Finite Group Databasebeta

← #49132886

Subgroups — #49132886

Order 36224.

498 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×#22583637C211yesmaximal, characteristic, complemented
2#31827126C211yesmaximal, characteristic, complemented
3#31827103C211yesmaximal, characteristic, complemented
4#31826850C211yesmaximal, characteristic, complemented
5#31826720C211yesmaximal, characteristic, complemented
6#31826718C211yesmaximal, characteristic, complemented
7#31826689C211yesmaximal, characteristic, complemented
8#31826682C211yesmaximal, characteristic, complemented
9#31826626C211yesmaximal, characteristic, complemented
10#31826614C211yesmaximal, characteristic, complemented
11#31826610C211yesmaximal, characteristic, complemented
12#31826604C211yesmaximal, characteristic, complemented
13SD16×Dic283C211yesmaximal, characteristic, complemented
14#31826541C211yesmaximal, characteristic, complemented
15C2×#137×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×D4×D283C2211yescharacteristic
18#22583637—24no—
19C2×#17985379C2211yescharacteristic
20C2×#17985379C2211yescharacteristic
21C2×#17985379C2211yescharacteristic, complemented
22C2×D2264C2211yescharacteristic
23C2×#17985377C2211yescharacteristic
24C2×#17985375C2211yescharacteristic
25#22583599—21no—
26#22583599C2211yescharacteristic
27#22583599C2211yescharacteristic, complemented
28#22583593—21no—
29#22583586—21no—
30#22583586C2211yescharacteristic
31#22583586C2211yescharacteristic
32#22583584—21no—
33#22583584C2211yescharacteristic
34#22583584C2211yescharacteristic, complemented
35#22583582—21no—
36#22583578—21no—
37Q8×Dic283C2211yescharacteristic, complemented
38D4×Dic283C2211yescharacteristic
39C4×C22⋊D283—21no—
40C4×C22⋊D283C2211yescharacteristic
41C2×(C2×D283)⋊C4—21no—
42#22583542C2211yescharacteristic, complemented
43Dic283⋊C8C2211yescharacteristic, complemented
44#22583534C2211yescharacteristic, complemented
45#22583534C2211yescharacteristic, complemented
46#22583534C2211yescharacteristic, complemented
47#22583532C2211yescharacteristic
48#22583532C2211yescharacteristic, complemented
49#22583531C2211yescharacteristic
50C22⋊(C283⋊C8)C2211yescharacteristic, complemented
51(C2×D283)⋊C8C2211yescharacteristic
52C8×Dic283C2211yescharacteristic, complemented
53C2×#41×C283C2211yescharacteristic
54C22×Q8×C283C2211yescharacteristic, complemented
55#137×C283—24no—
56C2×Q16×C283C2211yescharacteristic, complemented
57C2×Q16×C283C2211yescharacteristic, complemented
58C2×SD16×C283C2211yescharacteristic
59C2×SD16×C283C2211yescharacteristic
60C2×M16×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985393—22no—
63#17985393—41no—
64C2×D1132—21no—
65C2×D1132C2311yescharacteristic
66C2×D1132C2311yescharacteristic
67C2×C22⋊D283—21no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283D411yescharacteristic
71D4×D283—22no—
72D4×D283—41no—
73C22×Dic283—21no—
74C2×C4×D283—21no—
75C2×C4×D283C2311yescharacteristic
76#17985379—22no—
77#17985379—41no—
78#17985379—41no—
79D2264—22no—
80#17985377—22no—
81#17985375—22no—
82C2×C283⋊C8C2311yescharacteristic
83C2×C283⋊C8C2311yescharacteristic
84C4⋊Dic283—21no—
85C4⋊Dic283C2311yescharacteristic
86C4⋊Dic283C2311yescharacteristic
87Dic283⋊C4—21no—
88Dic283⋊C4—21no—
89Dic283⋊C4D411yescharacteristic
90C22⋊Dic283—21no—
91C22⋊Dic283D411yescharacteristic
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4—21no—
96(C2×D283)⋊C4—21no—
97(C2×D283)⋊C4—21no—
98(C2×D283)⋊C4—21no—
99(C2×D283)⋊C4—21no—
100(C2×D283)⋊C4D411yescharacteristic
101C4×Dic283—21no—
102C4×Dic283C2311yescharacteristic
103#41×C283—22no—
104#41×C283—41no—
105C2×Q8×C283—21no—
106C2×Q8×C283—22no—
107C2×Q8×C283—41no—
108C2×Q8×C283C2311yescharacteristic
109C2×Q8×C283C2311yescharacteristic
110C2×Q8×C283C2311yescharacteristic
111C2×D4×C283—21no—
112C2×D4×C283C2311yescharacteristic
113C22×C4×C283—21no—
114C22×C4×C283—21no—
115C22×C4×C283C2311yescharacteristic
116Q16×C283—22no—
117Q16×C283—22no—
118SD16×C283—22no—
119SD16×C283D414yescomplemented
120M16×C283—22no—
121C2×C8×C283C2311yescharacteristic
122C2×C8×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128C22×D283C2×D411yescharacteristic
129D1132—21no—
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—42no—
134C22⋊D283—41no—
135C22⋊D283—42no—
136C22⋊D283—42no—
137C22⋊D283—81no—
138C2×Dic283—21no—
139C2×Dic283—21no—
140C2×Dic283—81no—
141C2×Dic283C2×D411yescharacteristic
142C2×Dic283C2×D411yescharacteristic
143C2×Dic283C2×D411yescharacteristic
144C4×D283—21no—
145C4×D283—21no—
146C4×D283—42no—
147C283⋊C8—21no—
148C283⋊C8—21no—
149C23×C283—21no—
150C23×C283C2×D411yescharacteristic
151Q8×C283—21no—
152Q8×C283—21no—
153Q8×C283—21no—
154Q8×C283—21no—
155Q8×C283—21no—
156Q8×C283—42no—
157Q8×C283C2×D412yes—
158D4×C283—21no—
159D4×C283—42no—
160D4×C283C2×D412yes—
161C2×C4×C283—21no—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—42no—
167C2×C4×C283—42no—
168C2×C4×C283C2×D411yescharacteristic
169C2×C4×C283C2411yescharacteristic
170C8×C283—21no—
171C8×C283C2×D412yes—
172C22×C283—21no—
173C22×C283—22no—
174C22×C283—21no—
175C22×C283—21no—
176C22×C283—41no—
177C22×C283—41no—
178C22×C283C22×D411yescharacteristic
179C2×D283—21no—
180C2×D283—21no—
181C2×D283—21no—
182C2×D283—41no—
183C2×D283—41no—
184C2×D283—41no—
185C2×D283—41no—
186C2×D283—41no—
187C2×D283—41no—
188C2×D283—84no—
189C4×C283—21no—
190C4×C283—22no—
191C4×C283—41no—
192C4×C283—41no—
193C4×C283C22×D411yescharacteristic
194C4×C283H(5,𝔽2)11yescharacteristic
195Dic283—21no—
196Dic283—21no—
197Dic283—21no—
198Dic283—41no—
199Dic283—41no—
200C2×C283—42no—
201C2×C283—41no—
202C2×C283D4211yescharacteristic
203C2×C283#57511yescharacteristic
204C2×C283#57711yescharacteristic
205D283—41no—
206D283—81no—
207D283—81no—
208C283#327511yescharacteristic, complemented, Sylow
209#3275—2831nomaximal, Sylow
210C2×#137D28311yescharacteristic, complemented
211C2×AGL(1,ℤ/8)—2831no—
212#548—2831no—
213#546—2831no—
214#495—2831no—
215#470—2831no—
216#469—2831no—
217#464—2831no—
218#463—2831no—
219#452—2831no—
220#450—2831no—
221#449—2831no—
222#448—2831no—
223C4×SD16—2831no—
224#436—2831no—
225C2×#41—2831no—
226C2×#41C2×D28311yescharacteristic
227C22×Q8C2×D28311yescharacteristic, complemented
228C22×D4—2831no—
229#137—24no—
230AGL(1,ℤ/8)—5664no—
231C2×Q16C2×D28311yescharacteristic, complemented
232C2×Q16C2×D28311yescharacteristic, complemented
233C2×SD16—2831no—
234C2×SD16—2831no—
235C2×SD16—2831no—
236C2×SD16C2×D28311yescharacteristic
237C2×SD16C2×D28311yescharacteristic
238C2×D8—2831no—
239C2×D8—2831no—
240C2×M16—2831no—
241C2×M16C2×D28311yescharacteristic, complemented
242#124—2831no—
243#124—2831no—
244#124—5661no—
245#123—5661no—
246C22⋊Q8—2831no—
247C22⋊Q8—2831no—
248C22⋊Q8—5661no—
249#121—2831no—
250#121—2831no—
251#121—5661no—
252#121—5661no—
253#120—5661no—
254C4×Q8—2831no—
255C4×D4—2831no—
256C4×D4—2831no—
257C4×D4—5661no—
258C2×C22⋊C4—5661no—
259#106—2831no—
260C4⋊C8—2831no—
261#103—2831no—
262#103—2831no—
263#103—2831no—
264#102—2831no—
265#102—2831no—
266#102—2831no—
267C22⋊C8—2831no—
268C22⋊C8—2831no—
269C4×C8—2831no—
270C24—5661no—
271#41—22no—
272#41—41no—
273#41—5662no—
274#41—11321no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D28311yescharacteristic
279C2×Q8C22×D28311yescharacteristic
280C2×Q8C22×D28311yescharacteristic
281C2×D4—21no—
282C2×D4—2831no—
283C2×D4—2831no—
284C2×D4—2831no—
285C2×D4—5661no—
286C2×D4—5661no—
287C2×D4—5662no—
288C2×D4—5661no—
289C2×D4—5661no—
290C2×D4—11321no—
291C2×D4C22×D28311yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—2831no—
295C22×C4—5661no—
296C22×C4—5661no—
297C22×C4C22×D28311yescharacteristic
298Q16—22no—
299Q16—22no—
300SD16—22no—
301SD16—5662no—
302SD16—11321no—
303SD16—11321no—
304SD16C22⋊D28314yescomplemented
305D8—5662no—
306D8—5662no—
307M16—22no—
308M16—5662no—
309C2×C8—2831no—
310C2×C8—2831no—
311C2×C8C22×D28311yescharacteristic
312C2×C8C22×D28311yescharacteristic
313C4⋊C4—2831no—
314C4⋊C4—2831no—
315C4⋊C4—2831no—
316C4⋊C4—5661no—
317C4⋊C4—5661no—
318C4⋊C4—5661no—
319C22⋊C4—2831no—
320C22⋊C4—2831no—
321C22⋊C4—5661no—
322C22⋊C4—5661no—
323C22⋊C4—5661no—
324C22⋊C4—5661no—
325C22⋊C4—5661no—
326C22⋊C4—5661no—
327C22⋊C4—5661no—
328C22⋊C4—5661no—
329C22⋊C4—5661no—
330C42—2831no—
331C42—5661no—
332C23—21no—
333C23—2831no—
334C23—5661no—
335C23—5661no—
336C23—11322no—
337C23—11322no—
338C23—22641no—
339C23D4×D28311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—42no—
346Q8C2×C22⋊D28312yes—
347D4—21no—
348D4—42no—
349D4—5661no—
350D4—5661no—
351D4—5661no—
352D4—5661no—
353D4—11322no—
354D4—11321no—
355D4—11322no—
356D4—11322no—
357D4—22641no—
358D4C2×C22⋊D28312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—2831no—
367C2×C4—2831no—
368C2×C4—2831no—
369C2×C4—5661no—
370C2×C4—5661no—
371C2×C4—5661no—
372C2×C4—5661no—
373C2×C4—11322no—
374C2×C4—22641no—
375C2×C4D4×D28311yescharacteristic
376C2×C4C23×D28311yescharacteristic
377C8—21no—
378C8—5661no—
379C8—5661no—
380C8C2×C22⋊D28312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—5661no—
388C22—5661no—
389C22—5661no—
390C22—11321no—
391C22—11321no—
392C22—11321no—
393C22—11321no—
394C22—11321no—
395C22—11321no—
396C22—22644no—
397C22C2×D4×D28311yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—5661no—
403C4—5661no—
404C4—5661no—
405C4—11321no—
406C4—11321no—
407C4C22×C22⋊D28311yescharacteristic
408C4#2258365511yescharacteristic
409C2—41no—
410C2—42no—
411C2—11321no—
412C2—22641no—
413C2—22641no—
414C2D4×C22⋊D28311yescharacteristic
415C2#3182726411yescharacteristic
416C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×#137×C283C2
socleC22×C283C22×D4
hypercenterC2×#137D283
hypocenterC283#3275
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. #31827126C2semidirect
3. #31827103C2semidirect
4. #31826850C2semidirect
5. #31826720C2semidirect
6. #31826718C2semidirect
7. #31826689C2semidirect
8. #31826682C2semidirect
9. #31826626C2semidirect
10. #31826614C2semidirect
11. #31826610C2semidirect
12. #31826604C2semidirect
13. SD16×Dic283C2semidirect
14. #31826541C2semidirect
15. C2×#137×C283C2semidirect
21. C2×#17985379C22semidirect
27. #22583599C22semidirect
34. #22583584C22semidirect
37. Q8×Dic283C22semidirect
42. #22583542C22semidirect
43. Dic283⋊C8C22semidirect
44. #22583534C22semidirect
45. #22583534C22semidirect
46. #22583534C22semidirect
48. #22583532C22semidirect
50. C22⋊(C283⋊C8)C22semidirect
52. C8×Dic283C22semidirect
54. C22×Q8×C283C22semidirect
56. C2×Q16×C283C22semidirect
57. C2×Q16×C283C22semidirect
60. C2×M16×C283C22semidirect
119. SD16×C283D4semidirect
208. C283#3275semidirect
210. C2×#137D283semidirect
227. C22×Q8C2×D283semidirect
231. C2×Q16C2×D283semidirect
232. C2×Q16C2×D283semidirect
241. C2×M16C2×D283semidirect
304. SD16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
#318271262
#318271032
#318268502
#318267202
#318267182
#318266892
#318266822
#318266262
#318266142
#318266102
#318266042
SD16×Dic2832
#318265412
C2×#137×C2832
#3275283