Finite Group Databasebeta

← #16035861

Subgroups — #16035861

Order 3328.

2322 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719350C212yesmaximal, complemented
2#13123181C211yesmaximal, characteristic
3#13123168C212yesmaximal, complemented
4#13120770C212yesmaximal, complemented
5#13120581C212yesmaximal, complemented
6#13119987C214yesmaximal, complemented
7#13119957C214yesmaximal, complemented
8#13119418C212yesmaximal, complemented
9#13119354C212yesmaximal, complemented
10#13118682C212yesmaximal, complemented
11#13118603C212yesmaximal, complemented
12#13117536C212yesmaximal, complemented
13#13117342C212yesmaximal, complemented
14C2×C4×#6101C211yesmaximal, characteristic, complemented
15C13×#2857C211yesmaximal, characteristic, complemented
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8C2212yes—
18C22×C42×C13C2211yescharacteristic
19C2×#88200C2214yes—
20C2×#88184C2212yes—
21C2×#88184C2212yes—
22C2×#88184C2212yes—
23C22×#6101C2211yescharacteristic
24C22×#6101C2211yescharacteristic
25C22×C4×Dic13C2211yescharacteristic
26C22×C4×Dic13C2211yescharacteristic
27#11719350—216no—
28C2×#88152C2214yes—
29C2×#88148C2214yes—
30C2×#88147C2212yes—
31C2×#88144C2214yes—
32C2×#88140C2214yes—
33C2×#88135C2214yes—
34C2×#88128C2211yescharacteristic
35C13×#391C2212yes—
36C13×#389C2212yes—
37C13×#389C2212yes—
38C13×#384C2211yescharacteristic
39C13×#380C2212yes—
40C13×#380C2212yes—
41#11718892C2214yes—
42C2×#88071C2211yescharacteristic
43C2×#88071C2211yescharacteristic
44#11718655C2214yes—
45#11718655C2214yes—
46#11718654C2214yes—
47#11718652C2214yes—
48#11718651C2214yes—
49#11718650C2214yes—
50#11718649C2212yes—
51#11718649C2212yes—
52#11718649C2212yes—
53#11718647C2214yes—
54#11718646C2212yes—
55#11718646C2212yes—
56C4⋊C4×Dic13C2212yes—
57#11718634C2212yes—
58#11718619C2214yes—
59#11718619C2214yes—
60#11718600C2212yes—
61#11718600C2212yes—
62#11718600C2212yes—
63#11718596C2211yescharacteristic
64C4×#6101C2212yes—
65C4×#6101C2212yes—
66C4×#6101C2212yes—
67C4×#6101C2212yes—
68#11718333C2214yes—
69#11718331C2214yes—
70#11718331C2214yes—
71#11718330C2214yes—
72#11718327C2214yes—
73#11718326C2214yes—
74#11718322C2214yes—
75#11718315C2214yes—
76C22×Q8×C13C2312yes—
77C23×C4×C13C2311yescharacteristic
78C23×C4×C13C2311yescharacteristic
79C23×C4×C13C2311yescharacteristic
80C22×C22⋊D13C2314yes—
81C23×Dic13C2311yescharacteristic
82C23×Dic13C2311yescharacteristic
83C23×Dic13C2311yescharacteristic
84C23×Dic13C2311yescharacteristic
85C22×D52C2312yes—
86C22×D52C2312yes—
87C22×D52C2312yes—
88C22×C4×D13C2314yes—
89C13×C4⋊Q8—42no—
90C13×C22⋊Q8—28no—
91C2×C13×C4⋊C4C2312yes—
92C2×C13×C4⋊C4C2312yes—
93C2×C13×C4⋊C4C2312yes—
94C2×C13×C4⋊C4C2314yes—
95C2×C13×C4⋊C4C2312yes—
96C2×C13×C4⋊C4C2312yes—
97C2×C13×C22⋊C4C2314yes—
98C2×C42×C13—42no—
99C2×C42×C13C2311yescharacteristic
100C2×C42×C13C2311yescharacteristic
101C2×C42×C13C2311yescharacteristic
102C2×C42×C13C2311yescharacteristic
103#88200—216no—
104#88184—28no—
105#88184—28no—
106#88184—28no—
107C2×#6103C2314yes—
108C2×#6103C2314yes—
109C2×#6103C2314yes—
110C2×#6103C2314yes—
111C2×#6103C2314yes—
112C2×#6103C2314yes—
113C2×#6103C2314yes—
114C2×#6103C2314yes—
115C2×#6103C2314yes—
116C2×#6103C2314yes—
117C2×C4⋊Dic13C2314yes—
118C2×C4⋊Dic13C2314yes—
119C2×C4⋊Dic13C2312yes—
120C2×C4⋊Dic13C2312yes—
121C2×C4⋊Dic13C2312yes—
122C2×#6101—24no—
123C2×#6101D418yescomplemented
124C2×#6101C2314yes—
125C2×#6101C2312yes—
126C2×#6101C2312yes—
127C2×#6101C2312yes—
128C2×#6101C2312yes—
129C2×C4×Dic13—24no—
130C2×C4×Dic13—42no—
131C2×C4×Dic13C2314yes—
132C2×C4×Dic13C2312yes—
133C2×C4×Dic13C2312yes—
134C2×C4×Dic13C2312yes—
135C2×C4×Dic13C2312yes—
136#88152—216no—
137#88148—216no—
138#88147—44no—
139#88144—216no—
140#88140—216no—
141#88135—216no—
142#88128—42no—
143C13×#95C2314yes—
144C13×#95C2314yes—
145#88071C2312yes—
146#88071C2314yes—
147#88071C2314yes—
148#88071C2314yes—
149#88071C2314yes—
150#88071C2312yes—
151#88071C2312yes—
152#88071C2312yes—
153#35006—131nomaximal, Sylow
154C24×C13C2411yescharacteristic
155C23×D13C2414yes—
156C2×Q8×C13—24no—
157C2×Q8×C13—42no—
158C2×Q8×C13—42no—
159C22×C4×C13—22no—
160C22×C4×C13—22no—
161C22×C4×C13—41no—
162C22×C4×C13—41no—
163C22×C4×C13—41no—
164C22×C4×C13—41no—
165C22×C4×C13C2×D414yes—
166C22×C4×C13#4114yes—
167C22×C4×C13#4114yes—
168C22×C4×C13C2414yes—
169C22×C4×C13C2414yes—
170C22×C4×C13C2411yescharacteristic
171C22×C4×C13C2411yescharacteristic
172C22×C4×C13C2411yescharacteristic
173C22×C4×C13C2411yescharacteristic
174C22×C4×C13C2411yescharacteristic
175C22×C4×C13C2411yescharacteristic
176C2×C22⋊D13—28no—
177C2×C22⋊D13—28no—
178C2×C22⋊D13—28no—
179C22×Dic13—22no—
180C22×Dic13—22no—
181C22×Dic13—22no—
182C22×Dic13—22no—
183C22×Dic13—41no—
184C22×Dic13—41no—
185C22×Dic13—41no—
186C22×Dic13—41no—
187C22×Dic13—41no—
188C22×Dic13—41no—
189C22×Dic13C2×D414yes—
190C22×Dic13C2×D414yes—
191C22×Dic13C2412yes—
192C22×Dic13C2412yes—
193C22×Dic13C2414yes—
194C22×Dic13C2412yes—
195C22×Dic13C2412yes—
196C2×D52—24no—
197C2×D52—24no—
198C2×D52—24no—
199C2×D52—24no—
200C2×D52—24no—
201C2×D52—42no—
202C2×D52—42no—
203C2×D52—42no—
204C2×D52—42no—
205C2×C4×D13—28no—
206C2×C4×D13—44no—
207C2×C4×D13—44no—
208C13×C4⋊C4—28no—
209C13×C4⋊C4—24no—
210C13×C4⋊C4—24no—
211C13×C4⋊C4—24no—
212C13×C4⋊C4—42no—
213C13×C4⋊C4—42no—
214C13×C22⋊C4—28no—
215C42×C13—22no—
216C42×C13—22no—
217C42×C13—41no—
218C42×C13—41no—
219#6103—28no—
220#6103—28no—
221#6103—28no—
222#6103—28no—
223#6103—28no—
224#6103—28no—
225#6103—44no—
226#6103—44no—
227#6103—44no—
228#6103—44no—
229C4⋊Dic13—24no—
230C4⋊Dic13—24no—
231C4⋊Dic13—24no—
232C4⋊Dic13—28no—
233C4⋊Dic13—28no—
234#6101—24no—
235#6101—24no—
236#6101—28no—
237#6101C2×D418yes—
238#6101C2×D418yes—
239C4×Dic13—24no—
240C4×Dic13—24no—
241C4×Dic13—24no—
242C4×Dic13—28no—
243C4×Dic13—42no—
244C2×#539—132no—
245#2857D1311yescharacteristic, complemented
246#2856—131no—
247#2854—132no—
248#2639—132no—
249#2618—132no—
250#2560—134no—
251#2558—134no—
252#2512—132no—
253#2506—132no—
254#2452—132no—
255#2445—132no—
256#2325—132no—
257C4×#121—132no—
258C2×C4×C4⋊C4—131no—
259C23×C13—21no—
260C23×C13—21no—
261C23×C13—21no—
262C23×C13—21no—
263C23×C13C22×D412yes—
264C23×C13C2×#4112yes—
265C23×C13C2×#4112yes—
266C23×C13C2511yescharacteristic
267C22×D13—24no—
268C22×D13—24no—
269C22×D13—24no—
270C22×D13—24no—
271C22×D13—24no—
272C22×D13—24no—
273C22×D13—24no—
274Q8×C13—84no—
275C2×C4×C13—21no—
276C2×C4×C13—21no—
277C2×C4×C13—21no—
278C2×C4×C13—21no—
279C2×C4×C13—21no—
280C2×C4×C13—21no—
281C2×C4×C13—21no—
282C2×C4×C13—21no—
283C2×C4×C13—21no—
284C2×C4×C13—21no—
285C2×C4×C13—28no—
286C2×C4×C13—24no—
287C2×C4×C13—24no—
288C2×C4×C13—24no—
289C2×C4×C13—24no—
290C2×C4×C13—24no—
291C2×C4×C13—24no—
292C2×C4×C13—44no—
293C2×C4×C13—44no—
294C2×C4×C13C22×D412yes—
295C2×C4×C13C22×D412yes—
296C2×C4×C13C2×#4112yes—
297C2×C4×C13C2×#4112yes—
298C2×C4×C13C2×#4112yes—
299C2×C4×C13C2×#4112yes—
300C2×C4×C13H(5,𝔽2)12yes—
301C2×C4×C132−1+412yes—
302C22⋊D13—416no—
303C2×Dic13—28no—
304C2×Dic13—22no—
305C2×Dic13—22no—
306C2×Dic13—22no—
307C2×Dic13—22no—
308C2×Dic13—24no—
309C2×Dic13—22no—
310C2×Dic13—22no—
311C2×Dic13—24no—
312C2×Dic13—24no—
313C2×Dic13—22no—
314C2×Dic13—22no—
315C2×Dic13—22no—
316C2×Dic13—22no—
317C2×Dic13—44no—
318C2×Dic13—44no—
319C2×Dic13—44no—
320C2×Dic13C22×D414yes—
321C2×Dic13C22×D414yes—
322D52—48no—
323D52—84no—
324D52—84no—
325C4×D13—88no—
326#539—2616no—
327C2×C4⋊Q8C2×D1311yescharacteristic
328C2×#127—131no—
329C2×#126—134no—
330C2×#125—134no—
331C2×#123—134no—
332C2×#123—134no—
333C2×C22⋊Q8—134no—
334C2×C22⋊Q8C2×D1312yes—
335C2×#121—132no—
336C2×#121—132no—
337C2×#121—132no—
338C2×C4×D4—132no—
339C2×C4×D4—134no—
340C22×C4⋊C4—131no—
341C22×C4⋊C4—131no—
342C22×C42—131no—
343C22×C42—131no—
344C22×C42C2×D1311yescharacteristic
345#400—134no—
346#398—134no—
347#397—134no—
348#396—134no—
349#396—134no—
350#394—134no—
351#394—134no—
352#393—134no—
353#391C2×D1312yes—
354#390—132no—
355#390—132no—
356#390—132no—
357#390—132no—
358#390—132no—
359#390—132no—
360#389—132no—
361#389—132no—
362#389—132no—
363#389C2×D1312yes—
364#389C2×D1312yes—
365#388—134no—
366#388—134no—
367#388—134no—
368#388—134no—
369#387—134no—
370#386—134no—
371#386—134no—
372#386—134no—
373#385—134no—
374#384—131no—
375#384C2×D1311yescharacteristic
376#380C2×D1312yes—
377#380C2×D1312yes—
378C4×C4⋊C4—132no—
379C4×C4⋊C4—132no—
380C4×C4⋊C4—132no—
381C4×C4⋊C4—132no—
382C4×C4⋊C4—132no—
383C4×C22⋊C4—134no—
384C2×#95—131no—
385C2×#95—131no—
386C22×C13—22no—
387C22×C13—22no—
388C22×C13—22no—
389C22×C13—22no—
390C22×C13—22no—
391C22×C13—22no—
392C22×C13#53214yes—
393C22×C13C23×D411yescharacteristic
394C22×C13C22×#4111yescharacteristic
395C22×C13C22×#4111yescharacteristic
396C22×C13C2×H(5,𝔽2)11yescharacteristic
397C22×C13C2×2−1+411yescharacteristic
398C22×C13#58511yescharacteristic
399C22×C13#58511yescharacteristic
400C2×D13—44no—
401C2×D13—44no—
402C2×D13—44no—
403C2×D13—44no—
404C2×D13—44no—
405C2×D13—44no—
406C2×D13—44no—
407C4×C13—22no—
408C4×C13—22no—
409C4×C13—22no—
410C4×C13—22no—
411C4×C13—22no—
412C4×C13—44no—
413C4×C13—44no—
414C4×C13#53914yescomplemented
415Dic13—24no—
416Dic13—24no—
417Dic13—44no—
418Dic13—42no—
419Dic13—42no—
420C22×Q8C22×D1312yes—
421C22×D4—134no—
422C22×D4—132no—
423C22×D4—132no—
424C22×D4—132no—
425C23×C4—131no—
426C23×C4—131no—
427C23×C4—131no—
428C23×C4—131no—
429C23×C4—134no—
430C23×C4C22×D1311yescharacteristic
431C23×C4C22×D1311yescharacteristic
432C23×C4C22×D1311yescharacteristic
433C4⋊Q8—42no—
434#127—522no—
435#126—2616no—
436#125—2616no—
437#123—2616no—
438#123—2616no—
439C22⋊Q8—28no—
440C22⋊Q8—2616no—
441#121—268no—
442#121—268no—
443#121—268no—
444C4×D4—2616no—
445C4×D4—524no—
446C2×C4⋊C4—138no—
447C2×C4⋊C4—132no—
448C2×C4⋊C4—132no—
449C2×C4⋊C4—132no—
450C2×C4⋊C4—132no—
451C2×C4⋊C4—134no—
452C2×C4⋊C4—132no—
453C2×C4⋊C4—132no—
454C2×C4⋊C4—132no—
455C2×C4⋊C4—134no—
456C2×C4⋊C4—134no—
457C2×C4⋊C4—264no—
458C2×C4⋊C4C22×D1312yes—
459C2×C4⋊C4C22×D1314yes—
460C2×C4⋊C4C22×D1312yes—
461C2×C4⋊C4C22×D1312yes—
462C2×C4⋊C4C22×D1312yes—
463C2×C4⋊C4C22×D1312yes—
464C2×C22⋊C4—134no—
465C2×C22⋊C4—134no—
466C2×C22⋊C4—134no—
467C2×C22⋊C4—134no—
468C2×C22⋊C4—134no—
469C2×C22⋊C4—134no—
470C2×C22⋊C4—134no—
471C2×C22⋊C4—134no—
472C2×C22⋊C4—134no—
473C2×C22⋊C4—134no—
474C2×C22⋊C4C22×D1314yes—
475C2×C42—42no—
476C2×C42—134no—
477C2×C42—132no—
478C2×C42—132no—
479C2×C42—132no—
480C2×C42—132no—
481C2×C42—264no—
482C2×C42—522no—
483C2×C42C22×D1311yescharacteristic
484C2×C42C22×D1311yescharacteristic
485C2×C42C22×D1311yescharacteristic
486C2×C42C22×D1311yescharacteristic
487#95—132no—
488#95—132no—
489#95—132no—
490#95—132no—
491#95—134no—
492#95—134no—
493#95—134no—
494#95—134no—
495#95C22×D1314yes—
496#95C22×D1314yes—
497C2×C13—24no—
498C2×C13C2×#53211yescharacteristic
499C2×C13C2×#53911yescharacteristic
500C2×C13#345511yescharacteristic
501C2×C13#349611yescharacteristic
502C2×C13#349711yescharacteristic
503C2×C13#350511yescharacteristic
504C2×C13#350711yescharacteristic
505D13—84no—
506C24—134no—
507C24C23×D1311yescharacteristic
508C2×Q8—24no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×D4—268no—
512C2×D4—268no—
513C2×D4—268no—
514C2×D4—264no—
515C2×D4—264no—
516C2×D4—264no—
517C2×D4—264no—
518C2×D4—264no—
519C2×D4—522no—
520C2×D4—522no—
521C2×D4—522no—
522C2×D4—522no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—41no—
526C22×C4—41no—
527C22×C4—41no—
528C22×C4—41no—
529C22×C4—134no—
530C22×C4—132no—
531C22×C4—132no—
532C22×C4—134no—
533C22×C4—132no—
534C22×C4—132no—
535C22×C4—134no—
536C22×C4—268no—
537C22×C4—262no—
538C22×C4—262no—
539C22×C4—262no—
540C22×C4—262no—
541C22×C4—524no—
542C22×C4—524no—
543C22×C4—521no—
544C22×C4—521no—
545C22×C4—521no—
546C22×C4—521no—
547C22×C4—521no—
548C22×C4—521no—
549C22×C4D4×D1314yes—
550C22×C4#613114yes—
551C22×C4#613114yes—
552C22×C4C23×D1314yes—
553C22×C4C23×D1314yes—
554C22×C4C23×D1311yescharacteristic
555C22×C4C23×D1311yescharacteristic
556C22×C4C23×D1311yescharacteristic
557C22×C4C23×D1311yescharacteristic
558C22×C4C23×D1311yescharacteristic
559C22×C4C23×D1311yescharacteristic
560C4⋊C4—28no—
561C4⋊C4—24no—
562C4⋊C4—24no—
563C4⋊C4—24no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—138no—
567C4⋊C4—138no—
568C4⋊C4—268no—
569C4⋊C4—268no—
570C4⋊C4—268no—
571C4⋊C4—264no—
572C4⋊C4—264no—
573C4⋊C4—264no—
574C4⋊C4—264no—
575C4⋊C4—264no—
576C22⋊C4—28no—
577C22⋊C4—268no—
578C22⋊C4—268no—
579C22⋊C4—268no—
580C22⋊C4—268no—
581C22⋊C4—268no—
582C22⋊C4—268no—
583C22⋊C4—524no—
584C22⋊C4—524no—
585C22⋊C4—524no—
586C22⋊C4—524no—
587C42—22no—
588C42—22no—
589C42—41no—
590C42—41no—
591C42—268no—
592C42—264no—
593C42—264no—
594C42—264no—
595C42—522no—
596C13#3500611yescharacteristic, complemented, Sylow
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—264no—
602C23—264no—
603C23—264no—
604C23—264no—
605C23—264no—
606C23—264no—
607C23—264no—
608C23C2×D4×D1312yes—
609C23C2×#613112yes—
610C23C2×#613112yes—
611C23C24×D1311yescharacteristic
612Q8—84no—
613D4—5216no—
614D4—528no—
615D4—1044no—
616D4—1044no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—134no—
637C2×C4—134no—
638C2×C4—268no—
639C2×C4—262no—
640C2×C4—262no—
641C2×C4—262no—
642C2×C4—262no—
643C2×C4—264no—
644C2×C4—262no—
645C2×C4—262no—
646C2×C4—264no—
647C2×C4—264no—
648C2×C4—262no—
649C2×C4—262no—
650C2×C4—262no—
651C2×C4—262no—
652C2×C4—524no—
653C2×C4—524no—
654C2×C4—524no—
655C2×C4—1048no—
656C2×C4C2×D4×D1312yes—
657C2×C4C2×D4×D1312yes—
658C2×C4#8825112yes—
659C2×C4C2×#613112yes—
660C2×C4C2×#613112yes—
661C2×C4C2×#613112yes—
662C2×C4C2×#613112yes—
663C2×C4#8825712yes—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—524no—
671C22—524no—
672C22—524no—
673C22—524no—
674C22—524no—
675C22—524no—
676C22—524no—
677C22#1171943014yes—
678C22C22×D4×D1311yescharacteristic
679C22C2×#8825111yescharacteristic
680C22C22×#613111yescharacteristic
681C22C22×#613111yescharacteristic
682C22C2×#8825711yescharacteristic
683C22#1171964811yescharacteristic
684C22#1171964811yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—44no—
691C4—44no—
692C4—264no—
693C4—264no—
694C4—522no—
695C4—524no—
696C4—522no—
697C4#1171935014yescomplemented
698C2—24no—
699C2—1044no—
700C2C2×#1171943011yescharacteristic
701C2C2×#1171935011yescharacteristic
702C2#1312822811yescharacteristic
703C2#1312859311yescharacteristic
704C2#1312860211yescharacteristic
705C2#1312867711yescharacteristic
706C2#1312870311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D13
commutator, socle, cosocleC23×C13C25
FittingC13×#2857C2
prime radical#13123181C2
hypercenter#2857D13
hypocenterC13#35006
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C2×#11719350C2semidirect
3. #13123168C2semidirect
4. #13120770C2semidirect
5. #13120581C2semidirect
6. #13119987C2semidirect
7. #13119957C2semidirect
8. #13119418C2semidirect
9. #13119354C2semidirect
10. #13118682C2semidirect
11. #13118603C2semidirect
12. #13117536C2semidirect
13. #13117342C2semidirect
14. C2×C4×#6101C2semidirect
15. C13×#2857C2semidirect
123. C2×#6101D4semidirect
245. #2857D13semidirect
414. C4×C13#539semidirect
596. C13#35006semidirect
697. C4#11719350semidirect

Maximal subgroups

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

subgroupindex
C2×#117193502
#131231812
#131231682
#131207702
#131205812
#131199872
#131199572
#131194182
#131193542
#131186822
#131186032
#131175362
#131173422
C2×C4×#61012
C13×#28572
#3500613