Finite Group Databasebeta

← #30670890

Subgroups — #30670890

Order 16768.

900 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862231C211yesmaximal, characteristic, complemented
2C2×#16862228C211yesmaximal, characteristic, complemented
3C2×#16862219C211yesmaximal, characteristic, complemented
4C2×#16862216C211yesmaximal, characteristic, complemented
5C2×#16862214C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
7#22553248C211yesmaximal, characteristic, complemented
8#22553247C211yesmaximal, characteristic, complemented
9#22553221C211yesmaximal, characteristic, complemented
10#22553220C211yesmaximal, characteristic, complemented
11#22553214C211yesmaximal, characteristic, complemented
12#22553212C211yesmaximal, characteristic, complemented
13#22553212C211yesmaximal, characteristic, complemented
14C4×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
15#390×C131C211yesmaximal, characteristic, complemented
16C24×D131C2211yescharacteristic
17C22×D524C2211yescharacteristic
18C22×D524C2211yescharacteristic
19C22×C22⋊D131C2211yescharacteristic
20C22×C22⋊D131C2211yescharacteristic
21C22×C22⋊D131C2211yescharacteristic
22C22×C22⋊D131C2211yescharacteristic, complemented
23C22×C4×D131C2211yescharacteristic
24C22×C4×D131C2211yescharacteristic
25#16862231—24no—
26#16862228—24no—
27#16862219—24no—
28#16862216—24no—
29#16862214—24no—
30C2×C4⋊Dic131C2211yescharacteristic, complemented
31C2×Dic131⋊C4C2211yescharacteristic, complemented
32C2×Dic131⋊C4C2211yescharacteristic, complemented
33C2×Dic131⋊C4C2211yescharacteristic, complemented
34C2×C22⋊Dic131C2211yescharacteristic
35C2×C22⋊Dic131C2211yescharacteristic
36C2×C22⋊Dic131C2211yescharacteristic, complemented
37C2×(C2×D131)⋊C4C2211yescharacteristic
38C2×(C2×D131)⋊C4C2211yescharacteristic
39C2×(C2×D131)⋊C4C2211yescharacteristic
40C2×(C2×D131)⋊C4C2211yescharacteristic
41C2×(C2×D131)⋊C4C2211yescharacteristic
42C2×(C2×D131)⋊C4C2211yescharacteristic
43C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
46C22⋊C4×D131—24no—
47C2×C4×Dic131C2211yescharacteristic, complemented
48C131⋊#95C2211yescharacteristic, complemented
49C131⋊#95C2211yescharacteristic, complemented
50C22×D4×C131C2211yescharacteristic
51C2×C4⋊C4×C131C2211yescharacteristic, complemented
52C2×C22⋊C4×C131C2211yescharacteristic
53C2×C22⋊C4×C131C2211yescharacteristic
54C2×C22⋊C4×C131C2211yescharacteristic, complemented
55C2×C22⋊C4×C131C2211yescharacteristic, complemented
56C2×C42×C131C2211yescharacteristic, complemented
57C23×D131—22no—
58C23×D131—22no—
59C23×D131—22no—
60C23×D131—22no—
61C23×D131—22no—
62C23×D131—41no—
63C23×D131—41no—
64C23×D131C2311yescharacteristic
65C23×D131C2311yescharacteristic
66C23×D131C2311yescharacteristic
67C2×D524—22no—
68C2×D524—22no—
69C2×D524—22no—
70C2×D524—22no—
71C2×D524—22no—
72C2×D524—41no—
73C2×C22⋊D131—22no—
74C2×C22⋊D131—22no—
75C2×C22⋊D131—22no—
76C2×C22⋊D131—22no—
77C2×C22⋊D131—22no—
78C2×C22⋊D131—22no—
79C2×C22⋊D131—22no—
80C2×C22⋊D131—22no—
81C2×C22⋊D131—41no—
82C2×C22⋊D131—41no—
83C2×C22⋊D131—41no—
84C2×C22⋊D131—41no—
85C22×Dic131C2311yescharacteristic
86C22×Dic131C2311yescharacteristic
87C22×Dic131C2311yescharacteristic
88C22×Dic131C2311yescharacteristic
89C22×Dic131C2311yescharacteristic
90C2×C4×D131—22no—
91C2×C4×D131—22no—
92C2×C4×D131—22no—
93C2×C4×D131—41no—
94C2×C4×D131—41no—
95C2×C4×D131D414yescomplemented
96C4⋊Dic131—22no—
97Dic131⋊C4—22no—
98Dic131⋊C4—22no—
99Dic131⋊C4—22no—
100C22⋊Dic131—22no—
101C22⋊Dic131—22no—
102C22⋊Dic131—22no—
103(C2×D131)⋊C4—22no—
104(C2×D131)⋊C4—22no—
105(C2×D131)⋊C4—22no—
106(C2×D131)⋊C4—22no—
107(C2×D131)⋊C4—22no—
108(C2×D131)⋊C4—22no—
109(C2×D131)⋊C4—22no—
110(C2×D131)⋊C4D414yescomplemented
111(C2×D131)⋊C4D414yescomplemented
112C4×Dic131—41no—
113C24×C131C2311yescharacteristic
114C24×C131C2311yescharacteristic
115C2×D4×C131—22no—
116C2×D4×C131—22no—
117C2×D4×C131—41no—
118C22×C4×C131C2311yescharacteristic
119C22×C4×C131C2311yescharacteristic
120C22×C4×C131C2311yescharacteristic
121C22×C4×C131C2311yescharacteristic
122C22×C4×C131C2311yescharacteristic
123C4⋊C4×C131—22no—
124C22⋊C4×C131—22no—
125C22⋊C4×C131—22no—
126C22⋊C4×C131—22no—
127C22⋊C4×C131—22no—
128C42×C131—41no—
129C22×D131—21no—
130C22×D131—21no—
131C22×D131—21no—
132C22×D131—21no—
133C22×D131—28no—
134C22×D131—21no—
135C22×D131—21no—
136C22×D131—21no—
137C22×D131—21no—
138C22×D131—21no—
139C22×D131—21no—
140C22×D131—21no—
141C22×D131—21no—
142C22×D131—21no—
143C22×D131—21no—
144C22×D131—21no—
145C22×D131—21no—
146C22×D131—21no—
147C22×D131—44no—
148C22×D131—44no—
149C22×D131—44no—
150C22×D131—44no—
151C22×D131—44no—
152C22×D131—44no—
153C22×D131C2×D412yes—
154C22×D131C2×D412yes—
155C22×D131C2×D412yes—
156C22×D131#4112yes—
157D524—44no—
158D524—44no—
159C22⋊D131—44no—
160C22⋊D131—44no—
161C22⋊D131—44no—
162C22⋊D131—44no—
163C2×Dic131—21no—
164C2×Dic131—21no—
165C2×Dic131—21no—
166C2×Dic131—21no—
167C2×Dic131—21no—
168C2×Dic131—21no—
169C2×Dic131—21no—
170C2×Dic131—21no—
171C2×Dic131—21no—
172C2×Dic131—21no—
173C2×Dic131—21no—
174C2×Dic131—21no—
175C2×Dic131C2×D412yes—
176C2×Dic131C2×D412yes—
177C2×Dic131C2×D412yes—
178C4×D131—44no—
179C4×D131—44no—
180C23×C131—21no—
181C23×C131—21no—
182C23×C131—21no—
183C23×C131—21no—
184C23×C131—21no—
185C23×C131—21no—
186C23×C131—21no—
187C23×C131—21no—
188C23×C131—21no—
189C23×C131—21no—
190C23×C131—21no—
191C23×C131—21no—
192C23×C131—21no—
193C23×C131#4112yes—
194C23×C131C2411yescharacteristic
195D4×C131—44no—
196C2×C4×C131—21no—
197C2×C4×C131—21no—
198C2×C4×C131—21no—
199C2×C4×C131—21no—
200C2×C4×C131—21no—
201C2×C4×C131—21no—
202C2×C4×C131—21no—
203C2×C4×C131—21no—
204C2×C4×C131—21no—
205C2×C4×C131—21no—
206C2×C4×C131—21no—
207C2×C4×C131—21no—
208C2×C4×C131C2×D412yes—
209C2×C4×C131C2×D412yes—
210C2×C4×C131C2×D412yes—
211C22×C131—22no—
212C22×C131—22no—
213C22×C131—22no—
214C22×C131—41no—
215C22×C131—41no—
216C22×C131—41no—
217C22×C131—41no—
218C22×C131—41no—
219C22×C131—41no—
220C22×C131—41no—
221C22×C131—41no—
222C22×C131—41no—
223C22×C131—41no—
224C22×C131—41no—
225C22×C131C22×D411yescharacteristic
226C22×C131C22×D411yescharacteristic
227C22×C131C22×D411yescharacteristic
228C22×C131C2×#4111yescharacteristic
229C22×C131C2×#4111yescharacteristic
230C22×C131H(5,𝔽2)11yescharacteristic
231C22×C131H(5,𝔽2)11yescharacteristic
232C2×D131—22no—
233C2×D131—22no—
234C2×D131—22no—
235C2×D131—22no—
236C2×D131—22no—
237C2×D131—22no—
238C2×D131—22no—
239C2×D131—22no—
240C2×D131—22no—
241C2×D131—416no—
242C2×D131—41no—
243C2×D131—41no—
244C2×D131—41no—
245C2×D131—41no—
246C2×D131—41no—
247C2×D131—41no—
248C2×D131—41no—
249C2×D131—41no—
250C2×D131—41no—
251C2×D131—41no—
252C2×D131—41no—
253C2×D131#12714yes—
254C4×C131—22no—
255C4×C131—22no—
256C4×C131—22no—
257C4×C131—41no—
258C4×C131—41no—
259Dic131—22no—
260Dic131—22no—
261Dic131—22no—
262Dic131—41no—
263Dic131—41no—
264C2×C131—42no—
265C2×C131—81no—
266C2×C131#52511yescharacteristic
267C2×C131C2×#12711yescharacteristic
268C2×C131#53411yescharacteristic
269C2×C131#54611yescharacteristic
270C2×C131#54611yescharacteristic
271C2×C131#54611yescharacteristic
272C2×C131#54611yescharacteristic
273D131—24no—
274D131—42no—
275D131—81no—
276C131#253911yescharacteristic, complemented, Sylow
277#2539—1311nomaximal, Sylow
278C2×#123—1311no—
279C2×#123—1311no—
280C2×#121—1311no—
281C2×#120—1311no—
282C2×#120—1311no—
283C22×C22⋊C4—1311no—
284#399—1311no—
285#399—1311no—
286#392—1311no—
287#392—1311no—
288#390—1311no—
289#390—1311no—
290#390—1311no—
291#390D13111yescharacteristic, complemented
292C4×C22⋊C4—1311no—
293C25—1311no—
294C22×D4—1311no—
295C22×D4—1311no—
296C22×D4—1311no—
297C22×D4—1311no—
298C22×D4—1311no—
299C22×D4—1311no—
300C22×D4C2×D13111yescharacteristic
301C23×C4—1311no—
302C23×C4—1311no—
303#123—2624no—
304#123—2624no—
305#121—2624no—
306#120—2624no—
307#120—2624no—
308C2×C4⋊C4—1311no—
309C2×C4⋊C4—1311no—
310C2×C4⋊C4—1311no—
311C2×C4⋊C4—1311no—
312C2×C4⋊C4C2×D13111yescharacteristic, complemented
313C2×C22⋊C4—2624no—
314C2×C22⋊C4—1311no—
315C2×C22⋊C4—1311no—
316C2×C22⋊C4—1311no—
317C2×C22⋊C4—1311no—
318C2×C22⋊C4—1311no—
319C2×C22⋊C4—1311no—
320C2×C22⋊C4—1311no—
321C2×C22⋊C4—1311no—
322C2×C22⋊C4—1311no—
323C2×C22⋊C4—1311no—
324C2×C22⋊C4—1311no—
325C2×C22⋊C4—1311no—
326C2×C22⋊C4C2×D13111yescharacteristic
327C2×C22⋊C4C2×D13111yescharacteristic
328C2×C22⋊C4C2×D13111yescharacteristic, complemented
329C2×C22⋊C4C2×D13111yescharacteristic, complemented
330C2×C42—1311no—
331C2×C42C2×D13111yescharacteristic, complemented
332#95—1311no—
333#95—1311no—
334C24—2622no—
335C24—2622no—
336C24—2622no—
337C24—2622no—
338C24—2622no—
339C24—5241no—
340C24—5241no—
341C24—1311no—
342C24—1311no—
343C24—1311no—
344C24C22×D13111yescharacteristic
345C24C22×D13111yescharacteristic
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—2622no—
350C2×D4—2622no—
351C2×D4—2622no—
352C2×D4—2622no—
353C2×D4—2622no—
354C2×D4—2622no—
355C2×D4—2622no—
356C2×D4—2622no—
357C2×D4—2622no—
358C2×D4—2622no—
359C2×D4—2622no—
360C2×D4—2622no—
361C2×D4—2622no—
362C2×D4—5241no—
363C2×D4—5241no—
364C2×D4—5241no—
365C2×D4—5241no—
366C2×D4—5241no—
367C22×C4—2622no—
368C22×C4—2622no—
369C22×C4—2622no—
370C22×C4—5241no—
371C22×C4—5241no—
372C22×C4—1311no—
373C22×C4—1311no—
374C22×C4—1311no—
375C22×C4—1311no—
376C22×C4—1311no—
377C22×C4—1314no—
378C22×C4C22×D13111yescharacteristic
379C22×C4C22×D13111yescharacteristic
380C22×C4C22×D13111yescharacteristic
381C22×C4C22×D13111yescharacteristic
382C22×C4C22×D13111yescharacteristic
383C4⋊C4—22no—
384C4⋊C4—2622no—
385C4⋊C4—2622no—
386C4⋊C4—2622no—
387C4⋊C4—2622no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—2622no—
393C22⋊C4—2622no—
394C22⋊C4—2622no—
395C22⋊C4—2622no—
396C22⋊C4—2622no—
397C22⋊C4—2622no—
398C22⋊C4—2622no—
399C22⋊C4—2622no—
400C22⋊C4—2622no—
401C22⋊C4—2622no—
402C22⋊C4—1314no—
403C22⋊C4—1314no—
404C42—41no—
405C42—5241no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—2628no—
420C23—2621no—
421C23—2621no—
422C23—2621no—
423C23—2621no—
424C23—2621no—
425C23—2621no—
426C23—2621no—
427C23—2621no—
428C23—2621no—
429C23—2621no—
430C23—2621no—
431C23—2621no—
432C23—2621no—
433C23—2621no—
434C23—2621no—
435C23—2621no—
436C23—2621no—
437C23—5244no—
438C23—5244no—
439C23—5244no—
440C23—5244no—
441C23—5244no—
442C23—5244no—
443C23—1312no—
444C23—1312no—
445C23—1312no—
446C23—1312no—
447C23#1452351312yes—
448C23C23×D13111yescharacteristic
449D4—44no—
450D4—5244no—
451D4—5244no—
452D4—5244no—
453D4—5244no—
454D4—5244no—
455D4—5244no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—2621no—
469C2×C4—2621no—
470C2×C4—2621no—
471C2×C4—2621no—
472C2×C4—2621no—
473C2×C4—2621no—
474C2×C4—2621no—
475C2×C4—2621no—
476C2×C4—2621no—
477C2×C4—2621no—
478C2×C4—2621no—
479C2×C4—2621no—
480C2×C4—5244no—
481C2×C4—5244no—
482C2×C4—1312no—
483C2×C4—1312no—
484C2×C4—1312no—
485C2×C4D4×D13112yes—
486C2×C4D4×D13112yes—
487C2×C4D4×D13112yes—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—2622no—
503C22—2622no—
504C22—2622no—
505C22—2622no—
506C22—2622no—
507C22—2622no—
508C22—2622no—
509C22—2622no—
510C22—2622no—
511C22—52416no—
512C22—5241no—
513C22—5241no—
514C22—5241no—
515C22—5241no—
516C22—5241no—
517C22—5241no—
518C22—5241no—
519C22—5241no—
520C22—5241no—
521C22—5241no—
522C22—5241no—
523C22—1314no—
524C22C2×D4×D13111yescharacteristic
525C22C2×D4×D13111yescharacteristic
526C22C2×D4×D13111yescharacteristic
527C22#41×D13111yescharacteristic
528C22C2×#1452351311yescharacteristic
529C22#1686229111yescharacteristic
530C22#1686229211yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—41no—
535C4—41no—
536C4—2622no—
537C4—2622no—
538C4—2622no—
539C4—5241no—
540C4—5241no—
541C2—42no—
542C2—2624no—
543C2—81no—
544C2—5242no—
545C2—10481no—
546C2#2255382111yescharacteristic
547C2#127×D13111yescharacteristic
548C2#2255387011yescharacteristic
549C2#2255395111yescharacteristic
550C2#2255395211yescharacteristic
551C2#2255395511yescharacteristic
552C2#2255395811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#390×C131C2
hypercenter#390D131
hypocenterC131#2539
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16862231C2semidirect
2. C2×#16862228C2semidirect
3. C2×#16862219C2semidirect
4. C2×#16862216C2semidirect
5. C2×#16862214C2semidirect
6. C2×C22⋊C4×D131C2semidirect
7. #22553248C2semidirect
8. #22553247C2semidirect
9. #22553221C2semidirect
10. #22553220C2semidirect
11. #22553214C2semidirect
12. #22553212C2semidirect
13. #22553212C2semidirect
14. C4×(C2×D131)⋊C4C2semidirect
15. #390×C131C2semidirect
22. C22×C22⋊D131C22semidirect
30. C2×C4⋊Dic131C22semidirect
31. C2×Dic131⋊C4C22semidirect
32. C2×Dic131⋊C4C22semidirect
33. C2×Dic131⋊C4C22semidirect
36. C2×C22⋊Dic131C22semidirect
43. C2×(C2×D131)⋊C4C22semidirect
44. C2×(C2×D131)⋊C4C22semidirect
45. C2×(C2×D131)⋊C4C22semidirect
47. C2×C4×Dic131C22semidirect
48. C131⋊#95C22semidirect
49. C131⋊#95C22semidirect
51. C2×C4⋊C4×C131C22semidirect
54. C2×C22⋊C4×C131C22semidirect
55. C2×C22⋊C4×C131C22semidirect
56. C2×C42×C131C22semidirect
95. C2×C4×D131D4semidirect
110. (C2×D131)⋊C4D4semidirect
111. (C2×D131)⋊C4D4semidirect
276. C131#2539semidirect
291. #390D131semidirect
312. C2×C4⋊C4C2×D131semidirect
328. C2×C22⋊C4C2×D131semidirect
329. C2×C22⋊C4C2×D131semidirect
331. C2×C42C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622312
C2×#168622282
C2×#168622192
C2×#168622162
C2×#168622142
C2×C22⋊C4×D1312
#225532482
#225532472
#225532212
#225532202
#225532142
#225532122
C4×(C2×D131)⋊C42
#390×C1312
#2539131