Finite Group Databasebeta

← #30677498

Subgroups — #30677498

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×#16862270C211yesmaximal, characteristic, complemented
2C2×#16862268C211yesmaximal, characteristic, complemented
3#22553980C211yesmaximal, characteristic, complemented
4#22553959C211yesmaximal, characteristic, complemented
5#22553700C211yesmaximal, characteristic, complemented
6#22553559C211yesmaximal, characteristic, complemented
7#22553533C211yesmaximal, characteristic, complemented
8#22553506C211yesmaximal, characteristic, complemented
9#22553475C211yesmaximal, characteristic, complemented
10#22553455C211yesmaximal, characteristic, complemented
11#22553455C211yesmaximal, characteristic, complemented
12#22553447C211yesmaximal, characteristic, complemented
13#22553411C211yesmaximal, characteristic, complemented
14#22553383C211yesmaximal, characteristic, complemented
15#466×C131C211yesmaximal, characteristic, complemented
16C2×#14523513C2211yescharacteristic
17C2×#14523513C2211yescharacteristic, complemented
18C2×#14523512C2211yescharacteristic
19C2×D4×D131C2211yescharacteristic
20#16862270—24no—
21#16862268—24no—
22C2×Dic524C2211yescharacteristic, complemented
23C2×#14523502C2211yescharacteristic
24C2×#14523502C2211yescharacteristic, complemented
25C2×#14523501C2211yescharacteristic
26C2×#14523501C2211yescharacteristic, complemented
27C2×D1048C2211yescharacteristic
28C2×#14523498C2211yescharacteristic
29C2×#14523498C2211yescharacteristic
30C2×#14523496C2211yescharacteristic
31#16862242—21no—
32#16862242C2211yescharacteristic, complemented
33#16862233C2211yescharacteristic
34#16862228—21no—
35#16862226C2211yescharacteristic
36#16862221C2211yescharacteristic
37#16862219—21no—
38#16862218—21no—
39#16862217—21no—
40#16862214—21no—
41#16862211C2211yescharacteristic, complemented
42#16862208C2211yescharacteristic
43C4×C22⋊D131C2211yescharacteristic
44(C22⋊D131)⋊C4—21no—
45C22⋊C4×D131—21no—
46Dic131⋊C8C2211yescharacteristic, complemented
47#16862169C2211yescharacteristic, complemented
48#16862169C2211yescharacteristic, complemented
49#16862166C2211yescharacteristic
50#16862166C2211yescharacteristic, complemented
51C22⋊(C131⋊C8)C2211yescharacteristic, complemented
52(C2×D131)⋊C8C2211yescharacteristic
53#16862156C2211yescharacteristic, complemented
54C2×D8×C131C2211yescharacteristic
55C22×C8×C131C2211yescharacteristic, complemented
56#121×C131C2211yescharacteristic
57#121×C131C2211yescharacteristic, complemented
58#107×C131C2211yescharacteristic, complemented
59#102×C131C2211yescharacteristic
60#102×C131C2211yescharacteristic, complemented
61C23×D131—21no—
62#14523513—22no—
63#14523513—22no—
64#14523513—41no—
65#14523513—41no—
66#14523512—22no—
67#14523512—41no—
68C2×Dic262C2311yescharacteristic
69C2×Dic262C2311yescharacteristic
70C2×D524—21no—
71C2×D524C2311yescharacteristic
72C2×D524C2311yescharacteristic
73C2×C22⋊D131—21no—
74C2×C22⋊D131—21no—
75C2×C22⋊D131—21no—
76C2×C22⋊D131—21no—
77C2×C22⋊D131—21no—
78C2×C22⋊D131—21no—
79C2×C22⋊D131D411yescharacteristic
80D4×D131—22no—
81D4×D131—41no—
82C22×Dic131—21no—
83C2×C4×D131—21no—
84C2×C4×D131—21no—
85C2×C4×D131—21no—
86C2×C4×D131C2311yescharacteristic
87Dic524—22no—
88#14523502—22no—
89#14523502D414yescomplemented
90#14523501—22no—
91#14523501—41no—
92D1048—22no—
93#14523498—22no—
94#14523498—41no—
95#14523496—22no—
96C2×C131⋊C8C2311yescharacteristic
97C2×C131⋊C8C2311yescharacteristic
98C4⋊Dic131—21no—
99Dic131⋊C4—21no—
100Dic131⋊C4D411yescharacteristic
101C22⋊Dic131—21no—
102C22⋊Dic131—21no—
103C22⋊Dic131D411yescharacteristic
104(C2×D131)⋊C4—21no—
105(C2×D131)⋊C4—21no—
106(C2×D131)⋊C4—21no—
107(C2×D131)⋊C4D411yescharacteristic
108C4×Dic131—21no—
109C4×Dic131C2311yescharacteristic
110C2×D4×C131—21no—
111C2×D4×C131—21no—
112C2×D4×C131C2311yescharacteristic
113C2×D4×C131C2311yescharacteristic
114C22×C4×C131C2311yescharacteristic
115D8×C131—22no—
116C2×C8×C131—22no—
117C2×C8×C131C2311yescharacteristic
118C2×C8×C131C2311yescharacteristic
119C4⋊C4×C131C2311yescharacteristic
120C4⋊C4×C131C2311yescharacteristic
121C22⋊C4×C131—21no—
122C22⋊C4×C131—21no—
123C22×D131—21no—
124C22×D131—21no—
125C22×D131—42no—
126C22×D131—42no—
127C22×D131—81no—
128C22×D131C2×D411yescharacteristic
129Dic262—21no—
130Dic262—21no—
131Dic262—21no—
132Dic262C2×D412yes—
133D524—21no—
134D524—21no—
135D524—21no—
136D524—81no—
137D524C2×D412yes—
138C22⋊D131—41no—
139C22⋊D131—42no—
140C22⋊D131—42no—
141C22⋊D131—42no—
142C22⋊D131—42no—
143C22⋊D131—42no—
144C22⋊D131—81no—
145C2×Dic131—21no—
146C2×Dic131—21no—
147C2×Dic131—42no—
148C2×Dic131C2×D411yescharacteristic
149C2×Dic131C2×D411yescharacteristic
150C2×Dic131C2×D411yescharacteristic
151C4×D131—21no—
152C4×D131—21no—
153C4×D131—42no—
154C4×D131—42no—
155C4×D131—81no—
156C131⋊C8—21no—
157C131⋊C8—21no—
158C23×C131—21no—
159C23×C131—21no—
160C23×C131C2×D411yescharacteristic
161D4×C131—21no—
162D4×C131—21no—
163D4×C131—21no—
164D4×C131—21no—
165D4×C131—81no—
166D4×C131—81no—
167C2×C4×C131—21no—
168C2×C4×C131—21no—
169C2×C4×C131—21no—
170C2×C4×C131—21no—
171C2×C4×C131C2×D411yescharacteristic
172C2×C4×C131C2411yescharacteristic
173C8×C131—21no—
174C8×C131C2×D412yes—
175C22×C131—21no—
176C22×C131—21no—
177C22×C131—21no—
178C22×C131—41no—
179C22×C131—41no—
180C22×C131—41no—
181C22×C131—41no—
182C22×C131—41no—
183C22×C131—41no—
184C22×C131C22×D411yescharacteristic
185C2×D131—21no—
186C2×D131—22no—
187C2×D131—21no—
188C2×D131—21no—
189C2×D131—41no—
190C2×D131—41no—
191C2×D131—41no—
192C2×D131—41no—
193C2×D131—41no—
194C2×D131—84no—
195C4×C131—21no—
196C4×C131—41no—
197C4×C131—41no—
198C4×C131C22×D411yescharacteristic
199C4×C131H(5,𝔽2)11yescharacteristic
200Dic131—21no—
201Dic131—22no—
202Dic131—21no—
203Dic131—21no—
204Dic131—41no—
205C2×C131—41no—
206C2×C131—81no—
207C2×C131—81no—
208C2×C131D4211yescharacteristic
209C2×C131#57511yescharacteristic
210C2×C131#57611yescharacteristic
211D131—41no—
212D131—42no—
213D131—81no—
214C131#326811yescharacteristic, complemented, Sylow
215#3268—1311nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1311no—
217C2×#135—1311no—
218#550—1311no—
219#546—1311no—
220#496—1311no—
221#468—1311no—
222#466D13111yescharacteristic, complemented
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×#41—1311no—
234C22×D4—1311no—
235AGL(1,ℤ/8)—2624no—
236#135—2624no—
237C2×Q16—1311no—
238C2×SD16—1311no—
239C2×SD16—1311no—
240C2×SD16—1311no—
241C2×SD16—1311no—
242C2×D8—1311no—
243C2×D8—1311no—
244C2×D8—1311no—
245C2×D8C2×D13111yescharacteristic
246C2×M16—1311no—
247C22×C8C2×D13111yescharacteristic, complemented
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#121C2×D13111yescharacteristic
258#121C2×D13111yescharacteristic, complemented
259#120—2621no—
260C4×Q8—1311no—
261C4×D4—2621no—
262C4×D4—1311no—
263C4×D4—1311no—
264C2×C22⋊C4—2621no—
265#107C2×D13111yescharacteristic, complemented
266C4⋊C8—1311no—
267#103—1311no—
268#103—1311no—
269#102—1311no—
270#102—1311no—
271#102C2×D13111yescharacteristic
272#102C2×D13111yescharacteristic, complemented
273C22⋊C8—1311no—
274C22⋊C8—1311no—
275#97—1311no—
276C24—2621no—
277#41—2622no—
278#41—2622no—
279#41—2622no—
280#41—5241no—
281#41—5241no—
282#41—5241no—
283C2×Q8—1311no—
284C2×Q8—1311no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2621no—
288C2×D4—2622no—
289C2×D4—2621no—
290C2×D4—2621no—
291C2×D4—2621no—
292C2×D4—2621no—
293C2×D4—2621no—
294C2×D4—2621no—
295C2×D4—5241no—
296C2×D4—1311no—
297C2×D4—1311no—
298C2×D4—1311no—
299C2×D4C22×D13111yescharacteristic
300C2×D4C22×D13111yescharacteristic
301C22×C4—2621no—
302C22×C4—2621no—
303C22×C4—2621no—
304C22×C4—2621no—
305C22×C4—1311no—
306C22×C4C22×D13111yescharacteristic
307Q16—2622no—
308SD16—2622no—
309SD16—2622no—
310SD16—5241no—
311SD16—1314no—
312D8—22no—
313D8—2622no—
314D8—2622no—
315D8—5241no—
316M16—2622no—
317C2×C8—22no—
318C2×C8—1311no—
319C2×C8—1311no—
320C2×C8C22×D13111yescharacteristic
321C2×C8C22×D13111yescharacteristic
322C4⋊C4—2621no—
323C4⋊C4—2621no—
324C4⋊C4—1311no—
325C4⋊C4C22×D13111yescharacteristic
326C4⋊C4C22×D13111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
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—2621no—
341C23—2621no—
342C23—5242no—
343C23—5242no—
344C23—10481no—
345C23—1311no—
346C23D4×D13111yescharacteristic
347Q8—2621no—
348Q8—2621no—
349Q8—2621no—
350Q8—1312no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—2621no—
356D4—2621no—
357D4—2621no—
358D4—81no—
359D4—81no—
360D4—5241no—
361D4—5242no—
362D4—5242no—
363D4—5242no—
364D4—5242no—
365D4—5242no—
366D4—10481no—
367D4—10481no—
368D4—1312no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—2621no—
374C2×C4—2621no—
375C2×C4—2621no—
376C2×C4—2621no—
377C2×C4—5242no—
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—
389C8D4×D13112yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—2621no—
400C22—2621no—
401C22—2622no—
402C22—2621no—
403C22—5241no—
404C22—5241no—
405C22—5241no—
406C22—5241no—
407C22—5241no—
408C22—10484no—
409C22C2×D4×D13111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—2621no—
414C4—2621no—
415C4—2622no—
416C4—2621no—
417C4—5241no—
418C4C2×D4×D13111yescharacteristic
419C4#1686229211yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—5241no—
424C2—5242no—
425C2—10481no—
426C2#2255394611yescharacteristic
427C2#2255410911yescharacteristic
428C2#2255411311yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#466×C131C2
socleC22×C131C22×D4
hypercenter#466D131
hypocenterC131#3268
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16862270C2semidirect
2. C2×#16862268C2semidirect
3. #22553980C2semidirect
4. #22553959C2semidirect
5. #22553700C2semidirect
6. #22553559C2semidirect
7. #22553533C2semidirect
8. #22553506C2semidirect
9. #22553475C2semidirect
10. #22553455C2semidirect
11. #22553455C2semidirect
12. #22553447C2semidirect
13. #22553411C2semidirect
14. #22553383C2semidirect
15. #466×C131C2semidirect
17. C2×#14523513C22semidirect
22. C2×Dic524C22semidirect
24. C2×#14523502C22semidirect
26. C2×#14523501C22semidirect
32. #16862242C22semidirect
41. #16862211C22semidirect
46. Dic131⋊C8C22semidirect
47. #16862169C22semidirect
48. #16862169C22semidirect
50. #16862166C22semidirect
51. C22⋊(C131⋊C8)C22semidirect
53. #16862156C22semidirect
55. C22×C8×C131C22semidirect
57. #121×C131C22semidirect
58. #107×C131C22semidirect
60. #102×C131C22semidirect
89. #14523502D4semidirect
214. C131#3268semidirect
222. #466D131semidirect
247. C22×C8C2×D131semidirect
258. #121C2×D131semidirect
265. #107C2×D131semidirect
272. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622702
C2×#168622682
#225539802
#225539592
#225537002
#225535592
#225535332
#225535062
#225534752
#225534552
#225534472
#225534112
#225533832
#466×C1312
#3268131