Finite Group Databasebeta

← #46813864

Subgroups — #46813864

Order 34432.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575422C211yesmaximal, characteristic
2C2×C4×C22⋊D269C211yesmaximal, characteristic, complemented
3C2×#22575413C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic269C211yesmaximal, characteristic, complemented
5#31770505C211yesmaximal, characteristic, complemented
6#31770499C211yesmaximal, characteristic, complemented
7#31770493C211yesmaximal, characteristic, complemented
8#31770484C211yesmaximal, characteristic, complemented
9#31770482C211yesmaximal, characteristic, complemented
10#31770481C211yesmaximal, characteristic, complemented
11#31770478C211yesmaximal, characteristic, complemented
12#31770477C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic269C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic269C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C269C211yesmaximal, characteristic, complemented
16C22×C22⋊D269C2211yescharacteristic
17C22×C22⋊D269C2211yescharacteristic
18C22×C22⋊D269C2211yescharacteristic
19C23×Dic269C2211yescharacteristic
20C23×Dic269C2211yescharacteristic
21C22×C4×D269C2211yescharacteristic, complemented
22#22575422C418yescomplemented
23C4×C22⋊D269—24no—
24#22575413—24no—
25C2×#16864381C2211yescharacteristic, complemented
26C2×#16864381C2211yescharacteristic, complemented
27C2×#16864381C2211yescharacteristic, complemented
28C2×C22⋊Dic269—24no—
29C2×C22⋊Dic269C2211yescharacteristic
30C2×C22⋊Dic269C2211yescharacteristic
31C2×C22⋊Dic269C2211yescharacteristic
32C2×C22⋊Dic269C2211yescharacteristic
33C2×C22⋊Dic269C2211yescharacteristic
34C2×C22⋊Dic269C2211yescharacteristic
35C2×C22⋊Dic269C2211yescharacteristic
36C2×#16864379C2211yescharacteristic, complemented
37C2×#16864379C2211yescharacteristic, complemented
38C2×#16864379C2211yescharacteristic, complemented
39C2×C4×Dic269C2211yescharacteristic, complemented
40C2×C4×Dic269C2211yescharacteristic, complemented
41C2×C4×Dic269C2211yescharacteristic, complemented
42#22575360C2211yescharacteristic, complemented
43#22575360C2211yescharacteristic, complemented
44#22575360C2211yescharacteristic, complemented
45#22575360C2211yescharacteristic, complemented
46#22575360C2211yescharacteristic, complemented
47#22575360C2211yescharacteristic, complemented
48C25×C269C2211yescharacteristic
49C23×C4×C269C2211yescharacteristic
50C23×C4×C269C2211yescharacteristic
51C2×C22⋊C4×C269—24no—
52C2×C22⋊C4×C269C2211yescharacteristic
53C2×C22⋊C4×C269C2211yescharacteristic
54C2×C22⋊C4×C269C2211yescharacteristic
55C2×C22⋊C4×C269C2211yescharacteristic
56C23×D269C2311yescharacteristic
57C2×C22⋊D269—22no—
58C2×C22⋊D269—22no—
59C2×C22⋊D269—22no—
60C2×C22⋊D269—22no—
61C2×C22⋊D269—41no—
62C2×C22⋊D269—41no—
63C2×C22⋊D269C2×C414yescomplemented
64C2×C22⋊D269C2×C414yescomplemented
65C2×C22⋊D269C2×C414yescomplemented
66C22×Dic269—22no—
67C22×Dic269—22no—
68C22×Dic269—22no—
69C22×Dic269—22no—
70C22×Dic269—41no—
71C22×Dic269—41no—
72C22×Dic269C2311yescharacteristic
73C22×Dic269C2311yescharacteristic
74C22×Dic269C2311yescharacteristic
75C22×Dic269C2311yescharacteristic
76C22×Dic269C2311yescharacteristic
77C22×Dic269C2311yescharacteristic
78C22×Dic269C2311yescharacteristic
79C2×C4×D269—22no—
80C2×C4×D269—22no—
81C2×C4×D269—41no—
82#16864381—22no—
83#16864381—22no—
84#16864381—41no—
85C22⋊Dic269—22no—
86C22⋊Dic269—22no—
87C22⋊Dic269—22no—
88C22⋊Dic269C2×C414yescomplemented
89C22⋊Dic269C2×C414yescomplemented
90C22⋊Dic269C2×C414yescomplemented
91C22⋊Dic269D414yescomplemented
92#16864379—22no—
93#16864379—22no—
94#16864379—41no—
95C4×Dic269—22no—
96C4×Dic269—22no—
97C4×Dic269—41no—
98C24×C269—22no—
99C24×C269—22no—
100C24×C269—22no—
101C24×C269—22no—
102C24×C269—41no—
103C24×C269—41no—
104C24×C269C2×C414yescomplemented
105C24×C269C2311yescharacteristic
106C24×C269C2311yescharacteristic
107C24×C269C2311yescharacteristic
108C22×C4×C269—22no—
109C22×C4×C269—22no—
110C22×C4×C269—22no—
111C22×C4×C269—22no—
112C22×C4×C269—41no—
113C22×C4×C269—41no—
114C22×C4×C269C2311yescharacteristic
115C22×C4×C269C2311yescharacteristic
116C22×C4×C269C2311yescharacteristic
117C22×C4×C269C2311yescharacteristic
118C22⋊C4×C269—22no—
119C22⋊C4×C269—22no—
120C22⋊C4×C269—22no—
121C22⋊C4×C269D414yescomplemented
122C22×D269—21no—
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—21no—
127C22×D269—21no—
128C22×D269C22×C412yes—
129C22⋊D269—28no—
130C22⋊D269—28no—
131C22⋊D269—44no—
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—21no—
147C2×Dic269—21no—
148C2×Dic269—44no—
149C2×Dic269—44no—
150C2×Dic269C22×C412yes—
151C2×Dic269C22×C412yes—
152C2×Dic269C22×C412yes—
153C2×Dic269C2×D412yes—
154C2×Dic269C2×D412yes—
155C4×D269—44no—
156C23×C269—21no—
157C23×C269—21no—
158C23×C269—21no—
159C23×C269—21no—
160C23×C269—21no—
161C23×C269—21no—
162C23×C269—21no—
163C23×C269—21no—
164C23×C269—28no—
165C23×C269—28no—
166C23×C269—21no—
167C23×C269—21no—
168C23×C269—21no—
169C23×C269—21no—
170C23×C269—21no—
171C23×C269—21no—
172C23×C269—44no—
173C23×C269—44no—
174C23×C269—44no—
175C23×C269—44no—
176C23×C269—44no—
177C23×C269C22×C412yes—
178C23×C269C22×C412yes—
179C23×C269C22×C412yes—
180C23×C269C2×D412yes—
181C23×C269C2×D412yes—
182C23×C269#4112yes—
183C23×C269#4112yes—
184C23×C269C2411yescharacteristic
185C2×C4×C269—21no—
186C2×C4×C269—21no—
187C2×C4×C269—21no—
188C2×C4×C269—21no—
189C2×C4×C269—21no—
190C2×C4×C269—21no—
191C2×C4×C269—21no—
192C2×C4×C269—21no—
193C2×C4×C269—21no—
194C2×C4×C269—21no—
195C2×C4×C269—44no—
196C2×C4×C269—44no—
197C2×C4×C269C2×D412yes—
198C2×C4×C269C2×D412yes—
199C22×C269—22no—
200C22×C269—22no—
201C22×C269—22no—
202C22×C269—22no—
203C22×C269—22no—
204C22×C269—22no—
205C22×C269—22no—
206C22×C269—22no—
207C22×C269—22no—
208C22×C269—22no—
209C22×C269—22no—
210C22×C269—22no—
211C22×C269—22no—
212C22×C269—22no—
213C22×C269—22no—
214C22×C269—416no—
215C22×C269—41no—
216C22×C269—41no—
217C22×C269—41no—
218C22×C269—41no—
219C22×C269C4×D414yescomplemented
220C22×C269C4×D414yescomplemented
221C22×C269C23×C411yescharacteristic
222C22×C269C22×D411yescharacteristic
223C22×C269C22×D411yescharacteristic
224C22×C269C2×#4111yescharacteristic
225C22×C269C2×#4111yescharacteristic
226C22×C269H(5,𝔽2)11yescharacteristic
227C22×C269H(5,𝔽2)11yescharacteristic
228C2×D269—22no—
229C2×D269—22no—
230C2×D269—22no—
231C2×D269—41no—
232C2×D269—41no—
233C2×D269—41no—
234C2×D269—41no—
235C4×C269—22no—
236C4×C269—22no—
237C4×C269—41no—
238C4×C269—41no—
239Dic269—22no—
240Dic269—22no—
241Dic269—22no—
242Dic269—22no—
243Dic269—22no—
244Dic269—41no—
245Dic269—41no—
246C2×C269—24no—
247C2×C269—24no—
248C2×C269—42no—
249C2×C269C2×C4×D411yescharacteristic
250C2×C269C2×C4×D411yescharacteristic
251C2×C269#51811yescharacteristic
252C2×C269D4211yescharacteristic
253C2×C269#54611yescharacteristic
254C2×C269#54611yescharacteristic
255C2×C269#55111yescharacteristic
256D269—42no—
257C269#235811yescharacteristic, complemented, Sylow
258#2358—2691nomaximal, Sylow
259C2×#120—2691no—
260C2×C4×D4—2691no—
261C2×C4×D4—2691no—
262C22×C22⋊C4—2691no—
263C22×C22⋊C4D26911yescharacteristic, complemented
264#390—2691no—
265#388—2691no—
266#388—2691no—
267#386—2691no—
268#386—2691no—
269#386—2691no—
270#385—2691no—
271#385—2691no—
272C4×C22⋊C4—2691no—
273C4×C22⋊C4—2691no—
274C25C2×D26911yescharacteristic
275C22×D4—2691no—
276C22×D4—2691no—
277C22×D4—2691no—
278C23×C4—2691no—
279C23×C4—2691no—
280C23×C4—2691no—
281C23×C4C2×D26911yescharacteristic
282C23×C4C2×D26911yescharacteristic
283#120—2698no—
284C4×D4—5384no—
285C4×D4—5384no—
286C2×C4⋊C4—2691no—
287C2×C4⋊C4—2691no—
288C2×C4⋊C4—2691no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—2691no—
291C2×C22⋊C4—2691no—
292C2×C22⋊C4—2691no—
293C2×C22⋊C4—2691no—
294C2×C22⋊C4—2691no—
295C2×C22⋊C4—2691no—
296C2×C22⋊C4—2691no—
297C2×C22⋊C4—2691no—
298C2×C22⋊C4—2691no—
299C2×C22⋊C4—2691no—
300C2×C22⋊C4—5384no—
301C2×C22⋊C4C2×D26911yescharacteristic
302C2×C22⋊C4C2×D26911yescharacteristic
303C2×C22⋊C4C2×D26911yescharacteristic
304C2×C22⋊C4C2×D26911yescharacteristic
305C2×C42—2691no—
306C2×C42—2691no—
307C2×C42—2691no—
308#95—2691no—
309#95—2691no—
310#95—2691no—
311#95—2691no—
312#95—2691no—
313#95—2691no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—2691no—
321C24C4×D26914yescomplemented
322C24C22×D26911yescharacteristic
323C24C22×D26911yescharacteristic
324C24C22×D26911yescharacteristic
325C2×D4—2694no—
326C2×D4—2694no—
327C2×D4—2694no—
328C2×D4—5382no—
329C2×D4—5382no—
330C2×D4—5382no—
331C2×D4—5382no—
332C2×D4—10761no—
333C2×D4—10761no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—2691no—
341C22×C4—2691no—
342C22×C4—2691no—
343C22×C4—2691no—
344C22×C4—2691no—
345C22×C4—2691no—
346C22×C4—2691no—
347C22×C4—5382no—
348C22×C4—5382no—
349C22×C4—5382no—
350C22×C4—5382no—
351C22×C4—5382no—
352C22×C4—5382no—
353C22×C4—10761no—
354C22×C4—10761no—
355C22×C4—10761no—
356C22×C4C22×D26911yescharacteristic
357C22×C4C22×D26911yescharacteristic
358C22×C4C22×D26911yescharacteristic
359C22×C4C22×D26911yescharacteristic
360C4⋊C4—5382no—
361C4⋊C4—5382no—
362C4⋊C4—10761no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—2694no—
367C22⋊C4—2694no—
368C22⋊C4—2694no—
369C22⋊C4—2694no—
370C22⋊C4—5382no—
371C22⋊C4—5382no—
372C22⋊C4—5382no—
373C22⋊C4—5382no—
374C22⋊C4—5382no—
375C22⋊C4—10761no—
376C22⋊C4C22⋊D26914yescomplemented
377C42—5382no—
378C42—5382no—
379C42—10761no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—28no—
389C23—28no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—2692no—
402C23—5381no—
403C23—5381no—
404C23—5381no—
405C23—5381no—
406C23—5381no—
407C23—5381no—
408C23C2×C4×D26912yes—
409C23C2×C4×D26912yes—
410C23C2×C4×D26912yes—
411C23D4×D26912yes—
412C23C2×C22⋊D26912yes—
413C23#1686440112yes—
414C23#1686440212yes—
415C23C23×D26911yescharacteristic
416D4—5388no—
417D4—5388no—
418D4—10764no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—2692no—
432C2×C4—2692no—
433C2×C4—2692no—
434C2×C4—2692no—
435C2×C4—2692no—
436C2×C4—5381no—
437C2×C4—5381no—
438C2×C4—5381no—
439C2×C4—5381no—
440C2×C4—5381no—
441C2×C4—5381no—
442C2×C4—5381no—
443C2×C4—5381no—
444C2×C4—5381no—
445C2×C4—5381no—
446C2×C4—5381no—
447C2×C4—5381no—
448C2×C4—5381no—
449C2×C4—5381no—
450C2×C4—5381no—
451C2×C4—5381no—
452C2×C4—10764no—
453C2×C4—10764no—
454C2×C4—10764no—
455C2×C4C2×C22⋊D26912yes—
456C2×C4C2×C22⋊D26912yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—416no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—5382no—
478C22—5382no—
479C22—5382no—
480C22—10761no—
481C22—10761no—
482C22—10761no—
483C22—10761no—
484C22#2257541314yescomplemented
485C22C4×C22⋊D26914yescomplemented
486C22C22×C4×D26911yescharacteristic
487C22C2×D4×D26911yescharacteristic
488C22C22×C22⋊D26911yescharacteristic
489C22C2×#1686440111yescharacteristic
490C22C2×#1686440211yescharacteristic
491C22#2257549811yescharacteristic
492C22#2257549811yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—5382no—
498C4—5382no—
499C4—5382no—
500C4—5382no—
501C4—5382no—
502C4—10761no—
503C4—10761no—
504C2—24no—
505C2—24no—
506C2—42no—
507C2—10762no—
508C2C2×#2257541311yescharacteristic
509C2C2×C4×C22⋊D26911yescharacteristic
510C2#3177107811yescharacteristic
511C2D4×C22⋊D26911yescharacteristic
512C2#3177124511yescharacteristic
513C2#3177124911yescharacteristic
514C2#3177128111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
FittingC22×C22⋊C4×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575422C2
hypercenterC22×C22⋊C4D269
hypocenterC269#2358
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D269C2semidirect
3. C2×#22575413C2semidirect
4. C22×C22⋊Dic269C2semidirect
5. #31770505C2semidirect
6. #31770499C2semidirect
7. #31770493C2semidirect
8. #31770484C2semidirect
9. #31770482C2semidirect
10. #31770481C2semidirect
11. #31770478C2semidirect
12. #31770477C2semidirect
13. C22⋊C4×Dic269C2semidirect
14. C4×C22⋊Dic269C2semidirect
15. C22×C22⋊C4×C269C2semidirect
21. C22×C4×D269C22semidirect
22. #22575422C4semidirect
25. C2×#16864381C22semidirect
26. C2×#16864381C22semidirect
27. C2×#16864381C22semidirect
36. C2×#16864379C22semidirect
37. C2×#16864379C22semidirect
38. C2×#16864379C22semidirect
39. C2×C4×Dic269C22semidirect
40. C2×C4×Dic269C22semidirect
41. C2×C4×Dic269C22semidirect
42. #22575360C22semidirect
43. #22575360C22semidirect
44. #22575360C22semidirect
45. #22575360C22semidirect
46. #22575360C22semidirect
47. #22575360C22semidirect
63. C2×C22⋊D269C2×C4semidirect
64. C2×C22⋊D269C2×C4semidirect
65. C2×C22⋊D269C2×C4semidirect
88. C22⋊Dic269C2×C4semidirect
89. C22⋊Dic269C2×C4semidirect
90. C22⋊Dic269C2×C4semidirect
91. C22⋊Dic269D4semidirect
104. C24×C269C2×C4semidirect
121. C22⋊C4×C269D4semidirect
219. C22×C269C4×D4semidirect
220. C22×C269C4×D4semidirect
257. C269#2358semidirect
263. C22×C22⋊C4D269semidirect
321. C24C4×D269semidirect
376. C22⋊C4C22⋊D269semidirect
484. C22#22575413semidirect
485. C22C4×C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754222
C2×C4×C22⋊D2692
C2×#225754132
C22×C22⋊Dic2692
#317705052
#317704992
#317704932
#317704842
#317704822
#317704812
#317704782
#317704772
C22⋊C4×Dic2692
C4×C22⋊Dic2692
C22×C22⋊C4×C2692
#2358269