Finite Group Databasebeta

← #46813732

Subgroups — #46813732

Order 34432.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575411C211yesmaximal, characteristic, complemented
2C22×#16864379C211yesmaximal, characteristic
3C2×C22⋊C4×D269C211yesmaximal, characteristic
4#31770497C211yesmaximal, characteristic, complemented
5#31770497C211yesmaximal, characteristic, complemented
6#31770495C211yesmaximal, characteristic, complemented
7#31770495C211yesmaximal, characteristic, complemented
8#31770480C211yesmaximal, characteristic, complemented
9#31770467C211yesmaximal, characteristic, complemented
10#31770462C211yesmaximal, characteristic, complemented
11#31770460C211yesmaximal, characteristic
12C22⋊C4×Dic269C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic269C211yesmaximal, characteristic, complemented
14#31770452C211yesmaximal, characteristic, complemented
15#385×C269C211yesmaximal, characteristic, complemented
16C24×D269C2211yescharacteristic
17C23×Dic269C2211yescharacteristic
18C22×C4×D269C2211yescharacteristic
19C22×C4×D269C2211yescharacteristic
20#22575411—24no—
21C2×C4⋊Dic269C2211yescharacteristic, complemented
22C2×C4⋊Dic269C2211yescharacteristic, complemented
23C2×C22⋊Dic269C2211yescharacteristic
24C2×C22⋊Dic269C2211yescharacteristic
25C2×#16864379—24no—
26C2×#16864379C2211yescharacteristic
27C2×#16864379C2211yescharacteristic
28C2×#16864379C2211yescharacteristic
29C2×#16864379C2211yescharacteristic
30C2×#16864379C2211yescharacteristic
31C2×#16864379C2211yescharacteristic
32C2×#16864379C2211yescharacteristic
33C2×#16864379C2211yescharacteristic
34C2×#16864379C2211yescharacteristic
35C2×#16864379C2211yescharacteristic
36C22⋊C4×D269C418yescomplemented
37C2×C4×Dic269C2211yescharacteristic, complemented
38C2×C4×Dic269C2211yescharacteristic, complemented
39C2×C4×Dic269C2211yescharacteristic, complemented
40C2×C4×Dic269C2211yescharacteristic, complemented
41#22575360C2211yescharacteristic, complemented
42#22575360C2211yescharacteristic, complemented
43#22575360C2211yescharacteristic, complemented
44#22575360C2211yescharacteristic, complemented
45#22575360C2211yescharacteristic, complemented
46#22575360C2211yescharacteristic, complemented
47C23×C4×C269C2211yescharacteristic
48C2×C4⋊C4×C269C2211yescharacteristic, complemented
49C2×C4⋊C4×C269C2211yescharacteristic, complemented
50C2×C22⋊C4×C269C2211yescharacteristic
51C2×C22⋊C4×C269C2211yescharacteristic
52#95×C269C2211yescharacteristic, complemented
53#95×C269C2211yescharacteristic, complemented
54C23×D269—22no—
55C23×D269—22no—
56C23×D269—22no—
57C23×D269—22no—
58C23×D269—41no—
59C23×D269—41no—
60C23×D269C2×C414yes—
61C23×D269C2311yescharacteristic
62C23×D269C2311yescharacteristic
63C22×Dic269—22no—
64C22×Dic269—22no—
65C22×Dic269—41no—
66C22×Dic269C2311yescharacteristic
67C22×Dic269C2311yescharacteristic
68C22×Dic269C2311yescharacteristic
69C22×Dic269C2311yescharacteristic
70C22×Dic269C2311yescharacteristic
71C22×Dic269C2311yescharacteristic
72C2×C4×D269—22no—
73C2×C4×D269—22no—
74C2×C4×D269—41no—
75C2×C4×D269—41no—
76C2×C4×D269C2×C414yescomplemented
77C2×C4×D269C2×C414yescomplemented
78C4⋊Dic269—22no—
79C4⋊Dic269—41no—
80C22⋊Dic269—22no—
81C22⋊Dic269C2×C414yes—
82#16864379—22no—
83#16864379—22no—
84#16864379—22no—
85#16864379—22no—
86#16864379—22no—
87#16864379—41no—
88#16864379—41no—
89#16864379C2×C414yescomplemented
90#16864379C2×C414yescomplemented
91#16864379D414yes—
92C4×Dic269—22no—
93C4×Dic269—22no—
94C4×Dic269—41no—
95C4×Dic269—41no—
96C24×C269C2311yescharacteristic
97C22×C4×C269—22no—
98C22×C4×C269—22no—
99C22×C4×C269—41no—
100C22×C4×C269C2311yescharacteristic
101C22×C4×C269C2311yescharacteristic
102C22×C4×C269C2311yescharacteristic
103C22×C4×C269C2311yescharacteristic
104C22×C4×C269C2311yescharacteristic
105C22×C4×C269C2311yescharacteristic
106C4⋊C4×C269—22no—
107C4⋊C4×C269—41no—
108C22⋊C4×C269—22no—
109C22⋊C4×C269C2×C414yes—
110C22×D269—21no—
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—28no—
115C22×D269—21no—
116C22×D269—21no—
117C22×D269—21no—
118C22×D269—21no—
119C22×D269—21no—
120C22×D269—21no—
121C22×D269—28no—
122C22×D269—44no—
123C22×D269—44no—
124C22×D269—44no—
125C22×D269—44no—
126C22×D269—44no—
127C22×D269C22×C412yes—
128C22×D269C22×C412yes—
129C22×D269C2×D412yes—
130C22×D269#4112yes—
131C2×Dic269—21no—
132C2×Dic269—21no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C2×Dic269—21no—
138C2×Dic269—21no—
139C2×Dic269—21no—
140C2×Dic269—21no—
141C2×Dic269—21no—
142C2×Dic269—21no—
143C2×Dic269—21no—
144C2×Dic269—21no—
145C2×Dic269—21no—
146C2×Dic269—44no—
147C2×Dic269C22×C412yes—
148C2×Dic269C22×C412yes—
149C2×Dic269C2×D412yes—
150C4×D269—28no—
151C4×D269—44no—
152C23×C269—21no—
153C23×C269—21no—
154C23×C269—21no—
155C23×C269—21no—
156C23×C269C22×C412yes—
157C23×C269#4112yes—
158C23×C269#4112yes—
159C23×C269C2411yescharacteristic
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
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—44no—
176C2×C4×C269C22×C412yes—
177C2×C4×C269C22×C412yes—
178C2×C4×C269C2×D412yes—
179C22×C269—22no—
180C22×C269—22no—
181C22×C269—22no—
182C22×C269—22no—
183C22×C269—22no—
184C22×C269—22no—
185C22×C269#11714yescomplemented
186C22×C269C23×C411yescharacteristic
187C22×C269C22×D411yescharacteristic
188C22×C269C2×#4111yescharacteristic
189C22×C269C2×#4111yescharacteristic
190C22×C269C2×#4111yescharacteristic
191C22×C269H(5,𝔽2)11yescharacteristic
192C22×C269H(5,𝔽2)11yescharacteristic
193C2×D269—22no—
194C2×D269—22no—
195C2×D269—22no—
196C2×D269—22no—
197C2×D269—22no—
198C2×D269—22no—
199C2×D269—22no—
200C2×D269—22no—
201C2×D269—22no—
202C2×D269—416no—
203C2×D269—41no—
204C2×D269—41no—
205C2×D269—41no—
206C2×D269—41no—
207C2×D269C4×D414yes—
208C4×C269—22no—
209C4×C269—22no—
210C4×C269—22no—
211C4×C269—41no—
212C4×C269—41no—
213C4×C269—41no—
214Dic269—22no—
215Dic269—22no—
216Dic269—22no—
217Dic269—41no—
218Dic269—41no—
219Dic269—41no—
220C2×C269—24no—
221C2×C269C2×#11711yescharacteristic
222C2×C269C2×C4×D411yescharacteristic
223C2×C269#51811yescharacteristic
224C2×C269#54611yescharacteristic
225C2×C269#54611yescharacteristic
226C2×C269#55111yescharacteristic
227C2×C269#55111yescharacteristic
228D269—24no—
229D269—42no—
230C269#234211yescharacteristic, complemented, Sylow
231#2342—2691nomaximal, Sylow
232C2×#117—2691no—
233C22×C22⋊C4—2691no—
234C22×C22⋊C4—2691no—
235#388—2691no—
236#388—2691no—
237#388—2691no—
238#388—2691no—
239#385—2691no—
240#385D26911yescharacteristic, complemented
241#381—2691no—
242#380—2691no—
243#379—2691no—
244C4×C22⋊C4—2691no—
245C4×C22⋊C4—2691no—
246C4×C22⋊C4—2691no—
247C25—2691no—
248C23×C4—2691no—
249C23×C4—2691no—
250C23×C4—2691no—
251C23×C4C2×D26911yescharacteristic
252#117—5384no—
253C2×C4⋊C4—2691no—
254C2×C4⋊C4—2691no—
255C2×C4⋊C4C2×D26911yescharacteristic, complemented
256C2×C4⋊C4C2×D26911yescharacteristic, complemented
257C2×C22⋊C4—2691no—
258C2×C22⋊C4—2691no—
259C2×C22⋊C4—2691no—
260C2×C22⋊C4—2691no—
261C2×C22⋊C4—2698no—
262C2×C22⋊C4—2691no—
263C2×C22⋊C4—2691no—
264C2×C22⋊C4—2691no—
265C2×C22⋊C4—2691no—
266C2×C22⋊C4—2691no—
267C2×C22⋊C4—2691no—
268C2×C22⋊C4—2691no—
269C2×C22⋊C4—2691no—
270C2×C22⋊C4—5384no—
271C2×C22⋊C4C2×D26911yescharacteristic
272C2×C22⋊C4C2×D26911yescharacteristic
273C2×C42—2691no—
274C2×C42—2691no—
275C2×C42—2691no—
276C2×C42—2691no—
277#95—2691no—
278#95—2691no—
279#95—2691no—
280#95—2691no—
281#95—2691no—
282#95—2691no—
283#95C2×D26911yescharacteristic, complemented
284#95C2×D26911yescharacteristic, complemented
285C24—2691no—
286C24—2691no—
287C24—2694no—
288C24—5382no—
289C24—5382no—
290C24—5382no—
291C24—5382no—
292C24—10761no—
293C24—10761no—
294C24C22×D26911yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—2691no—
299C22×C4—2691no—
300C22×C4—2691no—
301C22×C4—2694no—
302C22×C4—2694no—
303C22×C4—2691no—
304C22×C4—2691no—
305C22×C4—2691no—
306C22×C4—5382no—
307C22×C4—5382no—
308C22×C4—5382no—
309C22×C4—5382no—
310C22×C4—10761no—
311C22×C4—10761no—
312C22×C4—10761no—
313C22×C4C22×D26911yescharacteristic
314C22×C4C22×D26911yescharacteristic
315C22×C4C22×D26911yescharacteristic
316C22×C4C22×D26911yescharacteristic
317C22×C4C22×D26911yescharacteristic
318C22×C4C22×D26911yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—5382no—
322C4⋊C4—10761no—
323C22⋊C4—22no—
324C22⋊C4—2694no—
325C22⋊C4—2694no—
326C22⋊C4—2694no—
327C22⋊C4—2694no—
328C22⋊C4—5382no—
329C22⋊C4—5382no—
330C22⋊C4—5382no—
331C22⋊C4—5382no—
332C22⋊C4—5382no—
333C22⋊C4—5382no—
334C22⋊C4—10761no—
335C22⋊C4—10761no—
336C22⋊C4C4×D26914yes—
337C42—5382no—
338C42—5382no—
339C42—10761no—
340C42—10761no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—2692no—
346C23—2692no—
347C23—2692no—
348C23—2692no—
349C23—5388no—
350C23—5381no—
351C23—5381no—
352C23—5381no—
353C23—5381no—
354C23—5388no—
355C23—5381no—
356C23—5381no—
357C23—5381no—
358C23—5381no—
359C23—5381no—
360C23—5381no—
361C23—10764no—
362C23—10764no—
363C23—10764no—
364C23—10764no—
365C23—10764no—
366C23C2×C4×D26912yes—
367C23#1686440112yes—
368C23#1686440312yes—
369C23C23×D26911yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—44no—
386C2×C4—2692no—
387C2×C4—2692no—
388C2×C4—2692no—
389C2×C4—5388no—
390C2×C4—5381no—
391C2×C4—5381no—
392C2×C4—5381no—
393C2×C4—5381no—
394C2×C4—5381no—
395C2×C4—5381no—
396C2×C4—5381no—
397C2×C4—5381no—
398C2×C4—5381no—
399C2×C4—5381no—
400C2×C4—5381no—
401C2×C4—5381no—
402C2×C4—5381no—
403C2×C4—5381no—
404C2×C4—5381no—
405C2×C4—10764no—
406C2×C4—10764no—
407C2×C4C2×C4×D26912yes—
408C2×C4C2×C4×D26912yes—
409C2×C4D4×D26912yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—2694no—
417C22—5382no—
418C22—5382no—
419C22—5382no—
420C22—5382no—
421C22—5382no—
422C22—5382no—
423C22—5382no—
424C22—5382no—
425C22—5382no—
426C22—107616no—
427C22—10761no—
428C22—10761no—
429C22—10761no—
430C22—10761no—
431C22#2257541114yescomplemented
432C22C22×C4×D26911yescharacteristic
433C22C2×D4×D26911yescharacteristic
434C22#41×D26911yescharacteristic
435C22C2×#1686440111yescharacteristic
436C22C2×#1686440311yescharacteristic
437C22#2257549811yescharacteristic
438C22#2257549911yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—5382no—
446C4—5382no—
447C4—5382no—
448C4—10761no—
449C4—10761no—
450C4—10761no—
451C2—24no—
452C2—5384no—
453C2—10762no—
454C2C2×#2257541111yescharacteristic
455C2C4×D4×D26911yescharacteristic
456C2#3177107711yescharacteristic
457C2#3177124211yescharacteristic
458C2#3177124611yescharacteristic
459C2#3177127811yescharacteristic
460C2#3177128211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#385×C269C2
socle, cosocleC23×C269C24
prime radicalC24×D269C22
hypercenter#385D269
hypocenterC269#2342
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#22575411C2semidirect
4. #31770497C2semidirect
5. #31770497C2semidirect
6. #31770495C2semidirect
7. #31770495C2semidirect
8. #31770480C2semidirect
9. #31770467C2semidirect
10. #31770462C2semidirect
12. C22⋊C4×Dic269C2semidirect
13. C4×C22⋊Dic269C2semidirect
14. #31770452C2semidirect
15. #385×C269C2semidirect
21. C2×C4⋊Dic269C22semidirect
22. C2×C4⋊Dic269C22semidirect
36. C22⋊C4×D269C4semidirect
37. C2×C4×Dic269C22semidirect
38. C2×C4×Dic269C22semidirect
39. C2×C4×Dic269C22semidirect
40. C2×C4×Dic269C22semidirect
41. #22575360C22semidirect
42. #22575360C22semidirect
43. #22575360C22semidirect
44. #22575360C22semidirect
45. #22575360C22semidirect
46. #22575360C22semidirect
48. C2×C4⋊C4×C269C22semidirect
49. C2×C4⋊C4×C269C22semidirect
52. #95×C269C22semidirect
53. #95×C269C22semidirect
76. C2×C4×D269C2×C4semidirect
77. C2×C4×D269C2×C4semidirect
89. #16864379C2×C4semidirect
90. #16864379C2×C4semidirect
185. C22×C269#117semidirect
230. C269#2342semidirect
240. #385D269semidirect
255. C2×C4⋊C4C2×D269semidirect
256. C2×C4⋊C4C2×D269semidirect
283. #95C2×D269semidirect
284. #95C2×D269semidirect
431. C22#22575411semidirect

Maximal subgroups

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

subgroupindex
C2×#225754112
C22×#168643792
C2×C22⋊C4×D2692
#317704972
#317704952
#317704802
#317704672
#317704622
#317704602
C22⋊C4×Dic2692
C4×C22⋊Dic2692
#317704522
#385×C2692
#2342269