Finite Group Databasebeta

← #30669278

Subgroups — #30669278

Order 16768.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523513C211yesmaximal, characteristic, complemented
2C2×#16862216C211yesmaximal, characteristic, complemented
3C2×#16862214C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
5#22553225C211yesmaximal, characteristic, complemented
6#22553224C211yesmaximal, characteristic, complemented
7#22553221C211yesmaximal, characteristic, complemented
8#22553220C211yesmaximal, characteristic, complemented
9#22553191C211yesmaximal, characteristic, complemented
10#22553190C211yesmaximal, characteristic, complemented
11#22553190C211yesmaximal, characteristic, complemented
12#22553188C211yesmaximal, characteristic, complemented
13#22553188C211yesmaximal, characteristic, complemented
14#22553174C211yesmaximal, characteristic, complemented
15#386×C131C211yesmaximal, characteristic, complemented
16C24×D131C2211yescharacteristic
17C2×#14523513—24no—
18C2×#14523513—24no—
19C2×#14523513—24no—
20C22×Dic262C2211yescharacteristic, complemented
21C22×D524C2211yescharacteristic
22C22×D524C2211yescharacteristic
23C22×C22⋊D131C2211yescharacteristic
24C22×C22⋊D131C2211yescharacteristic
25C22×C22⋊D131C2211yescharacteristic
26C22×C22⋊D131C2211yescharacteristic, complemented
27C22×C4×D131C2211yescharacteristic
28C22×C4×D131C2211yescharacteristic
29C22×C4×D131C2211yescharacteristic, complemented
30#16862216—24no—
31#16862214—24no—
32C2×C22⋊Dic131C2211yescharacteristic
33C2×C22⋊Dic131C2211yescharacteristic
34C2×C22⋊Dic131C2211yescharacteristic, complemented
35C2×(C2×D131)⋊C4C2211yescharacteristic
36C2×(C2×D131)⋊C4C2211yescharacteristic
37C2×(C2×D131)⋊C4C2211yescharacteristic
38C2×(C2×D131)⋊C4C2211yescharacteristic
39C2×(C2×D131)⋊C4C2211yescharacteristic
40C2×(C2×D131)⋊C4C2211yescharacteristic
41C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D131—24no—
44C131⋊#95C2211yescharacteristic, complemented
45C131⋊#95C2211yescharacteristic, complemented
46C131⋊#95C2211yescharacteristic, complemented
47C131⋊#95C2211yescharacteristic, complemented
48C131⋊#95C2211yescharacteristic, complemented
49C131⋊#95C2211yescharacteristic, complemented
50C22×D4×C131C2211yescharacteristic
51C23×C4×C131C2211yescharacteristic, complemented
52C2×C22⋊C4×C131C2211yescharacteristic
53C2×C22⋊C4×C131C2211yescharacteristic
54C2×C22⋊C4×C131C2211yescharacteristic, complemented
55#95×C131C2211yescharacteristic, complemented
56#95×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
67#14523513—232no—
68C2×Dic262—22no—
69C2×Dic262—22no—
70C2×Dic262—22no—
71C2×D524—22no—
72C2×D524—22no—
73C2×D524—22no—
74C2×D524—22no—
75C2×D524—41no—
76C2×D524D414yescomplemented
77C2×C22⋊D131—22no—
78C2×C22⋊D131—22no—
79C2×C22⋊D131—22no—
80C2×C22⋊D131—22no—
81C2×C22⋊D131—22no—
82C2×C22⋊D131—22no—
83C2×C22⋊D131—22no—
84C2×C22⋊D131—22no—
85C2×C22⋊D131—22no—
86C2×C22⋊D131—41no—
87C2×C22⋊D131—41no—
88C2×C22⋊D131D414yescomplemented
89C22×Dic131C2311yescharacteristic
90C22×Dic131C2311yescharacteristic
91C22×Dic131C2311yescharacteristic
92C22×Dic131C2311yescharacteristic
93C22×Dic131C2311yescharacteristic
94C2×C4×D131—22no—
95C2×C4×D131—22no—
96C2×C4×D131—22no—
97C2×C4×D131—22no—
98C2×C4×D131—22no—
99C2×C4×D131—22no—
100C2×C4×D131—41no—
101C2×C4×D131—41no—
102C2×C4×D131D414yescomplemented
103C22⋊Dic131—22no—
104C22⋊Dic131—22no—
105C22⋊Dic131—41no—
106(C2×D131)⋊C4—22no—
107(C2×D131)⋊C4—22no—
108(C2×D131)⋊C4—22no—
109(C2×D131)⋊C4—22no—
110(C2×D131)⋊C4—22no—
111(C2×D131)⋊C4—22no—
112(C2×D131)⋊C4—41no—
113(C2×D131)⋊C4—41no—
114C24×C131C2311yescharacteristic
115C24×C131C2311yescharacteristic
116C2×D4×C131—22no—
117C2×D4×C131—22no—
118C2×D4×C131—41no—
119C22×C4×C131—22no—
120C22×C4×C131—22no—
121C22×C4×C131—22no—
122C22×C4×C131C2311yescharacteristic
123C22×C4×C131C2311yescharacteristic
124C22×C4×C131C2311yescharacteristic
125C22×C4×C131C2311yescharacteristic
126C22×C4×C131C2311yescharacteristic
127C22⋊C4×C131—22no—
128C22⋊C4×C131—22no—
129C22⋊C4×C131—41no—
130C22×D131—21no—
131C22×D131—21no—
132C22×D131—21no—
133C22×D131—21no—
134C22×D131—28no—
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×D131C2×D412yes—
157C22×D131#4112yescomplemented
158Dic262—28no—
159D524—28no—
160D524—44no—
161C22⋊D131—28no—
162C22⋊D131—28no—
163C22⋊D131—44no—
164C22⋊D131—44no—
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×Dic131—21no—
176C2×Dic131—21no—
177C2×Dic131—21no—
178C2×Dic131C2×D412yes—
179C2×Dic131C2×D412yes—
180C4×D131—28no—
181C4×D131—28no—
182C4×D131—44no—
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—21no—
194C23×C131—21no—
195C23×C131C2×D412yes—
196C23×C131#4112yescomplemented
197C23×C131C2411yescharacteristic
198D4×C131—44no—
199C2×C4×C131—28no—
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×C131—21no—
209C2×C4×C131—21no—
210C2×C4×C131—21no—
211C2×C4×C131—21no—
212C2×C4×C131—21no—
213C2×C4×C131C2×D412yes—
214C2×C4×C131C2×D412yes—
215C22×C131—22no—
216C22×C131—22no—
217C22×C131—22no—
218C22×C131—22no—
219C22×C131—22no—
220C22×C131—22no—
221C22×C131—41no—
222C22×C131—41no—
223C22×C131—41no—
224C22×C131—41no—
225C22×C131—41no—
226C22×C131—41no—
227C22×C131—41no—
228C22×C131—41no—
229C22×C131C22×D411yescharacteristic
230C22×C131C22×D411yescharacteristic
231C22×C131C22×D411yescharacteristic
232C22×C131C2×#4111yescharacteristic
233C22×C131C2×#4111yescharacteristic
234C22×C131H(5,𝔽2)11yescharacteristic
235C22×C131H(5,𝔽2)11yescharacteristic
236C2×D131—22no—
237C2×D131—22no—
238C2×D131—22no—
239C2×D131—22no—
240C2×D131—22no—
241C2×D131—22no—
242C2×D131—22no—
243C2×D131—22no—
244C2×D131—22no—
245C2×D131—22no—
246C2×D131—22no—
247C2×D131—22no—
248C2×D131—416no—
249C2×D131—41no—
250C2×D131—41no—
251C2×D131—41no—
252C2×D131—41no—
253C2×D131—41no—
254C2×D131—41no—
255C2×D131—41no—
256C2×D131—41no—
257C2×D131#12014yes—
258C4×C131—22no—
259C4×C131—22no—
260C4×C131—41no—
261C4×C131—41no—
262C4×C131—41no—
263Dic131—22no—
264Dic131—22no—
265Dic131—41no—
266Dic131—41no—
267Dic131—41no—
268C2×C131—42no—
269C2×C131—42no—
270C2×C131C2×#12011yescharacteristic
271C2×C131#52511yescharacteristic
272C2×C131#53511yescharacteristic
273C2×C131#54611yescharacteristic
274C2×C131#54611yescharacteristic
275C2×C131#54611yescharacteristic
276C2×C131#54611yescharacteristic
277D131—24no—
278D131—42no—
279D131—42no—
280C131#240411yescharacteristic, complemented, Sylow
281#2404—1311nomaximal, Sylow
282C22×#41—1311no—
283C2×#120—1311no—
284C2×#120—1311no—
285C22×C22⋊C4—1311no—
286#393—1311no—
287#393—1311no—
288#392—1311no—
289#392—1311no—
290#386—1311no—
291#386—1311no—
292#386—1311no—
293#386—1311no—
294#386—1311no—
295#386D13111yescharacteristic, complemented
296#381—1311no—
297C25—1311no—
298C2×#41—2624no—
299C2×#41—2624no—
300C2×#41—2624no—
301C22×Q8—1311no—
302C22×D4—1311no—
303C22×D4—1311no—
304C22×D4—1311no—
305C22×D4—1311no—
306C22×D4—1311no—
307C22×D4—1311no—
308C22×D4C2×D13111yescharacteristic
309C23×C4—1311no—
310C23×C4—1311no—
311C23×C4—1311no—
312C23×C4C2×D13111yescharacteristic, complemented
313#120—2624no—
314#120—2624no—
315C2×C22⋊C4—2624no—
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⋊C4—1311no—
327C2×C22⋊C4C2×D13111yescharacteristic
328C2×C22⋊C4C2×D13111yescharacteristic
329C2×C22⋊C4C2×D13111yescharacteristic, complemented
330#95—1311no—
331#95—1311no—
332#95—1311no—
333#95—1311no—
334#95—1311no—
335#95—1311no—
336#95C2×D13111yescharacteristic, complemented
337#95C2×D13111yescharacteristic, complemented
338C24—2622no—
339C24—2622no—
340C24—2622no—
341C24—2622no—
342C24—2622no—
343C24—5241no—
344C24—5241no—
345C24—1311no—
346C24—1311no—
347C24—1311no—
348C24C22×D13111yescharacteristic
349C24C22×D13111yescharacteristic
350#41—26232no—
351C2×Q8—2622no—
352C2×Q8—2622no—
353C2×Q8—2622no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—2622no—
358C2×D4—2622no—
359C2×D4—2622no—
360C2×D4—2622no—
361C2×D4—2622no—
362C2×D4—2622no—
363C2×D4—2622no—
364C2×D4—2622no—
365C2×D4—2622no—
366C2×D4—2622no—
367C2×D4—2622no—
368C2×D4—2622no—
369C2×D4—2622no—
370C2×D4—5241no—
371C2×D4—5241no—
372C2×D4—5241no—
373C2×D4—1314no—
374C2×D4—1314no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—2622no—
379C22×C4—2622no—
380C22×C4—2622no—
381C22×C4—2622no—
382C22×C4—2622no—
383C22×C4—2622no—
384C22×C4—5241no—
385C22×C4—5241no—
386C22×C4—1311no—
387C22×C4—1311no—
388C22×C4—1311no—
389C22×C4—1311no—
390C22×C4—1311no—
391C22×C4—1314no—
392C22×C4C22×D13111yescharacteristic
393C22×C4C22×D13111yescharacteristic
394C22×C4C22×D13111yescharacteristic
395C22×C4C22×D13111yescharacteristic
396C22×C4C22×D13111yescharacteristic
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—2622no—
401C22⋊C4—2622no—
402C22⋊C4—2622no—
403C22⋊C4—2622no—
404C22⋊C4—2622no—
405C22⋊C4—2622no—
406C22⋊C4—2622no—
407C22⋊C4—2622no—
408C22⋊C4—5241no—
409C22⋊C4—5241no—
410C22⋊C4—5241no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—2628no—
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—2621no—
438C23—2621no—
439C23—2621no—
440C23—5244no—
441C23—5244no—
442C23—5244no—
443C23—5244no—
444C23—5244no—
445C23—5244no—
446C23—1312no—
447C23—1312no—
448C23—1312no—
449C23—1312no—
450C23—1312no—
451C23D4×D13112yes—
452C23#1452351312yescomplemented
453C23C23×D13111yescharacteristic
454Q8—2628no—
455D4—44no—
456D4—2628no—
457D4—2628no—
458D4—2628no—
459D4—5244no—
460D4—5244no—
461D4—5244no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—2628no—
477C2×C4—2628no—
478C2×C4—2621no—
479C2×C4—2621no—
480C2×C4—2621no—
481C2×C4—2621no—
482C2×C4—2621no—
483C2×C4—2621no—
484C2×C4—2621no—
485C2×C4—2621no—
486C2×C4—2621no—
487C2×C4—2621no—
488C2×C4—2621no—
489C2×C4—2621no—
490C2×C4—2621no—
491C2×C4—5244no—
492C2×C4—1312no—
493C2×C4—1312no—
494C2×C4D4×D13112yes—
495C2×C4D4×D13112yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—2622no—
511C22—2622no—
512C22—2622no—
513C22—2622no—
514C22—2622no—
515C22—2622no—
516C22—2622no—
517C22—2622no—
518C22—2622no—
519C22—2622no—
520C22—2622no—
521C22—2622no—
522C22—52416no—
523C22—5241no—
524C22—5241no—
525C22—5241no—
526C22—5241no—
527C22—5241no—
528C22—5241no—
529C22—5241no—
530C22—5241no—
531C22—1314no—
532C22C2×D4×D13111yescharacteristic
533C22C2×D4×D13111yescharacteristic
534C22C2×D4×D13111yescharacteristic
535C22#41×D13111yescharacteristic
536C22C2×#1452351311yescharacteristic
537C22#1686229111yescharacteristic
538C22#1686229211yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—2622no—
545C4—2622no—
546C4—5241no—
547C4—5241no—
548C4—5241no—
549C2—42no—
550C2—42no—
551C2—2624no—
552C2—5242no—
553C2—5242no—
554C2#120×D13111yescharacteristic
555C2#2255382111yescharacteristic
556C2#2255387811yescharacteristic
557C2#2255395211yescharacteristic
558C2#2255395411yescharacteristic
559C2#2255395811yescharacteristic
560C2#2255395911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#386×C131C2
hypercenter#386D131
hypocenterC131#2404
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#14523513C2semidirect
2. C2×#16862216C2semidirect
3. C2×#16862214C2semidirect
4. C2×C22⋊C4×D131C2semidirect
5. #22553225C2semidirect
6. #22553224C2semidirect
7. #22553221C2semidirect
8. #22553220C2semidirect
9. #22553191C2semidirect
10. #22553190C2semidirect
11. #22553190C2semidirect
12. #22553188C2semidirect
13. #22553188C2semidirect
14. #22553174C2semidirect
15. #386×C131C2semidirect
20. C22×Dic262C22semidirect
26. C22×C22⋊D131C22semidirect
29. C22×C4×D131C22semidirect
34. C2×C22⋊Dic131C22semidirect
41. C2×(C2×D131)⋊C4C22semidirect
42. C2×(C2×D131)⋊C4C22semidirect
44. C131⋊#95C22semidirect
45. C131⋊#95C22semidirect
46. C131⋊#95C22semidirect
47. C131⋊#95C22semidirect
48. C131⋊#95C22semidirect
49. C131⋊#95C22semidirect
51. C23×C4×C131C22semidirect
54. C2×C22⋊C4×C131C22semidirect
55. #95×C131C22semidirect
56. #95×C131C22semidirect
76. C2×D524D4semidirect
88. C2×C22⋊D131D4semidirect
102. C2×C4×D131D4semidirect
157. C22×D131#41semidirect
196. C23×C131#41semidirect
280. C131#2404semidirect
295. #386D131semidirect
312. C23×C4C2×D131semidirect
329. C2×C22⋊C4C2×D131semidirect
336. #95C2×D131semidirect
337. #95C2×D131semidirect
452. C23#14523513semidirect

Maximal subgroups

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

subgroupindex
C22×#145235132
C2×#168622162
C2×#168622142
C2×C22⋊C4×D1312
#225532252
#225532242
#225532212
#225532202
#225531912
#225531902
#225531882
#225531742
#386×C1312
#2404131