Finite Group Databasebeta

← #15914639

Subgroups — #15914639

Order 3328.

792 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×#11718726C211yesmaximal, characteristic, complemented
2C2×#11718669C211yesmaximal, characteristic
3#13117894C211yesmaximal, characteristic, complemented
4#13116318C211yesmaximal, characteristic, complemented
5#13116313C211yesmaximal, characteristic, complemented
6#13116224C211yesmaximal, characteristic
7#13116210C211yesmaximal, characteristic
8#13115794C211yesmaximal, characteristic, complemented
9#13115602C211yesmaximal, characteristic, complemented
10#13115597C211yesmaximal, characteristic, complemented
11#13115560C211yesmaximal, characteristic, complemented
12#13115460C211yesmaximal, characteristic
13#13115453C211yesmaximal, characteristic
14#13114661C211yesmaximal, characteristic
15C13×#2566C211yesmaximal, characteristic
16C2×C13×#125C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×#88141—21no—
19C2×#88123C2211yescharacteristic
20C13×#396—21no—
21C13×#396—21no—
22C13×#390C2211yescharacteristic
23C13×#390C2211yescharacteristic
24C13×#385—21no—
25C13×#385—21no—
26C13×#384C2211yescharacteristic
27C13×#382C2211yescharacteristic
28C4×C13×C4⋊C4C2211yescharacteristic
29C2×#88072C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33#11718726—24no—
34#11718669—24no—
35#11718635—21no—
36C2×#88048C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88048C2211yescharacteristic
39C2×#88048C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C4×C4⋊Dic13C2211yescharacteristic
43C4×C4⋊Dic13C2211yescharacteristic
44C4×C4⋊Dic13C2211yescharacteristic
45#11718589C2211yescharacteristic
46#11718589C2211yescharacteristic
47C2×C4⋊#1055C2211yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×#88043C2211yescharacteristic
51#11718314—21no—
52#11718314—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C22×Dic26—21no—
64C13×#125—22no—
65C13×#125—22no—
66C4×D4×C13—22no—
67C4×D4×C13—22no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4D411yescharacteristic
73C2×C13×C4⋊C4D411yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4—21no—
80C2×C13×C22⋊C4—21no—
81C2×C13×C22⋊C4—21no—
82C2×C42×C13C2311yescharacteristic
83C2×C42×C13C2311yescharacteristic
84C2×C42×C13C2311yescharacteristic
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×#6101—21no—
91C2×#6101—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99#88141—44no—
100#88123—22no—
101#88123D414yes—
102C13×#95—21no—
103C13×#95—21no—
104C13×#95D411yescharacteristic
105C13×#95D411yescharacteristic
106#88072—22no—
107#88072—22no—
108#88072—22no—
109#88072Q814yescomplemented
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88048—22no—
116#88048—22no—
117#88048—22no—
118#88048—22no—
119#88047—22no—
120#88047—22no—
121C4⋊#1055—22no—
122C4⋊#1055—22no—
123C4⋊#1055—22no—
124#88043—22no—
125#25461—131nomaximal, Sylow
126C24×C13—21no—
127C2×D4×C13—21no—
128C2×D4×C13—21no—
129C2×D4×C13—21no—
130C2×D4×C13—21no—
131C2×D4×C13—21no—
132C2×D4×C13C2×Q812yes—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—42no—
137C22×C4×C13—81no—
138C22×C4×C13—81no—
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2411yescharacteristic
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—21no—
150C2×Dic26—24no—
151C2×Dic26—42no—
152C2×Dic26—42no—
153C13×C4⋊C4—22no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—41no—
161C13×C4⋊C4—42no—
162C13×C4⋊C4—42no—
163C13×C4⋊C4—81no—
164C13×C4⋊C4—81no—
165C13×C22⋊C4—42no—
166C13×C22⋊C4—42no—
167C13×C22⋊C4—42no—
168C13×C22⋊C4—81no—
169C13×C22⋊C4—81no—
170C42×C13—21no—
171C42×C13—21no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13C2×D412yes—
175C42×C13#4112yes—
176C4⋊Dic13—24no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13C2×D412yes—
183C4⋊Dic13C2×D412yes—
184C4⋊Dic13C2×Q812yes—
185#6101—42no—
186#6101—42no—
187C4×Dic13—42no—
188C4×Dic13—42no—
189C4×Dic13—42no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055C2×Q812yes—
201C2×#1055#4112yes—
202C2×#501—131no—
203C2×#476—131no—
204#2566D1311yescharacteristic
205#2365—131no—
206#2085—131no—
207#2084—131no—
208#2067—131no—
209#2065—131no—
210#1989—131no—
211#1949—131no—
212#1948—131no—
213#1940—131no—
214#1926—131no—
215#1925—131no—
216#1762—131no—
217C23×C13—22no—
218C23×C13—41no—
219C23×C13—41no—
220C23×C13—41no—
221C23×C13—41no—
222C23×C13—41no—
223C23×C13—41no—
224C23×C13C22×D411yescharacteristic
225D4×C13—22no—
226D4×C13—22no—
227D4×C13—41no—
228D4×C13—41no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—84no—
254C2×C4×C13#12112yes—
255C2×C4×C13#12112yes—
256C2×C4×C13AGL(1,ℤ/8)12yes—
257C2×C4×C13C22×D411yescharacteristic
258C2×C4×C13C22×Q811yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×C4×C13C2×#4111yescharacteristic
261C2×C4×C132−1+411yescharacteristic
262C2×C4×C132−1+411yescharacteristic
263C2×Dic13—22no—
264C2×Dic13—22no—
265C2×Dic13—22no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275Dic26—48no—
276#1055—22no—
277#1055—22no—
278#1055—41no—
279#1055—41no—
280C2×C4⋊Q8—131no—
281C2×#125C2×D1311yescharacteristic
282C2×C4×Q8—261no—
283C2×C4×D4C2×D1311yescharacteristic
284#501—264no—
285#476—264no—
286C2×#107—131no—
287C2×#107—131no—
288C2×#106—131no—
289C2×#106—131no—
290C2×#106—131no—
291C2×#106—131no—
292C2×C4⋊C8—131no—
293C2×C4⋊C8—131no—
294C2×C4⋊C8—131no—
295C2×#102—131no—
296C2×#102—131no—
297C2×#102—131no—
298C2×#102—131no—
299C2×#97—131no—
300#396—21no—
301#396—21no—
302#391—261no—
303#390C2×D1311yescharacteristic
304#390C2×D1311yescharacteristic
305#389—131no—
306#389—131no—
307#387—261no—
308#387—261no—
309#385—21no—
310#385—21no—
311#384C2×D1311yescharacteristic
312#382C2×D1311yescharacteristic
313C4×C4⋊C4—131no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—131no—
316C4×C4⋊C4C2×D1311yescharacteristic
317#340—131no—
318#340—131no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#336—131no—
324#336—131no—
325C22×C13—42no—
326C22×C13—42no—
327C22×C13—42no—
328C22×C13—81no—
329C22×C13—81no—
330C22×C13—81no—
331C22×C13—81no—
332C22×C13C2×#12111yescharacteristic
333C22×C13#54711yescharacteristic
334C22×C13D4×Q811yescharacteristic
335C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
336C22×C13#57511yescharacteristic
337C22×C13#57711yescharacteristic
338C22×C13#57711yescharacteristic
339C4×C13—22no—
340C4×C13—22no—
341C4×C13—22no—
342C4×C13—41no—
343C4×C13—41no—
344C4×C13—41no—
345C4×C13—81no—
346C4×C13—81no—
347C4×C13—81no—
348C4×C13#53311yescharacteristic
349C4×C13#54111yescharacteristic
350C4×C13#54811yescharacteristic
351C4×C13D4×Q811yescharacteristic
352Dic13—42no—
353Dic13—42no—
354Dic13—42no—
355Dic13—81no—
356C22×Q8—261no—
357C22×D4C22×D1311yescharacteristic
358C23×C4—21no—
359C22×C8—131no—
360C22×C8—131no—
361C22×C8—131no—
362C22×C8—131no—
363C4⋊Q8—134no—
364C4⋊Q8—262no—
365#125—22no—
366#125—22no—
367C4×Q8—524no—
368C4×D4—22no—
369C4×D4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4—261no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4C22⋊D1311yescharacteristic
382C2×C4⋊C4C22⋊D1311yescharacteristic
383C2×C4⋊C4C22×D1311yescharacteristic
384C2×C4⋊C4C22×D1311yescharacteristic
385C2×C4⋊C4C22×D1311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—261no—
392C2×C42—261no—
393C2×C42—261no—
394C2×C42C22×D1311yescharacteristic
395C2×C42C22×D1311yescharacteristic
396C2×C42C22×D1311yescharacteristic
397#107—262no—
398#107—262no—
399#106—262no—
400#106—262no—
401#106—262no—
402#106—262no—
403C4⋊C8—262no—
404C4⋊C8—262no—
405C4⋊C8—262no—
406#102—134no—
407#102—262no—
408#102—262no—
409#102—262no—
410#97—262no—
411#95—21no—
412#95—21no—
413#95—261no—
414#95—261no—
415#95—261no—
416#95—261no—
417#95—261no—
418#95C22⋊D1311yescharacteristic
419#95C22⋊D1311yescharacteristic
420C2×C13—82no—
421C2×C13#242911yescharacteristic
422C2×C13#303611yescharacteristic
423C2×C13#304311yescharacteristic
424C2×C13#334511yescharacteristic
425C2×C13#335011yescharacteristic
426C2×C13#338411yescharacteristic
427C2×C13#338811yescharacteristic
428C24—21no—
429C2×Q8—264no—
430C2×Q8—522no—
431C2×Q8—522no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4Q8×D1312yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—261no—
448C22×C4C2×C22⋊D1311yescharacteristic
449C22×C4C2×C22⋊D1311yescharacteristic
450C22×C4C2×C22⋊D1311yescharacteristic
451C22×C4C2×C22⋊D1311yescharacteristic
452C22×C4C2×C22⋊D1311yescharacteristic
453C22×C4C2×C22⋊D1311yescharacteristic
454C22×C4C23×D1311yescharacteristic
455C2×C8—132no—
456C2×C8—132no—
457C2×C8—261no—
458C2×C8—261no—
459C2×C8—261no—
460C2×C8—261no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8—261no—
465C2×C8—261no—
466C2×C8—261no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—132no—
480C4⋊C4—132no—
481C4⋊C4—132no—
482C4⋊C4—264no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—522no—
489C4⋊C4—522no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—522no—
500C42—522no—
501C42—522no—
502C42D4×D1312yes—
503C42#612912yes—
504C13#2546111yescharacteristic, complemented, Sylow
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C22×C22⋊D1311yescharacteristic
513Q8—528no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—262no—
544C2×C4—262no—
545C2×C4—262no—
546C2×C4—521no—
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#8819212yes—
556C2×C4#8819212yes—
557C2×C4#8820312yes—
558C2×C4C2×D4×D1311yescharacteristic
559C2×C4C2×#612911yescharacteristic
560C2×C4C2×Q8×D1311yescharacteristic
561C2×C4#8825411yescharacteristic
562C2×C4#41×D1311yescharacteristic
563C2×C4#8825711yescharacteristic
564C8—262no—
565C8—262no—
566C8—521no—
567C8—521no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C2×#8819211yescharacteristic
576C22Q8×C22⋊D1311yescharacteristic
577C22C2×#8820311yescharacteristic
578C22#1171952711yescharacteristic
579C22#1171952911yescharacteristic
580C22#1171954111yescharacteristic
581C22#1171954111yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—522no—
592C4—522no—
593C4—522no—
594C4—1041no—
595C4#1171932711yescharacteristic
596C4#1171933011yescharacteristic
597C4#1171933711yescharacteristic
598C4#1171933811yescharacteristic
599C2—82no—
600C2#1311843111yescharacteristic
601C2#1312429611yescharacteristic
602C2#1312435311yescharacteristic
603C2#1312730011yescharacteristic
604C2#1312737511yescharacteristic
605C2#1312773511yescharacteristic
606C2#1312778111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2566C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
hypercenter#2566D13
hypocenterC13#25461
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#11718726C2semidirect
3. #13117894C2semidirect
4. #13116318C2semidirect
5. #13116313C2semidirect
8. #13115794C2semidirect
9. #13115602C2semidirect
10. #13115597C2semidirect
11. #13115560C2semidirect
109. #88072Q8semidirect
504. C13#25461semidirect

Maximal subgroups

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

subgroupindex
C2×#117187262
C2×#117186692
#131178942
#131163182
#131163132
#131162242
#131162102
#131157942
#131156022
#131155972
#131155602
#131154602
#131154532
#131146612
C13×#25662
#2546113