Finite Group Databasebeta

← #16023091

Subgroups — #16023091

Order 3328.

2470 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic13C211yesmaximal, characteristic
2#13117792C212yesmaximal, complemented
3#13117763C214yesmaximal, complemented
4D4×#6101C212yesmaximal, complemented
5#13117723C211yesmaximal, characteristic, complemented
6#13117719C211yesmaximal, characteristic, complemented
7#13117698C212yesmaximal, complemented
8#13117692C212yesmaximal, complemented
9#13117666C211yesmaximal, characteristic
10#13117481C212yesmaximal, complemented
11#13117462C214yesmaximal, complemented
12#13117238C211yesmaximal, characteristic, complemented
13#13117230C211yesmaximal, characteristic, complemented
14#13117227C212yesmaximal, complemented
15C2×#11718311C212yesmaximal, complemented
16#13117197C212yesmaximal, complemented
17C13×#2339C211yesmaximal, characteristic
18C22×C13×#41C2211yescharacteristic
19C2×C4×D4×C13C2212yes—
20#41×Dic13C4116yescomplemented
21C2×Q8×Dic13C2211yescharacteristic
22C2×Q8×Dic13C2211yescharacteristic
23C2×D4×Dic13C2212yes—
24C2×D4×Dic13C2211yescharacteristic
25C2×D4×Dic13C2211yescharacteristic
26C2×D4×Dic13C2212yes—
27C2×D4×Dic13C2212yes—
28C2×#88180C2212yes—
29C2×#88180C2211yescharacteristic
30C22×#6101C2214yes—
31C22×C4×Dic13C2212yes—
32C22×C4×Dic13C2211yescharacteristic
33C2×#88131C2212yes—
34C13×#391C2212yes—
35C13×#386C2214yes—
36C13×#381C2212yes—
37C13×#380C2211yescharacteristic
38C4×C13×C22⋊C4C2211yescharacteristic
39C2×C13×#95C2212yes—
40#11718891C2212yes—
41#11718881C2214yes—
42#11718873C2211yescharacteristic
43#11718872C2212yes—
44#11718871C2214yes—
45C4×C22⋊Dic13C2214yes—
46C4×C22⋊Dic13C2211yescharacteristic
47C2×#88071C2212yes—
48C2×#88071C2214yes—
49#11718637C2212yes—
50C4⋊C4×Dic13C2212yes—
51C4⋊C4×Dic13C2212yes—
52#11718634C2212yes—
53#11718607C2214yes—
54#11718606C2214yes—
55#11718605C2214yes—
56#11718604C2212yes—
57#11718603C2212yes—
58#11718603C2214yes—
59#11718603C2212yes—
60C22⋊C4×Dic13C2214yes—
61C22⋊C4×Dic13C2214yes—
62C4×C4⋊Dic13C2212yes—
63#11718593C2212yes—
64C4×#6101C2212yes—
65#11718589C2212yes—
66#11718315C2214yes—
67#11718315C2214yes—
68#11718314C2214yes—
69#11718314C2214yes—
70#11718312C2211yescharacteristic
71#11718312C2214yes—
72#11718312C2211yescharacteristic
73#11718311C2212yes—
74#11718311C2214yes—
75#11718311C2212yes—
76#11718311C2211yescharacteristic
77#11718311C2212yes—
78#11718311C2214yes—
79#11718311C2211yescharacteristic
80#11718310C2212yes—
81#11718310C2214yes—
82#11718309C2212yes—
83#11718309C2212yes—
84C2×C13×#41—28no—
85C2×C13×#41C2×C418yes—
86C22×Q8×C13C2311yescharacteristic
87C22×D4×C13C2312yes—
88C22×D4×C13C2311yescharacteristic
89C23×C4×C13C2311yescharacteristic
90C23×C4×C13C2312yes—
91C23×C4×C13C2314yes—
92C23×Dic13C2311yescharacteristic
93C23×Dic13C2312yes—
94C23×Dic13C2314yes—
95C23×Dic13C2312yes—
96C23×Dic13C2311yescharacteristic
97C4×D4×C13—28no—
98C2×C13×C4⋊C4C2312yes—
99C2×C13×C22⋊C4C2314yes—
100C2×C13×C22⋊C4C2312yes—
101C2×C13×C22⋊C4C2312yes—
102C2×C42×C13C2312yes—
103C2×C22⋊Dic13C2314yes—
104C2×C22⋊Dic13C2312yes—
105C2×C22⋊Dic13C2312yes—
106C2×C22⋊Dic13C2312yes—
107C2×C22⋊Dic13C2312yes—
108C2×C22⋊Dic13C2312yes—
109Q8×Dic13C2×C418yescomplemented
110Q8×Dic13C2×C418yescomplemented
111D4×Dic13—28no—
112D4×Dic13C2×C418yes—
113D4×Dic13C2×C418yes—
114D4×Dic13C2×C4116yescomplemented
115D4×Dic13C2×C4116yescomplemented
116#88180C2×C418yescomplemented
117#88180C2×C4116yescomplemented
118C2×C4⋊Dic13C2312yes—
119C2×C4⋊Dic13C2312yes—
120C2×C4⋊Dic13C2312yes—
121C2×C4⋊Dic13C2311yescharacteristic
122C2×C4⋊Dic13C2311yescharacteristic
123C2×#6101—48no—
124C2×#6101C2314yes—
125C2×#6101C2312yes—
126C2×#6101C2312yes—
127C2×#6101C2314yes—
128C2×#6101C2314yes—
129C2×#6101C2314yes—
130C2×C4×Dic13C2×C418yescomplemented
131C2×C4×Dic13C2×C4116yescomplemented
132C2×C4×Dic13C2312yes—
133C2×C4×Dic13C2314yes—
134C2×C4×Dic13C2314yes—
135C2×C4×Dic13C2314yes—
136C2×C4×Dic13C2311yescharacteristic
137C2×C4×Dic13C2312yes—
138C2×C4×Dic13C2311yescharacteristic
139C2×C4×Dic13C2312yes—
140C2×C4×Dic13C2312yes—
141C2×C4×Dic13C2311yescharacteristic
142C2×C4×Dic13C2311yescharacteristic
143#88131—44no—
144C13×#95C2311yescharacteristic
145C13×#95C2311yescharacteristic
146C13×#95C2312yes—
147C13×#95C2314yes—
148C13×#95C2314yes—
149#88071C2311yescharacteristic
150#88071C2312yes—
151#88071C2314yes—
152#88071C2314yes—
153#88071C2314yes—
154#88071C2311yescharacteristic
155#88071C2312yes—
156#88071C2314yes—
157#88071C2314yes—
158#88071C2314yes—
159#88071C2314yes—
160#88071C2312yes—
161#88071C2312yes—
162#88071C2312yes—
163#88071C2312yes—
164#88071C2312yes—
165#34191—131nomaximal, Sylow
166C24×C13C2411yescharacteristic
167C24×C13C2412yes—
168C13×#41—232no—
169C2×Q8×C13—24no—
170C2×Q8×C13C22×C414yes—
171C2×D4×C13—28no—
172C2×D4×C13C22×C418yes—
173C2×D4×C13C22×C414yes—
174C2×D4×C13#4118yes—
175C22×C4×C13—28no—
176C22×C4×C13—24no—
177C22×C4×C13—28no—
178C22×C4×C13—44no—
179C22×C4×C13—44no—
180C22×C4×C13C22×C418yes—
181C22×C4×C13C22×C414yes—
182C22×C4×C13C2412yes—
183C22×C4×C13C2412yes—
184C22×C4×C13C2414yes—
185C22×C4×C13C2411yescharacteristic
186C22×C4×C13C2412yes—
187C22×C4×C13C2411yescharacteristic
188C22×Dic13—22no—
189C22×Dic13—24no—
190C22×Dic13—24no—
191C22×Dic13—22no—
192C22×Dic13—28no—
193C22×Dic13—44no—
194C22×Dic13—42no—
195C22×Dic13—44no—
196C22×Dic13—44no—
197C22×Dic13C22×C418yes—
198C22×Dic13C22×C414yes—
199C22×Dic13C22×C418yes—
200C22×Dic13C22×C414yes—
201C22×Dic13C2412yes—
202C22×Dic13C2412yes—
203C22×Dic13C2414yes—
204C22×Dic13C2411yescharacteristic
205C22×Dic13C2411yescharacteristic
206C22×Dic13C2412yes—
207C22×Dic13C2412yes—
208C22×Dic13C2411yescharacteristic
209C22×Dic13C2411yescharacteristic
210C13×C4⋊C4—24no—
211C13×C22⋊C4—28no—
212C13×C22⋊C4—42no—
213C13×C22⋊C4—42no—
214C42×C13—24no—
215C22⋊Dic13—28no—
216C22⋊Dic13—42no—
217C22⋊Dic13—42no—
218C22⋊Dic13C22×C418yes—
219C22⋊Dic13C22×C418yes—
220C22⋊Dic13C22×C418yes—
221C4⋊Dic13—24no—
222C4⋊Dic13C22×C414yes—
223C4⋊Dic13C22×C414yes—
224C4⋊Dic13C22×C418yes—
225C4⋊Dic13C22×C418yes—
226#6101—28no—
227#6101—24no—
228#6101—24no—
229#6101—44no—
230#6101—44no—
231#6101—44no—
232C4×Dic13—24no—
233C4×Dic13—44no—
234C4×Dic13—44no—
235C4×Dic13—44no—
236C4×Dic13C22×C414yes—
237C4×Dic13C22×C418yes—
238C4×Dic13C22×C414yes—
239C4×Dic13C22×C414yes—
240C4×Dic13C22×C414yes—
241C4×Dic13C22×C418yes—
242C4×Dic13C22×C418yes—
243C2×C4×#41—131no—
244#2355—132no—
245#2352—134no—
246D4×C4⋊C4—132no—
247#2346—131no—
248#2346—131no—
249#2343—132no—
250#2343—132no—
251#2339—131no—
252#2339D1311yescharacteristic
253#2320—132no—
254#2318—134no—
255#2285—131no—
256#2283—132no—
257#2283—131no—
258C2×#382—132no—
259#2275—132no—
260C23×C13—22no—
261C23×C13—22no—
262C23×C13—24no—
263C23×C13—22no—
264C23×C13—21no—
265C23×C13—21no—
266C23×C13—22no—
267C23×C13C23×C412yes—
268C23×C13C23×C414yes—
269C23×C13C2×#4118yes—
270C23×C13C2511yescharacteristic
271Q8×C13—28no—
272D4×C13—216no—
273D4×C13#117116yescomplemented
274C2×C4×C13—216no—
275C2×C4×C13—22no—
276C2×C4×C13—24no—
277C2×C4×C13—28no—
278C2×C4×C13—22no—
279C2×C4×C13—22no—
280C2×C4×C13—24no—
281C2×C4×C13—24no—
282C2×C4×C13—22no—
283C2×C4×C13—22no—
284C2×C4×C13—22no—
285C2×C4×C13—24no—
286C2×C4×C13—22no—
287C2×C4×C13—88no—
288C2×C4×C13C23×C412yes—
289C2×C4×C13C23×C414yes—
290C2×C4×C13C23×C412yes—
291C2×C4×C13C2×#4114yes—
292C2×Dic13—21no—
293C2×Dic13—22no—
294C2×Dic13—21no—
295C2×Dic13—22no—
296C2×Dic13—24no—
297C2×Dic13—22no—
298C2×Dic13—22no—
299C2×Dic13—216no—
300C2×Dic13—28no—
301C2×Dic13—22no—
302C2×Dic13—22no—
303C2×Dic13—24no—
304C2×Dic13—24no—
305C2×Dic13—22no—
306C2×Dic13—22no—
307C2×Dic13—22no—
308C2×Dic13—24no—
309C2×Dic13—22no—
310C2×Dic13—48no—
311C2×Dic13—44no—
312C2×Dic13—88no—
313C2×Dic13C23×C412yes—
314C2×Dic13C23×C412yes—
315C2×Dic13C23×C414yes—
316C2×Dic13C23×C412yes—
317C2×Dic13C23×C414yes—
318C2×Dic13C23×C412yes—
319C2×Dic13H(5,𝔽2)12yes—
320C2×Dic132−1+412yes—
321C22×#41C2×D1311yescharacteristic
322C4×#41—1316no—
323C2×C4×Q8—131no—
324C2×C4×Q8—131no—
325C2×C4×D4—132no—
326C2×C4×D4—131no—
327C2×C4×D4—132no—
328C2×C4×D4—131no—
329C2×C4×D4—132no—
330C2×C4×D4C2×D1312yes—
331C2×#117—132no—
332C2×#117—131no—
333C2×#117—132no—
334C22×C4⋊C4—134no—
335C22×C42—132no—
336C22×C42—131no—
337#391—132no—
338#391C2×D1312yes—
339#389—132no—
340#389—132no—
341#388—134no—
342#388—134no—
343#387—134no—
344#387—134no—
345#387—134no—
346#387—134no—
347#386—134no—
348#386C2×D1314yes—
349#385—134no—
350#385—134no—
351#384—132no—
352#383—134no—
353#383—131no—
354#383—131no—
355#382—134no—
356#382—132no—
357#382—132no—
358#382—134no—
359#382—131no—
360#382—132no—
361#382—131no—
362#381—132no—
363#381—132no—
364#381C2×D1312yes—
365#380—131no—
366#380—132no—
367#380—134no—
368#380—132no—
369#380C2×D1311yescharacteristic
370C4×C4⋊C4—132no—
371C4×C4⋊C4—132no—
372C4×C4⋊C4—132no—
373C4×C4⋊C4—132no—
374C4×C4⋊C4—132no—
375C4×C4⋊C4—134no—
376C4×C22⋊C4—131no—
377C4×C22⋊C4—134no—
378C4×C22⋊C4—134no—
379C4×C22⋊C4—134no—
380C4×C22⋊C4C2×D1311yescharacteristic
381#376—132no—
382#376—132no—
383#376—132no—
384C2×#95—134no—
385C2×#95—132no—
386C2×#95C2×D1312yes—
387C22×C13—24no—
388C22×C13—24no—
389C22×C13—28no—
390C22×C13—28no—
391C22×C13—22no—
392C22×C13—22no—
393C22×C13—22no—
394C22×C13—42no—
395C22×C13—42no—
396C22×C13C2×#11718yes—
397C22×C13C24×C411yescharacteristic
398C22×C13C22×#4112yes—
399C22×C13C2×H(5,𝔽2)11yescharacteristic
400C22×C13C2×2−1+411yescharacteristic
401C22×C13#58512yes—
402C4×C13—22no—
403C4×C13—24no—
404C4×C13—44no—
405C4×C13—42no—
406C4×C13—42no—
407C4×C13C2×#11714yes—
408Dic13—22no—
409Dic13—24no—
410Dic13—22no—
411Dic13—24no—
412Dic13—22no—
413Dic13—44no—
414Dic13—42no—
415Dic13—42no—
416Dic13#52014yes—
417C2×#41—28no—
418C2×#41C4×D1318yes—
419C22×Q8C22×D1311yescharacteristic
420C22×D4C22×D1312yes—
421C22×D4C22×D1311yescharacteristic
422C23×C4—132no—
423C23×C4—131no—
424C23×C4—132no—
425C23×C4—131no—
426C23×C4—134no—
427C23×C4C22×D1312yes—
428C23×C4C22×D1311yescharacteristic
429C23×C4C22×D1314yes—
430C4×Q8—138no—
431C4×Q8—138no—
432C4×D4—28no—
433C4×D4—138no—
434C4×D4—138no—
435C4×D4—1316no—
436C4×D4—1316no—
437C4×D4—268no—
438#117—1316no—
439#117—138no—
440#117—524no—
441C2×C4⋊C4—131no—
442C2×C4⋊C4—132no—
443C2×C4⋊C4—132no—
444C2×C4⋊C4—131no—
445C2×C4⋊C4—132no—
446C2×C4⋊C4—134no—
447C2×C4⋊C4—132no—
448C2×C4⋊C4—132no—
449C2×C4⋊C4—134no—
450C2×C4⋊C4—134no—
451C2×C4⋊C4—134no—
452C2×C4⋊C4—528no—
453C2×C4⋊C4C22×D1312yes—
454C2×C22⋊C4—132no—
455C2×C22⋊C4—132no—
456C2×C22⋊C4—132no—
457C2×C22⋊C4—134no—
458C2×C22⋊C4—132no—
459C2×C22⋊C4—132no—
460C2×C22⋊C4C22×D1314yes—
461C2×C22⋊C4C22×D1312yes—
462C2×C22⋊C4C22×D1312yes—
463C2×C42—1316no—
464C2×C42—138no—
465C2×C42—132no—
466C2×C42—131no—
467C2×C42—131no—
468C2×C42—132no—
469C2×C42—131no—
470C2×C42—132no—
471C2×C42—131no—
472C2×C42—132no—
473C2×C42—134no—
474C2×C42—134no—
475C2×C42—134no—
476C2×C42C22×D1312yes—
477#95—132no—
478#95—131no—
479#95—132no—
480#95—134no—
481#95—134no—
482#95—134no—
483#95—134no—
484#95—131no—
485#95—132no—
486#95—132no—
487#95—132no—
488#95—134no—
489#95—134no—
490#95—132no—
491#95—132no—
492#95—134no—
493#95C22×D1311yescharacteristic
494#95C22×D1312yes—
495#95C22×D1314yes—
496#95C22×D1311yescharacteristic
497#95C22×D1314yes—
498C2×C13—28no—
499C2×C13—42no—
500C2×C13C22×#11711yescharacteristic
501C2×C13C2×#52011yescharacteristic
502C2×C13#342811yescharacteristic
503C2×C13#351412yes—
504C2×C13#351512yes—
505C24C23×D1311yescharacteristic
506C24C23×D1312yes—
507#41—232no—
508C2×Q8—24no—
509C2×Q8C2×C4×D1314yes—
510C2×D4—28no—
511C2×D4C2×C4×D1318yes—
512C2×D4C2×C4×D1314yes—
513C2×D4#612918yes—
514C22×C4—28no—
515C22×C4—24no—
516C22×C4—28no—
517C22×C4—44no—
518C22×C4—44no—
519C22×C4—131no—
520C22×C4—134no—
521C22×C4—138no—
522C22×C4—131no—
523C22×C4—132no—
524C22×C4—131no—
525C22×C4—131no—
526C22×C4—132no—
527C22×C4—134no—
528C22×C4—138no—
529C22×C4—132no—
530C22×C4—132no—
531C22×C4—134no—
532C22×C4—264no—
533C22×C4—262no—
534C22×C4—264no—
535C22×C4—262no—
536C22×C4—268no—
537C22×C4—524no—
538C22×C4—522no—
539C22×C4—524no—
540C22×C4—524no—
541C22×C4C2×C4×D1318yes—
542C22×C4C2×C4×D1314yes—
543C22×C4C23×D1312yes—
544C22×C4C23×D1312yes—
545C22×C4C23×D1314yes—
546C22×C4C23×D1311yescharacteristic
547C22×C4C23×D1312yes—
548C22×C4C23×D1311yescharacteristic
549C4⋊C4—24no—
550C4⋊C4—138no—
551C4⋊C4—138no—
552C4⋊C4—134no—
553C4⋊C4—134no—
554C4⋊C4—264no—
555C4⋊C4—268no—
556C4⋊C4—264no—
557C4⋊C4—264no—
558C4⋊C4—524no—
559C4⋊C4—524no—
560C4⋊C4—524no—
561C22⋊C4—28no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—138no—
565C22⋊C4—138no—
566C22⋊C4—138no—
567C22⋊C4—268no—
568C22⋊C4—522no—
569C22⋊C4—522no—
570C42—24no—
571C42—134no—
572C42—138no—
573C42—134no—
574C42—138no—
575C42—134no—
576C42—138no—
577C42—134no—
578C42—264no—
579C42—524no—
580C42—524no—
581C42—524no—
582C13#3419111yescharacteristic, complemented, Sylow
583C23—24no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—21no—
589C23—21no—
590C23C22×C4×D1312yes—
591C23C22×C4×D1314yes—
592C23C2×#612918yes—
593C23C24×D1311yescharacteristic
594Q8—28no—
595D4—216no—
596D4#88131116yescomplemented
597C2×C4—216no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—88no—
611C2×C4—132no—
612C2×C4—132no—
613C2×C4—132no—
614C2×C4—132no—
615C2×C4—132no—
616C2×C4—134no—
617C2×C4—134no—
618C2×C4—132no—
619C2×C4—2616no—
620C2×C4—268no—
621C2×C4—264no—
622C2×C4—262no—
623C2×C4—262no—
624C2×C4—261no—
625C2×C4—262no—
626C2×C4—261no—
627C2×C4—262no—
628C2×C4—262no—
629C2×C4—262no—
630C2×C4—264no—
631C2×C4—262no—
632C2×C4—264no—
633C2×C4—262no—
634C2×C4—264no—
635C2×C4—262no—
636C2×C4—262no—
637C2×C4—528no—
638C2×C4—524no—
639C2×C4—1048no—
640C2×C4C22×C4×D1312yes—
641C2×C4C22×C4×D1314yes—
642C2×C4C22×C4×D1312yes—
643C2×C4C2×#612914yes—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—42no—
652C22—42no—
653C22C2×#8813118yes—
654C22C23×C4×D1311yescharacteristic
655C22C22×#612912yes—
656C22#1171964812yes—
657C22D13×H(5,𝔽2)11yescharacteristic
658C22D13×2−1+411yescharacteristic
659C4—22no—
660C4—24no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—134no—
665C4—262no—
666C4—264no—
667C4—264no—
668C4—262no—
669C4—262no—
670C4—524no—
671C4—522no—
672C4—522no—
673C4C2×#8813114yes—
674C2—28no—
675C2—42no—
676C2C22×#8813111yescharacteristic
677C2D13×#52011yescharacteristic
678C2#1312803511yescharacteristic
679C2#1312878412yes—
680C2#1312880112yes—
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC13×#2339C2
socle, cosocleC23×C13C25
prime radicalC22×C13×#41C22
hypercenter#2339D13
hypocenterC13#34191
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2339

Complements

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

NH ≅ G/Ntype
2. #13117792C2semidirect
3. #13117763C2semidirect
4. D4×#6101C2semidirect
5. #13117723C2semidirect
6. #13117719C2semidirect
7. #13117698C2semidirect
8. #13117692C2semidirect
10. #13117481C2semidirect
11. #13117462C2semidirect
12. #13117238C2semidirect
13. #13117230C2semidirect
14. #13117227C2semidirect
15. C2×#11718311C2semidirect
16. #13117197C2semidirect
20. #41×Dic13C4semidirect
109. Q8×Dic13C2×C4semidirect
110. Q8×Dic13C2×C4semidirect
114. D4×Dic13C2×C4semidirect
115. D4×Dic13C2×C4semidirect
116. #88180C2×C4semidirect
117. #88180C2×C4semidirect
130. C2×C4×Dic13C2×C4semidirect
131. C2×C4×Dic13C2×C4semidirect
273. D4×C13#117semidirect
582. C13#34191semidirect
596. D4#88131semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic132
#131177922
#131177632
D4×#61012
#131177232
#131177192
#131176982
#131176922
#131176662
#131174812
#131174622
#131172382
#131172302
#131172272
C2×#117183112
#131171972
C13×#23392
#3419113