Finite Group Databasebeta

← #49132832

Subgroups — #49132832

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×#22583641C211yesmaximal, characteristic, complemented
2C22×#17985382C211yesmaximal, characteristic, complemented
3#31827124C211yesmaximal, characteristic
4#31826857C211yesmaximal, characteristic, complemented
5#31826721C211yesmaximal, characteristic, complemented
6#31826696C211yesmaximal, characteristic, complemented
7#31826687C211yesmaximal, characteristic, complemented
8#31826672C211yesmaximal, characteristic, complemented
9#31826625C211yesmaximal, characteristic, complemented
10#31826621C211yesmaximal, characteristic, complemented
11#31826620C211yesmaximal, characteristic, complemented
12#31826604C211yesmaximal, characteristic, complemented
13#31826570C211yesmaximal, characteristic, complemented
14#31826537C211yesmaximal, characteristic, complemented
15D4×Q8×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×Dic566C2211yescharacteristic
18#22583641—24no—
19C2×#17985382—24no—
20C2×#17985382C2211yescharacteristic
21C2×#17985382C2211yescharacteristic
22C2×#17985382C2211yescharacteristic
23C2×#17985382C2211yescharacteristic
24C2×#17985382C2211yescharacteristic
25C2×#17985382C2211yescharacteristic
26C2×#17985379C2211yescharacteristic
27C2×#17985379C2211yescharacteristic
28C2×#17985379C2211yescharacteristic
29C2×#17985376C2211yescharacteristic
30C22×C283⋊C8C2211yescharacteristic
31#22583598C2211yescharacteristic
32#22583596—21no—
33#22583589C2211yescharacteristic
34#22583588—21no—
35#22583584C2211yescharacteristic
36#22583582—21no—
37C4×Dic566C2211yescharacteristic
38(C22⋊D283)⋊C4—21no—
39#22583541C2211yescharacteristic
40C4⋊(C283⋊C8)C2211yescharacteristic
41#22583534C2211yescharacteristic
42#22583534C2211yescharacteristic
43#22583534C2211yescharacteristic
44#22583533C2211yescharacteristic
45#22583533C2211yescharacteristic
46#22583530C2211yescharacteristic
47C22⋊(C283⋊C8)C2211yescharacteristic
48C22⋊(C283⋊C8)C2211yescharacteristic
49#22583521C2211yescharacteristic
50C22×Q8×C283C2211yescharacteristic
51C22×Q8×C283C2211yescharacteristic
52C4⋊Q8×C283—21no—
53C4⋊Q8×C283C2211yescharacteristic
54C22⋊Q8×C283—21no—
55C22⋊Q8×C283—21no—
56C22⋊Q8×C283C2211yescharacteristic
57C22⋊Q8×C283C2211yescharacteristic
58C4×Q8×C283C2211yescharacteristic
59C4×D4×C283—21no—
60C4×D4×C283C2211yescharacteristic
61#17985392—22no—
62#17985392—41no—
63C2×Dic566—22no—
64C2×Dic566—21no—
65C2×Dic566—41no—
66C2×Dic566C2311yescharacteristic
67C2×Dic566C2311yescharacteristic
68C2×D1132C2311yescharacteristic
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C22×Dic283—21no—
72C2×C4×D283—21no—
73#17985382—22no—
74#17985382—22no—
75#17985382—22no—
76#17985382—22no—
77#17985382—41no—
78#17985382D414yescomplemented
79#17985379—22no—
80#17985379—22no—
81#17985379—41no—
82#17985376—22no—
83C2×C283⋊C8—22no—
84C2×C283⋊C8C2311yescharacteristic
85C2×C283⋊C8C2311yescharacteristic
86C2×C283⋊C8C2311yescharacteristic
87C2×C283⋊C8C2311yescharacteristic
88C4⋊Dic283C2311yescharacteristic
89Dic283⋊C4—21no—
90Dic283⋊C4—21no—
91C22⋊Dic283—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95C4×Dic283—21no—
96C2×Q8×C283—21no—
97C2×Q8×C283—21no—
98C2×Q8×C283—22no—
99C2×Q8×C283—22no—
100C2×Q8×C283—41no—
101C2×Q8×C283—41no—
102C2×Q8×C283C2311yescharacteristic
103C2×Q8×C283C2311yescharacteristic
104C2×Q8×C283C2311yescharacteristic
105C2×D4×C283D411yescharacteristic
106C22×C4×C283—21no—
107C22×C4×C283—21no—
108C22×C4×C283C2311yescharacteristic
109C22×C4×C283C2311yescharacteristic
110C4⋊C4×C283—21no—
111C4⋊C4×C283—21no—
112C4⋊C4×C283—21no—
113C4⋊C4×C283—21no—
114C4⋊C4×C283—21no—
115C4⋊C4×C283D411yescharacteristic
116C4⋊C4×C283C2311yescharacteristic
117C22⋊C4×C283—21no—
118C22⋊C4×C283—21no—
119C22⋊C4×C283D411yescharacteristic
120C22⋊C4×C283D411yescharacteristic
121C42×C283—21no—
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124Dic566—21no—
125Dic566—21no—
126Dic566—21no—
127Dic566—42no—
128Dic566C2×D412yes—
129D1132—21no—
130D1132—21no—
131C22⋊D283—42no—
132C22⋊D283—42no—
133C2×Dic283—21no—
134C2×Dic283—21no—
135C2×Dic283—21no—
136C2×Dic283—42no—
137C4×D283—42no—
138C283⋊C8—21no—
139C283⋊C8—21no—
140C283⋊C8—21no—
141C283⋊C8C2×D412yes—
142C23×C283C2×D411yescharacteristic
143C23×C283C2×D411yescharacteristic
144Q8×C283—21no—
145Q8×C283—21no—
146Q8×C283—21no—
147Q8×C283—21no—
148Q8×C283—21no—
149Q8×C283—42no—
150Q8×C283—42no—
151Q8×C283C2×D412yes—
152D4×C283—22no—
153C2×C4×C283—21no—
154C2×C4×C283—21no—
155C2×C4×C283—21no—
156C2×C4×C283—21no—
157C2×C4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283—21no—
161C2×C4×C283—42no—
162C2×C4×C283—42no—
163C2×C4×C283C2×D411yescharacteristic
164C2×C4×C283C2×D411yescharacteristic
165C2×C4×C283C2×D411yescharacteristic
166C2×C4×C283C2×D411yescharacteristic
167C2×C4×C283C2411yescharacteristic
168C22×C283—21no—
169C22×C283—21no—
170C22×C283—21no—
171C22×C283—21no—
172C22×C283—21no—
173C22×C283#13712yescomplemented
174C22×C283C22×D411yescharacteristic
175C2×D283—41no—
176C2×D283—41no—
177C2×D283—41no—
178C4×C283—21no—
179C4×C283—21no—
180C4×C283—22no—
181C4×C283—21no—
182C4×C283—21no—
183C4×C283—41no—
184C4×C283—41no—
185C4×C283—41no—
186C4×C283C22×D411yescharacteristic
187C4×C283H(5,𝔽2)11yescharacteristic
188Dic283—22no—
189Dic283—41no—
190Dic283—41no—
191C2×C283—22no—
192C2×C283—41no—
193C2×C283D4211yescharacteristic
194C2×C283C2×#13711yescharacteristic
195C2×C283#57811yescharacteristic
196D283—81no—
197C283#327011yescharacteristic, complemented, Sylow
198#3270—2831nomaximal, Sylow
199C2×#137—2831no—
200C22×Q16—2831no—
201D4×Q8D28311yescharacteristic, complemented
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—2831no—
220C2×Q16—2831no—
221C2×Q16—2831no—
222C2×Q16—2831no—
223C2×Q16—2831no—
224C2×Q16—2831no—
225C2×Q16—5664no—
226C2×SD16—2831no—
227C2×SD16—2831no—
228C2×SD16—2831no—
229C2×M16—2831no—
230C22×C8—2831no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D28311yescharacteristic
233#124—2831no—
234#124—5661no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—2831no—
238C22⋊Q8—5661no—
239C22⋊Q8C2×D28311yescharacteristic
240C22⋊Q8C2×D28311yescharacteristic
241#121—2831no—
242#121—5661no—
243C4×Q8—2831no—
244C4×Q8C2×D28311yescharacteristic
245C4×D4—21no—
246C4×D4—5661no—
247C4×D4C2×D28311yescharacteristic
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—2831no—
269C2×Q8—5662no—
270C2×Q8—5661no—
271C2×Q8—11321no—
272C2×Q8C22×D28311yescharacteristic
273C2×Q8C22×D28311yescharacteristic
274C2×Q8C22×D28311yescharacteristic
275C2×D4—2831no—
276C2×D4—5661no—
277C2×D4—5661no—
278C2×D4C22⋊D28311yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—5661no—
282C22×C4—5661no—
283C22×C4C22×D28311yescharacteristic
284C22×C4C22×D28311yescharacteristic
285Q16—2834no—
286Q16—5662no—
287Q16—5662no—
288Q16—5662no—
289Q16—5662no—
290Q16—11321no—
291SD16—5662no—
292SD16—5662no—
293SD16—11321no—
294M16—5662no—
295C2×C8—2831no—
296C2×C8—2831no—
297C2×C8—2831no—
298C2×C8—2831no—
299C2×C8—5662no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—2831no—
306C4⋊C4—5661no—
307C4⋊C4—5661no—
308C4⋊C4C22⋊D28311yescharacteristic
309C4⋊C4C22×D28311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—5661no—
313C22⋊C4—5661no—
314C22⋊C4—5661no—
315C22⋊C4—5661no—
316C22⋊C4C22⋊D28311yescharacteristic
317C22⋊C4C22⋊D28311yescharacteristic
318C42—21no—
319C42—5661no—
320C42C22×D28311yescharacteristic
321C23—5661no—
322C23C2×C22⋊D28311yescharacteristic
323C23C2×C22⋊D28311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—2832no—
332Q8—5661no—
333Q8—5661no—
334Q8—5661no—
335Q8—11322no—
336Q8D4×D28312yes—
337D4—22no—
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—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—5661no—
353C2×C4—5661no—
354C2×C4—5661no—
355C2×C4—11322no—
356C2×C4—11322no—
357C2×C4C2×C22⋊D28311yescharacteristic
358C2×C4C2×C22⋊D28311yescharacteristic
359C2×C4C2×C22⋊D28311yescharacteristic
360C2×C4C2×C22⋊D28311yescharacteristic
361C2×C4C23×D28311yescharacteristic
362C8—2832no—
363C8—5661no—
364C8—5661no—
365C8—5661no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—11321no—
372C22—11321no—
373C22—11321no—
374C22#2258364112yescomplemented
375C22C22×C22⋊D28311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—5662no—
385C4—11321no—
386C4—11321no—
387C4C2×D4×D28311yescharacteristic
388C4#2258365511yescharacteristic
389C2—22no—
390C2—41no—
391C2—22641no—
392C2D4×C22⋊D28311yescharacteristic
393C2C2×#2258364111yescharacteristic
394C2#3182728011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingD4×Q8×C283C2
socleC22×C283C22×D4
prime radical#31827124C2
hypercenterD4×Q8D283
hypocenterC283#3270
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22583641C2semidirect
2. C22×#17985382C2semidirect
4. #31826857C2semidirect
5. #31826721C2semidirect
6. #31826696C2semidirect
7. #31826687C2semidirect
8. #31826672C2semidirect
9. #31826625C2semidirect
10. #31826621C2semidirect
11. #31826620C2semidirect
12. #31826604C2semidirect
13. #31826570C2semidirect
14. #31826537C2semidirect
15. D4×Q8×C283C2semidirect
78. #17985382D4semidirect
173. C22×C283#137semidirect
197. C283#3270semidirect
201. D4×Q8D283semidirect
374. C22#22583641semidirect

Maximal subgroups

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

subgroupindex
C2×#225836412
C22×#179853822
#318271242
#318268572
#318267212
#318266962
#318266872
#318266722
#318266252
#318266212
#318266202
#318266042
#318265702
#318265372
D4×Q8×C2832
#3270283