Finite Group Databasebeta

← #49132831

Subgroups — #49132831

Order 36224.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583642C211yesmaximal, characteristic, complemented
2Q8×C22⋊D283C211yesmaximal, characteristic, complemented
3#31827126C211yesmaximal, characteristic
4#31826858C211yesmaximal, characteristic, complemented
5#31826720C211yesmaximal, characteristic, complemented
6#31826694C211yesmaximal, characteristic, complemented
7#31826689C211yesmaximal, characteristic, complemented
8#31826673C211yesmaximal, characteristic, complemented
9#31826626C211yesmaximal, characteristic, complemented
10#31826621C211yesmaximal, characteristic, complemented
11#31826619C211yesmaximal, characteristic, complemented
12#31826604C211yesmaximal, characteristic, complemented
13#31826572C211yesmaximal, characteristic, complemented
14#31826536C211yesmaximal, characteristic, complemented
15C22×Q16×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18#22583642—24no—
19C2×#17985382C2211yescharacteristic
20C2×#17985382C2211yescharacteristic
21C2×#17985381C2211yescharacteristic
22C2×#17985379C2211yescharacteristic
23C2×#17985379C2211yescharacteristic
24C2×#17985375C2211yescharacteristic
25#22583608—21no—
26#22583608C2211yescharacteristic
27#22583599—21no—
28#22583599C2211yescharacteristic
29#22583589—21no—
30#22583589C2211yescharacteristic
31#22583586—21no—
32#22583586—21no—
33#22583586C2211yescharacteristic
34#22583586C2211yescharacteristic
35#22583584—21no—
36#22583584C2211yescharacteristic
37Q8×Dic283C2211yescharacteristic
38Q8×Dic283C2211yescharacteristic
39C4×C22⋊D283—21no—
40C4×C22⋊D283—21no—
41C4×C22⋊D283C2211yescharacteristic
42#22583542C2211yescharacteristic
43Dic283⋊C8C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583534C2211yescharacteristic
48#22583534C2211yescharacteristic
49#22583532C2211yescharacteristic
50C22⋊(C283⋊C8)C2211yescharacteristic
51(C2×D283)⋊C8C2211yescharacteristic
52#22583520C2211yescharacteristic
53C22×Q8×C283C2211yescharacteristic
54C22×Q8×C283C2211yescharacteristic
55C2×Q16×C283—24no—
56C2×Q16×C283C2211yescharacteristic
57C2×Q16×C283C2211yescharacteristic
58C2×Q16×C283C2211yescharacteristic
59C2×Q16×C283C2211yescharacteristic
60C22×C8×C283C2211yescharacteristic
61#17985393—22no—
62#17985393—41no—
63C2×Dic566—21no—
64C2×Dic566C2311yescharacteristic
65Q8×D283—22no—
66Q8×D283—41no—
67C2×D1132—21no—
68C2×D1132C2311yescharacteristic
69C2×C22⋊D283—21no—
70C2×C22⋊D283D411yescharacteristic
71C2×C4×D283—21no—
72C2×C4×D283—21no—
73C2×C4×D283C2311yescharacteristic
74#17985382—22no—
75#17985382—41no—
76#17985381—22no—
77#17985379—22no—
78#17985379—41no—
79#17985375—22no—
80C2×C283⋊C8C2311yescharacteristic
81C2×C283⋊C8C2311yescharacteristic
82C4⋊Dic283—21no—
83C4⋊Dic283—21no—
84C4⋊Dic283C2311yescharacteristic
85C4⋊Dic283C2311yescharacteristic
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4—21no—
89Dic283⋊C4—21no—
90Dic283⋊C4—21no—
91Dic283⋊C4D411yescharacteristic
92C22⋊Dic283—21no—
93C22⋊Dic283D411yescharacteristic
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)⋊C4D411yescharacteristic
100C4×Dic283—21no—
101C4×Dic283—21no—
102C4×Dic283C2311yescharacteristic
103C2×Q8×C283—21no—
104C2×Q8×C283—21no—
105C2×Q8×C283—22no—
106C2×Q8×C283—22no—
107C2×Q8×C283—41no—
108C2×Q8×C283—41no—
109C2×Q8×C283C2311yescharacteristic
110C2×Q8×C283C2311yescharacteristic
111C2×Q8×C283C2311yescharacteristic
112C2×Q8×C283C2311yescharacteristic
113C22×C4×C283—21no—
114C22×C4×C283—21no—
115C22×C4×C283C2311yescharacteristic
116Q16×C283—22no—
117Q16×C283—22no—
118Q16×C283—22no—
119Q16×C283D414yescomplemented
120C2×C8×C283—22no—
121C2×C8×C283C2311yescharacteristic
122C2×C8×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283C2×D411yescharacteristic
125Dic566—21no—
126Dic566—21no—
127Dic566—42no—
128D1132—21no—
129D1132—21no—
130D1132—42no—
131C22⋊D283—22no—
132C22⋊D283—42no—
133C2×Dic283—21no—
134C2×Dic283—21no—
135C2×Dic283—21no—
136C2×Dic283C2×D411yescharacteristic
137C2×Dic283C2×D411yescharacteristic
138C2×Dic283C2×D411yescharacteristic
139C4×D283—21no—
140C4×D283—21no—
141C4×D283—42no—
142C4×D283—42no—
143C283⋊C8—21no—
144C283⋊C8—21no—
145C23×C283C2×D411yescharacteristic
146Q8×C283—21no—
147Q8×C283—21no—
148Q8×C283—21no—
149Q8×C283—21no—
150Q8×C283—21no—
151Q8×C283—21no—
152Q8×C283—42no—
153Q8×C283—42no—
154Q8×C283C2×D412yes—
155Q8×C283C2×D412yes—
156C2×C4×C283—21no—
157C2×C4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283—21no—
161C2×C4×C283—21no—
162C2×C4×C283—42no—
163C2×C4×C283—42no—
164C2×C4×C283C2×D411yescharacteristic
165C2×C4×C283C2411yescharacteristic
166C8×C283—21no—
167C8×C283C2×D412yes—
168C22×C283—21no—
169C22×C283—21no—
170C22×C283#13712yescomplemented
171C22×C283C22×D411yescharacteristic
172C2×D283—21no—
173C2×D283—21no—
174C2×D283—21no—
175C2×D283—41no—
176C2×D283—41no—
177C2×D283—41no—
178C4×C283—21no—
179C4×C283—22no—
180C4×C283—22no—
181C4×C283—41no—
182C4×C283—41no—
183C4×C283C22×D411yescharacteristic
184C4×C283H(5,𝔽2)11yescharacteristic
185Dic283—21no—
186Dic283—21no—
187Dic283—21no—
188Dic283—41no—
189Dic283—41no—
190Dic283—41no—
191C2×C283—22no—
192C2×C283D4211yescharacteristic
193C2×C283C2×#13711yescharacteristic
194C2×C283#57811yescharacteristic
195D283—41no—
196D283—81no—
197C283#327011yescharacteristic, complemented, Sylow
198#3270—2831nomaximal, Sylow
199C2×#137—2831no—
200C22×Q16D28311yescharacteristic, complemented
201D4×Q8—2831no—
202#548—2831no—
203#497—2831no—
204#470—2831no—
205#465—2831no—
206#464—2831no—
207#462—2831no—
208#452—2831no—
209#451—2831no—
210#451—2831no—
211#448—2831no—
212#441—2831no—
213#435—2831no—
214C2×#41—2831no—
215C22×Q8—2831no—
216C22×Q8C2×D28311yescharacteristic
217C22×Q8C2×D28311yescharacteristic
218#137—5664no—
219C2×Q16—24no—
220C2×Q16—2831no—
221C2×Q16—2831no—
222C2×Q16C2×D28311yescharacteristic
223C2×Q16C2×D28311yescharacteristic
224C2×Q16C2×D28311yescharacteristic
225C2×Q16C2×D28311yescharacteristic
226C2×SD16—2831no—
227C2×SD16—2831no—
228C2×SD16—2831no—
229C2×M16—2831no—
230C22×C8C2×D28311yescharacteristic
231C4⋊Q8—2831no—
232C4⋊Q8—5661no—
233#124—2831no—
234#124—5661no—
235C22⋊Q8—2831no—
236C22⋊Q8—2831no—
237C22⋊Q8—2831no—
238C22⋊Q8—5661no—
239C22⋊Q8—5661no—
240C22⋊Q8—5661no—
241#121—2831no—
242#121—5661no—
243C4×Q8—2831no—
244C4×Q8—2831no—
245C4×D4—2831no—
246C4×D4—5661no—
247C4×D4—5661no—
248#106—2831no—
249C4⋊C8—2831no—
250#103—2831no—
251#103—2831no—
252#103—2831no—
253#103—2831no—
254#103—2831no—
255#102—2831no—
256C22⋊C8—2831no—
257C22⋊C8—2831no—
258#97—2831no—
259#41—5662no—
260#41—11321no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—2831no—
268C2×Q8—5661no—
269C2×Q8—5662no—
270C2×Q8—11321no—
271C2×Q8C22×D28311yescharacteristic
272C2×Q8C22×D28311yescharacteristic
273C2×Q8C22×D28311yescharacteristic
274C2×Q8C22×D28311yescharacteristic
275C2×D4—2831no—
276C2×D4—2831no—
277C2×D4—5661no—
278C2×D4—5661no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2831no—
282C22×C4—5661no—
283C22×C4—5661no—
284C22×C4C22×D28311yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—5662no—
289Q16—11321no—
290Q16C22⋊D28314yescomplemented
291SD16—5662no—
292SD16—5662no—
293SD16—11321no—
294M16—5662no—
295C2×C8—22no—
296C2×C8—2831no—
297C2×C8—2831no—
298C2×C8C22×D28311yescharacteristic
299C2×C8C22×D28311yescharacteristic
300C4⋊C4—2831no—
301C4⋊C4—2831no—
302C4⋊C4—2831no—
303C4⋊C4—5661no—
304C4⋊C4—5661no—
305C4⋊C4—5661no—
306C4⋊C4—5661no—
307C4⋊C4—5661no—
308C4⋊C4—5661no—
309C4⋊C4—5661no—
310C22⋊C4—2831no—
311C22⋊C4—2831no—
312C22⋊C4—5661no—
313C22⋊C4—5661no—
314C22⋊C4—5661no—
315C22⋊C4—5661no—
316C22⋊C4—5661no—
317C22⋊C4—5661no—
318C42—2831no—
319C42—5661no—
320C42—5661no—
321C23—2831no—
322C23—5661no—
323C23D4×D28311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—5661no—
333Q8—5661no—
334Q8—11322no—
335Q8C2×C22⋊D28312yes—
336Q8C2×C22⋊D28312yes—
337D4—5662no—
338D4—5661no—
339D4—5661no—
340D4—11322no—
341D4—11322no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—2831no—
351C2×C4—2831no—
352C2×C4—2831no—
353C2×C4—5661no—
354C2×C4—5661no—
355C2×C4—5661no—
356C2×C4—5661no—
357C2×C4—5661no—
358C2×C4—11322no—
359C2×C4—11322no—
360C2×C4D4×D28311yescharacteristic
361C2×C4C23×D28311yescharacteristic
362C8—21no—
363C8—5661no—
364C8—5661no—
365C8C2×C22⋊D28312yes—
366C22—21no—
367C22—21no—
368C22—5661no—
369C22—5661no—
370C22—5661no—
371C22—11321no—
372C22—11321no—
373C22—11321no—
374C22#2258364212yescomplemented
375C22C2×D4×D28311yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—5661no—
382C4—5661no—
383C4—5661no—
384C4—11321no—
385C4—11321no—
386C4—11321no—
387C4C22×C22⋊D28311yescharacteristic
388C4#2258365511yescharacteristic
389C2—22no—
390C2—11321no—
391C2—22641no—
392C2D4×C22⋊D28311yescharacteristic
393C2C2×#2258364211yescharacteristic
394C2#3182727811yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC22×Q16×C283C2
socleC22×C283C22×D4
prime radical#31827126C2
hypercenterC22×Q16D283
hypocenterC283#3270
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#22583642C2semidirect
2. Q8×C22⋊D283C2semidirect
4. #31826858C2semidirect
5. #31826720C2semidirect
6. #31826694C2semidirect
7. #31826689C2semidirect
8. #31826673C2semidirect
9. #31826626C2semidirect
10. #31826621C2semidirect
11. #31826619C2semidirect
12. #31826604C2semidirect
13. #31826572C2semidirect
14. #31826536C2semidirect
15. C22×Q16×C283C2semidirect
119. Q16×C283D4semidirect
170. C22×C283#137semidirect
197. C283#3270semidirect
200. C22×Q16D283semidirect
290. Q16C22⋊D283semidirect
374. C22#22583642semidirect

Maximal subgroups

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

subgroupindex
C2×#225836422
Q8×C22⋊D2832
#318271262
#318268582
#318267202
#318266942
#318266892
#318266732
#318266262
#318266212
#318266192
#318266042
#318265722
#318265362
C22×Q16×C2832
#3270283