Finite Group Databasebeta

← #46813538

Subgroups — #46813538

Order 34432.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575423C211yesmaximal, characteristic
2C2×#22575415C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D269C211yesmaximal, characteristic, complemented
4#31770507C211yesmaximal, characteristic, complemented
5#31770505C211yesmaximal, characteristic, complemented
6#31770498C211yesmaximal, characteristic, complemented
7#31770495C211yesmaximal, characteristic, complemented
8#31770494C211yesmaximal, characteristic, complemented
9#31770483C211yesmaximal, characteristic, complemented
10#31770483C211yesmaximal, characteristic, complemented
11#31770481C211yesmaximal, characteristic, complemented
12#31770479C211yesmaximal, characteristic, complemented
13#31770460C211yesmaximal, characteristic, complemented
14C4×#16864379C211yesmaximal, characteristic, complemented
15#388×C269C211yesmaximal, characteristic, complemented
16C24×D269C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C22×D1076C2211yescharacteristic
19C22×C22⋊D269C2211yescharacteristic
20C22×C4×D269C2211yescharacteristic
21C22×C4×D269C2211yescharacteristic
22C22×C4×D269C2211yescharacteristic, complemented
23#22575423C418yescomplemented
24#22575415—24no—
25C2×C4⋊Dic269C2211yescharacteristic, complemented
26C2×#16864381C2211yescharacteristic, complemented
27C2×C22⋊Dic269C2211yescharacteristic
28C2×C22⋊Dic269C2211yescharacteristic
29C2×#16864379C2211yescharacteristic
30C2×#16864379C2211yescharacteristic
31C2×#16864379C2211yescharacteristic
32C2×#16864379C2211yescharacteristic
33C2×#16864379C2211yescharacteristic
34C2×#16864379C2211yescharacteristic
35C2×#16864379C2211yescharacteristic
36C2×#16864379C2211yescharacteristic
37C2×#16864379C2211yescharacteristic, complemented
38C2×#16864379C2211yescharacteristic, complemented
39C2×#16864379C2211yescharacteristic, complemented
40C22⋊C4×D269—24no—
41C2×C4×Dic269C2211yescharacteristic, complemented
42C2×C4×Dic269C2211yescharacteristic, complemented
43#22575360C2211yescharacteristic, complemented
44#22575360C2211yescharacteristic, complemented
45#22575360C2211yescharacteristic, complemented
46#22575360C2211yescharacteristic, complemented
47C2×C4⋊C4×C269C2211yescharacteristic, complemented
48C2×C22⋊C4×C269C2211yescharacteristic
49C2×C22⋊C4×C269C2211yescharacteristic
50C2×C22⋊C4×C269C2211yescharacteristic
51C2×C42×C269C2211yescharacteristic, complemented
52#95×C269C2211yescharacteristic, complemented
53#95×C269C2211yescharacteristic, complemented
54C23×D269—22no—
55C23×D269—22no—
56C23×D269—41no—
57C23×D269—41no—
58C23×D269—41no—
59C23×D269—41no—
60C23×D269C2×C414yescomplemented
61C23×D269C2311yescharacteristic
62C23×D269C2311yescharacteristic
63C23×D269C2311yescharacteristic
64C2×D1076—22no—
65C2×D1076—22no—
66C2×D1076—41no—
67C2×D1076—41no—
68C2×D1076C2×C414yes—
69C2×D1076C2×C414yescomplemented
70C2×C22⋊D269—41no—
71C2×C22⋊D269—41no—
72C2×C22⋊D269C2×C414yescomplemented
73C22×Dic269C2311yescharacteristic
74C22×Dic269C2311yescharacteristic
75C22×Dic269C2311yescharacteristic
76C22×Dic269C2311yescharacteristic
77C22×Dic269C2311yescharacteristic
78C2×C4×D269—22no—
79C2×C4×D269—22no—
80C2×C4×D269—41no—
81C2×C4×D269—41no—
82C2×C4×D269—41no—
83C2×C4×D269—41no—
84C2×C4×D269—41no—
85C2×C4×D269—41no—
86C2×C4×D269D414yescomplemented
87C4⋊Dic269—41no—
88#16864381—41no—
89C22⋊Dic269—22no—
90C22⋊Dic269—41no—
91#16864379—22no—
92#16864379—22no—
93#16864379—22no—
94#16864379—22no—
95#16864379—22no—
96#16864379—41no—
97#16864379—41no—
98#16864379—41no—
99#16864379—41no—
100#16864379C2×C414yescomplemented
101#16864379C2×C414yescomplemented
102C4×Dic269—22no—
103C4×Dic269—41no—
104C24×C269C2311yescharacteristic
105C22×C4×C269C2311yescharacteristic
106C22×C4×C269C2311yescharacteristic
107C22×C4×C269C2311yescharacteristic
108C22×C4×C269C2311yescharacteristic
109C22×C4×C269C2311yescharacteristic
110C22×C4×C269C2311yescharacteristic
111C4⋊C4×C269—22no—
112C22⋊C4×C269—22no—
113C22⋊C4×C269—41no—
114C22⋊C4×C269C2×C414yescomplemented
115C42×C269—41no—
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—21no—
129C22×D269—21no—
130C22×D269—21no—
131C22×D269—21no—
132C22×D269—28no—
133C22×D269—44no—
134C22×D269—44no—
135C22×D269—44no—
136C22×D269—44no—
137C22×D269—44no—
138C22×D269—44no—
139C22×D269C22×C412yes—
140C22×D269C22×C412yes—
141C22×D269C22×C412yes—
142C22×D269C2×D412yes—
143C22×D269#4112yes—
144D1076—28no—
145D1076—44no—
146C22⋊D269—44no—
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×Dic269—21no—
156C2×Dic269—21no—
157C2×Dic269—21no—
158C2×Dic269—21no—
159C2×Dic269—21no—
160C2×Dic269C22×C412yes—
161C2×Dic269C2×D412yes—
162C4×D269—44no—
163C4×D269—44no—
164C4×D269—44no—
165C23×C269—21no—
166C23×C269—21no—
167C23×C269—21no—
168C23×C269—21no—
169C23×C269—21no—
170C23×C269—21no—
171C23×C269C22×C412yes—
172C23×C269C2411yescharacteristic
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×C269—21no—
182C2×C4×C269—21no—
183C2×C4×C269—21no—
184C2×C4×C269—21no—
185C2×C4×C269—21no—
186C2×C4×C269—21no—
187C2×C4×C269—21no—
188C2×C4×C269C22×C412yes—
189C2×C4×C269C22×C412yes—
190C2×C4×C269C2×D412yes—
191C22×C269—22no—
192C22×C269—22no—
193C22×C269—22no—
194C22×C269—41no—
195C22×C269—41no—
196C22×C269—41no—
197C22×C269—41no—
198C22×C269C23×C411yescharacteristic
199C22×C269C22×D411yescharacteristic
200C22×C269C2×#4111yescharacteristic
201C22×C269H(5,𝔽2)11yescharacteristic
202C22×C269H(5,𝔽2)11yescharacteristic
203C22×C269H(5,𝔽2)11yescharacteristic
204C22×C269H(5,𝔽2)11yescharacteristic
205C2×D269—22no—
206C2×D269—22no—
207C2×D269—22no—
208C2×D269—22no—
209C2×D269—22no—
210C2×D269—22no—
211C2×D269—22no—
212C2×D269—22no—
213C2×D269—22no—
214C2×D269—22no—
215C2×D269—22no—
216C2×D269—22no—
217C2×D269—416no—
218C2×D269—41no—
219C2×D269—41no—
220C2×D269—41no—
221C2×D269—41no—
222C2×D269—41no—
223C2×D269—41no—
224C2×D269—41no—
225C2×D269—41no—
226C2×D269C4×D414yes—
227C4×C269—22no—
228C4×C269—22no—
229C4×C269—22no—
230C4×C269—41no—
231C4×C269—41no—
232C4×C269—41no—
233Dic269—22no—
234Dic269—22no—
235Dic269—41no—
236Dic269—41no—
237Dic269—41no—
238C2×C269—42no—
239C2×C269C2×C4×D411yescharacteristic
240C2×C269#51811yescharacteristic
241C2×C269#51811yescharacteristic
242C2×C269#53411yescharacteristic
243C2×C269#53511yescharacteristic
244C2×C269#53811yescharacteristic
245C2×C269#53811yescharacteristic
246D269—24no—
247D269—42no—
248D269—42no—
249C269#232111yescharacteristic, complemented, Sylow
250#2321—2691nomaximal, Sylow
251C2×#120—2691no—
252C2×C4×D4—2691no—
253C22×C22⋊C4—2691no—
254#390—2691no—
255#390—2691no—
256#388—2691no—
257#388—2691no—
258#388—2691no—
259#388D26911yescharacteristic, complemented
260#386—2691no—
261#386—2691no—
262#386—2691no—
263#385—2691no—
264#379—2691no—
265C4×C22⋊C4—2691no—
266C25—2691no—
267C22×D4—2691no—
268C22×D4—2691no—
269C22×D4—2691no—
270C23×C4—2691no—
271C23×C4—2691no—
272C23×C4—2691no—
273#120—2698no—
274C4×D4—5384no—
275C2×C4⋊C4—2691no—
276C2×C4⋊C4—2691no—
277C2×C4⋊C4C2×D26911yescharacteristic, complemented
278C2×C22⋊C4—2691no—
279C2×C22⋊C4—2691no—
280C2×C22⋊C4—2691no—
281C2×C22⋊C4—2691no—
282C2×C22⋊C4—2691no—
283C2×C22⋊C4—2691no—
284C2×C22⋊C4—2691no—
285C2×C22⋊C4—2691no—
286C2×C22⋊C4—2691no—
287C2×C22⋊C4—2691no—
288C2×C22⋊C4—2691no—
289C2×C22⋊C4—2691no—
290C2×C22⋊C4—2691no—
291C2×C22⋊C4—5384no—
292C2×C22⋊C4C2×D26911yescharacteristic
293C2×C22⋊C4C2×D26911yescharacteristic
294C2×C22⋊C4C2×D26911yescharacteristic
295C2×C42—2691no—
296C2×C42—2691no—
297C2×C42C2×D26911yescharacteristic, complemented
298#95—2691no—
299#95—2691no—
300#95—2691no—
301#95—2691no—
302#95C2×D26911yescharacteristic, complemented
303#95C2×D26911yescharacteristic, complemented
304C24—2691no—
305C24—2691no—
306C24—2691no—
307C24—2694no—
308C24—5382no—
309C24—5382no—
310C24—10761no—
311C24—10761no—
312C24—10761no—
313C24—10761no—
314C24C22×D26911yescharacteristic
315C2×D4—2694no—
316C2×D4—2694no—
317C2×D4—2694no—
318C2×D4—5382no—
319C2×D4—5382no—
320C2×D4—10761no—
321C2×D4—10761no—
322C2×D4—10761no—
323C2×D4—10761no—
324C22×C4—2691no—
325C22×C4—2691no—
326C22×C4—2694no—
327C22×C4—2691no—
328C22×C4—2691no—
329C22×C4—2691no—
330C22×C4—5382no—
331C22×C4—5382no—
332C22×C4—10761no—
333C22×C4—10761no—
334C22×C4—10761no—
335C22×C4—10761no—
336C22×C4—10761no—
337C22×C4—10761no—
338C22×C4C22×D26911yescharacteristic
339C22×C4C22×D26911yescharacteristic
340C22×C4C22×D26911yescharacteristic
341C22×C4C22×D26911yescharacteristic
342C22×C4C22×D26911yescharacteristic
343C22×C4C22×D26911yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—10761no—
346C4⋊C4—10761no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—2694no—
350C22⋊C4—2694no—
351C22⋊C4—5382no—
352C22⋊C4—5382no—
353C22⋊C4—5382no—
354C22⋊C4—5382no—
355C22⋊C4—5382no—
356C22⋊C4—5382no—
357C22⋊C4—10761no—
358C22⋊C4—10761no—
359C22⋊C4—10761no—
360C22⋊C4—10761no—
361C22⋊C4—10761no—
362C22⋊C4C4×D26914yescomplemented
363C42—41no—
364C42—5382no—
365C42—10761no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—2692no—
373C23—2692no—
374C23—2692no—
375C23—2692no—
376C23—2692no—
377C23—5388no—
378C23—5381no—
379C23—5381no—
380C23—5381no—
381C23—5381no—
382C23—5381no—
383C23—5381no—
384C23—5381no—
385C23—5381no—
386C23—5381no—
387C23—5381no—
388C23—5381no—
389C23—5381no—
390C23—5381no—
391C23—5381no—
392C23—5381no—
393C23—5381no—
394C23—10764no—
395C23—10764no—
396C23—10764no—
397C23—10764no—
398C23—10764no—
399C23—10764no—
400C23C2×C4×D26912yes—
401C23C23×D26911yescharacteristic
402D4—5388no—
403D4—10764no—
404D4—10764no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—2692no—
421C2×C4—2692no—
422C2×C4—5381no—
423C2×C4—5381no—
424C2×C4—5381no—
425C2×C4—5381no—
426C2×C4—5381no—
427C2×C4—5381no—
428C2×C4—5381no—
429C2×C4—5381no—
430C2×C4—5381no—
431C2×C4—5381no—
432C2×C4—5381no—
433C2×C4—5381no—
434C2×C4—5381no—
435C2×C4—10764no—
436C2×C4—10764no—
437C2×C4—10764no—
438C2×C4C2×C4×D26912yes—
439C2×C4C2×C4×D26912yes—
440C2×C4D4×D26912yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—2694no—
449C22—5382no—
450C22—5382no—
451C22—5382no—
452C22—5382no—
453C22—5382no—
454C22—5382no—
455C22—5382no—
456C22—5382no—
457C22—5382no—
458C22—5382no—
459C22—5382no—
460C22—5382no—
461C22—107616no—
462C22—10761no—
463C22—10761no—
464C22—10761no—
465C22—10761no—
466C22—10761no—
467C22—10761no—
468C22—10761no—
469C22—10761no—
470C22C22×C4×D26911yescharacteristic
471C22C2×D4×D26911yescharacteristic
472C22#41×D26911yescharacteristic
473C22#2257549811yescharacteristic
474C22#2257549911yescharacteristic
475C22#2257549911yescharacteristic
476C22#2257549911yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—5382no—
484C4—5382no—
485C4—10761no—
486C4—10761no—
487C4—10761no—
488C2—42no—
489C2—5384no—
490C2—10762no—
491C2—10762no—
492C2C4×D4×D26911yescharacteristic
493C2#3177107711yescharacteristic
494C2#3177107911yescharacteristic
495C2#3177116311yescharacteristic
496C2#3177117311yescharacteristic
497C2#3177119111yescharacteristic
498C2#3177119411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#388×C269C2
socle, cosocleC23×C269C24
prime radicalC2×#22575423C2
hypercenter#388D269
hypocenterC269#2321
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22575415C2semidirect
3. C2×C22⋊C4×D269C2semidirect
4. #31770507C2semidirect
5. #31770505C2semidirect
6. #31770498C2semidirect
7. #31770495C2semidirect
8. #31770494C2semidirect
9. #31770483C2semidirect
10. #31770483C2semidirect
11. #31770481C2semidirect
12. #31770479C2semidirect
13. #31770460C2semidirect
14. C4×#16864379C2semidirect
15. #388×C269C2semidirect
22. C22×C4×D269C22semidirect
23. #22575423C4semidirect
25. C2×C4⋊Dic269C22semidirect
26. C2×#16864381C22semidirect
37. C2×#16864379C22semidirect
38. C2×#16864379C22semidirect
39. C2×#16864379C22semidirect
41. C2×C4×Dic269C22semidirect
42. C2×C4×Dic269C22semidirect
43. #22575360C22semidirect
44. #22575360C22semidirect
45. #22575360C22semidirect
46. #22575360C22semidirect
47. C2×C4⋊C4×C269C22semidirect
51. C2×C42×C269C22semidirect
52. #95×C269C22semidirect
53. #95×C269C22semidirect
60. C23×D269C2×C4semidirect
69. C2×D1076C2×C4semidirect
72. C2×C22⋊D269C2×C4semidirect
86. C2×C4×D269D4semidirect
100. #16864379C2×C4semidirect
101. #16864379C2×C4semidirect
114. C22⋊C4×C269C2×C4semidirect
249. C269#2321semidirect
259. #388D269semidirect
277. C2×C4⋊C4C2×D269semidirect
297. C2×C42C2×D269semidirect
302. #95C2×D269semidirect
303. #95C2×D269semidirect
362. 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
C2×#225754152
C2×C22⋊C4×D2692
#317705072
#317705052
#317704982
#317704952
#317704942
#317704832
#317704812
#317704792
#317704602
C4×#168643792
#388×C2692
#2321269