Finite Group Databasebeta

← #15876858

Subgroups — #15876858

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#88132C211yesmaximal, characteristic
2#13115771C211yesmaximal, characteristic, complemented
3#13115770C211yesmaximal, characteristic, complemented
4#13115552C211yesmaximal, characteristic, complemented
5#13115551C211yesmaximal, characteristic, complemented
6#13115394C211yesmaximal, characteristic, complemented
7#13115391C211yesmaximal, characteristic, complemented
8#13115391C211yesmaximal, characteristic, complemented
9#13115390C211yesmaximal, characteristic, complemented
10#13115014C211yesmaximal, characteristic
11#13114814C211yesmaximal, characteristic
12#13114610C211yesmaximal, characteristic
13C22⋊C4×#1055C211yesmaximal, characteristic
14C2×#11718142C211yesmaximal, characteristic
15C13×#1850C211yesmaximal, characteristic
16C23×C8×C13C2211yescharacteristic
17C22×C4×Dic13C2211yescharacteristic
18C23×#1055C2211yescharacteristic
19C2×#88132C411yescharacteristic
20C2×C4×Dic26C2211yescharacteristic
21C2×C13×C4⋊C8C2211yescharacteristic
22C2×C13×C4⋊C8C2211yescharacteristic
23C2×C13×C22⋊C8C2211yescharacteristic
24C4×C13×C22⋊C4C2211yescharacteristic
25C4×C22⋊Dic13C2211yescharacteristic
26C2×C22⋊#1055C2211yescharacteristic
27C2×#88054C2211yescharacteristic
28C2×#88054C2211yescharacteristic
29C2×#88054C2211yescharacteristic
30C2×#88054C2211yescharacteristic
31C2×C8×Dic13C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33#11718611C411yescharacteristic
34#11718607C411yescharacteristic
35#11718603C411yescharacteristic
36C4×C4⋊Dic13C2211yescharacteristic
37C4×#6101C2211yescharacteristic
38C4×#6101C2211yescharacteristic
39C2×C4×#1055C2211yescharacteristic
40C2×C4×#1055C2211yescharacteristic
41#11718316C411yescharacteristic
42#11718315C411yescharacteristic
43#11718308C411yescharacteristic
44#11718307C411yescharacteristic
45C13×#336C2211yescharacteristic
46C13×#336C2211yescharacteristic
47#11718142C2211yescharacteristic
48#11718142C2211yescharacteristic
49#11718142C2211yescharacteristic
50#11718142C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C23×Dic13C2×C411yescharacteristic
61C23×Dic13C2×C411yescharacteristic
62C22×Dic26C2×C411yescharacteristic
63C22×C8×C13—22no—
64C22×C8×C13—41no—
65C22×C8×C13—41no—
66C22×C8×C13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C2×C13×C22⋊C4C2×C411yescharacteristic
71C2×C13×C22⋊C4C2×C411yescharacteristic
72C2×C42×C13C2311yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×C22⋊Dic13C2×C411yescharacteristic
75C2×C22⋊Dic13C2×C411yescharacteristic
76C2×C4⋊Dic13C2×C411yescharacteristic
77C2×C4⋊Dic13C2×C411yescharacteristic
78C2×C4⋊Dic13C2×C411yescharacteristic
79C2×#6101C2×C411yescharacteristic
80C2×#6101C2×C411yescharacteristic
81C2×#6101C2×C411yescharacteristic
82C2×#6101C2×C411yescharacteristic
83C2×#6101C2×C411yescharacteristic
84C2×#6101C2×C411yescharacteristic
85C2×C4×Dic13—41no—
86C2×C4×Dic13—41no—
87C2×C4×Dic13C2×C411yescharacteristic
88C2×C4×Dic13C2×C411yescharacteristic
89C2×C4×Dic13C2311yescharacteristic
90C2×C4×Dic13C2311yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93C22×#1055—41no—
94C22×#1055—41no—
95C22×#1055Q814yescomplemented
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100#88132C818yescomplemented
101C4×Dic26—41no—
102C4×Dic26—41no—
103C13×C4⋊C8—41no—
104C13×C4⋊C8—41no—
105C13×C22⋊C8—22no—
106C13×#95C2×C411yescharacteristic
107C13×#95C2×C411yescharacteristic
108#88071C2×C411yescharacteristic
109#88071C2×C411yescharacteristic
110#88071C2×C411yescharacteristic
111#88071C2×C411yescharacteristic
112#88071C2×C411yescharacteristic
113#88071C2×C411yescharacteristic
114#88071C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116C22⋊#1055—22no—
117#88054—41no—
118#88054—41no—
119#88054—41no—
120#88054—41no—
121C8×Dic13—41no—
122C8×Dic13—41no—
123C4×#1055—41no—
124C4×#1055—41no—
125#22287—131nomaximal, Sylow
126C24×C13C22×C411yescharacteristic
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13C22×C411yescharacteristic
132C22×C4×C13C22×C411yescharacteristic
133C22×C4×C13C22×C411yescharacteristic
134C22×C4×C13C22×C411yescharacteristic
135C22×C4×C13C22×C411yescharacteristic
136C22×C4×C13C2×Q812yes—
137C22×C4×C13#4112yes—
138C22×C4×C13C2411yescharacteristic
139C22×Dic13—22no—
140C22×Dic13—41no—
141C22×Dic13—41no—
142C22×Dic13—41no—
143C22×Dic13—41no—
144C22×Dic13C2×C814yes—
145C22×Dic13C22×C411yescharacteristic
146C22×Dic13C22×C411yescharacteristic
147C22×Dic13C22×C411yescharacteristic
148C22×Dic13C22×C411yescharacteristic
149C22×Dic13C22×C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C22×Dic13C22×C411yescharacteristic
153C2×Dic26—41no—
154C2×Dic26—41no—
155C2×Dic26C2×C814yescomplemented
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13—44no—
169C13×C22⋊C4—22no—
170C13×C22⋊C4C2×C814yes—
171C42×C13—21no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13—21no—
175C22⋊Dic13—22no—
176C22⋊Dic13C2×C814yes—
177C4⋊Dic13—41no—
178C4⋊Dic13—41no—
179C4⋊Dic13C2×C814yescomplemented
180#6101—41no—
181#6101—41no—
182#6101—41no—
183#6101—41no—
184#6101C2×C814yescomplemented
185#6101C2×C814yescomplemented
186C4×Dic13—21no—
187C4×Dic13—21no—
188C4×Dic13—21no—
189C4×Dic13—21no—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—41no—
195C4×Dic13—41no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—44no—
207C2×#1055C2×Q812yes—
208C2×#1055C2×Q812yes—
209C4×C22⋊Q8—131no—
210#1984—131no—
211#1984—131no—
212#1939—131no—
213#1939—131no—
214#1916—131no—
215#1916—131no—
216#1916—131no—
217#1916—131no—
218#1850—131no—
219#1850D1311yescharacteristic
220#1797—131no—
221C8×C22⋊C4—131no—
222C8×C22⋊C4—131no—
223C2×#336—131no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13C4×Q812yes—
229C23×C13C22×C812yes—
230C23×C13#13112yes—
231C23×C13C23×C411yescharacteristic
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—22no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13C4×Q812yes—
249C2×C4×C13C22×C812yes—
250C2×C4×C13C22×C812yes—
251C2×C4×C13C22×Q811yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C2×C4×C13H(5,𝔽2)11yescharacteristic
254C2×C4×C13H(5,𝔽2)11yescharacteristic
255C2×C4×C13H(5,𝔽2)11yescharacteristic
256C2×C4×C132−1+411yescharacteristic
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—44no—
278C2×Dic13—44no—
279C2×Dic13C22×C812yes—
280C2×Dic13C22×C812yes—
281C2×Dic13C22×C812yes—
282C2×Dic13C22×C812yes—
283Dic26—44no—
284C8×C13—41no—
285C8×C13—41no—
286C8×C13—41no—
287C8×C13—41no—
288#1055—22no—
289#1055—22no—
290#1055—41no—
291#1055—41no—
292C23×C8—131no—
293C23×C8C2×D1311yescharacteristic
294C2×C22⋊Q8—131no—
295C2×C4×Q8—131no—
296C22×C42—131no—
297C2×C4⋊C8—131no—
298C2×C4⋊C8—131no—
299C2×C4⋊C8—131no—
300C2×C4⋊C8—131no—
301C2×C4⋊C8C2×D1311yescharacteristic
302C2×C4⋊C8C2×D1311yescharacteristic
303C2×C22⋊C8—131no—
304C2×C22⋊C8C2×D1311yescharacteristic
305C2×C4×C8—131no—
306C2×C4×C8—131no—
307C2×C4×C8—131no—
308C2×C4×C8—131no—
309#391—131no—
310#389—131no—
311#387—131no—
312#387—131no—
313#385—131no—
314#385—131no—
315#380—131no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4—131no—
319C4×C22⋊C4—131no—
320C4×C22⋊C4C2×D1311yescharacteristic
321#336—131no—
322#336—131no—
323#336—131no—
324#336—131no—
325#336—131no—
326#336—131no—
327#336—131no—
328#336—131no—
329#336—131no—
330#336—131no—
331#336—131no—
332#336—131no—
333#336C2×D1311yescharacteristic
334#336C2×D1311yescharacteristic
335C22×C13—22no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—22no—
340C22×C13—22no—
341C22×C13C8×Q814yes—
342C22×C13C2×C4×Q811yescharacteristic
343C22×C13#51811yescharacteristic
344C22×C13#52011yescharacteristic
345C22×C13C23×C811yescharacteristic
346C22×C13C2×#13111yescharacteristic
347C22×C13#56811yescharacteristic
348C22×C13#56811yescharacteristic
349C4×C13—22no—
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—41no—
353C4×C13—41no—
354C4×C13#53811yescharacteristic
355C4×C13#53911yescharacteristic
356C4×C13#54311yescharacteristic
357C4×C13#54411yescharacteristic
358Dic13—22no—
359Dic13—22no—
360Dic13—22no—
361Dic13—22no—
362Dic13—41no—
363Dic13—41no—
364Dic13—41no—
365Dic13—41no—
366C22×Q8—131no—
367C23×C4—131no—
368C23×C4—131no—
369C23×C4C22×D1311yescharacteristic
370C22×C8—22no—
371C22×C8—41no—
372C22×C8—41no—
373C22×C8—131no—
374C22×C8—131no—
375C22×C8—134no—
376C22×C8—131no—
377C22×C8—131no—
378C22×C8—521no—
379C22×C8—521no—
380C22×C8C22×D1311yescharacteristic
381C22×C8C22×D1311yescharacteristic
382C22×C8C22×D1311yescharacteristic
383C22×C8C22×D1311yescharacteristic
384C22⋊Q8—138no—
385C4×Q8—521no—
386C4×Q8—521no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—131no—
391C2×C4⋊C4—131no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—131no—
394C2×C4⋊C4—131no—
395C2×C4⋊C4—131no—
396C2×C22⋊C4—131no—
397C2×C22⋊C4—131no—
398C2×C22⋊C4C4×D1311yescharacteristic
399C2×C22⋊C4C4×D1311yescharacteristic
400C2×C42—131no—
401C2×C42—131no—
402C2×C42—131no—
403C2×C42—131no—
404C2×C42—131no—
405C2×C42—131no—
406C2×C42—521no—
407C2×C42—521no—
408C2×C42C22×D1311yescharacteristic
409C2×C42C22×D1311yescharacteristic
410C4⋊C8—41no—
411C4⋊C8—41no—
412C4⋊C8—521no—
413C4⋊C8—521no—
414C4⋊C8—521no—
415C4⋊C8—521no—
416C22⋊C8—22no—
417C22⋊C8—262no—
418C4×C8—521no—
419C4×C8—521no—
420C4×C8—521no—
421C4×C8—521no—
422#95—131no—
423#95—131no—
424#95—131no—
425#95—131no—
426#95—131no—
427#95—131no—
428#95—131no—
429#95—131no—
430#95C4×D1311yescharacteristic
431#95C4×D1311yescharacteristic
432C2×C13—24no—
433C2×C13#232911yescharacteristic
434C2×C13C2×C8×Q811yescharacteristic
435C2×C13#296111yescharacteristic
436C2×C13#297211yescharacteristic
437C2×C13#297611yescharacteristic
438C2×C13#298811yescharacteristic
439C2×C13#299211yescharacteristic
440C24C2×C4×D1311yescharacteristic
441C2×Q8—134no—
442C2×Q8—521no—
443C2×Q8—521no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—131no—
449C22×C4—131no—
450C22×C4—131no—
451C22×C4—131no—
452C22×C4—134no—
453C22×C4—131no—
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—131no—
457C22×C4—262no—
458C22×C4—521no—
459C22×C4—521no—
460C22×C4—521no—
461C22×C4—521no—
462C22×C4C2×C4×D1311yescharacteristic
463C22×C4C2×C4×D1311yescharacteristic
464C22×C4C2×C4×D1311yescharacteristic
465C22×C4C2×C4×D1311yescharacteristic
466C22×C4C2×C4×D1311yescharacteristic
467C22×C4#612912yes—
468C22×C4Q8×D1312yes—
469C22×C4C23×D1311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—44no—
483C2×C8—132no—
484C2×C8—132no—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C2×C8—524no—
496C4⋊C4—134no—
497C4⋊C4—134no—
498C4⋊C4—134no—
499C4⋊C4—521no—
500C4⋊C4—521no—
501C4⋊C4—521no—
502C4⋊C4—521no—
503C4⋊C4—521no—
504C4⋊C4—521no—
505C22⋊C4—22no—
506C22⋊C4—134no—
507C22⋊C4—262no—
508C22⋊C4C8×D1314yes—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—261no—
514C42—261no—
515C42—261no—
516C42—261no—
517C42—261no—
518C42—261no—
519C42—261no—
520C42—261no—
521C42—521no—
522C42—521no—
523C13#2228711yescharacteristic, complemented, Sylow
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#8814112yes—
529C23C2×C8×D1312yes—
530C23#8816112yes—
531C23C22×C4×D1311yescharacteristic
532Q8—524no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—132no—
550C2×C4—132no—
551C2×C4—132no—
552C2×C4—132no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—261no—
557C2×C4—261no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—261no—
569C2×C4—261no—
570C2×C4—261no—
571C2×C4—261no—
572C2×C4—261no—
573C2×C4—524no—
574C2×C4—524no—
575C2×C4#8814112yes—
576C2×C4C2×C8×D1312yes—
577C2×C4C2×C8×D1312yes—
578C2×C4C2×#612911yescharacteristic
579C2×C4#8825111yescharacteristic
580C2×C4#8825111yescharacteristic
581C2×C4C2×Q8×D1311yescharacteristic
582C2×C4#8825611yescharacteristic
583C2×C4#8825711yescharacteristic
584C8—41no—
585C8—41no—
586C8—41no—
587C8—41no—
588C8—262no—
589C8—262no—
590C8—521no—
591C8—521no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1171849114yes—
599C22C2×#8814111yescharacteristic
600C22#1171925011yescharacteristic
601C22#1171925111yescharacteristic
602C22C22×C8×D1311yescharacteristic
603C22C2×#8816111yescharacteristic
604C22#1171945511yescharacteristic
605C22#1171945511yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—262no—
612C4—262no—
613C4—262no—
614C4—262no—
615C4—521no—
616C4—521no—
617C4—521no—
618C4—521no—
619C4#1171929811yescharacteristic
620C4#1171932811yescharacteristic
621C4#1171933911yescharacteristic
622C4#1171937011yescharacteristic
623C2—24no—
624C2#1311756711yescharacteristic
625C2C2×#1171849111yescharacteristic
626C2#1312377511yescharacteristic
627C2#1312385711yescharacteristic
628C2#1312388511yescharacteristic
629C2#1312394311yescharacteristic
630C2#1312397611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutatorC22×C13C23×C8
FittingC13×#1850C2
socleC23×C13C23×C4
prime radicalC24×C13C22×C4
cosocleC22×C4×C13C24
hypercenter#1850D13
hypocenterC13#22287
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
2. #13115771C2semidirect
3. #13115770C2semidirect
4. #13115552C2semidirect
5. #13115551C2semidirect
6. #13115394C2semidirect
7. #13115391C2semidirect
8. #13115391C2semidirect
9. #13115390C2semidirect
95. C22×#1055Q8semidirect
100. #88132C8semidirect
155. C2×Dic26C2×C8semidirect
179. C4⋊Dic13C2×C8semidirect
184. #6101C2×C8semidirect
185. #6101C2×C8semidirect
523. C13#22287semidirect

Maximal subgroups

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

subgroupindex
C4×#881322
#131157712
#131157702
#131155522
#131155512
#131153942
#131153912
#131153902
#131150142
#131148142
#131146102
C22⋊C4×#10552
C2×#117181422
C13×#18502
#2228713