Finite Group Databasebeta

← #46814025

Subgroups — #46814025

Order 34432.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575423C211yesmaximal, characteristic
2#31770507C211yesmaximal, characteristic, complemented
3#31770506C211yesmaximal, characteristic, complemented
4#31770505C211yesmaximal, characteristic, complemented
5#31770498C211yesmaximal, characteristic, complemented
6#31770497C211yesmaximal, characteristic, complemented
7#31770496C211yesmaximal, characteristic, complemented
8#31770495C211yesmaximal, characteristic, complemented
9#31770494C211yesmaximal, characteristic, complemented
10#31770483C211yesmaximal, characteristic, complemented
11#31770483C211yesmaximal, characteristic, complemented
12#31770481C211yesmaximal, characteristic, complemented
13#31770460C211yesmaximal, characteristic, complemented
14#31770460C211yesmaximal, characteristic, complemented
15#388×C269C211yesmaximal, characteristic, complemented
16C24×D269C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C22×D1076C2211yescharacteristic
19C22×C22⋊D269C2211yescharacteristic
20C22×C4×D269C2211yescharacteristic, complemented
21#22575423C418yescomplemented
22C2×C4⋊Dic269C2211yescharacteristic, complemented
23C2×#16864381C2211yescharacteristic, complemented
24C2×C22⋊Dic269C2211yescharacteristic
25C2×C22⋊Dic269C2211yescharacteristic
26C2×#16864379C2211yescharacteristic
27C2×#16864379C2211yescharacteristic
28C2×#16864379C2211yescharacteristic
29C2×#16864379C2211yescharacteristic
30C2×#16864379C2211yescharacteristic
31C2×#16864379C2211yescharacteristic
32C2×#16864379C2211yescharacteristic
33C2×#16864379C2211yescharacteristic
34C2×#16864379C2211yescharacteristic
35C2×#16864379C2211yescharacteristic
36C2×#16864379C2211yescharacteristic, complemented
37C2×#16864379C2211yescharacteristic, complemented
38C2×#16864379C2211yescharacteristic, complemented
39C2×C4×Dic269C2211yescharacteristic, complemented
40C2×C4×Dic269C2211yescharacteristic, complemented
41#22575360C2211yescharacteristic, complemented
42#22575360C2211yescharacteristic, complemented
43#22575360C2211yescharacteristic, complemented
44#22575360C2211yescharacteristic, complemented
45C2×C4⋊C4×C269C2211yescharacteristic, complemented
46C2×C22⋊C4×C269C2211yescharacteristic
47C2×C22⋊C4×C269C2211yescharacteristic
48C2×C22⋊C4×C269C2211yescharacteristic
49C2×C42×C269C2211yescharacteristic, complemented
50#95×C269C2211yescharacteristic, complemented
51#95×C269C2211yescharacteristic, complemented
52C23×D269—41no—
53C23×D269—41no—
54C23×D269—41no—
55C23×D269—41no—
56C23×D269—41no—
57C23×D269—41no—
58C23×D269C2×C414yescomplemented
59C23×D269C2311yescharacteristic
60C23×D269C2311yescharacteristic
61C23×D269C2311yescharacteristic
62C2×D1076—41no—
63C2×D1076—41no—
64C2×D1076—41no—
65C2×D1076—41no—
66C2×D1076C2×C414yes—
67C2×D1076C2×C414yes—
68C2×C22⋊D269—41no—
69C2×C22⋊D269—41no—
70C2×C22⋊D269C2×C414yes—
71C22×Dic269C2311yescharacteristic
72C22×Dic269C2311yescharacteristic
73C22×Dic269C2311yescharacteristic
74C22×Dic269C2311yescharacteristic
75C22×Dic269C2311yescharacteristic
76C2×C4×D269—41no—
77C2×C4×D269—41no—
78C2×C4×D269—41no—
79C4⋊Dic269—41no—
80#16864381—41no—
81C22⋊Dic269—41no—
82C22⋊Dic269—41no—
83#16864379—41no—
84#16864379—41no—
85#16864379—41no—
86#16864379—41no—
87#16864379—41no—
88#16864379—41no—
89#16864379—41no—
90#16864379—41no—
91#16864379—41no—
92#16864379—41no—
93#16864379—41no—
94#16864379C2×C414yescomplemented
95#16864379C2×C414yescomplemented
96C4×Dic269—41no—
97C4×Dic269—41no—
98C24×C269C2311yescharacteristic
99C22×C4×C269C2311yescharacteristic
100C22×C4×C269C2311yescharacteristic
101C22×C4×C269C2311yescharacteristic
102C22×C4×C269C2311yescharacteristic
103C22×C4×C269C2311yescharacteristic
104C22×C4×C269C2311yescharacteristic
105C4⋊C4×C269—41no—
106C22⋊C4×C269—41no—
107C22⋊C4×C269—41no—
108C22⋊C4×C269C2×C414yescomplemented
109C42×C269—41no—
110C22×D269—21no—
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—21no—
115C22×D269—21no—
116C22×D269—21no—
117C22×D269—21no—
118C22×D269—21no—
119C22×D269—21no—
120C22×D269—21no—
121C22×D269—21no—
122C22×D269—21no—
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—21no—
127C22×D269—21no—
128C22×D269—44no—
129C22×D269—44no—
130C22×D269—44no—
131C22×D269—44no—
132C22×D269—44no—
133C22×D269—44no—
134C22×D269—44no—
135C22×D269C22×C412yes—
136C22×D269C22×C412yes—
137C22×D269C22×C412yes—
138D1076—44no—
139D1076—44no—
140C22⋊D269—44no—
141C2×Dic269—21no—
142C2×Dic269—21no—
143C2×Dic269—21no—
144C2×Dic269—21no—
145C2×Dic269—21no—
146C2×Dic269—21no—
147C2×Dic269—21no—
148C2×Dic269—21no—
149C2×Dic269—21no—
150C2×Dic269—21no—
151C2×Dic269—21no—
152C2×Dic269—21no—
153C2×Dic269—21no—
154C2×Dic269—21no—
155C2×Dic269C22×C412yes—
156C4×D269—44no—
157C23×C269—21no—
158C23×C269—21no—
159C23×C269—21no—
160C23×C269—21no—
161C23×C269—21no—
162C23×C269—21no—
163C23×C269C22×C412yes—
164C23×C269C2411yescharacteristic
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269—21no—
168C2×C4×C269—21no—
169C2×C4×C269—21no—
170C2×C4×C269—21no—
171C2×C4×C269—21no—
172C2×C4×C269—21no—
173C2×C4×C269—21no—
174C2×C4×C269—21no—
175C2×C4×C269—21no—
176C2×C4×C269—21no—
177C2×C4×C269—21no—
178C2×C4×C269—21no—
179C2×C4×C269—21no—
180C2×C4×C269—21no—
181C2×C4×C269C22×C412yes—
182C2×C4×C269C22×C412yes—
183C22×C269—22no—
184C22×C269—22no—
185C22×C269—22no—
186C22×C269—41no—
187C22×C269—41no—
188C22×C269—41no—
189C22×C269—41no—
190C22×C269C23×C411yescharacteristic
191C22×C269H(5,𝔽2)11yescharacteristic
192C22×C269H(5,𝔽2)11yescharacteristic
193C22×C269H(5,𝔽2)11yescharacteristic
194C22×C269H(5,𝔽2)11yescharacteristic
195C22×C269H(5,𝔽2)11yescharacteristic
196C22×C269H(5,𝔽2)11yescharacteristic
197C2×D269—22no—
198C2×D269—22no—
199C2×D269—22no—
200C2×D269—22no—
201C2×D269—22no—
202C2×D269—22no—
203C2×D269—22no—
204C2×D269—22no—
205C2×D269—22no—
206C2×D269—416no—
207C2×D269—41no—
208C2×D269—41no—
209C2×D269—41no—
210C2×D269—41no—
211C2×D269—41no—
212C2×D269—41no—
213C2×D269—41no—
214C2×D269—41no—
215C2×D269—41no—
216C2×D269—41no—
217C2×D269—41no—
218C2×D269—41no—
219C4×C269—22no—
220C4×C269—22no—
221C4×C269—41no—
222C4×C269—41no—
223C4×C269—41no—
224C4×C269—41no—
225Dic269—22no—
226Dic269—41no—
227Dic269—41no—
228Dic269—41no—
229Dic269—41no—
230C2×C269—42no—
231C2×C269#51811yescharacteristic
232C2×C269#51811yescharacteristic
233C2×C269#51811yescharacteristic
234C2×C269#56011yescharacteristic
235C2×C269#56011yescharacteristic
236C2×C269#56011yescharacteristic
237C2×C269H(3,𝔽4)11yescharacteristic
238D269—42no—
239D269—42no—
240D269—42no—
241C269#237511yescharacteristic, complemented, Sylow
242#2375—2691nomaximal, Sylow
243C2×#120—2691no—
244#390—2691no—
245#390—2691no—
246#390—2691no—
247#388—2691no—
248#388—2691no—
249#388—2691no—
250#388—2691no—
251#388—2691no—
252#388D26911yescharacteristic, complemented
253#386—2691no—
254#386—2691no—
255#386—2691no—
256#379—2691no—
257#379—2691no—
258C25—2691no—
259C22×D4—2691no—
260C22×D4—2691no—
261C22×D4—2691no—
262C23×C4—2691no—
263#120—2698no—
264C2×C4⋊C4—2691no—
265C2×C4⋊C4—2691no—
266C2×C4⋊C4C2×D26911yescharacteristic, complemented
267C2×C22⋊C4—2691no—
268C2×C22⋊C4—2691no—
269C2×C22⋊C4—2691no—
270C2×C22⋊C4—2691no—
271C2×C22⋊C4—2691no—
272C2×C22⋊C4—2691no—
273C2×C22⋊C4—2691no—
274C2×C22⋊C4—2691no—
275C2×C22⋊C4—2691no—
276C2×C22⋊C4—2691no—
277C2×C22⋊C4—2691no—
278C2×C22⋊C4—2691no—
279C2×C22⋊C4—2691no—
280C2×C22⋊C4—2691no—
281C2×C22⋊C4—2691no—
282C2×C22⋊C4C2×D26911yescharacteristic
283C2×C22⋊C4C2×D26911yescharacteristic
284C2×C22⋊C4C2×D26911yescharacteristic
285C2×C42—2691no—
286C2×C42—2691no—
287C2×C42C2×D26911yescharacteristic, complemented
288#95—2691no—
289#95—2691no—
290#95—2691no—
291#95—2691no—
292#95C2×D26911yescharacteristic, complemented
293#95C2×D26911yescharacteristic, complemented
294C24—2691no—
295C24—2691no—
296C24—2691no—
297C24—2694no—
298C24—10761no—
299C24—10761no—
300C24—10761no—
301C24—10761no—
302C24—10761no—
303C24—10761no—
304C24C22×D26911yescharacteristic
305C2×D4—2694no—
306C2×D4—2694no—
307C2×D4—2694no—
308C2×D4—10761no—
309C2×D4—10761no—
310C2×D4—10761no—
311C2×D4—10761no—
312C2×D4—10761no—
313C2×D4—10761no—
314C22×C4—2691no—
315C22×C4—2691no—
316C22×C4—2691no—
317C22×C4—2691no—
318C22×C4—2691no—
319C22×C4—10761no—
320C22×C4—10761no—
321C22×C4—10761no—
322C22×C4C22×D26911yescharacteristic
323C22×C4C22×D26911yescharacteristic
324C22×C4C22×D26911yescharacteristic
325C22×C4C22×D26911yescharacteristic
326C22×C4C22×D26911yescharacteristic
327C22×C4C22×D26911yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—10761no—
330C4⋊C4—10761no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—2694no—
334C22⋊C4—2694no—
335C22⋊C4—10761no—
336C22⋊C4—10761no—
337C22⋊C4—10761no—
338C22⋊C4—10761no—
339C22⋊C4—10761no—
340C22⋊C4—10761no—
341C22⋊C4—10761no—
342C22⋊C4—10761no—
343C22⋊C4—10761no—
344C22⋊C4—10761no—
345C22⋊C4—10761no—
346C22⋊C4—10761no—
347C22⋊C4—10761no—
348C22⋊C4C4×D26914yescomplemented
349C42—41no—
350C42—10761no—
351C42—10761no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—2692no—
359C23—2692no—
360C23—2692no—
361C23—5381no—
362C23—5381no—
363C23—5381no—
364C23—5381no—
365C23—5381no—
366C23—5381no—
367C23—5381no—
368C23—5381no—
369C23—5381no—
370C23—5381no—
371C23—5381no—
372C23—5381no—
373C23—5381no—
374C23—5381no—
375C23—5381no—
376C23—5381no—
377C23—5381no—
378C23—5381no—
379C23—10764no—
380C23—10764no—
381C23—10764no—
382C23—10764no—
383C23—10764no—
384C23—10764no—
385C23—10764no—
386C23C2×C4×D26912yes—
387C23C23×D26911yescharacteristic
388D4—10764no—
389D4—10764no—
390D4—10764no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—2692no—
408C2×C4—5381no—
409C2×C4—5381no—
410C2×C4—5381no—
411C2×C4—5381no—
412C2×C4—5381no—
413C2×C4—5381no—
414C2×C4—5381no—
415C2×C4—5381no—
416C2×C4—5381no—
417C2×C4—5381no—
418C2×C4—5381no—
419C2×C4—5381no—
420C2×C4—5381no—
421C2×C4—5381no—
422C2×C4—10764no—
423C2×C4C2×C4×D26912yes—
424C2×C4C2×C4×D26912yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—5382no—
433C22—5382no—
434C22—5382no—
435C22—5382no—
436C22—5382no—
437C22—5382no—
438C22—5382no—
439C22—5382no—
440C22—5382no—
441C22—107616no—
442C22—10761no—
443C22—10761no—
444C22—10761no—
445C22—10761no—
446C22—10761no—
447C22—10761no—
448C22—10761no—
449C22—10761no—
450C22—10761no—
451C22—10761no—
452C22—10761no—
453C22—10761no—
454C22C22×C4×D26911yescharacteristic
455C22#2257549811yescharacteristic
456C22#2257549811yescharacteristic
457C22#2257549911yescharacteristic
458C22#2257549911yescharacteristic
459C22#2257549911yescharacteristic
460C22#2257549911yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—5382no—
468C4—10761no—
469C4—10761no—
470C4—10761no—
471C4—10761no—
472C2—42no—
473C2—10762no—
474C2—10762no—
475C2—10762no—
476C2#3177107711yescharacteristic
477C2#3177107711yescharacteristic
478C2#3177107911yescharacteristic
479C2#3177134311yescharacteristic
480C2#3177134511yescharacteristic
481C2#3177134611yescharacteristic
482C2#3177134711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#388×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575423C2
hypercenter#388D269
hypocenterC269#2375
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #31770507C2semidirect
3. #31770506C2semidirect
4. #31770505C2semidirect
5. #31770498C2semidirect
6. #31770497C2semidirect
7. #31770496C2semidirect
8. #31770495C2semidirect
9. #31770494C2semidirect
10. #31770483C2semidirect
11. #31770483C2semidirect
12. #31770481C2semidirect
13. #31770460C2semidirect
14. #31770460C2semidirect
15. #388×C269C2semidirect
20. C22×C4×D269C22semidirect
21. #22575423C4semidirect
22. C2×C4⋊Dic269C22semidirect
23. C2×#16864381C22semidirect
36. C2×#16864379C22semidirect
37. C2×#16864379C22semidirect
38. C2×#16864379C22semidirect
39. C2×C4×Dic269C22semidirect
40. C2×C4×Dic269C22semidirect
41. #22575360C22semidirect
42. #22575360C22semidirect
43. #22575360C22semidirect
44. #22575360C22semidirect
45. C2×C4⋊C4×C269C22semidirect
49. C2×C42×C269C22semidirect
50. #95×C269C22semidirect
51. #95×C269C22semidirect
58. C23×D269C2×C4semidirect
94. #16864379C2×C4semidirect
95. #16864379C2×C4semidirect
108. C22⋊C4×C269C2×C4semidirect
241. C269#2375semidirect
252. #388D269semidirect
266. C2×C4⋊C4C2×D269semidirect
287. C2×C42C2×D269semidirect
292. #95C2×D269semidirect
293. #95C2×D269semidirect
348. C22⋊C4C4×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754232
#317705072
#317705062
#317705052
#317704982
#317704972
#317704962
#317704952
#317704942
#317704832
#317704812
#317704602
#388×C2692
#2375269