Finite Group Databasebeta

← #49130522

Subgroups — #49130522

Order 36224.

754 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×#41×D283C211yesmaximal, characteristic
2C2×#22583643C211yesmaximal, characteristic, complemented
3C2×#22583640C211yesmaximal, characteristic, complemented
4C2×#22583590C211yesmaximal, characteristic, complemented
5#31826630C212yesmaximal, complemented
6#31826628C212yesmaximal, complemented
7#31826622C212yesmaximal, complemented
8#31826617C212yesmaximal, complemented
9#31826464C211yesmaximal, characteristic, complemented
10#31826428C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2211yescharacteristic
13C2×#17985392—21no—
14C2×#17985392C2211yescharacteristic
15C2×#17985391—21no—
16C2×#17985391C2211yescharacteristic
17#41×D283—24no—
18#41×D283—24no—
19C22×Dic566C2211yescharacteristic, complemented
20C2×Q8×D283C2211yescharacteristic
21C2×D4×D283—21no—
22C2×D4×D283C2211yescharacteristic
23C22×C4×D283—21no—
24C22×C4×D283C2211yescharacteristic
25#22583643—24no—
26#22583640—24no—
27C2×Dic1132C2212yescomplemented
28C2×#17985382C2212yescomplemented
29C2×#17985381C2212yescomplemented
30C2×#17985380C2212yescomplemented
31C2×#17985376C2211yescharacteristic, complemented
32#22583590—21no—
33#22583590—21no—
34#22583590C2212yes—
35#22583590C2212yescomplemented
36C2×C4⋊Dic283—21no—
37C2×(C2×D283)⋊C4—21no—
38#22583533C2212yescomplemented
39#22583530C2212yescomplemented
40(C2×D283)⋊C8C2212yes—
41(C2×D283)⋊C8C2212yescomplemented
42C2×#41×C283C2211yescharacteristic
43C2×M16×C283C2211yescharacteristic, complemented
44C2×C4⋊C4×C283C2211yescharacteristic, complemented
45#103×C283C2212yescomplemented
46#102×C283C2212yescomplemented
47C23×D283—21no—
48C23×D283D411yescharacteristic
49#17985393—22no—
50#17985393—22no—
51#17985392—28no—
52#17985392—22no—
53#17985392D414yes—
54#17985391—28no—
55#17985391—22no—
56#17985391—22no—
57C2×Dic566—21no—
58C2×Dic566—21no—
59C2×Dic566—22no—
60C2×Dic566—22no—
61C2×Dic566C2312yes—
62C2×Dic566C2311yescharacteristic
63Q8×D283—22no—
64Q8×D283—22no—
65C2×D1132—21no—
66C2×D1132C2311yescharacteristic
67C2×C22⋊D283—21no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70D4×D283—28no—
71D4×D283—22no—
72D4×D283—22no—
73C22×Dic283—21no—
74C22×Dic283—21no—
75C22×Dic283D411yescharacteristic
76C2×C4×D283—28no—
77C2×C4×D283—21no—
78C2×C4×D283—21no—
79C2×C4×D283—21no—
80C2×C4×D283—22no—
81C2×C4×D283—22no—
82C2×C4×D283D411yescharacteristic
83C2×C4×D283D411yescharacteristic
84C2×C4×D283C2311yescharacteristic
85C2×C4×D283C2311yescharacteristic
86Dic1132—24no—
87#17985382—24no—
88#17985381—24no—
89#17985380—24no—
90#17985376—22no—
91C2×C283⋊C8C2312yes—
92C4⋊Dic283—22no—
93C4⋊Dic283—22no—
94(C2×D283)⋊C4—22no—
95(C2×D283)⋊C4—22no—
96#41×C283—22no—
97#41×C283D414yes—
98C2×Q8×C283C2311yescharacteristic
99C2×D4×C283—21no—
100C2×D4×C283C2311yescharacteristic
101C22×C4×C283—21no—
102C22×C4×C283—21no—
103C22×C4×C283C2311yescharacteristic
104M16×C283—22no—
105C2×C8×C283C2312yes—
106C4⋊C4×C283—21no—
107C4⋊C4×C283C2312yes—
108C22×D283—24no—
109C22×D283—22no—
110C22×D283—21no—
111C22×D283—22no—
112C22×D283—41no—
113C22×D283—42no—
114C22×D283—42no—
115C22×D283C2×D411yescharacteristic
116C22×D283C2×D411yescharacteristic
117Dic566—24no—
118Dic566—21no—
119Dic566—22no—
120Dic566—22no—
121Dic566—42no—
122Dic566C2×D412yes—
123D1132—24no—
124D1132—21no—
125D1132C2×D412yes—
126C22⋊D283—24no—
127C22⋊D283—24no—
128C22⋊D283—24no—
129C2×Dic283—24no—
130C2×Dic283—22no—
131C2×Dic283—21no—
132C2×Dic283—22no—
133C2×Dic283—42no—
134C2×Dic283C2×D411yescharacteristic
135C2×Dic283C2×D411yescharacteristic
136C4×D283—24no—
137C4×D283—24no—
138C4×D283—22no—
139C4×D283—24no—
140C4×D283—22no—
141C4×D283—21no—
142C4×D283—21no—
143C4×D283—21no—
144C4×D283—21no—
145C283⋊C8—22no—
146C23×C283—21no—
147C23×C283C2×D411yescharacteristic
148Q8×C283—21no—
149Q8×C283C2×D412yes—
150D4×C283—24no—
151D4×C283—21no—
152D4×C283C2×D412yes—
153C2×C4×C283—21no—
154C2×C4×C283—24no—
155C2×C4×C283—22no—
156C2×C4×C283—42no—
157C2×C4×C283C2×D411yescharacteristic
158C2×C4×C283C2×D412yes—
159C2×C4×C283C2×D412yes—
160C2×C4×C283C2411yescharacteristic
161C8×C283—22no—
162C22×C283—21no—
163C22×C283—21no—
164C22×C283—22no—
165C22×C283—41no—
166C22×C283—41no—
167C22×C283#12012yes—
168C22×C283C22×D411yescharacteristic
169C2×D283—21no—
170C2×D283—21no—
171C2×D283—21no—
172C2×D283—22no—
173C2×D283—21no—
174C2×D283—21no—
175C2×D283—44no—
176C2×D283—48no—
177C2×D283—41no—
178C2×D283—41no—
179C2×D283#13712yescomplemented
180C4×C283—22no—
181C4×C283—42no—
182C4×C283#12012yes—
183C4×C283C22×D411yescharacteristic
184C4×C283C22×D411yescharacteristic
185Dic283—21no—
186Dic283—22no—
187Dic283—21no—
188Dic283—42no—
189C2×C283—22no—
190C2×C283—42no—
191C2×C283C2×#12011yescharacteristic
192C2×C283C2×#13711yescharacteristic
193C2×C283#57511yescharacteristic
194D283—22no—
195D283—41no—
196D283—42no—
197C283#300711yescharacteristic, complemented, Sylow
198#3007—2831nomaximal, Sylow
199C22×#41—2831no—
200C2×#137—2831no—
201C2×#137—2831no—
202C2×C22⋊Q8—2831no—
203#452—2832no—
204#452—2832no—
205#451—2832no—
206#450—2832no—
207#417—2831no—
208#417D28311yescharacteristic, complemented
209#407—2831no—
210C2×#41—2831no—
211C2×#41—2831no—
212C2×#41—2831no—
213C2×#41—5661no—
214C2×#41—5661no—
215C2×#41—5664no—
216C2×#41—5664no—
217C2×#41C2×D28311yescharacteristic
218C22×Q8—2831no—
219C22×Q8—2831no—
220C22×D4—2831no—
221C22×D4—5661no—
222C23×C4—2831no—
223C23×C4—5661no—
224#137—5664no—
225#137—5664no—
226C2×Q16—2832no—
227C2×Q16—2832no—
228C2×SD16—2832no—
229C2×SD16—2832no—
230C2×M16—2831no—
231C2×M16C2×D28311yescharacteristic, complemented
232C22⋊Q8—2832no—
233C22⋊Q8—2832no—
234C22⋊Q8—5661no—
235C22⋊Q8—5661no—
236C2×C4⋊C4—5661no—
237C2×C4⋊C4C2×D28311yescharacteristic, complemented
238C2×C22⋊C4—5661no—
239#103—2832no—
240#103C2×D28312yescomplemented
241#102—2832no—
242#102C2×D28312yescomplemented
243C22⋊C8—2832no—
244C22⋊C8—2832no—
245C24—2831no—
246C24—5661no—
247#41—22no—
248#41—2834no—
249#41—5668no—
250#41—5662no—
251#41—5668no—
252#41—5662no—
253#41—5662no—
254#41—5662no—
255#41—5662no—
256#41D113214yes—
257C2×Q8—2831no—
258C2×Q8—2832no—
259C2×Q8—5661no—
260C2×Q8—5662no—
261C2×Q8—5662no—
262C2×Q8—5661no—
263C2×Q8—5662no—
264C2×Q8—5662no—
265C2×Q8C22×D28311yescharacteristic
266C2×D4—21no—
267C2×D4—2831no—
268C2×D4—5661no—
269C2×D4—5661no—
270C2×D4—5662no—
271C2×D4—5668no—
272C2×D4—5661no—
273C2×D4—5661no—
274C2×D4—5662no—
275C2×D4C22×D28311yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—2831no—
279C22×C4—2831no—
280C22×C4—2831no—
281C22×C4—2831no—
282C22×C4—2831no—
283C22×C4—5661no—
284C22×C4—5661no—
285C22×C4—5668no—
286C22×C4—5661no—
287C22×C4—5662no—
288C22×C4—5661no—
289C22×C4—5662no—
290C22×C4—5661no—
291C22×C4C22×D28311yescharacteristic
292Q16—5664no—
293Q16—5664no—
294SD16—5664no—
295SD16—5664no—
296M16—22no—
297M16—5662no—
298C2×C8—2832no—
299C2×C8C22×D28312yes—
300C4⋊C4—21no—
301C4⋊C4—5662no—
302C4⋊C4—5662no—
303C4⋊C4C22×D28312yes—
304C22⋊C4—5662no—
305C22⋊C4—5662no—
306C23—21no—
307C23—2831no—
308C23—2831no—
309C23—5662no—
310C23—5664no—
311C23—5662no—
312C23—5661no—
313C23—11321no—
314C23—11322no—
315C23—11322no—
316C23D4×D28311yescharacteristic
317Q8—21no—
318Q8—2832no—
319Q8—5664no—
320Q8—5661no—
321Q8—5662no—
322Q8—5662no—
323Q8—11322no—
324Q8C2×D113212yes—
325D4—24no—
326D4—21no—
327D4—2832no—
328D4—5664no—
329D4—5664no—
330D4—5664no—
331D4—5664no—
332D4—5661no—
333D4C2×D113212yes—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—2831no—
339C2×C4—2831no—
340C2×C4—5664no—
341C2×C4—5664no—
342C2×C4—5664no—
343C2×C4—5662no—
344C2×C4—5661no—
345C2×C4—5662no—
346C2×C4—5662no—
347C2×C4—5664no—
348C2×C4—5661no—
349C2×C4—5661no—
350C2×C4—5661no—
351C2×C4—5661no—
352C2×C4—5662no—
353C2×C4—11322no—
354C2×C4D4×D28311yescharacteristic
355C2×C4D4×D28312yes—
356C2×C4C2×D113212yes—
357C2×C4C23×D28311yescharacteristic
358C8—22no—
359C8—5662no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—2832no—
366C22—5661no—
367C22—5661no—
368C22—5662no—
369C22—5661no—
370C22—5661no—
371C22—5661no—
372C22—11324no—
373C22—11328no—
374C22—11321no—
375C22—11321no—
376C22#2258358012yes—
377C22C2×D4×D28311yescharacteristic
378C4—22no—
379C4—42no—
380C4—5661no—
381C4—5661no—
382C4—5662no—
383C4—11322no—
384C4#2258358012yes—
385C4C2×D4×D28311yescharacteristic
386C4C22×D113211yescharacteristic
387C2—22no—
388C2—42no—
389C2—5662no—
390C2—11321no—
391C2—11322no—
392C2C2×#2258358011yescharacteristic
393C2#137×D28311yescharacteristic
394C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
prime radicalC2×#41×D283C2
hypercenter#417D283
hypocenterC283#3007
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#22583643C2semidirect
3. C2×#22583640C2semidirect
4. C2×#22583590C2semidirect
5. #31826630C2semidirect
6. #31826628C2semidirect
7. #31826622C2semidirect
8. #31826617C2semidirect
9. #31826464C2semidirect
10. #31826428C2semidirect
11. #417×C283C2semidirect
19. C22×Dic566C22semidirect
27. C2×Dic1132C22semidirect
28. C2×#17985382C22semidirect
29. C2×#17985381C22semidirect
30. C2×#17985380C22semidirect
31. C2×#17985376C22semidirect
35. #22583590C22semidirect
38. #22583533C22semidirect
39. #22583530C22semidirect
41. (C2×D283)⋊C8C22semidirect
43. C2×M16×C283C22semidirect
44. C2×C4⋊C4×C283C22semidirect
45. #103×C283C22semidirect
46. #102×C283C22semidirect
179. C2×D283#137semidirect
197. C283#3007semidirect
208. #417D283semidirect
231. C2×M16C2×D283semidirect
237. C2×C4⋊C4C2×D283semidirect
240. #103C2×D283semidirect
242. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2832
C2×#225836432
C2×#225836402
C2×#225835902
#318266302
#318266282
#318266222
#318266172
#318264642
#318264282
#417×C2832
#3007283