Finite Group Databasebeta

← #13074991

Subgroups — #13074991

Order 1408.

688 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66990C211yesmaximal, characteristic
2C2×#66938C211yesmaximal, characteristic, complemented
3C11×#394C211yesmaximal, characteristic, complemented
4#10622833C211yesmaximal, characteristic, complemented
5#10622606C211yesmaximal, characteristic, complemented
6#10622603C211yesmaximal, characteristic, complemented
7#10622574C211yesmaximal, characteristic, complemented
8#10622571C211yesmaximal, characteristic, complemented
9#10622571C211yesmaximal, characteristic, complemented
10#10622568C211yesmaximal, characteristic, complemented
11#10622568C211yesmaximal, characteristic, complemented
12#10622564C211yesmaximal, characteristic, complemented
13#10622560C211yesmaximal, characteristic, complemented
14#10622287C211yesmaximal, characteristic, complemented
15#10622286C211yesmaximal, characteristic, complemented
16C24×D11C2211yescharacteristic
17C22×D4×C11C2211yescharacteristic
18C22×C22⋊D11C2211yescharacteristic
19C22×C22⋊D11C2211yescharacteristic
20C23×Dic11C2211yescharacteristic, complemented
21C22×Dic22C2211yescharacteristic, complemented
22C2×C11×C4⋊C4C2211yescharacteristic, complemented
23C2×C11×C22⋊C4C2211yescharacteristic
24C2×C11×C22⋊C4C2211yescharacteristic
25C2×C11×C22⋊C4C2211yescharacteristic, complemented
26C2×C11×C22⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic11C2211yescharacteristic
28C2×C22⋊Dic11C2211yescharacteristic
29C2×C22⋊Dic11C2211yescharacteristic
30C2×C22⋊Dic11C2211yescharacteristic, complemented
31#66990—24no—
32C2×(C2×D11)⋊C4C2211yescharacteristic
33C2×(C2×D11)⋊C4C2211yescharacteristic
34C2×(C2×D11)⋊C4C2211yescharacteristic
35C2×(C2×D11)⋊C4C2211yescharacteristic
36C2×(C2×D11)⋊C4C2211yescharacteristic
37C2×(C2×D11)⋊C4C2211yescharacteristic
38C2×(C2×D11)⋊C4C2211yescharacteristic
39C2×(C2×D11)⋊C4C2211yescharacteristic
40C2×(C2×D11)⋊C4C2211yescharacteristic
41C2×(C2×D11)⋊C4C2211yescharacteristic
42C2×C4⋊Dic11C2211yescharacteristic, complemented
43C2×C4×Dic11C2211yescharacteristic, complemented
44#66938—24no—
45C11×#95C2211yescharacteristic, complemented
46C11⋊#95C2211yescharacteristic, complemented
47C11⋊#95C2211yescharacteristic, complemented
48C11⋊#95C2211yescharacteristic, complemented
49C11⋊#95C2211yescharacteristic, complemented
50C11⋊#95C2211yescharacteristic, complemented
51C11⋊#95C2211yescharacteristic, complemented
52C11⋊#95C2211yescharacteristic, complemented
53C24×C11C2311yescharacteristic
54C24×C11C2311yescharacteristic
55C23×D11—22no—
56C23×D11—22no—
57C23×D11—22no—
58C23×D11—41no—
59C23×D11—41no—
60C23×D11—41no—
61C23×D11—41no—
62C23×D11C2311yescharacteristic
63C23×D11C2311yescharacteristic
64C2×D4×C11—22no—
65C2×D4×C11—22no—
66C2×D4×C11—41no—
67C22×C4×C11C2311yescharacteristic
68C22×C4×C11C2311yescharacteristic
69C22×C4×C11C2311yescharacteristic
70C22×C4×C11C2311yescharacteristic
71C22×C4×C11C2311yescharacteristic
72C2×C22⋊D11—22no—
73C2×C22⋊D11—22no—
74C2×C22⋊D11—41no—
75C2×C22⋊D11—41no—
76C2×C22⋊D11—41no—
77C2×C22⋊D11D414yes—
78C22×Dic11—41no—
79C22×Dic11—41no—
80C22×Dic11—41no—
81C22×Dic11C2311yescharacteristic
82C22×Dic11C2311yescharacteristic
83C22×Dic11C2311yescharacteristic
84C22×Dic11C2311yescharacteristic
85C22×Dic11C2311yescharacteristic
86C22×Dic11C2311yescharacteristic
87C2×Dic22—22no—
88C2×Dic22—41no—
89C2×Dic22—41no—
90C11×C4⋊C4—41no—
91C11×C22⋊C4—22no—
92C11×C22⋊C4—22no—
93C11×C22⋊C4—41no—
94C11×C22⋊C4—41no—
95C22⋊Dic11—22no—
96C22⋊Dic11—22no—
97C22⋊Dic11—41no—
98C22⋊Dic11—41no—
99(C2×D11)⋊C4—22no—
100(C2×D11)⋊C4—22no—
101(C2×D11)⋊C4—22no—
102(C2×D11)⋊C4—22no—
103(C2×D11)⋊C4—22no—
104(C2×D11)⋊C4—41no—
105(C2×D11)⋊C4—41no—
106(C2×D11)⋊C4—41no—
107(C2×D11)⋊C4—41no—
108(C2×D11)⋊C4—41no—
109C4⋊Dic11—41no—
110C4×Dic11—22no—
111#2684—111nomaximal, Sylow
112C23×C11—21no—
113C23×C11—21no—
114C23×C11—21no—
115C23×C11—21no—
116C23×C11—21no—
117C23×C11—21no—
118C23×C11—21no—
119C23×C11—21no—
120C23×C11—21no—
121C23×C11—21no—
122C23×C11—21no—
123C23×C11—21no—
124C23×C11C2×D412yes—
125C23×C11#4112yes—
126C23×C11C2411yescharacteristic
127C22×D11—21no—
128C22×D11—21no—
129C22×D11—21no—
130C22×D11—21no—
131C22×D11—21no—
132C22×D11—21no—
133C22×D11—21no—
134C22×D11—21no—
135C22×D11—21no—
136C22×D11—21no—
137C22×D11—21no—
138C22×D11—21no—
139C22×D11—44no—
140C22×D11—44no—
141C22×D11—44no—
142C22×D11—44no—
143C22×D11—44no—
144C22×D11—44no—
145C22×D11—44no—
146C22×D11C2×D412yes—
147C22×D11#4112yes—
148D4×C11—44no—
149C2×C4×C11—21no—
150C2×C4×C11—21no—
151C2×C4×C11—21no—
152C2×C4×C11—21no—
153C2×C4×C11—21no—
154C2×C4×C11—21no—
155C2×C4×C11—21no—
156C2×C4×C11—21no—
157C2×C4×C11—21no—
158C2×C4×C11—21no—
159C2×C4×C11—21no—
160C2×C4×C11—21no—
161C2×C4×C11—21no—
162C2×C4×C11—21no—
163C2×C4×C11—21no—
164C22⋊D11—44no—
165C22⋊D11—44no—
166C2×Dic11—21no—
167C2×Dic11—21no—
168C2×Dic11—21no—
169C2×Dic11—21no—
170C2×Dic11—21no—
171C2×Dic11—21no—
172C2×Dic11—21no—
173C2×Dic11—21no—
174C2×Dic11—21no—
175C2×Dic11—21no—
176C2×Dic11—21no—
177C2×Dic11—21no—
178C2×Dic11—21no—
179C2×Dic11—21no—
180C2×Dic11—21no—
181C2×Dic11—21no—
182C2×Dic11—21no—
183C2×Dic11—44no—
184C2×Dic11C2×D412yes—
185Dic22—44no—
186C2×#124—111no—
187C2×#120—111no—
188#397—111no—
189#397—111no—
190#394—111no—
191#394D1111yescharacteristic, complemented
192#393—111no—
193#393—111no—
194#388—111no—
195#388—111no—
196#386—111no—
197#386—111no—
198#386—111no—
199#379—111no—
200#379—111no—
201C22×C11—22no—
202C22×C11—22no—
203C22×C11—22no—
204C22×C11—22no—
205C22×C11—22no—
206C22×C11—22no—
207C22×C11—41no—
208C22×C11—41no—
209C22×C11—41no—
210C22×C11—41no—
211C22×C11—41no—
212C22×C11—41no—
213C22×C11—41no—
214C22×C11—41no—
215C22×C11C22×D411yescharacteristic
216C22×C11C2×#4111yescharacteristic
217C22×C11C2×#4111yescharacteristic
218C22×C11H(5,𝔽2)11yescharacteristic
219C22×C11H(5,𝔽2)11yescharacteristic
220C22×C11H(5,𝔽2)11yescharacteristic
221C22×C11H(5,𝔽2)11yescharacteristic
222C2×D11—22no—
223C2×D11—22no—
224C2×D11—22no—
225C2×D11—22no—
226C2×D11—22no—
227C2×D11—22no—
228C2×D11—416no—
229C2×D11—41no—
230C2×D11—41no—
231C2×D11—41no—
232C2×D11—41no—
233C2×D11—41no—
234C2×D11—41no—
235C2×D11—41no—
236C2×D11—41no—
237C4×C11—41no—
238C4×C11—41no—
239C4×C11—41no—
240C4×C11—41no—
241C4×C11—41no—
242Dic11—22no—
243Dic11—41no—
244Dic11—41no—
245Dic11—41no—
246Dic11—41no—
247Dic11—41no—
248C25—111no—
249C22×Q8—111no—
250C22×D4—111no—
251C22×D4—111no—
252C22×D4C2×D1111yescharacteristic
253C23×C4—111no—
254#124—224no—
255#120—224no—
256C2×C4⋊C4—111no—
257C2×C4⋊C4C2×D1111yescharacteristic, complemented
258C2×C22⋊C4—111no—
259C2×C22⋊C4—111no—
260C2×C22⋊C4—111no—
261C2×C22⋊C4—111no—
262C2×C22⋊C4—111no—
263C2×C22⋊C4—111no—
264C2×C22⋊C4—111no—
265C2×C22⋊C4—111no—
266C2×C22⋊C4—111no—
267C2×C22⋊C4—111no—
268C2×C22⋊C4—111no—
269C2×C22⋊C4—111no—
270C2×C22⋊C4—111no—
271C2×C22⋊C4—111no—
272C2×C22⋊C4C2×D1111yescharacteristic
273C2×C22⋊C4C2×D1111yescharacteristic
274C2×C22⋊C4C2×D1111yescharacteristic, complemented
275C2×C22⋊C4C2×D1111yescharacteristic, complemented
276C2×C42—111no—
277#95—111no—
278#95—111no—
279#95—111no—
280#95—111no—
281#95—111no—
282#95—111no—
283#95—111no—
284#95C2×D1111yescharacteristic, complemented
285C2×C11—42no—
286C2×C11—42no—
287C2×C11#53411yescharacteristic
288C2×C11#53811yescharacteristic
289C2×C11#53811yescharacteristic
290C2×C11#54611yescharacteristic
291C2×C11#54611yescharacteristic
292C2×C11#55111yescharacteristic
293C2×C11H(3,𝔽4)11yescharacteristic
294D11—42no—
295D11—42no—
296C24—111no—
297C24—111no—
298C24—222no—
299C24—222no—
300C24—222no—
301C24—441no—
302C24—441no—
303C24—441no—
304C24—441no—
305C24C22×D1111yescharacteristic
306C24C22×D1111yescharacteristic
307C2×Q8—222no—
308C2×Q8—441no—
309C2×Q8—441no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—41no—
313C2×D4—114no—
314C2×D4—222no—
315C2×D4—222no—
316C2×D4—441no—
317C2×D4—441no—
318C2×D4—441no—
319C22×C4—111no—
320C22×C4—111no—
321C22×C4—111no—
322C22×C4—111no—
323C22×C4—111no—
324C22×C4—111no—
325C22×C4—441no—
326C22×C4—441no—
327C22×C4—441no—
328C22×C4C22×D1111yescharacteristic
329C22×C4C22×D1111yescharacteristic
330C22×C4C22×D1111yescharacteristic
331C22×C4C22×D1111yescharacteristic
332C22×C4C22×D1111yescharacteristic
333C4⋊C4—41no—
334C4⋊C4—441no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—222no—
340C22⋊C4—222no—
341C22⋊C4—222no—
342C22⋊C4—222no—
343C22⋊C4—222no—
344C22⋊C4—222no—
345C22⋊C4—222no—
346C22⋊C4—441no—
347C22⋊C4—441no—
348C22⋊C4—441no—
349C22⋊C4—441no—
350C22⋊C4—441no—
351C22⋊C4—441no—
352C22⋊C4—441no—
353C42—222no—
354C11#268411yescharacteristic, complemented, Sylow
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—112no—
368C23—112no—
369C23—221no—
370C23—221no—
371C23—221no—
372C23—221no—
373C23—221no—
374C23—221no—
375C23—221no—
376C23—221no—
377C23—221no—
378C23—221no—
379C23—221no—
380C23—221no—
381C23—444no—
382C23—444no—
383C23—444no—
384C23—444no—
385C23—444no—
386C23—444no—
387C23—444no—
388C23D4×D1112yes—
389C23#434112yes—
390C23C23×D1111yescharacteristic
391Q8—444no—
392D4—44no—
393D4—444no—
394D4—444no—
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—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—112no—
411C2×C4—221no—
412C2×C4—221no—
413C2×C4—221no—
414C2×C4—221no—
415C2×C4—221no—
416C2×C4—221no—
417C2×C4—221no—
418C2×C4—221no—
419C2×C4—221no—
420C2×C4—221no—
421C2×C4—221no—
422C2×C4—221no—
423C2×C4—221no—
424C2×C4—221no—
425C2×C4—221no—
426C2×C4—221no—
427C2×C4—221no—
428C2×C4—444no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—222no—
444C22—222no—
445C22—222no—
446C22—222no—
447C22—222no—
448C22—222no—
449C22—4416no—
450C22—441no—
451C22—441no—
452C22—441no—
453C22—441no—
454C22—441no—
455C22—441no—
456C22—441no—
457C22—441no—
458C22C2×D4×D1111yescharacteristic
459C22C2×#434111yescharacteristic
460C22#6703711yescharacteristic
461C22#6703711yescharacteristic
462C22#41×D1111yescharacteristic
463C22#6704211yescharacteristic
464C22#6704211yescharacteristic
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—222no—
471C4—441no—
472C4—441no—
473C4—441no—
474C4—441no—
475C4—441no—
476C2—42no—
477C2—42no—
478C2—442no—
479C2—442no—
480C2#1062312211yescharacteristic
481C2#1062313611yescharacteristic
482C2#1062314411yescharacteristic
483C2#1062314811yescharacteristic
484C2#1062318511yescharacteristic
485C2#1062318911yescharacteristic
486C2#1062321411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#394C2
prime radicalC2×#66990C2
hypercenter#394D11
hypocenterC11#2684
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#66938C2semidirect
3. C11×#394C2semidirect
4. #10622833C2semidirect
5. #10622606C2semidirect
6. #10622603C2semidirect
7. #10622574C2semidirect
8. #10622571C2semidirect
9. #10622571C2semidirect
10. #10622568C2semidirect
11. #10622568C2semidirect
12. #10622564C2semidirect
13. #10622560C2semidirect
14. #10622287C2semidirect
15. #10622286C2semidirect
20. C23×Dic11C22semidirect
21. C22×Dic22C22semidirect
22. C2×C11×C4⋊C4C22semidirect
25. C2×C11×C22⋊C4C22semidirect
26. C2×C11×C22⋊C4C22semidirect
30. C2×C22⋊Dic11C22semidirect
42. C2×C4⋊Dic11C22semidirect
43. C2×C4×Dic11C22semidirect
45. C11×#95C22semidirect
46. C11⋊#95C22semidirect
47. C11⋊#95C22semidirect
48. C11⋊#95C22semidirect
49. C11⋊#95C22semidirect
50. C11⋊#95C22semidirect
51. C11⋊#95C22semidirect
52. C11⋊#95C22semidirect
191. #394D11semidirect
257. C2×C4⋊C4C2×D11semidirect
274. C2×C22⋊C4C2×D11semidirect
275. C2×C22⋊C4C2×D11semidirect
284. #95C2×D11semidirect
354. C11#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#669902
C2×#669382
C11×#3942
#106228332
#106226062
#106226032
#106225742
#106225712
#106225682
#106225642
#106225602
#106222872
#106222862
#268411