Finite Group Databasebeta

← #49132795

Subgroups — #49132795

Order 36224.

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×#22583631C211yesmaximal, characteristic, complemented
2#31827134C211yesmaximal, characteristic, complemented
3#31827109C211yesmaximal, characteristic, complemented
4#31826853C211yesmaximal, characteristic, complemented
5#31826711C211yesmaximal, characteristic, complemented
6#31826698C211yesmaximal, characteristic, complemented
7#31826685C211yesmaximal, characteristic, complemented
8#31826658C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826609C211yesmaximal, characteristic, complemented
11#31826607C211yesmaximal, characteristic, complemented
12#31826599C211yesmaximal, characteristic, complemented
13#31826566C211yesmaximal, characteristic, complemented
14#31826534C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C2×#17985391C2211yescharacteristic
18#22583631—24no—
19C2×#17985382C2211yescharacteristic, complemented
20C2×#17985381C2211yescharacteristic, complemented
21C2×#17985380C2211yescharacteristic, complemented
22C2×#17985377C2211yescharacteristic
23C2×#17985377C2211yescharacteristic
24C2×C8×D283C2211yescharacteristic, complemented
25#22583606—21no—
26#22583606C2211yescharacteristic, complemented
27#22583597C2211yescharacteristic
28#22583591—21no—
29#22583589C2211yescharacteristic
30#22583584—21no—
31#22583584C2211yescharacteristic
32#22583582—21no—
33#22583581—21no—
34#22583581C2211yescharacteristic
35#22583581C2211yescharacteristic, complemented
36#22583579—21no—
37Q8×Dic283C2211yescharacteristic, complemented
38D4×Dic283C2211yescharacteristic
39C4×C22⋊D283—21no—
40C4×C22⋊D283C2211yescharacteristic
41C2×C22⋊Dic283—21no—
42#22583544C2211yescharacteristic, complemented
43Dic283⋊C8C2211yescharacteristic, complemented
44#22583535C2211yescharacteristic, complemented
45#22583534C2211yescharacteristic, complemented
46#22583532C2211yescharacteristic, complemented
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49#22583531C2211yescharacteristic, complemented
50C22⋊(C283⋊C8)C2211yescharacteristic
51(C2×D283)⋊C8C2211yescharacteristic, complemented
52#22583520C2211yescharacteristic, complemented
53C2×#41×C283C2211yescharacteristic
54C22×D4×C283C2211yescharacteristic
55AGL(1,ℤ/8)×C283—24no—
56C2×SD16×C283C2211yescharacteristic
57C2×SD16×C283C2211yescharacteristic
58C2×D8×C283C2211yescharacteristic
59C2×D8×C283C2211yescharacteristic
60C2×M16×C283C2211yescharacteristic
61#17985393—22no—
62#17985393—41no—
63#17985391—22no—
64#17985391—41no—
65C2×Dic566C2311yescharacteristic
66C2×D1132—21no—
67C2×D1132C2311yescharacteristic
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283D411yescharacteristic
75C22×Dic283—21no—
76C22×Dic283—21no—
77C2×C4×D283—21no—
78C2×C4×D283C2311yescharacteristic
79#17985382—22no—
80#17985381—22no—
81#17985380—41no—
82#17985377—22no—
83#17985377—41no—
84C2×C283⋊C8C2311yescharacteristic
85C2×C283⋊C8C2311yescharacteristic
86C8×D283—22no—
87C4⋊Dic283—21no—
88C4⋊Dic283C2311yescharacteristic
89C4⋊Dic283C2311yescharacteristic
90Dic283⋊C4—21no—
91Dic283⋊C4D411yescharacteristic
92C22⋊Dic283—21no—
93C22⋊Dic283—21no—
94C22⋊Dic283—21no—
95C22⋊Dic283—21no—
96C22⋊Dic283—21no—
97C22⋊Dic283—21no—
98C22⋊Dic283D411yescharacteristic
99(C2×D283)⋊C4—21no—
100(C2×D283)⋊C4D411yescharacteristic
101C4×Dic283—21no—
102C4×Dic283C2311yescharacteristic
103C24×C283—21no—
104#41×C283—22no—
105#41×C283—41no—
106C2×Q8×C283C2311yescharacteristic
107C2×D4×C283—22no—
108C2×D4×C283—21no—
109C2×D4×C283—21no—
110C2×D4×C283—41no—
111C2×D4×C283C2311yescharacteristic
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115C22×C4×C283C2311yescharacteristic
116SD16×C283—22no—
117SD16×C283D414yescomplemented
118D8×C283—22no—
119D8×C283—22no—
120M16×C283—22no—
121C2×C8×C283C2311yescharacteristic
122C2×C8×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283C2×D411yescharacteristic
125Dic566—21no—
126Dic566—21no—
127D1132—21no—
128D1132—21no—
129D1132—42no—
130C22⋊D283—41no—
131C22⋊D283—42no—
132C22⋊D283—42no—
133C22⋊D283—81no—
134C22⋊D283—81no—
135C22⋊D283—81no—
136C22⋊D283—81no—
137C2×Dic283—21no—
138C2×Dic283—21no—
139C2×Dic283—21no—
140C2×Dic283—42no—
141C2×Dic283—81no—
142C2×Dic283C2×D411yescharacteristic
143C2×Dic283C2×D411yescharacteristic
144C2×Dic283C2×D411yescharacteristic
145C4×D283—21no—
146C4×D283—21no—
147C4×D283—42no—
148C283⋊C8—21no—
149C283⋊C8—21no—
150C23×C283—21no—
151C23×C283—21no—
152C23×C283—21no—
153C23×C283—42no—
154C23×C283—42no—
155C23×C283—81no—
156C23×C283C2×D411yescharacteristic
157Q8×C283—21no—
158Q8×C283C2×D412yes—
159D4×C283—21no—
160D4×C283—21no—
161D4×C283—21no—
162D4×C283—21no—
163D4×C283—21no—
164D4×C283—42no—
165D4×C283—42no—
166D4×C283C2×D412yes—
167C2×C4×C283—21no—
168C2×C4×C283—21no—
169C2×C4×C283—21no—
170C2×C4×C283—42no—
171C2×C4×C283C2×D411yescharacteristic
172C2×C4×C283C2411yescharacteristic
173C8×C283—21no—
174C8×C283C2×D412yes—
175C22×C283—21no—
176C22×C283—22no—
177C22×C283—21no—
178C22×C283—21no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—84no—
188C22×C283C22×D411yescharacteristic
189C2×D283—21no—
190C2×D283—21no—
191C2×D283—21no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—41no—
195C4×C283—21no—
196C4×C283—22no—
197C4×C283C22×D411yescharacteristic
198C4×C283H(5,𝔽2)11yescharacteristic
199Dic283—21no—
200Dic283—21no—
201Dic283—21no—
202Dic283—41no—
203Dic283—41no—
204Dic283—41no—
205C2×C283—41no—
206C2×C283—42no—
207C2×C283—81no—
208C2×C283—81no—
209C2×C283D4211yescharacteristic
210C2×C283#57511yescharacteristic
211C2×C283#57611yescharacteristic
212D283—41no—
213D283—81no—
214C283#326811yescharacteristic, complemented, Sylow
215#3268—2831nomaximal, Sylow
216C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
217C2×#135—2831no—
218#550—2831no—
219#546—2831no—
220#496—2831no—
221#468—2831no—
222#466—2831no—
223#464—2831no—
224#459—2831no—
225#452—2831no—
226#449—2831no—
227#449—2831no—
228#447—2831no—
229#440—2831no—
230#435—2831no—
231C2×#41—2831no—
232C2×#41—2831no—
233C2×#41C2×D28311yescharacteristic
234C22×D4C2×D28311yescharacteristic
235AGL(1,ℤ/8)—24no—
236#135—5664no—
237C2×Q16—2831no—
238C2×SD16—2831no—
239C2×SD16—2831no—
240C2×SD16C2×D28311yescharacteristic
241C2×SD16C2×D28311yescharacteristic
242C2×D8—2831no—
243C2×D8—2831no—
244C2×D8C2×D28311yescharacteristic
245C2×D8C2×D28311yescharacteristic
246C2×M16C2×D28311yescharacteristic
247C22×C8—2831no—
248#127—2831no—
249#127—5661no—
250#124—2831no—
251#123—5661no—
252C22⋊Q8—2831no—
253#121—2831no—
254#121—2831no—
255#121—2831no—
256#121—5661no—
257#121—5661no—
258#121—5661no—
259#120—5661no—
260C4×Q8—2831no—
261C4×D4—2831no—
262C4×D4—2831no—
263C4×D4—5661no—
264C2×C22⋊C4—5661no—
265#107—2831no—
266C4⋊C8—2831no—
267#103—2831no—
268#103—2831no—
269#102—2831no—
270#102—2831no—
271#102—2831no—
272#102—2831no—
273C22⋊C8—2831no—
274C22⋊C8—2831no—
275#97—2831no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—5662no—
280#41—5662no—
281#41—11321no—
282#41—11321no—
283C2×Q8—2831no—
284C2×Q8C22×D28311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2831no—
290C2×D4—2831no—
291C2×D4—5661no—
292C2×D4—5661no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—5661no—
298C2×D4C22×D28311yescharacteristic
299C2×D4C22×D28311yescharacteristic
300C2×D4C22×D28311yescharacteristic
301C22×C4—21no—
302C22×C4—2831no—
303C22×C4—5661no—
304C22×C4—5661no—
305C22×C4—5661no—
306C22×C4C22×D28311yescharacteristic
307Q16—5662no—
308SD16—22no—
309SD16—5662no—
310SD16—11321no—
311SD16C22⋊D28314yescomplemented
312D8—22no—
313D8—22no—
314D8—5662no—
315D8—11321no—
316M16—22no—
317C2×C8—2831no—
318C2×C8—2831no—
319C2×C8—5662no—
320C2×C8C22×D28311yescharacteristic
321C2×C8C22×D28311yescharacteristic
322C4⋊C4—2831no—
323C4⋊C4—2831no—
324C4⋊C4—2831no—
325C4⋊C4—5661no—
326C4⋊C4—5661no—
327C22⋊C4—2831no—
328C22⋊C4—2831no—
329C22⋊C4—5661no—
330C22⋊C4—5661no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C22⋊C4—5661no—
335C22⋊C4—5661no—
336C42—2831no—
337C42—5661no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—2831no—
345C23—5661no—
346C23D4×D28311yescharacteristic
347Q8—21no—
348Q8—5661no—
349Q8—5661no—
350Q8C2×C22⋊D28312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—5661no—
359D4—5661no—
360D4—11322no—
361D4—11321no—
362D4—11322no—
363D4—11322no—
364D4—22641no—
365D4—22641no—
366D4—22641no—
367D4—22641no—
368D4C2×C22⋊D28312yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—2831no—
374C2×C4—2831no—
375C2×C4—2831no—
376C2×C4—5661no—
377C2×C4—5661no—
378C2×C4—5661no—
379C2×C4—5661no—
380C2×C4—5661no—
381C2×C4—11322no—
382C2×C4—11322no—
383C2×C4—22641no—
384C2×C4D4×D28311yescharacteristic
385C2×C4C23×D28311yescharacteristic
386C8—21no—
387C8—5661no—
388C8—5661no—
389C8C2×C22⋊D28312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—5661no—
404C22—5661no—
405C22—5661no—
406C22—11321no—
407C22—11321no—
408C22—11321no—
409C22C2×D4×D28311yescharacteristic
410C4—21no—
411C4—22no—
412C4—5661no—
413C4—5661no—
414C4—5661no—
415C4—11321no—
416C4—11321no—
417C4—11321no—
418C4C22×C22⋊D28311yescharacteristic
419C4#2258365511yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—11321no—
425C2—22641no—
426C2D4×C22⋊D28311yescharacteristic
427C2#3182726111yescharacteristic
428C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3268
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
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×#22583631C2semidirect
2. #31827134C2semidirect
3. #31827109C2semidirect
4. #31826853C2semidirect
5. #31826711C2semidirect
6. #31826698C2semidirect
7. #31826685C2semidirect
8. #31826658C2semidirect
9. #31826627C2semidirect
10. #31826609C2semidirect
11. #31826607C2semidirect
12. #31826599C2semidirect
13. #31826566C2semidirect
14. #31826534C2semidirect
15. C2×AGL(1,ℤ/8)×C283C2semidirect
16. C2×#17985393C22semidirect
19. C2×#17985382C22semidirect
20. C2×#17985381C22semidirect
21. C2×#17985380C22semidirect
24. C2×C8×D283C22semidirect
26. #22583606C22semidirect
35. #22583581C22semidirect
37. Q8×Dic283C22semidirect
42. #22583544C22semidirect
43. Dic283⋊C8C22semidirect
44. #22583535C22semidirect
45. #22583534C22semidirect
46. #22583532C22semidirect
49. #22583531C22semidirect
51. (C2×D283)⋊C8C22semidirect
52. #22583520C22semidirect
117. SD16×C283D4semidirect
214. C283#3268semidirect
216. C2×AGL(1,ℤ/8)D283semidirect
311. SD16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836312
#318271342
#318271092
#318268532
#318267112
#318266982
#318266852
#318266582
#318266272
#318266092
#318266072
#318265992
#318265662
#318265342
C2×AGL(1,ℤ/8)×C2832
#3268283