Finite Group Databasebeta

← #49132901

Subgroups — #49132901

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×#22583639C211yesmaximal, characteristic, complemented
2Q8×C22⋊D283C211yesmaximal, characteristic, complemented
3#31827114C211yesmaximal, characteristic
4#31826843C211yesmaximal, characteristic, complemented
5#31826723C211yesmaximal, characteristic, complemented
6#31826720C211yesmaximal, characteristic, complemented
7#31826673C211yesmaximal, characteristic, complemented
8#31826668C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826624C211yesmaximal, characteristic, complemented
11#31826621C211yesmaximal, characteristic, complemented
12#31826619C211yesmaximal, characteristic, complemented
13SD16×Dic283C211yesmaximal, characteristic, complemented
14#31826541C211yesmaximal, characteristic, complemented
15C2×#137×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18#22583639—24no—
19C2×Dic1132C2211yescharacteristic
20C2×#17985382C2211yescharacteristic
21C2×#17985382C2211yescharacteristic
22C2×#17985382C2211yescharacteristic
23C2×#17985380C2211yescharacteristic
24C2×#17985375C2211yescharacteristic
25#22583608—21no—
26#22583608C2211yescharacteristic
27#22583608C2211yescharacteristic
28#22583593—21no—
29#22583591—21no—
30#22583589—21no—
31#22583589C2211yescharacteristic
32#22583589C2211yescharacteristic
33#22583586—21no—
34#22583586C2211yescharacteristic
35#22583586C2211yescharacteristic
36#22583582—21no—
37Q8×Dic283C2211yescharacteristic
38D4×Dic283C2211yescharacteristic
39C4×C22⋊D283—21no—
40C4×C22⋊D283C2211yescharacteristic
41C2×Dic283⋊C4—21no—
42#22583542C2211yescharacteristic
43Dic283⋊C8C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583535C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583534C2211yescharacteristic
48#22583534C2211yescharacteristic
49#22583531C2211yescharacteristic
50C22⋊(C283⋊C8)C2211yescharacteristic
51(C2×D283)⋊C8C2211yescharacteristic
52C8×Dic283C2211yescharacteristic
53C2×#41×C283C2211yescharacteristic
54C22×Q8×C283C2211yescharacteristic
55#137×C283—24no—
56C2×Q16×C283C2211yescharacteristic
57C2×Q16×C283C2211yescharacteristic
58C2×SD16×C283C2211yescharacteristic
59C2×SD16×C283C2211yescharacteristic
60C2×M16×C283C2211yescharacteristic
61#17985391—22no—
62#17985391—41no—
63C2×Dic566—21no—
64C2×Dic566C2311yescharacteristic
65C2×Dic566C2311yescharacteristic
66Q8×D283—22no—
67Q8×D283—41no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283D411yescharacteristic
70C22×Dic283—21no—
71C22×Dic283—21no—
72C2×C4×D283—21no—
73C2×C4×D283C2311yescharacteristic
74Dic1132—22no—
75#17985382—22no—
76#17985382—41no—
77#17985382—41no—
78#17985380—22no—
79#17985375—22no—
80C2×C283⋊C8C2311yescharacteristic
81C2×C283⋊C8C2311yescharacteristic
82C4⋊Dic283—21no—
83C4⋊Dic283C2311yescharacteristic
84C4⋊Dic283C2311yescharacteristic
85Dic283⋊C4—21no—
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4—21no—
89Dic283⋊C4—21no—
90Dic283⋊C4—21no—
91Dic283⋊C4—21no—
92Dic283⋊C4—21no—
93Dic283⋊C4D411yescharacteristic
94C22⋊Dic283—21no—
95C22⋊Dic283—21no—
96C22⋊Dic283—21no—
97C22⋊Dic283D411yescharacteristic
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×D283C2×D411yescharacteristic
124Dic566—21no—
125Dic566—21no—
126Dic566—21no—
127Dic566—21no—
128Dic566—42no—
129C22⋊D283—22no—
130C22⋊D283—42no—
131C2×Dic283—21no—
132C2×Dic283—21no—
133C2×Dic283—21no—
134C2×Dic283—21no—
135C2×Dic283—42no—
136C2×Dic283—42no—
137C2×Dic283C2×D411yescharacteristic
138C2×Dic283C2×D411yescharacteristic
139C2×Dic283C2×D411yescharacteristic
140C4×D283—21no—
141C4×D283—21no—
142C4×D283—42no—
143C283⋊C8—21no—
144C283⋊C8—21no—
145C23×C283—21no—
146C23×C283C2×D411yescharacteristic
147Q8×C283—21no—
148Q8×C283—21no—
149Q8×C283—21no—
150Q8×C283—21no—
151Q8×C283—21no—
152Q8×C283—42no—
153Q8×C283C2×D412yes—
154D4×C283—21no—
155D4×C283—42no—
156D4×C283C2×D412yes—
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—22no—
170C22×C283—21no—
171C22×C283—21no—
172C22×C283—41no—
173C22×C283—41no—
174C22×C283C22×D411yescharacteristic
175C2×D283—21no—
176C2×D283—21no—
177C2×D283—21no—
178C4×C283—21no—
179C4×C283—22no—
180C4×C283—41no—
181C4×C283—41no—
182C4×C283C22×D411yescharacteristic
183C4×C283H(5,𝔽2)11yescharacteristic
184Dic283—21no—
185Dic283—21no—
186Dic283—41no—
187Dic283—41no—
188Dic283—41no—
189Dic283—41no—
190Dic283#13712yescomplemented
191C2×C283—41no—
192C2×C283—42no—
193C2×C283D4211yescharacteristic
194C2×C283C2×#13711yescharacteristic
195C2×C283#57811yescharacteristic
196D283—41no—
197C283#327611yescharacteristic, complemented, Sylow
198#3276—2831nomaximal, Sylow
199C2×#137—2831no—
200C2×#137D28311yescharacteristic, complemented
201D4×Q8—2831no—
202#547—2831no—
203#494—2831no—
204#470—2831no—
205#470—2831no—
206#462—2831no—
207#461—2831no—
208#452—2831no—
209#452—2831no—
210#451—2831no—
211#451—2831no—
212C4×SD16—2831no—
213#436—2831no—
214C2×#41—2831no—
215C2×#41C2×D28311yescharacteristic
216C22×Q8—2831no—
217C22×Q8C2×D28311yescharacteristic
218#137—24no—
219#137—5664no—
220C2×Q16—2831no—
221C2×Q16—2831no—
222C2×Q16—2831no—
223C2×Q16—2831no—
224C2×Q16C2×D28311yescharacteristic
225C2×Q16C2×D28311yescharacteristic
226C2×SD16—2831no—
227C2×SD16C2×D28311yescharacteristic
228C2×SD16C2×D28311yescharacteristic
229C2×M16—2831no—
230C2×M16C2×D28311yescharacteristic
231C4⋊Q8—2831no—
232C4⋊Q8—2831no—
233C4⋊Q8—5661no—
234#123—5661no—
235#123—5661no—
236C22⋊Q8—2831no—
237C22⋊Q8—2831no—
238C22⋊Q8—2831no—
239C22⋊Q8—2831no—
240C22⋊Q8—5661no—
241C22⋊Q8—5661no—
242#121—5661no—
243C4×Q8—2831no—
244C4×D4—2831no—
245C4×D4—2831no—
246C4×D4—5661no—
247C2×C4⋊C4—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—
258C4×C8—2831no—
259#41—22no—
260#41—41no—
261#41—5662no—
262#41—11321no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—2831no—
267C2×Q8—2831no—
268C2×Q8—5661no—
269C2×Q8—5662no—
270C2×Q8—11321no—
271C2×Q8C22×D28311yescharacteristic
272C2×Q8C22×D28311yescharacteristic
273C2×Q8C22×D28311yescharacteristic
274C2×D4—21no—
275C2×D4—2831no—
276C2×D4—5661no—
277C2×D4C22×D28311yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—2831no—
281C22×C4—5661no—
282C22×C4—5661no—
283C22×C4—5661no—
284C22×C4C22×D28311yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—5662no—
288Q16—5662no—
289Q16—11321no—
290Q16—11321no—
291SD16—22no—
292SD16—5662no—
293SD16C22⋊D28314yescomplemented
294M16—22no—
295M16—5662no—
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—
310C4⋊C4—5661no—
311C4⋊C4—5661no—
312C22⋊C4—2831no—
313C22⋊C4—2831no—
314C22⋊C4—5661no—
315C22⋊C4—5661no—
316C22⋊C4—5661no—
317C22⋊C4—5661no—
318C22⋊C4—5661no—
319C42—2831no—
320C42—5661no—
321C23—21no—
322C23—2831no—
323C23D4×D28311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—5661no—
331Q8—5661no—
332Q8—5661no—
333Q8—5661no—
334Q8—11322no—
335Q8C2×C22⋊D28312yes—
336D4—21no—
337D4—42no—
338D4—5662no—
339D4—11322no—
340D4C2×C22⋊D28312yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—2831no—
349C2×C4—2831no—
350C2×C4—2831no—
351C2×C4—5661no—
352C2×C4—5661no—
353C2×C4—5661no—
354C2×C4—5661no—
355C2×C4—5661no—
356C2×C4—5661no—
357C2×C4—11322no—
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—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—5661no—
373C22—5661no—
374C22—5661no—
375C22C2×D4×D28311yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—2832no—
381C4—5661no—
382C4—5661no—
383C4—11321no—
384C4—11321no—
385C4—11321no—
386C4—11321no—
387C4C22×C22⋊D28311yescharacteristic
388C4#2258365511yescharacteristic
389C2—41no—
390C2—42no—
391C2—11321no—
392C2D4×C22⋊D28311yescharacteristic
393C2#137×D28311yescharacteristic
394C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×#137×C283C2
socleC22×C283C22×D4
prime radical#31827114C2
hypercenterC2×#137D283
hypocenterC283#3276
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×#22583639C2semidirect
2. Q8×C22⋊D283C2semidirect
4. #31826843C2semidirect
5. #31826723C2semidirect
6. #31826720C2semidirect
7. #31826673C2semidirect
8. #31826668C2semidirect
9. #31826627C2semidirect
10. #31826624C2semidirect
11. #31826621C2semidirect
12. #31826619C2semidirect
13. SD16×Dic283C2semidirect
14. #31826541C2semidirect
15. C2×#137×C283C2semidirect
119. SD16×C283D4semidirect
190. Dic283#137semidirect
197. C283#3276semidirect
200. C2×#137D283semidirect
293. SD16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836392
Q8×C22⋊D2832
#318271142
#318268432
#318267232
#318267202
#318266732
#318266682
#318266272
#318266242
#318266212
#318266192
SD16×Dic2832
#318265412
C2×#137×C2832
#3276283