Finite Group Databasebeta

← #16023020

Subgroups — #16023020

Order 3328.

3206 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719209C211yesmaximal, characteristic
2C2×#11719228C211yesmaximal, characteristic
3C2×#11719227C211yesmaximal, characteristic
4#13117750C214yesmaximal, complemented
5#13117737C214yesmaximal, complemented
6#13117734C218yesmaximal, complemented
7#13117461C212yesmaximal, complemented
8#13117459C211yesmaximal, characteristic, complemented
9#13117457C212yesmaximal, complemented
10#13117242C212yesmaximal, complemented
11#13117239C211yesmaximal, characteristic, complemented
12C2×#11718637C211yesmaximal, characteristic, complemented
13D13×#384C212yesmaximal, complemented
14C13×#2317C211yesmaximal, characteristic, complemented
15C22×#6127C2211yescharacteristic
16C2×C13×#117C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C2×C4×C22⋊D13C2214yes—
20C2×#88180C2211yescharacteristic
21C22×C4⋊Dic13C2211yescharacteristic
22C22×#6101C2214yes—
23C22×C4×Dic13C2211yescharacteristic
24#11719228—28no—
25#11719227C4116yescomplemented
26#11719209—28no—
27C2×#88147C2212yes—
28C2×#88147C2212yes—
29C2×#88146C2212yes—
30C2×#88146C2212yes—
31C2×C4⋊C4×D13C2212yes—
32C2×C4⋊C4×D13C2212yes—
33C2×C4⋊C4×D13C2218yes—
34C2×#88141C2212yes—
35C2×#88141C2212yes—
36C2×#88135C2214yes—
37C2×#88135C2214yes—
38C2×C4×D52C2212yes—
39C2×#88126C2212yes—
40C2×C42×D13C2212yes—
41C2×C4×Dic26C2212yes—
42C13×#389C2214yes—
43C13×#385C2214yes—
44C13×#384C2212yes—
45C13×#382C2212yes—
46#11718650C2218yes—
47#11718646C2214yes—
48#11718638C2211yescharacteristic
49#11718638C2211yescharacteristic
50#11718637C2212yes—
51#11718637C2218yes—
52#11718637C2214yes—
53#11718637C2211yescharacteristic
54#11718637C2212yes—
55#11718637C2211yescharacteristic
56#11718633C2218yes—
57#11718611C2214yes—
58#11718603C2214yes—
59#11718597C2212yes—
60#11718596C2212yes—
61C4×#6101C2214yes—
62#11718329C2218yes—
63#11718328C2218yes—
64#11718310C2218yes—
65#11718308C2218yes—
66C23×C4×C13C2312yes—
67C23×C4×C13C2311yescharacteristic
68C22×C22⋊D13C2312yes—
69C23×Dic13C2312yes—
70C23×Dic13C2312yes—
71C2×#6127C2×C418yes—
72C2×#6127D418yes—
73C2×#6127Q818yes—
74C22×D52C2311yescharacteristic
75C22×C4×D13C2314yes—
76C22×C4×D13C2314yes—
77C22×C4×D13C2314yes—
78C22×C4×D13C2312yes—
79C22×Dic26C2311yescharacteristic
80C13×#117—24no—
81C13×#117C2×C418yescomplemented
82C2×C13×C4⋊C4—24no—
83C2×C13×C4⋊C4C2311yescharacteristic
84C2×C13×C4⋊C4C2312yes—
85C2×C13×C4⋊C4C2311yescharacteristic
86C2×C13×C4⋊C4C2312yes—
87C2×C13×C4⋊C4C2318yes—
88C2×C13×C4⋊C4C2312yes—
89C2×C13×C22⋊C4C2312yes—
90C2×C13×C22⋊C4C2312yes—
91C2×C42×C13C2312yes—
92C2×C42×C13C2312yes—
93C2×C22⋊Dic13C2312yes—
94C4×C22⋊D13—216no—
95C2×#6103C2314yes—
96C2×#6103C2314yes—
97C2×#6103C2314yes—
98#88180—24no—
99C2×C4⋊Dic13—24no—
100C2×C4⋊Dic13C2312yes—
101C2×C4⋊Dic13C2311yescharacteristic
102C2×C4⋊Dic13C2312yes—
103C2×C4⋊Dic13C2311yescharacteristic
104C2×C4⋊Dic13C2318yes—
105C2×#6101—48no—
106C2×#6101C2314yes—
107C2×#6101C2314yes—
108C2×#6101C2314yes—
109C2×#6101C2318yes—
110C2×#6101C2314yes—
111C2×#6101C2318yes—
112C2×#6101C2314yes—
113C2×C4×Dic13C2×C418yescomplemented
114C2×C4×Dic13C2314yes—
115C2×C4×Dic13C2314yes—
116C2×C4×Dic13C2312yes—
117C2×C4×Dic13C2311yescharacteristic
118C2×C4×Dic13C2314yes—
119C2×C4×Dic13C2311yescharacteristic
120C2×C4×Dic13C2312yes—
121#88147—28no—
122#88147C2×C4116yescomplemented
123#88146—28no—
124#88146—28no—
125C4⋊C4×D13—28no—
126C4⋊C4×D13—28no—
127C4⋊C4×D13—416no—
128#88141—28no—
129#88141C2×C4116yescomplemented
130#88135—216no—
131#88135C2×C4132yescomplemented
132C4×D52—28no—
133#88126C2×C4116yescomplemented
134C42×D13C2×C4116yescomplemented
135C4×Dic26—28no—
136C13×#95C2318yes—
137#88071C2314yes—
138#88071C2318yes—
139#88071C2314yes—
140#34183—131nomaximal, Sylow
141C24×C13C2411yescharacteristic
142C23×D13C2412yes—
143C22×C4×C13—24no—
144C22×C4×C13—44no—
145C22×C4×C13C22×C414yes—
146C22×C4×C13C2×D414yes—
147C22×C4×C13C2×Q814yes—
148C22×C4×C13C2412yes—
149C22×C4×C13C2412yes—
150C22×C4×C13C2414yes—
151C22×C4×C13C2414yes—
152C22×C4×C13C2411yescharacteristic
153C22×C4×C13C2411yescharacteristic
154C2×C22⋊D13C22×C418yes—
155C2×C22⋊D13C2×D418yes—
156C2×C22⋊D13C2×Q818yes—
157C22×Dic13—24no—
158C22×Dic13—24no—
159C22×Dic13—24no—
160C22×Dic13—44no—
161C22×Dic13C22×C418yes—
162C22×Dic13C2412yes—
163C22×Dic13C2414yes—
164C22×Dic13C2412yes—
165C22×Dic13C2412yes—
166C22×Dic13C2412yes—
167C22×Dic13C2412yes—
168#6127C4⋊C4164yescomplemented
169C2×D52C22×C414yes—
170C2×D52C2×D414yes—
171C2×D52C2×Q814yes—
172C2×C4×D13—28no—
173C2×C4×D13—28no—
174C2×C4×D13—28no—
175C2×C4×D13—28no—
176C2×C4×D13—48no—
177C2×C4×D13—48no—
178C2×C4×D13C22×C4116yes—
179C2×C4×D13C22×C418yes—
180C2×C4×D13C2×D418yes—
181C2×C4×D13C2×Q818yes—
182C2×Dic26C22×C414yes—
183C2×Dic26C2×D414yes—
184C2×Dic26C2×Q814yes—
185C13×C4⋊C4—24no—
186C13×C4⋊C4—22no—
187C13×C4⋊C4—24no—
188C13×C4⋊C4—22no—
189C13×C4⋊C4—48no—
190C13×C4⋊C4C22×C418yes—
191C13×C22⋊C4—24no—
192C13×C22⋊C4C22×C418yes—
193C42×C13—24no—
194C42×C13C22×C418yes—
195C22⋊Dic13—24no—
196#6103—28no—
197#6103—28no—
198#6103C22×C4116yes—
199C4⋊Dic13—24no—
200C4⋊Dic13—22no—
201C4⋊Dic13—24no—
202C4⋊Dic13—22no—
203C4⋊Dic13—48no—
204#6101—28no—
205#6101—28no—
206#6101—28no—
207#6101—216no—
208#6101—28no—
209#6101—48no—
210#6101C22×C4116yes—
211C4×Dic13—24no—
212C4×Dic13—28no—
213C4×Dic13—28no—
214C4×Dic13C22×C414yes—
215C4×Dic13C22×C4116yes—
216C4×Dic13C22×C414yes—
217C4×Dic13C22×C418yes—
218C2×#520—131no—
219C2×#520—131no—
220C2×C4×#41—131no—
221Q8×C4⋊C4—134no—
222D4×C4⋊C4—134no—
223D4×C4⋊C4—138no—
224#2317—132no—
225#2317—132no—
226#2317—131no—
227#2317D1311yescharacteristic, complemented
228#2285—132no—
229#2285—131no—
230C2×#384—132no—
231C2×#384—131no—
232C23×C13—22no—
233C23×C13—22no—
234C23×C13C23×C412yes—
235C23×C13C22×D412yes—
236C23×C13C22×Q812yes—
237C23×C13C2511yescharacteristic
238C22×D13—24no—
239C22×D13—24no—
240C22×D13C23×C414yes—
241C22×D13C22×D414yes—
242C22×D13C22×Q814yes—
243C2×C4×C13—24no—
244C2×C4×C13—24no—
245C2×C4×C13—22no—
246C2×C4×C13—24no—
247C2×C4×C13—22no—
248C2×C4×C13—28no—
249C2×C4×C13—24no—
250C2×C4×C13—24no—
251C2×C4×C13—84no—
252C2×C4×C13C2×C4⋊C4116yescomplemented
253C2×C4×C13C23×C418yes—
254C2×C4×C13C23×C412yes—
255C2×C4×C13C23×C412yes—
256C2×C4×C13C22×D412yes—
257C2×C4×C13C22×D412yes—
258C2×C4×C13C22×Q812yes—
259C2×C4×C13C22×Q812yes—
260C22⋊D13C2×C4⋊C4132yescomplemented
261C2×Dic13—22no—
262C2×Dic13—22no—
263C2×Dic13—216no—
264C2×Dic13—22no—
265C2×Dic13—22no—
266C2×Dic13—24no—
267C2×Dic13—28no—
268C2×Dic13—24no—
269C2×Dic13—24no—
270C2×Dic13—84no—
271C2×Dic13C23×C414yes—
272C2×Dic13C23×C414yes—
273C2×Dic13C23×C414yes—
274C2×Dic13C22×D414yes—
275C2×Dic13C22×Q814yes—
276C2×Dic13H(5,𝔽2)14yes—
277C2×Dic132−1+414yes—
278D52C2×C4⋊C4116yescomplemented
279C4×D13—232no—
280C4×D13—88no—
281C4×D13—88no—
282C4×D13C2×C4⋊C4132yescomplemented
283Dic26C2×C4⋊C4116yescomplemented
284C22×#41—131no—
285#520—268no—
286#520—268no—
287C4×#41—1316no—
288C2×C4×Q8—132no—
289C2×C4×Q8—132no—
290C2×C4×Q8—132no—
291C2×C4×D4—134no—
292C2×C4×D4—132no—
293C2×C4×D4—132no—
294C2×C4×D4—134no—
295C2×C4×D4—134no—
296C2×C4×D4—132no—
297C2×#117—131no—
298C2×#117—132no—
299C2×#117—132no—
300C2×#117—132no—
301C2×#117C2×D1311yescharacteristic
302C2×#117C2×D1311yescharacteristic
303C22×C4⋊C4—131no—
304C22×C4⋊C4—134no—
305C22×C4⋊C4—132no—
306C22×C4⋊C4—132no—
307C22×C4⋊C4—138no—
308C22×C4⋊C4C2×D1311yescharacteristic
309C22×C42—131no—
310C22×C42—132no—
311#389—138no—
312#389—134no—
313#389—138no—
314#389C2×D1314yes—
315#385—138no—
316#385—134no—
317#385—138no—
318#385C2×D1314yes—
319#384—132no—
320#384—138no—
321#384—134no—
322#384—131no—
323#384—132no—
324#384—131no—
325#384—132no—
326#384C2×D1312yes—
327#382—131no—
328#382—131no—
329#382—132no—
330#382C2×D1312yes—
331#380—134no—
332#380—138no—
333C4×C4⋊C4—134no—
334C4×C4⋊C4—138no—
335C22×C13—24no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—24no—
339C22×C13C22×C4⋊C414yes—
340C22×C13C24×C411yescharacteristic
341C22×C13C23×D411yescharacteristic
342C22×C13C23×Q811yescharacteristic
343C22×C13C2×H(5,𝔽2)12yes—
344C22×C13C2×2−1+412yes—
345C2×D13—28no—
346C2×D13—24no—
347C2×D13—24no—
348C2×D13—28no—
349C2×D13C22×C4⋊C418yes—
350C4×C13—28no—
351C4×C13—42no—
352C4×C13—44no—
353C4×C13—42no—
354C4×C13C22×C4⋊C414yes—
355C4×C13C22×C4⋊C414yes—
356Dic13—24no—
357Dic13—42no—
358Dic13—44no—
359Dic13—42no—
360Dic13C22×C4⋊C418yes—
361Dic13#52018yescomplemented
362C2×#41—138no—
363C2×#41—138no—
364C2×#41—138no—
365C22×Q8—131no—
366C22×D4—132no—
367C22×D4—131no—
368C23×C4—132no—
369C23×C4—132no—
370C23×C4—134no—
371C23×C4—132no—
372C23×C4—134no—
373C23×C4—134no—
374C23×C4C22×D1312yes—
375C23×C4C22×D1311yescharacteristic
376C4×Q8—1316no—
377C4×Q8—268no—
378C4×Q8—268no—
379C4×D4—1316no—
380C4×D4—1332no—
381C4×D4—268no—
382C4×D4—2616no—
383C4×D4—268no—
384C4×D4—2616no—
385#117—24no—
386#117—1316no—
387#117—268no—
388#117—268no—
389#117—264no—
390#117C4×D1318yescomplemented
391C2×C4⋊C4—24no—
392C2×C4⋊C4—134no—
393C2×C4⋊C4—132no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—134no—
396C2×C4⋊C4—138no—
397C2×C4⋊C4—134no—
398C2×C4⋊C4—134no—
399C2×C4⋊C4—132no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—134no—
402C2×C4⋊C4—138no—
403C2×C4⋊C4—138no—
404C2×C4⋊C4—268no—
405C2×C4⋊C4—268no—
406C2×C4⋊C4—264no—
407C2×C4⋊C4—5216no—
408C2×C4⋊C4—528no—
409C2×C4⋊C4C22×D1311yescharacteristic
410C2×C4⋊C4C22×D1312yes—
411C2×C4⋊C4C22×D1312yes—
412C2×C4⋊C4C22×D1311yescharacteristic
413C2×C4⋊C4C22×D1318yes—
414C2×C4⋊C4C22×D1312yes—
415C2×C22⋊C4—134no—
416C2×C22⋊C4—132no—
417C2×C22⋊C4—134no—
418C2×C22⋊C4—134no—
419C2×C22⋊C4C22×D1312yes—
420C2×C22⋊C4C22×D1312yes—
421C2×C42—138no—
422C2×C42—131no—
423C2×C42—134no—
424C2×C42—132no—
425C2×C42—131no—
426C2×C42—1316no—
427C2×C42—134no—
428C2×C42—134no—
429C2×C42—132no—
430C2×C42C22×D1312yes—
431C2×C42C22×D1312yes—
432#95—134no—
433#95—134no—
434#95—138no—
435#95C22×D1318yes—
436C2×C13—24no—
437C2×C13C23×C4⋊C411yescharacteristic
438C2×C13C2×#52012yes—
439C2×C13#349811yescharacteristic
440C2×C13#349911yescharacteristic
441C2×C13#350312yes—
442D13—28no—
443C24—132no—
444C24C23×D1311yescharacteristic
445#41—1364no—
446C2×Q8—134no—
447C2×Q8—134no—
448C2×Q8—134no—
449C2×D4—138no—
450C2×D4—138no—
451C2×D4—138no—
452C2×D4—134no—
453C2×D4—134no—
454C2×D4—134no—
455C22×C4—24no—
456C22×C4—44no—
457C22×C4—132no—
458C22×C4—132no—
459C22×C4—138no—
460C22×C4—138no—
461C22×C4—138no—
462C22×C4—138no—
463C22×C4—132no—
464C22×C4—1316no—
465C22×C4—132no—
466C22×C4—132no—
467C22×C4—134no—
468C22×C4—268no—
469C22×C4—268no—
470C22×C4—268no—
471C22×C4—268no—
472C22×C4—264no—
473C22×C4—264no—
474C22×C4—264no—
475C22×C4—528no—
476C22×C4—528no—
477C22×C4—524no—
478C22×C4C2×C4×D1314yes—
479C22×C4D4×D1314yes—
480C22×C4Q8×D1314yes—
481C22×C4C23×D1312yes—
482C22×C4C23×D1312yes—
483C22×C4C23×D1314yes—
484C22×C4C23×D1314yes—
485C22×C4C23×D1311yescharacteristic
486C22×C4C23×D1311yescharacteristic
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—48no—
492C4⋊C4—1316no—
493C4⋊C4—268no—
494C4⋊C4—268no—
495C4⋊C4—268no—
496C4⋊C4—2616no—
497C4⋊C4—268no—
498C4⋊C4—262no—
499C4⋊C4—264no—
500C4⋊C4—264no—
501C4⋊C4—262no—
502C4⋊C4—528no—
503C4⋊C4—528no—
504C4⋊C4C2×C4×D1318yes—
505C22⋊C4—24no—
506C22⋊C4—1316no—
507C22⋊C4—268no—
508C22⋊C4—268no—
509C22⋊C4—264no—
510C22⋊C4C2×C4×D1318yes—
511C42—24no—
512C42—134no—
513C42—134no—
514C42—1316no—
515C42—138no—
516C42—268no—
517C42—268no—
518C42—264no—
519C42C2×C4×D1318yes—
520C13#3418311yescharacteristic, complemented, Sylow
521C23—22no—
522C23—22no—
523C23—134no—
524C23—134no—
525C23—134no—
526C23—264no—
527C23—264no—
528C23C22×C4×D1312yes—
529C23C2×D4×D1312yes—
530C23C2×Q8×D1312yes—
531C23C24×D1311yescharacteristic
532Q8—1316no—
533D4—1332no—
534D4—1316no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—84no—
544C2×C4—134no—
545C2×C4—134no—
546C2×C4—134no—
547C2×C4—134no—
548C2×C4—134no—
549C2×C4—134no—
550C2×C4—1332no—
551C2×C4—134no—
552C2×C4—2632no—
553C2×C4—2616no—
554C2×C4—262no—
555C2×C4—262no—
556C2×C4—262no—
557C2×C4—264no—
558C2×C4—262no—
559C2×C4—268no—
560C2×C4—264no—
561C2×C4—264no—
562C2×C4—1048no—
563C2×C4—1048no—
564C2×C4—1044no—
565C2×C4C4⋊C4×D13116yescomplemented
566C2×C4C22×C4×D1318yes—
567C2×C4C22×C4×D1312yes—
568C2×C4C22×C4×D1312yes—
569C2×C4C2×D4×D1312yes—
570C2×C4C2×D4×D1312yes—
571C2×C4C2×Q8×D1312yes—
572C2×C4C2×Q8×D1312yes—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—24no—
577C22—138no—
578C22—268no—
579C22—264no—
580C22—264no—
581C22—268no—
582C22C2×C4⋊C4×D1314yes—
583C22C23×C4×D1311yescharacteristic
584C22C22×D4×D1311yescharacteristic
585C22C22×Q8×D1311yescharacteristic
586C22D13×H(5,𝔽2)12yes—
587C22D13×2−1+412yes—
588C4—28no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—138no—
593C4—138no—
594C4—264no—
595C4—522no—
596C4—524no—
597C4—522no—
598C4C2×C4⋊C4×D1314yes—
599C4C2×C4⋊C4×D1314yes—
600C2—24no—
601C2—268no—
602C2C22×C4⋊C4×D1311yescharacteristic
603C2D13×#52012yes—
604C2#1312861111yescharacteristic
605C2#1312861811yescharacteristic
606C2#1312865112yes—
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC13×#2317C2
socle, cosocleC23×C13C25
prime radicalC22×#6127C22
hypercenter#2317D13
hypocenterC13#34183
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. #13117750C2semidirect
5. #13117737C2semidirect
6. #13117734C2semidirect
7. #13117461C2semidirect
8. #13117459C2semidirect
9. #13117457C2semidirect
10. #13117242C2semidirect
11. #13117239C2semidirect
12. C2×#11718637C2semidirect
13. D13×#384C2semidirect
14. C13×#2317C2semidirect
25. #11719227C4semidirect
81. C13×#117C2×C4semidirect
113. C2×C4×Dic13C2×C4semidirect
122. #88147C2×C4semidirect
129. #88141C2×C4semidirect
131. #88135C2×C4semidirect
133. #88126C2×C4semidirect
134. C42×D13C2×C4semidirect
168. #6127C4⋊C4semidirect
227. #2317D13semidirect
252. C2×C4×C13C2×C4⋊C4semidirect
260. C22⋊D13C2×C4⋊C4semidirect
278. D52C2×C4⋊C4semidirect
282. C4×D13C2×C4⋊C4semidirect
283. Dic26C2×C4⋊C4semidirect
361. Dic13#520semidirect
390. #117C4×D13semidirect
520. C13#34183semidirect
565. C2×C4C4⋊C4×D13semidirect

Maximal subgroups

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

subgroupindex
C2×#117192092
C2×#117192282
C2×#117192272
#131177502
#131177372
#131177342
#131174612
#131174592
#131174572
#131172422
#131172392
C2×#117186372
D13×#3842
C13×#23172
#3418313