Finite Group Databasebeta

← #16822154

Subgroups — #16822154

Order 3712.

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×#11745244C211yesmaximal, characteristic
2C2×C4×D116C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D29C211yesmaximal, characteristic, complemented
4#14271064C211yesmaximal, characteristic, complemented
5#14271063C211yesmaximal, characteristic, complemented
6#14271056C211yesmaximal, characteristic, complemented
7#14271055C211yesmaximal, characteristic, complemented
8#14271054C211yesmaximal, characteristic, complemented
9#14271041C211yesmaximal, characteristic, complemented
10#14271041C211yesmaximal, characteristic, complemented
11#14271039C211yesmaximal, characteristic, complemented
12#14271034C211yesmaximal, characteristic, complemented
13#14271018C211yesmaximal, characteristic, complemented
14#14271010C211yesmaximal, characteristic, complemented
15C29×#388C211yesmaximal, characteristic, complemented
16C24×D29C2211yescharacteristic
17C22×C22⋊D29C2211yescharacteristic
18C22×D116C2211yescharacteristic
19C22×D116C2211yescharacteristic
20C22×C4×D29C2211yescharacteristic
21C22×C4×D29C2211yescharacteristic
22C22×C4×D29C2211yescharacteristic, complemented
23C2×C4⋊C4×C29C2211yescharacteristic, complemented
24C2×C22⋊C4×C29C2211yescharacteristic
25C2×C22⋊C4×C29C2211yescharacteristic
26C2×C22⋊C4×C29C2211yescharacteristic
27C2×C42×C29C2211yescharacteristic, complemented
28C2×C22⋊Dic29C2211yescharacteristic
29C2×C22⋊Dic29C2211yescharacteristic
30C2×#90781C2211yescharacteristic
31C2×#90781C2211yescharacteristic
32C2×#90781C2211yescharacteristic
33C2×#90781C2211yescharacteristic
34C2×#90781C2211yescharacteristic
35C2×#90781C2211yescharacteristic
36C2×#90781C2211yescharacteristic
37C2×#90781C2211yescharacteristic
38C2×#90781C2211yescharacteristic, complemented
39C2×#90781C2211yescharacteristic, complemented
40C2×#90781C2211yescharacteristic, complemented
41C2×C4⋊Dic29C2211yescharacteristic, complemented
42C2×#90779C2211yescharacteristic, complemented
43C2×C4×Dic29C2211yescharacteristic, complemented
44C2×C4×Dic29C2211yescharacteristic, complemented
45#11745244C418yescomplemented
46C22⋊C4×D29—24no—
47C4×D116—24no—
48C29×#95C2211yescharacteristic, complemented
49C29×#95C2211yescharacteristic, complemented
50#11745179C2211yescharacteristic, complemented
51#11745179C2211yescharacteristic, complemented
52#11745179C2211yescharacteristic, complemented
53#11745179C2211yescharacteristic, complemented
54C24×C29C2311yescharacteristic
55C23×D29—22no—
56C23×D29—22no—
57C23×D29—41no—
58C23×D29—41no—
59C23×D29—41no—
60C23×D29—41no—
61C23×D29C2×C414yescomplemented
62C23×D29C2311yescharacteristic
63C23×D29C2311yescharacteristic
64C23×D29C2311yescharacteristic
65C22×C4×C29C2311yescharacteristic
66C22×C4×C29C2311yescharacteristic
67C22×C4×C29C2311yescharacteristic
68C22×C4×C29C2311yescharacteristic
69C22×C4×C29C2311yescharacteristic
70C22×C4×C29C2311yescharacteristic
71C2×C22⋊D29—41no—
72C2×C22⋊D29—41no—
73C2×C22⋊D29C2×C414yescomplemented
74C22×Dic29C2311yescharacteristic
75C22×Dic29C2311yescharacteristic
76C22×Dic29C2311yescharacteristic
77C22×Dic29C2311yescharacteristic
78C22×Dic29C2311yescharacteristic
79C2×D116—22no—
80C2×D116—22no—
81C2×D116—41no—
82C2×D116—41no—
83C2×D116C2×C414yes—
84C2×D116C2×C414yescomplemented
85C2×C4×D29—22no—
86C2×C4×D29—22no—
87C2×C4×D29—41no—
88C2×C4×D29—41no—
89C2×C4×D29—41no—
90C2×C4×D29—41no—
91C2×C4×D29—41no—
92C2×C4×D29—41no—
93C2×C4×D29D414yescomplemented
94C4⋊C4×C29—41no—
95C22⋊C4×C29—22no—
96C22⋊C4×C29—41no—
97C22⋊C4×C29C2×C414yescomplemented
98C42×C29—22no—
99C22⋊Dic29—22no—
100C22⋊Dic29—41no—
101#90781—22no—
102#90781—22no—
103#90781—22no—
104#90781—22no—
105#90781—22no—
106#90781—41no—
107#90781—41no—
108#90781—41no—
109#90781—41no—
110#90781C2×C414yescomplemented
111#90781C2×C414yescomplemented
112C4⋊Dic29—22no—
113#90779—41no—
114C4×Dic29—41no—
115C4×Dic29—41no—
116C23×C29—21no—
117C23×C29—21no—
118C23×C29—21no—
119C23×C29—21no—
120C23×C29—21no—
121C23×C29—21no—
122C23×C29C22×C412yes—
123C23×C29C2411yescharacteristic
124C22×D29—21no—
125C22×D29—21no—
126C22×D29—21no—
127C22×D29—21no—
128C22×D29—21no—
129C22×D29—21no—
130C22×D29—21no—
131C22×D29—21no—
132C22×D29—21no—
133C22×D29—21no—
134C22×D29—21no—
135C22×D29—21no—
136C22×D29—21no—
137C22×D29—21no—
138C22×D29—21no—
139C22×D29—21no—
140C22×D29—28no—
141C22×D29—44no—
142C22×D29—44no—
143C22×D29—44no—
144C22×D29—44no—
145C22×D29—44no—
146C22×D29—44no—
147C22×D29C22×C412yes—
148C22×D29C22×C412yes—
149C22×D29C22×C412yes—
150C22×D29C2×D412yes—
151C22×D29#4112yes—
152C2×C4×C29—21no—
153C2×C4×C29—21no—
154C2×C4×C29—21no—
155C2×C4×C29—21no—
156C2×C4×C29—21no—
157C2×C4×C29—21no—
158C2×C4×C29—21no—
159C2×C4×C29—21no—
160C2×C4×C29—21no—
161C2×C4×C29—21no—
162C2×C4×C29—21no—
163C2×C4×C29—21no—
164C2×C4×C29—21no—
165C2×C4×C29—21no—
166C2×C4×C29—21no—
167C2×C4×C29C22×C412yes—
168C2×C4×C29C22×C412yes—
169C2×C4×C29C2×D412yes—
170C22⋊D29—44no—
171C2×Dic29—21no—
172C2×Dic29—21no—
173C2×Dic29—21no—
174C2×Dic29—21no—
175C2×Dic29—21no—
176C2×Dic29—21no—
177C2×Dic29—21no—
178C2×Dic29—21no—
179C2×Dic29—21no—
180C2×Dic29—21no—
181C2×Dic29—21no—
182C2×Dic29—21no—
183C2×Dic29—21no—
184C2×Dic29C22×C412yes—
185C2×Dic29C2×D412yes—
186D116—28no—
187D116—44no—
188C4×D29—44no—
189C4×D29—44no—
190C4×D29—44no—
191#2321—291nomaximal, Sylow
192C22×C29—22no—
193C22×C29—22no—
194C22×C29—22no—
195C22×C29—41no—
196C22×C29—41no—
197C22×C29—41no—
198C22×C29—41no—
199C22×C29C23×C411yescharacteristic
200C22×C29C22×D411yescharacteristic
201C22×C29C2×#4111yescharacteristic
202C22×C29H(5,𝔽2)11yescharacteristic
203C22×C29H(5,𝔽2)11yescharacteristic
204C22×C29H(5,𝔽2)11yescharacteristic
205C22×C29H(5,𝔽2)11yescharacteristic
206C2×D29—22no—
207C2×D29—22no—
208C2×D29—22no—
209C2×D29—22no—
210C2×D29—22no—
211C2×D29—22no—
212C2×D29—22no—
213C2×D29—22no—
214C2×D29—22no—
215C2×D29—22no—
216C2×D29—22no—
217C2×D29—22no—
218C2×D29—416no—
219C2×D29—41no—
220C2×D29—41no—
221C2×D29—41no—
222C2×D29—41no—
223C2×D29—41no—
224C2×D29—41no—
225C2×D29—41no—
226C2×D29—41no—
227C2×D29C4×D414yes—
228C4×C29—22no—
229C4×C29—22no—
230C4×C29—22no—
231C4×C29—41no—
232C4×C29—41no—
233C4×C29—41no—
234Dic29—22no—
235Dic29—22no—
236Dic29—41no—
237Dic29—41no—
238Dic29—41no—
239C2×#120—291no—
240C2×C4×D4—291no—
241C22×C22⋊C4—291no—
242#390—291no—
243#390—291no—
244#388—291no—
245#388—291no—
246#388—291no—
247#388D2911yescharacteristic, complemented
248#386—291no—
249#386—291no—
250#386—291no—
251#385—291no—
252#379—291no—
253C4×C22⋊C4—291no—
254C2×C29—42no—
255C2×C29C2×C4×D411yescharacteristic
256C2×C29#51811yescharacteristic
257C2×C29#51811yescharacteristic
258C2×C29#53411yescharacteristic
259C2×C29#53511yescharacteristic
260C2×C29#53811yescharacteristic
261C2×C29#53811yescharacteristic
262D29—24no—
263D29—42no—
264D29—42no—
265C25—291no—
266C22×D4—291no—
267C22×D4—291no—
268C22×D4—291no—
269C23×C4—291no—
270C23×C4—291no—
271C23×C4—291no—
272#120—298no—
273C4×D4—584no—
274C2×C4⋊C4—291no—
275C2×C4⋊C4—291no—
276C2×C4⋊C4C2×D2911yescharacteristic, complemented
277C2×C22⋊C4—291no—
278C2×C22⋊C4—291no—
279C2×C22⋊C4—291no—
280C2×C22⋊C4—291no—
281C2×C22⋊C4—291no—
282C2×C22⋊C4—291no—
283C2×C22⋊C4—291no—
284C2×C22⋊C4—291no—
285C2×C22⋊C4—291no—
286C2×C22⋊C4—291no—
287C2×C22⋊C4—291no—
288C2×C22⋊C4—291no—
289C2×C22⋊C4—291no—
290C2×C22⋊C4—584no—
291C2×C22⋊C4C2×D2911yescharacteristic
292C2×C22⋊C4C2×D2911yescharacteristic
293C2×C22⋊C4C2×D2911yescharacteristic
294C2×C42—291no—
295C2×C42—291no—
296C2×C42C2×D2911yescharacteristic, complemented
297#95—291no—
298#95—291no—
299#95—291no—
300#95—291no—
301#95C2×D2911yescharacteristic, complemented
302#95C2×D2911yescharacteristic, complemented
303C29#232111yescharacteristic, complemented, Sylow
304C24—291no—
305C24—291no—
306C24—291no—
307C24—294no—
308C24—582no—
309C24—582no—
310C24—1161no—
311C24—1161no—
312C24—1161no—
313C24—1161no—
314C24C22×D2911yescharacteristic
315C2×D4—294no—
316C2×D4—294no—
317C2×D4—294no—
318C2×D4—582no—
319C2×D4—582no—
320C2×D4—1161no—
321C2×D4—1161no—
322C2×D4—1161no—
323C2×D4—1161no—
324C22×C4—291no—
325C22×C4—291no—
326C22×C4—294no—
327C22×C4—291no—
328C22×C4—291no—
329C22×C4—291no—
330C22×C4—582no—
331C22×C4—582no—
332C22×C4—1161no—
333C22×C4—1161no—
334C22×C4—1161no—
335C22×C4—1161no—
336C22×C4—1161no—
337C22×C4—1161no—
338C22×C4C22×D2911yescharacteristic
339C22×C4C22×D2911yescharacteristic
340C22×C4C22×D2911yescharacteristic
341C22×C4C22×D2911yescharacteristic
342C22×C4C22×D2911yescharacteristic
343C22×C4C22×D2911yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—582no—
346C4⋊C4—1161no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—294no—
350C22⋊C4—294no—
351C22⋊C4—582no—
352C22⋊C4—582no—
353C22⋊C4—582no—
354C22⋊C4—582no—
355C22⋊C4—582no—
356C22⋊C4—582no—
357C22⋊C4—1161no—
358C22⋊C4—1161no—
359C22⋊C4—1161no—
360C22⋊C4—1161no—
361C22⋊C4—1161no—
362C22⋊C4C4×D2914yescomplemented
363C42—22no—
364C42—1161no—
365C42—1161no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—292no—
373C23—292no—
374C23—292no—
375C23—292no—
376C23—292no—
377C23—588no—
378C23—581no—
379C23—581no—
380C23—581no—
381C23—581no—
382C23—581no—
383C23—581no—
384C23—581no—
385C23—581no—
386C23—581no—
387C23—581no—
388C23—581no—
389C23—581no—
390C23—581no—
391C23—581no—
392C23—581no—
393C23—581no—
394C23—1164no—
395C23—1164no—
396C23—1164no—
397C23—1164no—
398C23—1164no—
399C23—1164no—
400C23C2×C4×D2912yes—
401C23C23×D2911yescharacteristic
402D4—588no—
403D4—1164no—
404D4—1164no—
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—292no—
421C2×C4—292no—
422C2×C4—581no—
423C2×C4—581no—
424C2×C4—581no—
425C2×C4—581no—
426C2×C4—581no—
427C2×C4—581no—
428C2×C4—581no—
429C2×C4—581no—
430C2×C4—581no—
431C2×C4—581no—
432C2×C4—581no—
433C2×C4—581no—
434C2×C4—581no—
435C2×C4—1164no—
436C2×C4—1164no—
437C2×C4—1164no—
438C2×C4C2×C4×D2912yes—
439C2×C4C2×C4×D2912yes—
440C2×C4D4×D2912yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—294no—
449C22—582no—
450C22—582no—
451C22—582no—
452C22—582no—
453C22—582no—
454C22—582no—
455C22—582no—
456C22—582no—
457C22—582no—
458C22—582no—
459C22—582no—
460C22—582no—
461C22—11616no—
462C22—1161no—
463C22—1161no—
464C22—1161no—
465C22—1161no—
466C22—1161no—
467C22—1161no—
468C22—1161no—
469C22—1161no—
470C22C22×C4×D2911yescharacteristic
471C22C2×D4×D2911yescharacteristic
472C22#1174535911yescharacteristic
473C22#41×D2911yescharacteristic
474C22#1174536411yescharacteristic
475C22#1174536411yescharacteristic
476C22#1174536411yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—582no—
484C4—582no—
485C4—1161no—
486C4—1161no—
487C4—1161no—
488C2—42no—
489C2—584no—
490C2—1162no—
491C2—1162no—
492C2C4×D4×D2911yescharacteristic
493C2#1427163511yescharacteristic
494C2#1427163711yescharacteristic
495C2#1427172411yescharacteristic
496C2#1427172911yescharacteristic
497C2#1427174711yescharacteristic
498C2#1427175211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutatorC22×C29C23×C4
FittingC29×#388C2
socle, cosocleC23×C29C24
prime radicalC2×#11745244C2
hypercenter#388D29
hypocenterC29#2321
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D116C2semidirect
3. C2×C22⋊C4×D29C2semidirect
4. #14271064C2semidirect
5. #14271063C2semidirect
6. #14271056C2semidirect
7. #14271055C2semidirect
8. #14271054C2semidirect
9. #14271041C2semidirect
10. #14271041C2semidirect
11. #14271039C2semidirect
12. #14271034C2semidirect
13. #14271018C2semidirect
14. #14271010C2semidirect
15. C29×#388C2semidirect
22. C22×C4×D29C22semidirect
23. C2×C4⋊C4×C29C22semidirect
27. C2×C42×C29C22semidirect
38. C2×#90781C22semidirect
39. C2×#90781C22semidirect
40. C2×#90781C22semidirect
41. C2×C4⋊Dic29C22semidirect
42. C2×#90779C22semidirect
43. C2×C4×Dic29C22semidirect
44. C2×C4×Dic29C22semidirect
45. #11745244C4semidirect
48. C29×#95C22semidirect
49. C29×#95C22semidirect
50. #11745179C22semidirect
51. #11745179C22semidirect
52. #11745179C22semidirect
53. #11745179C22semidirect
61. C23×D29C2×C4semidirect
73. C2×C22⋊D29C2×C4semidirect
84. C2×D116C2×C4semidirect
93. C2×C4×D29D4semidirect
97. C22⋊C4×C29C2×C4semidirect
110. #90781C2×C4semidirect
111. #90781C2×C4semidirect
247. #388D29semidirect
276. C2×C4⋊C4C2×D29semidirect
296. C2×C42C2×D29semidirect
301. #95C2×D29semidirect
302. #95C2×D29semidirect
303. C29#2321semidirect
362. C22⋊C4C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452442
C2×C4×D1162
C2×C22⋊C4×D292
#142710642
#142710632
#142710562
#142710552
#142710542
#142710412
#142710392
#142710342
#142710182
#142710102
C29×#3882
#232129