Finite Group Databasebeta

← #16026396

Subgroups — #16026396

Order 3328.

2286 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719530C211yesmaximal, characteristic
2#13118015C212yesmaximal, complemented
3#13117979C212yesmaximal, complemented
4#13117969C212yesmaximal, complemented
5#13117924C212yesmaximal, complemented
6#13117863C212yesmaximal, complemented
7#13117841C212yesmaximal, complemented
8#13117731C212yesmaximal, complemented
9#13117668C212yesmaximal, complemented
10#13117645C214yesmaximal, complemented
11#13117623C214yesmaximal, complemented
12#13117514C212yesmaximal, complemented
13#13117364C212yesmaximal, complemented
14#13117327C211yesmaximal, characteristic
15C13×#2311C211yesmaximal, characteristic
16C22×C13×#41C2211yescharacteristic
17C23×Dic26C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C22×C13×C4⋊C4C2211yescharacteristic
20#11719530C4116yescomplemented
21C2×#88198C2212yes—
22C2×#88190C2212yes—
23C2×#88189C2212yes—
24C2×#88189C2212yes—
25C2×#88180C2211yescharacteristic
26C2×#88180C2211yescharacteristic
27C2×#88178C2212yes—
28C2×#88178C2212yes—
29C22×C4×Dic13C2211yescharacteristic
30C2×#88141C2212yes—
31C2×#88141C2212yes—
32C13×#391C2212yes—
33C13×#390C2212yes—
34C13×#386C2212yes—
35C13×#386C2212yes—
36C13×#380C2212yes—
37C13×#380C2212yes—
38#11718891C2211yescharacteristic
39#11718891C2211yescharacteristic
40#11718882C2211yescharacteristic
41#11718882C2211yescharacteristic
42#11718881C2212yes—
43#11718881C2212yes—
44#11718871C2214yes—
45C4×C22⋊Dic13C2214yes—
46C4×C22⋊Dic13C2212yes—
47C4×C22⋊Dic13C2212yes—
48#11718639C2214yes—
49C4⋊C4×Dic13C2212yes—
50C4⋊C4×Dic13C2212yes—
51#11718635C2212yes—
52#11718635C2212yes—
53#11718634C2212yes—
54#11718634C2212yes—
55#11718633C2214yes—
56#11718607C2214yes—
57#11718606C2214yes—
58#11718606C2214yes—
59#11718605C2214yes—
60#11718605C2214yes—
61#11718605C2214yes—
62#11718605C2214yes—
63#11718603C2214yes—
64C22⋊C4×Dic13C2214yes—
65C22⋊C4×Dic13C2214yes—
66#11718594C2214yes—
67C4×#6101C2214yes—
68#11718316C2214yes—
69#11718313C2214yes—
70#11718312C2214yes—
71#11718310C2214yes—
72#11718310C2214yes—
73#11718308C2214yes—
74#11718307C2214yes—
75#11718307C2214yes—
76C2×C13×#41—42no—
77C2×C13×#41—42no—
78C2×C13×#41C2×C418yes—
79C22×Q8×C13C2311yescharacteristic
80C22×D4×C13C2311yescharacteristic
81C22×D4×C13C2311yescharacteristic
82C22×D4×C13C2311yescharacteristic
83C23×C4×C13C2311yescharacteristic
84C23×C4×C13C2311yescharacteristic
85C23×C4×C13C2311yescharacteristic
86C23×C4×C13C2312yes—
87C23×Dic13C2312yes—
88C23×Dic13C2312yes—
89C22×Dic26—42no—
90C22×Dic26—42no—
91C22×Dic26C2×C418yescomplemented
92C22×Dic26C2312yes—
93C22×Dic26C2312yes—
94C13×#117—42no—
95C2×C13×C4⋊C4—42no—
96C2×C13×C4⋊C4C2312yes—
97C2×C13×C4⋊C4C2312yes—
98C2×C13×C4⋊C4C2312yes—
99C2×C13×C22⋊C4C2314yes—
100C2×C13×C22⋊C4C2314yes—
101C2×C13×C22⋊C4C2312yes—
102C2×C42×C13C2312yes—
103C2×C22⋊Dic13C2312yes—
104C2×C22⋊Dic13C2312yes—
105C2×C22⋊Dic13C2312yes—
106C2×C22⋊Dic13C2312yes—
107C2×C22⋊Dic13C2312yes—
108C2×C22⋊Dic13C2314yes—
109C2×C22⋊Dic13C2314yes—
110C2×C22⋊Dic13C2312yes—
111#88198C2×C4116yescomplemented
112#88190C2×C4116yes—
113#88189C2×C4116yescomplemented
114#88189C2×C4116yescomplemented
115#88180—42no—
116#88180C2×C418yescomplemented
117C2×C4⋊Dic13C2311yescharacteristic
118C2×C4⋊Dic13C2311yescharacteristic
119C2×C4⋊Dic13C2312yes—
120#88178C2×C4116yescomplemented
121#88178C2×C4116yescomplemented
122C2×#6101C2314yes—
123C2×#6101C2314yes—
124C2×#6101C2314yes—
125C2×#6101C2314yes—
126C2×#6101C2314yes—
127C2×#6101C2314yes—
128C2×C4×Dic13—42no—
129C2×C4×Dic13C2312yes—
130C2×C4×Dic13C2312yes—
131C2×C4×Dic13C2314yes—
132C2×C4×Dic13C2314yes—
133C2×C4×Dic13C2311yescharacteristic
134C2×C4×Dic13C2311yescharacteristic
135C2×C4×Dic13C2314yes—
136C2×C4×Dic13C2314yes—
137C2×C4×Dic13C2312yes—
138#88141—44no—
139#88141—44no—
140C13×#95C2314yes—
141C13×#95C2314yes—
142#88071C2312yes—
143#88071C2312yes—
144#88071C2314yes—
145#88071C2314yes—
146#88071C2314yes—
147#88071C2314yes—
148#88071C2314yes—
149#88071C2314yes—
150#88071C2314yes—
151#88071C2314yes—
152#88071C2312yes—
153#88071C2312yes—
154#34413—131nomaximal, Sylow
155C24×C13C2411yescharacteristic
156C24×C13C2411yescharacteristic
157C24×C13C2411yescharacteristic
158C13×#41—416no—
159C2×Q8×C13—41no—
160C2×Q8×C13—41no—
161C2×Q8×C13C22×C414yes—
162C2×D4×C13—41no—
163C2×D4×C13—41no—
164C2×D4×C13—41no—
165C2×D4×C13—41no—
166C2×D4×C13—41no—
167C2×D4×C13—41no—
168C2×D4×C13C22×C414yes—
169C2×D4×C13C22×C414yes—
170C2×D4×C13C22×C414yes—
171C22×C4×C13—22no—
172C22×C4×C13—22no—
173C22×C4×C13—22no—
174C22×C4×C13—22no—
175C22×C4×C13—22no—
176C22×C4×C13—22no—
177C22×C4×C13—42no—
178C22×C4×C13—44no—
179C22×C4×C13C22×C414yes—
180C22×C4×C13C22×C414yes—
181C22×C4×C13C22×C414yes—
182C22×C4×C13C2414yes—
183C22×C4×C13C2414yes—
184C22×C4×C13C2411yescharacteristic
185C22×C4×C13C2411yescharacteristic
186C22×C4×C13C2411yescharacteristic
187C22×C4×C13C2411yescharacteristic
188C22×Dic13—24no—
189C22×Dic13—42no—
190C22×Dic13—42no—
191C22×Dic13—42no—
192C22×Dic13—42no—
193C22×Dic13C22×C418yes—
194C22×Dic13C2412yes—
195C22×Dic13C2412yes—
196C22×Dic13C2412yes—
197C22×Dic13C2414yes—
198C22×Dic13C2414yes—
199C22×Dic13C2412yes—
200C2×Dic26—24no—
201C2×Dic26—24no—
202C2×Dic26—416no—
203C2×Dic26—42no—
204C2×Dic26—42no—
205C2×Dic26C22×C418yes—
206C2×Dic26C22×C418yes—
207C13×C4⋊C4—24no—
208C13×C4⋊C4—24no—
209C13×C4⋊C4—42no—
210C13×C22⋊C4—44no—
211C13×C22⋊C4—44no—
212C13×C22⋊C4—42no—
213C42×C13—42no—
214C22⋊Dic13—24no—
215C22⋊Dic13—24no—
216C22⋊Dic13—42no—
217C22⋊Dic13—42no—
218C22⋊Dic13—42no—
219C22⋊Dic13C22×C4116yes—
220C22⋊Dic13C22×C4116yes—
221C22⋊Dic13C22×C418yes—
222C4⋊Dic13—41no—
223C4⋊Dic13—41no—
224C4⋊Dic13C22×C418yes—
225#6101—28no—
226#6101—44no—
227#6101—44no—
228#6101—44no—
229#6101C22×C4116yes—
230#6101C22×C4116yes—
231C4×Dic13—24no—
232C4×Dic13—24no—
233C4×Dic13—44no—
234C4×Dic13—44no—
235C4×Dic13—41no—
236C4×Dic13—41no—
237C4×Dic13—44no—
238C4×Dic13—44no—
239C4×Dic13C22×C418yes—
240C2×#536—131no—
241#2381—132no—
242#2376—132no—
243#2374—132no—
244#2369—132no—
245#2362—132no—
246#2360—132no—
247#2347—132no—
248Q8×C22⋊C4—132no—
249#2337—134no—
250#2335—134no—
251#2323—132no—
252#2311D1311yescharacteristic
253C4×C22⋊Q8—132no—
254#2297—131no—
255C23×C13—21no—
256C23×C13—21no—
257C23×C13—21no—
258C23×C13—21no—
259C23×C13—21no—
260C23×C13—21no—
261C23×C13—22no—
262C23×C13—22no—
263C23×C13—22no—
264C23×C13—22no—
265C23×C13—22no—
266C23×C13—22no—
267C23×C13C23×C412yes—
268C23×C13C23×C412yes—
269C23×C13C23×C412yes—
270C23×C13C2511yescharacteristic
271Q8×C13—44no—
272D4×C13—44no—
273D4×C13—44no—
274D4×C13—44no—
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—28no—
282C2×C4×C13—28no—
283C2×C4×C13—28no—
284C2×C4×C13—24no—
285C2×C4×C13—24no—
286C2×C4×C13—24no—
287C2×C4×C13—24no—
288C2×C4×C13—24no—
289C2×C4×C13—24no—
290C2×C4×C13—84no—
291C2×C4×C13C23×C412yes—
292C2×C4×C13C23×C412yes—
293C2×C4×C13C23×C412yes—
294C2×C4×C13C23×C412yes—
295C2×C4×C13H(5,𝔽2)12yes—
296C2×C4×C13H(5,𝔽2)12yes—
297C2×Dic13—22no—
298C2×Dic13—24no—
299C2×Dic13—24no—
300C2×Dic13—24no—
301C2×Dic13—24no—
302C2×Dic13—22no—
303C2×Dic13—22no—
304C2×Dic13—22no—
305C2×Dic13—22no—
306C2×Dic13—22no—
307C2×Dic13—22no—
308C2×Dic13—22no—
309C2×Dic13—22no—
310C2×Dic13—22no—
311C2×Dic13—22no—
312C2×Dic13—48no—
313C2×Dic13—48no—
314C2×Dic13C23×C418yes—
315C2×Dic13C23×C418yes—
316C2×Dic132−1+414yes—
317Dic26—216no—
318Dic26—48no—
319C22×#41C2×D1311yescharacteristic
320C23×Q8—131no—
321#536—1316no—
322C2×C4⋊Q8—132no—
323C2×#124—132no—
324C2×#123—132no—
325C2×#123—132no—
326C2×C22⋊Q8—132no—
327C2×C22⋊Q8—132no—
328C2×C4×Q8—132no—
329C2×C4×Q8—132no—
330C2×#117—131no—
331C2×#117—131no—
332C2×#117C2×D1311yescharacteristic
333C22×C4⋊C4C2×D1311yescharacteristic
334C22×C42—131no—
335#391—131no—
336#391—131no—
337#391—132no—
338#391—132no—
339#391—134no—
340#391—134no—
341#391C2×D1312yes—
342#390—131no—
343#390—131no—
344#390C2×D1312yes—
345#389—132no—
346#389—134no—
347#389—132no—
348#389—134no—
349#388—134no—
350#388—134no—
351#388—134no—
352#388—134no—
353#388—134no—
354#388—134no—
355#387—134no—
356#387—134no—
357#387—134no—
358#387—134no—
359#386—132no—
360#386—132no—
361#386C2×D1312yes—
362#386C2×D1312yes—
363#385—134no—
364#385—134no—
365#383—134no—
366#380—134no—
367#380C2×D1312yes—
368#380C2×D1312yes—
369C4×C4⋊C4—132no—
370C4×C4⋊C4—132no—
371C4×C4⋊C4—134no—
372C4×C4⋊C4—134no—
373C4×C4⋊C4—134no—
374C4×C22⋊C4—134no—
375C4×C22⋊C4—132no—
376C4×C22⋊C4—132no—
377C4×C22⋊C4—134no—
378C4×C22⋊C4—134no—
379C22×C13—22no—
380C22×C13—22no—
381C22×C13—22no—
382C22×C13—24no—
383C22×C13—24no—
384C22×C13—24no—
385C22×C13—42no—
386C22×C13—42no—
387C22×C13—42no—
388C22×C13—41no—
389C22×C13—41no—
390C22×C13—41no—
391C22×C13—41no—
392C22×C13—41no—
393C22×C13—41no—
394C22×C13C24×C411yescharacteristic
395C22×C13C2×H(5,𝔽2)11yescharacteristic
396C22×C13C2×H(5,𝔽2)11yescharacteristic
397C22×C13C2×2−1+412yes—
398C22×C13#58512yes—
399C4×C13—22no—
400C4×C13—22no—
401C4×C13—22no—
402C4×C13—44no—
403C4×C13—44no—
404C4×C13#51814yes—
405Dic13—28no—
406Dic13—28no—
407Dic13—24no—
408Dic13—42no—
409Dic13—42no—
410Dic13—42no—
411C2×#41—42no—
412C2×#41—42no—
413C2×#41C4×D1318yes—
414C22×Q8—138no—
415C22×Q8—132no—
416C22×Q8—132no—
417C22×Q8—522no—
418C22×Q8—522no—
419C22×Q8C22×D1311yescharacteristic
420C22×D4C22×D1311yescharacteristic
421C22×D4C22×D1311yescharacteristic
422C22×D4C22×D1311yescharacteristic
423C23×C4—132no—
424C23×C4—132no—
425C23×C4C22×D1311yescharacteristic
426C23×C4C22×D1311yescharacteristic
427C23×C4C22×D1311yescharacteristic
428C23×C4C22×D1312yes—
429C4⋊Q8—1316no—
430#124—1316no—
431#123—1316no—
432#123—1316no—
433C22⋊Q8—1316no—
434C22⋊Q8—1316no—
435C4×Q8—524no—
436C4×Q8—524no—
437#117—42no—
438#117—138no—
439#117—522no—
440C2×C4⋊C4—42no—
441C2×C4⋊C4—132no—
442C2×C4⋊C4—134no—
443C2×C4⋊C4—134no—
444C2×C4⋊C4—134no—
445C2×C4⋊C4—131no—
446C2×C4⋊C4—131no—
447C2×C4⋊C4—134no—
448C2×C4⋊C4—134no—
449C2×C4⋊C4—134no—
450C2×C4⋊C4C22×D1312yes—
451C2×C4⋊C4C22×D1312yes—
452C2×C4⋊C4C22×D1312yes—
453C2×C22⋊C4—132no—
454C2×C22⋊C4—134no—
455C2×C22⋊C4—134no—
456C2×C22⋊C4—132no—
457C2×C22⋊C4—132no—
458C2×C22⋊C4—132no—
459C2×C22⋊C4—132no—
460C2×C22⋊C4—132no—
461C2×C22⋊C4C22×D1312yes—
462C2×C22⋊C4C22×D1314yes—
463C2×C22⋊C4C22×D1314yes—
464C2×C42—132no—
465C2×C42—132no—
466C2×C42—132no—
467C2×C42—134no—
468C2×C42—134no—
469C2×C42—134no—
470C2×C42—131no—
471C2×C42—134no—
472C2×C42—131no—
473C2×C42—522no—
474C2×C42C22×D1312yes—
475#95—134no—
476#95—132no—
477#95—132no—
478#95—134no—
479#95—132no—
480#95—134no—
481#95—134no—
482#95—132no—
483#95—134no—
484#95—134no—
485#95—134no—
486#95—134no—
487#95C22×D1314yes—
488#95C22×D1314yes—
489C2×C13—42no—
490C2×C13—42no—
491C2×C13—42no—
492C2×C13C2×#51811yescharacteristic
493C2×C13C4×2−1+412yes—
494C2×C13#354112yes—
495C2×C13#355011yescharacteristic
496C2×C13#355111yescharacteristic
497C24C23×D1311yescharacteristic
498C24C23×D1311yescharacteristic
499C24C23×D1311yescharacteristic
500#41—416no—
501C2×Q8—41no—
502C2×Q8—41no—
503C2×Q8—138no—
504C2×Q8—138no—
505C2×Q8—264no—
506C2×Q8—264no—
507C2×Q8—522no—
508C2×Q8—522no—
509C2×Q8—5216no—
510C2×Q8C2×C4×D1314yes—
511C2×D4—41no—
512C2×D4—41no—
513C2×D4—41no—
514C2×D4—41no—
515C2×D4—41no—
516C2×D4—41no—
517C2×D4C2×C4×D1314yes—
518C2×D4C2×C4×D1314yes—
519C2×D4C2×C4×D1314yes—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—42no—
527C22×C4—44no—
528C22×C4—132no—
529C22×C4—134no—
530C22×C4—134no—
531C22×C4—138no—
532C22×C4—132no—
533C22×C4—132no—
534C22×C4—132no—
535C22×C4—264no—
536C22×C4—522no—
537C22×C4—522no—
538C22×C4—522no—
539C22×C4—522no—
540C22×C4C2×C4×D1314yes—
541C22×C4C2×C4×D1314yes—
542C22×C4C2×C4×D1314yes—
543C22×C4C23×D1314yes—
544C22×C4C23×D1314yes—
545C22×C4C23×D1311yescharacteristic
546C22×C4C23×D1311yescharacteristic
547C22×C4C23×D1311yescharacteristic
548C22×C4C23×D1311yescharacteristic
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—42no—
552C4⋊C4—138no—
553C4⋊C4—1316no—
554C4⋊C4—1316no—
555C4⋊C4—268no—
556C4⋊C4—521no—
557C4⋊C4—521no—
558C4⋊C4—524no—
559C4⋊C4—524no—
560C4⋊C4—524no—
561C22⋊C4—42no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—138no—
565C22⋊C4—1316no—
566C22⋊C4—1316no—
567C22⋊C4—264no—
568C22⋊C4—264no—
569C22⋊C4—522no—
570C22⋊C4—522no—
571C22⋊C4—522no—
572C42—42no—
573C42—138no—
574C42—264no—
575C42—264no—
576C42—521no—
577C42—521no—
578C42—524no—
579C42—524no—
580C42—524no—
581C42—524no—
582C13#3441311yescharacteristic, complemented, Sylow
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23C22×C4×D1312yes—
596C23C22×C4×D1312yes—
597C23C22×C4×D1312yes—
598C23C24×D1311yescharacteristic
599Q8—44no—
600Q8—2616no—
601Q8—528no—
602D4—44no—
603D4—44no—
604D4—44no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—84no—
621C2×C4—138no—
622C2×C4—138no—
623C2×C4—134no—
624C2×C4—262no—
625C2×C4—264no—
626C2×C4—264no—
627C2×C4—264no—
628C2×C4—262no—
629C2×C4—264no—
630C2×C4—262no—
631C2×C4—262no—
632C2×C4—262no—
633C2×C4—262no—
634C2×C4—262no—
635C2×C4—262no—
636C2×C4—262no—
637C2×C4—262no—
638C2×C4—262no—
639C2×C4—528no—
640C2×C4—528no—
641C2×C4C22×C4×D1312yes—
642C2×C4C22×C4×D1312yes—
643C2×C4C22×C4×D1312yes—
644C2×C4C22×C4×D1312yes—
645C2×C4#8825112yes—
646C2×C4#8825112yes—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22C23×C4×D1311yescharacteristic
663C22C2×#8825111yescharacteristic
664C22C2×#8825111yescharacteristic
665C22#1171965012yes—
666C22D13×2−1+412yes—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—44no—
672C4—268no—
673C4—268no—
674C4—264no—
675C4—522no—
676C4—522no—
677C4—522no—
678C4#1171919114yes—
679C2—42no—
680C2—42no—
681C2—42no—
682C2C2×#1171919111yescharacteristic
683C2#1312804712yes—
684C2#1312911712yes—
685C2#1312924511yescharacteristic
686C2#1312926011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC13×#2311C2
socle, cosocleC23×C13C25
prime radicalC22×C13×#41C22
hypercenter#2311D13
hypocenterC13#34413
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #13118015C2semidirect
3. #13117979C2semidirect
4. #13117969C2semidirect
5. #13117924C2semidirect
6. #13117863C2semidirect
7. #13117841C2semidirect
8. #13117731C2semidirect
9. #13117668C2semidirect
10. #13117645C2semidirect
11. #13117623C2semidirect
12. #13117514C2semidirect
13. #13117364C2semidirect
20. #11719530C4semidirect
91. C22×Dic26C2×C4semidirect
111. #88198C2×C4semidirect
113. #88189C2×C4semidirect
114. #88189C2×C4semidirect
116. #88180C2×C4semidirect
120. #88178C2×C4semidirect
121. #88178C2×C4semidirect
582. C13#34413semidirect

Maximal subgroups

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

subgroupindex
C2×#117195302
#131180152
#131179792
#131179692
#131179242
#131178632
#131178412
#131177312
#131176682
#131176452
#131176232
#131175142
#131173642
#131173272
C13×#23112
#3441313