Finite Group Databasebeta

← #46824538

Subgroups — #46824538

Order 34432.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575500C211yesmaximal, characteristic, complemented
2C2×#41×D269C211yesmaximal, characteristic, complemented
3#31771275C213yesmaximal, complemented
4#31771274C211yesmaximal, characteristic, complemented
5#31771273C213yesmaximal, complemented
6#31771267C211yesmaximal, characteristic, complemented
7#31771264C213yesmaximal, complemented
8#31771262C211yesmaximal, characteristic, complemented
9#31771261C213yesmaximal, complemented
10#31771171C213yesmaximal, complemented
11#31771168C213yesmaximal, complemented
12#31771115C213yesmaximal, complemented
13#31771114C213yesmaximal, complemented
14#31771081C211yesmaximal, characteristic, complemented
15#550×C269C211yesmaximal, characteristic, complemented
16#22575500—28no—
17C2×#16864403C2211yescharacteristic
18C2×#16864402C2213yes—
19C2×#16864402C2213yes—
20C2×#16864401C2213yes—
21C2×#16864401C2213yes—
22C2×#16864401C2213yescomplemented
23#41×D269—28no—
24C22×Dic538C2213yescomplemented
25C2×Q8×D269C2211yescharacteristic
26C2×Q8×D269C2211yescharacteristic, complemented
27C2×D4×D269C2213yes—
28C22×C4×D269C2213yes—
29#22575449C2213yescomplemented
30#22575449C2216yescomplemented
31#22575441C2213yescomplemented
32#22575440C2213yescomplemented
33#22575439C2216yescomplemented
34#22575438C2211yescharacteristic
35#22575435C2213yes—
36#22575433C2213yescomplemented
37#22575432C2213yescomplemented
38#22575431C2213yescomplemented
39#22575430C2213yescomplemented
40#22575428C2213yescomplemented
41#22575427C2213yescomplemented
42#22575426C2213yescomplemented
43#22575425C2216yescomplemented
44#22575425C2216yescomplemented
45#22575425C2213yescomplemented
46#22575424C2213yescomplemented
47#22575424C2213yescomplemented
48#22575423C2213yes—
49#22575421C2213yes—
50#22575421C2216yes—
51Q8×Dic269C2211yescharacteristic, complemented
52C4×Dic538C2213yescomplemented
53#22575418C2213yescomplemented
54C4×D1076C2213yescomplemented
55D4×Dic269C2216yescomplemented
56C4×C22⋊D269C2213yescomplemented
57C4×C22⋊D269C2213yescomplemented
58#22575413C2216yescomplemented
59#22575411C2213yes—
60#22575410C2213yes—
61C2×#41×C269C2211yescharacteristic
62C2×#41×C269C2211yescharacteristic, complemented
63#127×C269C2213yescomplemented
64#121×C269C2213yescomplemented
65#121×C269C2213yescomplemented
66C4×Q8×C269C2211yescharacteristic, complemented
67C4×D4×C269C2213yescomplemented
68C23×D269C2313yes—
69#16864403—24no—
70#16864402—212no—
71#16864402—212no—
72#16864401—212no—
73#16864401—212no—
74#16864401—212no—
75C2×Dic538—212no—
76C2×Dic538C2313yes—
77C2×Dic538C2313yes—
78C2×Dic538C2313yes—
79Q8×D269—24no—
80Q8×D269D418yescomplemented
81C2×D1076C2311yescharacteristic
82C2×D1076C2313yes—
83C2×D1076C2313yes—
84C2×C22⋊D269C2313yes—
85C2×C22⋊D269C2316yes—
86C2×C22⋊D269C2313yes—
87C2×C22⋊D269C2316yes—
88C2×C22⋊D269C2313yes—
89C2×C22⋊D269C2313yes—
90D4×D269—212no—
91C22×Dic269C2316yes—
92C22×Dic269C2313yes—
93C2×C4×D269—212no—
94C2×C4×D269C2313yes—
95C2×C4×D269C2313yes—
96C2×C4×D269C2313yes—
97C2×C4×D269C2311yescharacteristic
98C2×C4×D269C2313yes—
99C4⋊Dic269C2313yes—
100C4⋊Dic269C2313yes—
101#16864381C2313yes—
102#16864381C2313yes—
103#16864381C2313yes—
104#16864381C2311yescharacteristic
105C22⋊Dic269C2313yes—
106C22⋊Dic269C2313yes—
107C22⋊Dic269C2316yes—
108#16864379C2311yescharacteristic
109#16864379C2313yes—
110#16864379C2316yes—
111#16864379C2311yescharacteristic
112#16864379C2313yes—
113#16864379C2311yescharacteristic
114#16864379C2313yes—
115C4×Dic269C2313yes—
116C4×Dic269C2316yes—
117#41×C269—24no—
118#41×C269—24no—
119C2×Q8×C269C2311yescharacteristic
120C2×D4×C269C2313yes—
121C2×D4×C269C2313yes—
122C2×D4×C269C2316yes—
123C2×D4×C269C2313yes—
124C22×C4×C269C2313yes—
125C22×C4×C269C2313yes—
126C4⋊C4×C269C2313yes—
127C4⋊C4×C269C2311yescharacteristic
128C22⋊C4×C269C2313yes—
129C22⋊C4×C269C2313yes—
130C42×C269C2313yes—
131C22×D269—26no—
132C22×D269—43no—
133C22×D269—43no—
134C22×D269C2413yes—
135C22×D269C2411yescharacteristic
136C22×D269C2411yescharacteristic
137Dic538—26no—
138Dic538—26no—
139Dic538C2×D4112yes—
140D1076—22no—
141D1076—26no—
142D1076—43no—
143C22⋊D269—26no—
144C22⋊D269—212no—
145C22⋊D269—26no—
146C22⋊D269—43no—
147C22⋊D269—46no—
148C22⋊D269—43no—
149C2×Dic269—26no—
150C2×Dic269—212no—
151C2×Dic269C2411yescharacteristic
152C2×Dic269C2413yes—
153C2×Dic269C2413yes—
154C2×Dic269C2411yescharacteristic
155C2×Dic269C2413yes—
156C4×D269—22no—
157C4×D269—26no—
158C4×D269—26no—
159C4×D269—26no—
160C4×D269C2×D4112yes—
161C23×C269C2413yes—
162C23×C269C2413yes—
163Q8×C269C2×D414yes—
164D4×C269—26no—
165D4×C269—26no—
166D4×C269—43no—
167D4×C269—46no—
168C2×C4×C269—26no—
169C2×C4×C269—26no—
170C2×C4×C269C2411yescharacteristic
171C2×C4×C269C2413yes—
172C2×C4×C269C2411yescharacteristic
173C2×C4×C269C2411yescharacteristic
174C2×C4×C269C2413yes—
175C22×C269—23no—
176C22×C269—23no—
177C22×C269—23no—
178C22×C269—23no—
179C22×C269—23no—
180C22×C269—23no—
181C22×C269C2511yescharacteristic
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—21no—
185C2×D269—23no—
186C2×D269—21no—
187C2×D269—23no—
188C2×D269—23no—
189C2×D269—21no—
190C2×D269—412no—
191C2×D269C22×D412yes—
192C4×C269—21no—
193C4×C269—21no—
194C4×C269—21no—
195C4×C269—23no—
196C4×C269C22×D416yes—
197Dic269—21no—
198Dic269—23no—
199Dic269—21no—
200Dic269—23no—
201Dic269C22×D416yes—
202C2×C269—43no—
203C2×C269—43no—
204C2×C269C23×D411yescharacteristic
205C2×C269#58511yescharacteristic
206C2×C269#58511yescharacteristic
207D269—22no—
208D269—41no—
209D269—43no—
210C269#348711yescharacteristic, complemented, Sylow
211#3487—2691nomaximal, Sylow
212C2×2−1+4—2691no—
213C22×#41—2691no—
214#550—2691no—
215#550—2693no—
216#550—2693no—
217#550D26911yescharacteristic, complemented
218#548—2691no—
219#548—2693no—
220#548—2691no—
221#548—2693no—
222#535—2693no—
223#535—2693no—
224#525—2693no—
225#525—2693no—
226#519—2691no—
2272−1+4—5388no—
228C2×#41—2691no—
229C2×#41—2693no—
230C2×#41—2693no—
231C2×#41—2693no—
232C2×#41—2693no—
233C2×#41—2693no—
234C2×#41—5388no—
235C2×#41C2×D26911yescharacteristic
236C2×#41C2×D26911yescharacteristic, complemented
237C22×Q8—2691no—
238C22×Q8—2691no—
239C22×Q8—2693no—
240C22×D4—2693no—
241C23×C4—2693no—
242#127—2696no—
243#127—2693no—
244#127C2×D26913yescomplemented
245#124—2693no—
246#124—2696no—
247#124—2693no—
248#123—2691no—
249#123—2693no—
250C22⋊Q8—2693no—
251C22⋊Q8—2693no—
252C22⋊Q8—2693no—
253C22⋊Q8—2693no—
254#121—2693no—
255#121—2693no—
256#121—2693no—
257#121—2696no—
258#121—2696no—
259#121—2693no—
260#121—2693no—
261#121—2693no—
262#121C2×D26913yescomplemented
263#121C2×D26913yescomplemented
264#120—2696no—
265#120—2693no—
266#120—2693no—
267C4×Q8—2691no—
268C4×Q8—2693no—
269C4×Q8—2693no—
270C4×Q8C2×D26911yescharacteristic, complemented
271C4×D4—2696no—
272C4×D4—2693no—
273C4×D4—2693no—
274C4×D4—2696no—
275C4×D4—2693no—
276C4×D4C2×D26913yescomplemented
277#117—2693no—
278#117—2693no—
279C24—2693no—
280#41—24no—
281#41—24no—
282#41—53812no—
283#41—53812no—
284#41—53812no—
285#41—53812no—
286#41—53812no—
287#41—5384no—
288C2×Q8—2698no—
289C2×Q8—2693no—
290C2×Q8—2693no—
291C2×Q8—2693no—
292C2×Q8—53812no—
293C2×Q8—5384no—
294C2×Q8C22×D26911yescharacteristic
295C2×D4—2693no—
296C2×D4—2696no—
297C2×D4—2693no—
298C2×D4—2693no—
299C2×D4—2696no—
300C2×D4—2693no—
301C2×D4—2691no—
302C2×D4—2693no—
303C2×D4—2693no—
304C2×D4—53812no—
305C2×D4C22×D26913yes—
306C2×D4C22×D26913yes—
307C2×D4C22×D26916yes—
308C2×D4C22×D26913yes—
309C22×C4—2693no—
310C22×C4—2696no—
311C22×C4—2693no—
312C22×C4—2693no—
313C22×C4—2693no—
314C22×C4—2693no—
315C22×C4—2691no—
316C22×C4—53812no—
317C22×C4C22×D26913yes—
318C22×C4C22×D26913yes—
319C4⋊C4—2693no—
320C4⋊C4—2693no—
321C4⋊C4—2693no—
322C4⋊C4—2693no—
323C4⋊C4—2691no—
324C4⋊C4—2693no—
325C4⋊C4C22×D26913yes—
326C4⋊C4C22×D26911yescharacteristic
327C22⋊C4—2693no—
328C22⋊C4—2696no—
329C22⋊C4—2693no—
330C22⋊C4—2693no—
331C22⋊C4—2691no—
332C22⋊C4—2691no—
333C22⋊C4—2691no—
334C22⋊C4—2693no—
335C22⋊C4—2696no—
336C22⋊C4—2693no—
337C22⋊C4C22×D26913yes—
338C22⋊C4C22×D26913yes—
339C42—2696no—
340C42—2693no—
341C42C22×D26913yes—
342C23—2691no—
343C23—2693no—
344C23—2691no—
345C23—5386no—
346C23—10763no—
347C23—10763no—
348C23C23×D26913yes—
349C23C23×D26913yes—
350Q8—26912no—
351Q8—5386no—
352Q8—5386no—
353Q8D4×D26914yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—5382no—
359D4—53812no—
360D4—5386no—
361D4—5386no—
362D4—5386no—
363D4—10763no—
364D4—10763no—
365D4—10766no—
366D4—10763no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—2693no—
370C2×C4—26912no—
371C2×C4—2691no—
372C2×C4—2693no—
373C2×C4—2693no—
374C2×C4—2691no—
375C2×C4—5386no—
376C2×C4—5382no—
377C2×C4—5386no—
378C2×C4—53812no—
379C2×C4—5386no—
380C2×C4—5386no—
381C2×C4C23×D26913yes—
382C2×C4C23×D26911yescharacteristic
383C2×C4C23×D26911yescharacteristic
384C2×C4C23×D26911yescharacteristic
385C2×C4C23×D26913yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—2692no—
393C22—5381no—
394C22—5381no—
395C22—5381no—
396C22—5383no—
397C22—5381no—
398C22—5383no—
399C22—5381no—
400C22—5383no—
401C22—107612no—
402C22C24×D26911yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—2696no—
408C4—5381no—
409C4—5383no—
410C4—5381no—
411C4—5383no—
412C4C2×D4×D26916yes—
413C2—43no—
414C2—43no—
415C2—5382no—
416C2—10763no—
417C2—10761no—
418C2C22×D4×D26911yescharacteristic
419C2#3177144011yescharacteristic
420C2#3177144111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#550×C269C2
hypercenter#550D269
hypocenterC269#3487
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22575500C2semidirect
2. C2×#41×D269C2semidirect
3. #31771275C2semidirect
4. #31771274C2semidirect
5. #31771273C2semidirect
6. #31771267C2semidirect
7. #31771264C2semidirect
8. #31771262C2semidirect
9. #31771261C2semidirect
10. #31771171C2semidirect
11. #31771168C2semidirect
12. #31771115C2semidirect
13. #31771114C2semidirect
14. #31771081C2semidirect
15. #550×C269C2semidirect
22. C2×#16864401C22semidirect
24. C22×Dic538C22semidirect
26. C2×Q8×D269C22semidirect
29. #22575449C22semidirect
30. #22575449C22semidirect
31. #22575441C22semidirect
32. #22575440C22semidirect
33. #22575439C22semidirect
36. #22575433C22semidirect
37. #22575432C22semidirect
38. #22575431C22semidirect
39. #22575430C22semidirect
40. #22575428C22semidirect
41. #22575427C22semidirect
42. #22575426C22semidirect
43. #22575425C22semidirect
44. #22575425C22semidirect
45. #22575425C22semidirect
46. #22575424C22semidirect
47. #22575424C22semidirect
51. Q8×Dic269C22semidirect
52. C4×Dic538C22semidirect
53. #22575418C22semidirect
54. C4×D1076C22semidirect
55. D4×Dic269C22semidirect
56. C4×C22⋊D269C22semidirect
57. C4×C22⋊D269C22semidirect
58. #22575413C22semidirect
62. C2×#41×C269C22semidirect
63. #127×C269C22semidirect
64. #121×C269C22semidirect
65. #121×C269C22semidirect
66. C4×Q8×C269C22semidirect
67. C4×D4×C269C22semidirect
80. Q8×D269D4semidirect
210. C269#3487semidirect
217. #550D269semidirect
236. C2×#41C2×D269semidirect
244. #127C2×D269semidirect
262. #121C2×D269semidirect
263. #121C2×D269semidirect
270. C4×Q8C2×D269semidirect
276. C4×D4C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225755002
C2×#41×D2692
#317712752
#317712742
#317712732
#317712672
#317712642
#317712622
#317712612
#317711712
#317711682
#317711152
#317711142
#317710812
#550×C2692
#3487269