yescharacteristic
60C2×M16×C131C2211yescharacteristic
61#14523514—22no—
62#14523514—41no—
63#14523512—22no—
64#14523512—41no—
65C2×Dic262C2311yescharacteristic
66C2×D524—21no—
67C2×D524C2311yescharacteristic
68C2×C22⋊D131—21no—
69C2×C22⋊D131—21no—
70C2×C22⋊D131—21no—
71C2×C22⋊D131—21no—
72C2×C22⋊D131—21no—
73C2×C22⋊D131—21no—
74C2×C22⋊D131D411yescharacteristic
75C22×Dic131—21no—
76C22×Dic131—21no—
77C2×C4×D131—21no—
78C2×C4×D131C2311yescharacteristic
79#14523503—22no—
80#14523502—22no—
81#14523501—41no—
82#14523498—22no—
83#14523498—41no—
84C2×C131⋊C8C2311yescharacteristic
85C2×C131⋊C8C2311yescharacteristic
86C8×D131—22no—
87C4⋊Dic131—21no—
88C4⋊Dic131C2311yescharacteristic
89C4⋊Dic131C2311yescharacteristic
90Dic131⋊C4—21no—
91Dic131⋊C4D411yescharacteristic
92C22⋊Dic131—21no—
93C22⋊Dic131—21no—
94C22⋊Dic131—21no—
95C22⋊Dic131—21no—
96C22⋊Dic131—21no—
97C22⋊Dic131—21no—
98C22⋊Dic131D411yescharacteristic
99(C2×D131)⋊C4—21no—
100(C2×D131)⋊C4D411yescharacteristic
101C4×Dic131—21no—
102C4×Dic131C2311yescharacteristic
103C24×C131—21no—
104#41×C131—22no—
105#41×C131—41no—
106C2×Q8×C131C2311yescharacteristic
107C2×D4×C131—22no—
108C2×D4×C131—21no—
109C2×D4×C131—21no—
110C2×D4×C131—41no—
111C2×D4×C131C2311yescharacteristic
112C2×D4×C131C2311yescharacteristic
113C2×D4×C131C2311yescharacteristic
114C22×C4×C131—21no—
115C22×C4×C131C2311yescharacteristic
116SD16×C131—22no—
117SD16×C131D414yescomplemented
118D8×C131—22no—
119D8×C131—22no—
120M16×C131—22no—
121C2×C8×C131C2311yescharacteristic
122C2×C8×C131C2311yescharacteristic
123C22×D131—21no—
124C22×D131C2×D411yescharacteristic
125Dic262—21no—
126Dic262—21no—
127D524—21no—
128D524—21no—
129D524—42no—
130C22⋊D131—41no—
131C22⋊D131—42no—
132C22⋊D131—42no—
133C22⋊D131—81no—
134C22⋊D131—81no—
135C22⋊D131—81no—
136C22⋊D131—81no—
137C2×Dic131—21no—
138C2×Dic131—21no—
139C2×Dic131—21no—
140C2×Dic131—42no—
141C2×Dic131—81no—
142C2×Dic131C2×D411yescharacteristic
143C2×Dic131C2×D411yescharacteristic
144C2×Dic131C2×D411yescharacteristic
145C4×D131—21no—
146C4×D131—21no—
147C4×D131—42no—
148C131⋊C8—21no—
149C131⋊C8—21no—
150C23×C131—21no—
151C23×C131—21no—
152C23×C131—21no—
153C23×C131—42no—
154C23×C131—42no—
155C23×C131—81no—
156C23×C131C2×D411yescharacteristic
157Q8×C131—21no—
158Q8×C131C2×D412yes—
159D4×C131—21no—
160D4×C131—21no—
161D4×C131—21no—
162D4×C131—21no—
163D4×C131—21no—
164D4×C131—42no—
165D4×C131—42no—
166D4×C131C2×D412yes—
167C2×C4×C131—21no—
168C2×C4×C131—21no—
169C2×C4×C131—21no—
170C2×C4×C131—42no—
171C2×C4×C131C2×D411yescharacteristic
172C2×C4×C131C2411yescharacteristic
173C8×C131—21no—
174C8×C131C2×D412yes—
175C22×C131—21no—
176C22×C131—22no—
177C22×C131—21no—
178C22×C131—21no—
179C22×C131—41no—
180C22×C131—41no—
181C22×C131—41no—
182C22×C131—41no—
183C22×C131—41no—
184C22×C131—41no—
185C22×C131—41no—
186C22×C131—41no—
187C22×C131—84no—
188C22×C131C22×D411yescharacteristic
189C2×D131—21no—
190C2×D131—21no—
191C2×D131—21no—
192C2×D131—41no—
193C2×D131—41no—
194C2×D131—41no—
195C4×C131—21no—
196C4×C131—22no—
197C4×C131C22×D411yescharacteristic
198C4×C131H(5,𝔽2)11yescharacteristic
199Dic131—21no—
200Dic131—21no—
201Dic131—21no—
202Dic131—41no—
203Dic131—41no—
204Dic131—41no—
205C2×C131—41no—
206C2×C131—42no—
207C2×C131—81no—
208C2×C131—81no—
209C2×C131D4211yescharacteristic
210C2×C131#57511yescharacteristic
211C2×C131#57611yescharacteristic
212D131—41no—
213D131—81no—
214C131#326811yescharacteristic, complemented, Sylow
215#3268—1311nomaximal, Sylow
216C2×AGL(1,ℤ/8)D13111yescharacteristic, complemented
217C2×#135—1311no—
218#550—1311no—
219#546—1311no—
220#496—1311no—
221#468—1311no—
222#466—1311no—
223#464—1311no—
224#459—1311no—
225#452—1311no—
226#449—1311no—
227#449—1311no—
228#447—1311no—
229#440—1311no—
230#435—1311no—
231C2×#41—1311no—
232C2×#41—1311no—
233C2×#41C2×D13111yescharacteristic
234C22×D4C2×D13111yescharacteristic
235AGL(1,ℤ/8)—24no—
236#135—2624no—
237C2×Q16—1311no—
238C2×SD16—1311no—
239C2×SD16—1311no—
240C2×SD16C2×D13111yescharacteristic
241C2×SD16C2×D13111yescharacteristic
242C2×D8—1311no—
243C2×D8—1311no—
244C2×D8C2×D13111yescharacteristic
245C2×D8C2×D13111yescharacteristic
246C2×M16C2×D13111yescharacteristic
247C22×C8—1311no—
248#127—2621no—
249#127—1311no—
250#124—1311no—
251#123—2621no—
252C22⋊Q8—1311no—
253#121—2621no—
254#121—2621no—
255#121—2621no—
256#121—1311no—
257#121—1311no—
258#121—1311no—
259#120—2621no—
260C4×Q8—1311no—
261C4×D4—2621no—
262C4×D4—1311no—
263C4×D4—1311no—
264C2×C22⋊C4—2621no—
265#107—1311no—
266C4⋊C8—1311no—
267#103—1311no—
268#103—1311no—
269#102—1311no—
270#102—1311no—
271#102—1311no—
272#102—1311no—
273C22⋊C8—1311no—
274C22⋊C8—1311no—
275#97—1311no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—2622no—
280#41—2622no—
281#41—5241no—
282#41—5241no—
283C2×Q8—1311no—
284C2×Q8C22×D13111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2621no—
290C2×D4—2621no—
291C2×D4—2621no—
292C2×D4—2621no—
293C2×D4—2621no—
294C2×D4—2621no—
295C2×D4—2621no—
296C2×D4—1311no—
297C2×D4—1311no—
298C2×D4C22×D13111yescharacteristic
299C2×D4C22×D13111yescharacteristic
300C2×D4C22×D13111yescharacteristic
301C22×C4—21no—
302C22×C4—2621no—
303C22×C4—2621no—
304C22×C4—2621no—
305C22×C4—1311no—
306C22×C4C22×D13111yescharacteristic
307Q16—2622no—
308SD16—22no—
309SD16—2622no—
310SD16—5241no—
311SD16C22⋊D13114yescomplemented
312D8—22no—
313D8—22no—
314D8—2622no—
315D8—5241no—
316M16—22no—
317C2×C8—2622no—
318C2×C8—1311no—
319C2×C8—1311no—
320C2×C8C22×D13111yescharacteristic
321C2×C8C22×D13111yescharacteristic
322C4⋊C4—2621no—
323C4⋊C4—2621no—
324C4⋊C4—1311no—
325C4⋊C4—1311no—
326C4⋊C4—1311no—
327C22⋊C4—2621no—
328C22⋊C4—2621no—
329C22⋊C4—2621no—
330C22⋊C4—2621no—
331C22⋊C4—2621no—
332C22⋊C4—2621no—
333C22⋊C4—2621no—
334C22⋊C4—1311no—
335C22⋊C4—1311no—
336C42—2621no—
337C42—1311no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—2621no—
344C23—81no—
345C23—1311no—
346C23D4×D13111yescharacteristic
347Q8—21no—
348Q8—2621no—
349Q8—2621no—
350Q8C2×C22⋊D13112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—2621no—
359D4—2621no—
360D4—5242no—
361D4—5241no—
362D4—5242no—
363D4—5242no—
364D4—10481no—
365D4—10481no—
366D4—10481no—
367D4—10481no—
368D4C2×C22⋊D13112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—2621no—
374C2×C4—2621no—
375C2×C4—2621no—
376C2×C4—2621no—
377C2×C4—2621no—
378C2×C4—5242no—
379C2×C4—5242no—
380C2×C4—10481no—
381C2×C4—1311no—
382C2×C4—1311no—
383C2×C4—1311no—
384C2×C4D4×D13111yescharacteristic
385C2×C4C23×D13111yescharacteristic
386C8—21no—
387C8—2621no—
388C8—2621no—
389C8C2×C22⋊D13112yes—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—2621no—
403C22—2621no—
404C22—2621no—
405C22—84no—
406C22—5241no—
407C22—5241no—
408C22—5241no—
409C22C2×D4×D13111yescharacteristic
410C4—21no—
411C4—22no—
412C4—2621no—
413C4—2621no—
414C4—2621no—
415C4—5241no—
416C4—5241no—
417C4—5241no—
418C4C22×C22⋊D13111yescharacteristic
419C4#1686229111yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—5241no—
425C2—10481no—
426C2D4×C22⋊D13111yescharacteristic
427C2#2255410811yescharacteristic
428C2#2255411611yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×AGL(1,ℤ/8)×C131C2
socleC22×C131C22×D4
hypercenterC2×AGL(1,ℤ/8)D131
hypocenterC131#3268
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16862267C2semidirect
2. #22553981C2semidirect
3. #22553956C2semidirect
4. #22553700C2semidirect
5. #22553558C2semidirect
6. #22553545C2semidirect
7. #22553532C2semidirect
8. #22553505C2semidirect
9. #22553474C2semidirect
10. #22553456C2semidirect
11. #22553454C2semidirect
12. #22553446C2semidirect
13. #22553413C2semidirect
14. #22553381C2semidirect
15. C2×AGL(1,ℤ/8)×C131C2semidirect
16. C2×#14523514C22semidirect
19. C2×#14523503C22semidirect
20. C2×#14523502C22semidirect
21. C2×#14523501C22semidirect
24. C2×C8×D131C22semidirect
26. #16862242C22semidirect
35. #16862217C22semidirect
37. Q8×Dic131C22semidirect
42. #16862180C22semidirect
43. Dic131⋊C8C22semidirect
44. #16862171C22semidirect
45. #16862170C22semidirect
46. #16862168C22semidirect
49. #16862167C22semidirect
51. (C2×D131)⋊C8C22semidirect
52. #16862156C22semidirect
117. SD16×C131D4semidirect
214. C131#3268semidirect
216. C2×AGL(1,ℤ/8)D131semidirect
311. SD16C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622672
#225539812
#225539562
#225537002
#225535582
#225535452
#225535322
#225535052
#225534742
#225534562
#225534542
#225534462
#225534132
#225533812
C2×AGL(1,ℤ/8)×C1312
#3268131