Finite Group Databasebeta

← #15867798

Subgroups — #15867798

Order 3328.

894 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×#11718380C211yesmaximal, characteristic
2#13117238C211yesmaximal, characteristic
3#117×Dic13C211yesmaximal, characteristic
4#13115598C211yesmaximal, characteristic, complemented
5#13115596C211yesmaximal, characteristic, complemented
6#13115593C211yesmaximal, characteristic, complemented
7#13115591C211yesmaximal, characteristic, complemented
8#13115037C211yesmaximal, characteristic
9#13115035C211yesmaximal, characteristic
10#13115033C211yesmaximal, characteristic
11#13114736C211yesmaximal, characteristic, complemented
12#13114735C211yesmaximal, characteristic, complemented
13#13114732C211yesmaximal, characteristic, complemented
14#13114732C211yesmaximal, characteristic, complemented
15C13×#1854C211yesmaximal, characteristic
16C2×C13×#117C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C22×C4×Dic13C2211yescharacteristic
19C2×#88131—21no—
20C2×C13×#107C2211yescharacteristic
21C2×C13×#106C2211yescharacteristic
22C2×C13×C22⋊C8C2211yescharacteristic
23C13×#380C2211yescharacteristic
24#11718873C2211yescharacteristic
25C2×C22⋊#1055C2211yescharacteristic
26C2×#88058C2211yescharacteristic
27C2×#88057C2211yescharacteristic
28C2×#88055C2211yescharacteristic
29C2×#88055C2211yescharacteristic
30C2×#88054C2211yescharacteristic
31C2×#88054C2211yescharacteristic
32#11718637C2211yescharacteristic
33#11718637C2211yescharacteristic
34C4⋊C4×Dic13C2211yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38C4⋊C4×Dic13C2211yescharacteristic
39C4⋊C4×Dic13C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44#11718603—21no—
45C22⋊C4×Dic13—21no—
46C22⋊C4×Dic13—21no—
47#11718593—21no—
48C42×Dic13—21no—
49#11718380C418yescomplemented
50#11718311—21no—
51#11718310—21no—
52C13×#340C2211yescharacteristic
53C13×#340C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C23×Dic13D411yescharacteristic
62C23×Dic13D411yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C13×#117—22no—
66C13×#117—22no—
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C22⋊C4—21no—
72C2×C13×C22⋊C4—21no—
73C2×C42×C13—21no—
74C2×C22⋊Dic13—21no—
75C2×C22⋊Dic13—21no—
76#88180—22no—
77#88180—22no—
78C2×C4⋊Dic13C2311yescharacteristic
79C2×C4⋊Dic13C2311yescharacteristic
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×#6101—21no—
83C2×#6101—21no—
84C2×#6101—21no—
85C2×#6101—21no—
86C2×C4×Dic13—21no—
87C2×C4×Dic13—21no—
88C2×C4×Dic13—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13C2×C414yes—
94C2×C4×Dic13D411yescharacteristic
95C2×C4×Dic13D411yescharacteristic
96C2×C4×Dic13Q814yes—
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101#88131—44no—
102C13×#107—22no—
103C13×#106—22no—
104C13×C22⋊C8C2×C414yes—
105C13×#95—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113C22⋊#1055C2×C414yes—
114#88058—22no—
115#88057—22no—
116#88055C2×C414yes—
117#88055C2×C414yes—
118#88054C2×C414yes—
119#88054C2×C414yes—
120#88048—22no—
121#88048—22no—
122#88047—22no—
123#88047—22no—
124#21364—131nomaximal, Sylow
125C24×C13C2×D411yescharacteristic
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13C22×C412yes—
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×Q812yes—
137C22×C4×C13C2411yescharacteristic
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—41no—
145C22×Dic13—41no—
146C22×Dic13C4⋊C414yes—
147C22×Dic13C4⋊C414yes—
148C22×Dic13C2×D411yescharacteristic
149C22×Dic13C2×D411yescharacteristic
150C22×Dic13C2×D411yescharacteristic
151C22×Dic13C2×D411yescharacteristic
152C2×C8×C13—21no—
153C2×C8×C13—21no—
154C2×C8×C13—21no—
155C2×C8×C13—21no—
156C2×C8×C13C22×C412yes—
157C2×C8×C13C22×C412yes—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4#4112yes—
165C13×C4⋊C4#4112yes—
166C13×C22⋊C4—24no—
167C13×C22⋊C4—42no—
168C42×C13—24no—
169C22⋊Dic13—24no—
170C22⋊Dic13—42no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13#4112yes—
178C4⋊Dic13#4112yes—
179#6101—42no—
180#6101—42no—
181#6101—42no—
182#6101—42no—
183C4×Dic13—24no—
184C4×Dic13—42no—
185C4×Dic13—42no—
186C4×Dic13—42no—
187C4×Dic13—42no—
188C4×Dic13—42no—
189C4×Dic13—42no—
190C4×Dic13C4⋊C414yes—
191C4×Dic13C4⋊C414yes—
192C4×Dic13C22×C412yes—
193C4×Dic13C22×C412yes—
194C4×Dic13C2×Q812yes—
195C4×Dic13C2×Q812yes—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055C22×C412yes—
201C2×#1055C22×C412yes—
202C2×#432—131no—
203#2285—131no—
204C4×#117—131no—
205#1948—131no—
206#1948—131no—
207#1947—131no—
208#1947—131no—
209#1854—131no—
210#1854—131no—
211#1854—131no—
212#1854D1311yescharacteristic
213#1776—131no—
214#1776—131no—
215#1776—131no—
216#1776—131no—
217C23×C13—21no—
218C23×C13—21no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13C2×C4⋊C412yes—
224C23×C13C22×D411yescharacteristic
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—22no—
228C2×C4×C13—22no—
229C2×C4×C13—22no—
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13C2×C4⋊C412yes—
244C2×C4×C13C23×C411yescharacteristic
245C2×C4×C13C22×Q811yescharacteristic
246C2×C4×C13C2×#4111yescharacteristic
247C2×C4×C13C2×#4111yescharacteristic
248C2×C4×C13C2×#4111yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—28no—
257C2×Dic13—22no—
258C2×Dic13—22no—
259C2×Dic13—44no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13C2×C4⋊C412yes—
271C2×Dic13C2×C4⋊C412yes—
272C2×Dic13C2×C4⋊C412yes—
273C2×Dic13C2×C4⋊C412yes—
274C2×Dic13AGL(1,ℤ/8)12yes—
275C2×Dic13#13712yes—
276C8×C13—22no—
277C8×C13—22no—
278#1055—22no—
279#1055—22no—
280C2×#117—131no—
281C2×#117—261no—
282C2×#117C2×D1311yescharacteristic
283C22×C42—131no—
284#432—138no—
285C2×#107—131no—
286C2×#107—131no—
287C2×#107—131no—
288C2×#107C2×D1311yescharacteristic
289C2×#106—131no—
290C2×#106—131no—
291C2×#106—131no—
292C2×#106C2×D1311yescharacteristic
293C2×C4⋊C8—131no—
294C2×C4⋊C8—131no—
295C2×C22⋊C8—131no—
296C2×C22⋊C8C2×D1311yescharacteristic
297C2×#97—131no—
298C2×#97—131no—
299#384—131no—
300#384—131no—
301#382—261no—
302#380—131no—
303#380—261no—
304#380C2×D1311yescharacteristic
305C4×C4⋊C4—131no—
306C4×C4⋊C4—131no—
307C4×C4⋊C4—131no—
308C4×C4⋊C4—131no—
309C4×C4⋊C4—131no—
310C4×C4⋊C4—131no—
311C4×C4⋊C4—261no—
312C4×C22⋊C4—261no—
313C4×C22⋊C4—261no—
314#376—261no—
315C43—261no—
316#340—131no—
317#340—131no—
318#340—131no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340C2×D1311yescharacteristic
323#340C2×D1311yescharacteristic
324C22×C13—22no—
325C22×C13—22no—
326C22×C13—22no—
327C22×C13—41no—
328C22×C13—41no—
329C22×C13—41no—
330C22×C13—41no—
331C22×C13C22×C4⋊C411yescharacteristic
332C22×C13#52511yescharacteristic
333C22×C13#52511yescharacteristic
334C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
335C22×C13C2×#13711yescharacteristic
336C22×C13#57511yescharacteristic
337C22×C13#57511yescharacteristic
338C4×C13—22no—
339C4×C13—42no—
340C4×C13—42no—
341C4×C13—81no—
342C4×C13—81no—
343C4×C13C4×#4111yescharacteristic
344C4×C13C4×#4111yescharacteristic
345C4×C13#53311yescharacteristic
346C4×C13#53311yescharacteristic
347Dic13—22no—
348Dic13—22no—
349Dic13—22no—
350Dic13—42no—
351Dic13—42no—
352Dic13—81no—
353Dic13—81no—
354Dic13#42814yes—
355C23×C4—131no—
356C23×C4—131no—
357C23×C4C22×D1311yescharacteristic
358C22×C8—131no—
359C22×C8—131no—
360C22×C8C22×D1311yescharacteristic
361C22×C8C22×D1311yescharacteristic
362#117—22no—
363#117—22no—
364#117—262no—
365#117—262no—
366#117—524no—
367C2×C4⋊C4—131no—
368C2×C4⋊C4—131no—
369C2×C4⋊C4—131no—
370C2×C4⋊C4—131no—
371C2×C4⋊C4—261no—
372C2×C4⋊C4—261no—
373C2×C4⋊C4—261no—
374C2×C4⋊C4—261no—
375C2×C4⋊C4C22×D1311yescharacteristic
376C2×C4⋊C4C22×D1311yescharacteristic
377C2×C4⋊C4C22×D1311yescharacteristic
378C2×C4⋊C4C22×D1311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—261no—
382C2×C22⋊C4—261no—
383C2×C42—21no—
384C2×C42—134no—
385C2×C42—131no—
386C2×C42—131no—
387C2×C42—134no—
388C2×C42—131no—
389C2×C42—131no—
390C2×C42—261no—
391C2×C42—261no—
392C2×C42—261no—
393C2×C42—261no—
394C2×C42—261no—
395C2×C42—261no—
396C2×C42—261no—
397#107—22no—
398#107—262no—
399#107—262no—
400#107—262no—
401#106—22no—
402#106—262no—
403#106—262no—
404#106—262no—
405C4⋊C8—134no—
406C4⋊C8—134no—
407C22⋊C8—134no—
408C22⋊C8C4×D1314yes—
409#97—134no—
410#97—134no—
411#95—21no—
412#95—261no—
413#95—261no—
414#95—261no—
415#95—261no—
416#95—261no—
417#95—261no—
418#95—261no—
419C2×C13—42no—
420C2×C13#229611yescharacteristic
421C2×C13C2×#42811yescharacteristic
422C2×C13#291111yescharacteristic
423C2×C13#311511yescharacteristic
424C2×C13#311511yescharacteristic
425C2×C13#311611yescharacteristic
426C2×C13#311611yescharacteristic
427C24D4×D1311yescharacteristic
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—131no—
437C22×C4—134no—
438C22×C4—131no—
439C22×C4—131no—
440C22×C4—131no—
441C22×C4—134no—
442C22×C4—262no—
443C22×C4—262no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—261no—
448C22×C4—521no—
449C22×C4—521no—
450C22×C4C2×C4×D1312yes—
451C22×C4D4×D1311yescharacteristic
452C22×C4Q8×D1312yes—
453C22×C4C23×D1311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—132no—
459C2×C8—132no—
460C2×C8—261no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8C2×C4×D1312yes—
465C2×C8C2×C4×D1312yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—132no—
473C4⋊C4—132no—
474C4⋊C4—261no—
475C4⋊C4—261no—
476C4⋊C4—261no—
477C4⋊C4—261no—
478C4⋊C4—261no—
479C4⋊C4—261no—
480C4⋊C4—522no—
481C4⋊C4—522no—
482C4⋊C4—522no—
483C4⋊C4—522no—
484C4⋊C4#612912yes—
485C4⋊C4#612912yes—
486C22⋊C4—24no—
487C22⋊C4—42no—
488C22⋊C4—264no—
489C22⋊C4—522no—
490C42—24no—
491C42—134no—
492C42—134no—
493C42—132no—
494C42—132no—
495C42—132no—
496C42—132no—
497C42—264no—
498C42—522no—
499C42—522no—
500C42—522no—
501C42—522no—
502C42—522no—
503C42—522no—
504C13#2136411yescharacteristic, complemented, Sylow
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4⋊C4×D1312yes—
512C23C2×D4×D1311yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—132no—
532C2×C4—132no—
533C2×C4—132no—
534C2×C4—132no—
535C2×C4—132no—
536C2×C4—132no—
537C2×C4—268no—
538C2×C4—261no—
539C2×C4—261no—
540C2×C4—261no—
541C2×C4—261no—
542C2×C4—261no—
543C2×C4—261no—
544C2×C4—262no—
545C2×C4—262no—
546C2×C4—524no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4—521no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—521no—
557C2×C4C4⋊C4×D1312yes—
558C2×C4C22×C4×D1311yescharacteristic
559C2×C4C2×#612911yescharacteristic
560C2×C4C2×#612911yescharacteristic
561C2×C4C2×Q8×D1311yescharacteristic
562C2×C4#41×D1311yescharacteristic
563C2×C4#41×D1311yescharacteristic
564C8—22no—
565C8—22no—
566C8—262no—
567C8—262no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×C4⋊C4×D1311yescharacteristic
576C22#1171929111yescharacteristic
577C22#1171936111yescharacteristic
578C22#1171947311yescharacteristic
579C22#1171947311yescharacteristic
580C22D13×AGL(1,ℤ/8)11yescharacteristic
581C22D13×#13711yescharacteristic
582C4—22no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—134no—
588C4—262no—
589C4—262no—
590C4—262no—
591C4—522no—
592C4—522no—
593C4—1041no—
594C4—1041no—
595C4C4×#612911yescharacteristic
596C4C4×#612911yescharacteristic
597C4#1171932711yescharacteristic
598C4#1171932711yescharacteristic
599C2—42no—
600C2#1311731511yescharacteristic
601C2D13×#42811yescharacteristic
602C2#1312342411yescharacteristic
603C2#1312498011yescharacteristic
604C2#1312498211yescharacteristic
605C2#1312499111yescharacteristic
606C2#1312499311yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1854C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1854D13
hypocenterC13#21364
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #13115598C2semidirect
5. #13115596C2semidirect
6. #13115593C2semidirect
7. #13115591C2semidirect
11. #13114736C2semidirect
12. #13114735C2semidirect
13. #13114732C2semidirect
14. #13114732C2semidirect
49. #11718380C4semidirect
504. C13#21364semidirect

Maximal subgroups

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

subgroupindex
C2×#117183802
#131172382
#117×Dic132
#131155982
#131155962
#131155932
#131155912
#131150372
#131150352
#131150332
#131147362
#131147352
#131147322
C13×#18542
#2136413