Finite Group Databasebeta

← #15920608

Subgroups — #15920608

Order 3328.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117711C211yesmaximal, characteristic
2#13116398C211yesmaximal, characteristic, complemented
3#13116332C211yesmaximal, characteristic, complemented
4#13116332C211yesmaximal, characteristic, complemented
5#13116161C211yesmaximal, characteristic, complemented
6#13115785C211yesmaximal, characteristic, complemented
7#13115592C211yesmaximal, characteristic, complemented
8#13115425C211yesmaximal, characteristic, complemented
9#13115425C211yesmaximal, characteristic, complemented
10#13115024C211yesmaximal, characteristic
11#13115016C211yesmaximal, characteristic
12#13114924C211yesmaximal, characteristic
13#13114863C211yesmaximal, characteristic
14C2×#11718121C211yesmaximal, characteristic
15C13×#2440C211yesmaximal, characteristic
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8—21no—
18C22×C13×C4⋊C4C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C23×#1055C2211yescharacteristic
21C13×#398—21no—
22C13×#395—21no—
23C13×#389C2211yescharacteristic
24C13×#389C2211yescharacteristic
25C13×#385—21no—
26C13×#384C2211yescharacteristic
27C13×#380C2211yescharacteristic
28C4×C13×C22⋊C4C2211yescharacteristic
29#11718873C2211yescharacteristic
30C2×C22⋊#1055C2211yescharacteristic
31C2×#88047C2211yescharacteristic
32C2×#88047C2211yescharacteristic
33C2×#88047C2211yescharacteristic
34C2×#88047C2211yescharacteristic
35#11718603—21no—
36C22⋊C4×Dic13—21no—
37C4×C4⋊Dic13C2211yescharacteristic
38C4×C4⋊Dic13C2211yescharacteristic
39#11718589C2211yescharacteristic
40#11718589C2211yescharacteristic
41C2×C4⋊#1055C2211yescharacteristic
42C2×C4⋊#1055C2211yescharacteristic
43#11718315—21no—
44#11718315—21no—
45#11718121C2211yescharacteristic
46#11718121C2211yescharacteristic
47#11718121C2211yescharacteristic
48#11718121C2211yescharacteristic
49#11718121C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C22×Q8×C13—21no—
60C23×C4×C13—21no—
61C23×C4×C13C2311yescharacteristic
62C23×Dic13—21no—
63C13×C4⋊Q8—22no—
64C13×C4⋊Q8—41no—
65C13×C22⋊Q8—44no—
66C2×C13×C4⋊C4—22no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—41no—
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4D411yescharacteristic
80C2×C13×C22⋊C4D411yescharacteristic
81C2×C42×C13C2311yescharacteristic
82C2×C42×C13C2311yescharacteristic
83C2×C22⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—22no—
86C2×C4⋊Dic13Q814yescomplemented
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×#6101—21no—
92C2×#6101—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C22×#1055—22no—
96C22×#1055—22no—
97C22×#1055—41no—
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C13×#95—21no—
103C13×#95—21no—
104C13×#95—21no—
105C13×#95D411yescharacteristic
106C13×#95D411yescharacteristic
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112C22⋊#1055—22no—
113#88047—22no—
114#88047—22no—
115#88047—41no—
116#88047—41no—
117C4⋊#1055—22no—
118C4⋊#1055—41no—
119#25994—131nomaximal, Sylow
120C24×C13C2×D411yescharacteristic
121C2×Q8×C13—42no—
122C2×Q8×C13—42no—
123C2×Q8×C13—81no—
124C22×C4×C13—21no—
125C22×C4×C13—21no—
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—42no—
133C22×C4×C13—42no—
134C22×C4×C13—81no—
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×Q812yes—
141C22×C4×C13#4112yes—
142C22×C4×C13C2411yescharacteristic
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—24no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—81no—
149C22×Dic13—81no—
150C13×C4⋊C4—21no—
151C13×C4⋊C4—21no—
152C13×C4⋊C4—21no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—42no—
158C13×C4⋊C4—42no—
159C13×C4⋊C4—42no—
160C13×C4⋊C4—42no—
161C13×C4⋊C4—42no—
162C13×C4⋊C4—81no—
163C13×C4⋊C4#4112yes—
164C13×C22⋊C4—22no—
165C13×C22⋊C4—42no—
166C13×C22⋊C4Q1614yes—
167C42×C13—21no—
168C42×C13—21no—
169C42×C13—21no—
170C42×C13—21no—
171C22⋊Dic13—42no—
172C4⋊Dic13—24no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13C2×Q812yes—
180C4⋊Dic13C2×Q812yes—
181#6101—81no—
182#6101—81no—
183C4×Dic13—81no—
184C4×Dic13—81no—
185C2×#1055—21no—
186C2×#1055—21no—
187C2×#1055—21no—
188C2×#1055—21no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—44no—
197C2×#1055#4112yes—
198#2440D1311yescharacteristic
199#2345—131no—
200#2100—131no—
201#2087—131no—
202#2087—131no—
203#2057—131no—
204#1987—131no—
205#1947—131no—
206#1921—131no—
207#1921—131no—
208#1852—131no—
209#1850—131no—
210#1824—131no—
211#1808—131no—
212C2×#340—131no—
213C23×C13—21no—
214C23×C13—21no—
215C23×C13—21no—
216C23×C13—21no—
217C23×C13C22⋊Q812yes—
218C23×C13C2×Q1612yes—
219C23×C13AGL(1,ℤ/8)12yes—
220C23×C13C22×D411yescharacteristic
221Q8×C13—84no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—22no—
229C2×C4×C13—22no—
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—21no—
233C2×C4×C13—22no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
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—84no—
251C2×C4×C13C22⋊Q812yes—
252C2×C4×C13C2×Q1612yes—
253C2×C4×C13C2×Q1612yes—
254C2×C4×C13C22×Q811yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×C4×C13H(5,𝔽2)11yescharacteristic
259C2×C4×C132−1+411yescharacteristic
260C2×Dic13—22no—
261C2×Dic13—22no—
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×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—84no—
273#1055—22no—
274#1055—41no—
275#1055—41no—
276#1055—41no—
277C23×C8—131no—
278C2×C4⋊Q8C2×D1311yescharacteristic
279C2×C22⋊Q8—21no—
280C22×C4⋊C4—131no—
281C22×C4⋊C4C2×D1311yescharacteristic
282C2×#107—131no—
283C2×#107—131no—
284C2×#107—131no—
285C2×#107—131no—
286C2×C4⋊C8—131no—
287C2×C4⋊C8—131no—
288C2×C22⋊C8—131no—
289#398—21no—
290#395—21no—
291#389—131no—
292#389—131no—
293#389C2×D1311yescharacteristic
294#389C2×D1311yescharacteristic
295#387—261no—
296#387—261no—
297#385—21no—
298#384C2×D1311yescharacteristic
299#380—131no—
300#380—261no—
301#380C2×D1311yescharacteristic
302C4×C4⋊C4—131no—
303C4×C4⋊C4—131no—
304C4×C22⋊C4—261no—
305C4×C22⋊C4C2×D1311yescharacteristic
306#340—131no—
307#340—131no—
308#340—131no—
309#340—131no—
310#340—131no—
311#340—131no—
312#340—131no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318#336—131no—
319#336—131no—
320C22×C13—22no—
321C22×C13—22no—
322C22×C13—22no—
323C22×C13—22no—
324C22×C13—22no—
325C22×C13—22no—
326C22×C13#47714yes—
327C22×C13C2×C22⋊Q811yescharacteristic
328C22×C13#54611yescharacteristic
329C22×C13#54711yescharacteristic
330C22×C13C22×Q1611yescharacteristic
331C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
332C22×C13#57611yescharacteristic
333C22×C13#57711yescharacteristic
334C4×C13—22no—
335C4×C13—22no—
336C4×C13—22no—
337C4×C13—41no—
338C4×C13—41no—
339C4×C13—42no—
340C4×C13—81no—
341C4×C13—81no—
342C4×C13—81no—
343C4×C13#53311yescharacteristic
344C4×C13#54411yescharacteristic
345C4×C13#55111yescharacteristic
346C4×C13#55211yescharacteristic
347Dic13—42no—
348Dic13—42no—
349Dic13—81no—
350Dic13—81no—
351C22×Q8—21no—
352C23×C4—21no—
353C23×C4—261no—
354C23×C4C22×D1311yescharacteristic
355C22×C8—131no—
356C22×C8—131no—
357C22×C8—131no—
358C22×C8—131no—
359C22×C8—262no—
360C22×C8—262no—
361C22×C8—521no—
362C4⋊Q8—22no—
363C4⋊Q8—41no—
364C22⋊Q8—44no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—134no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—262no—
379C2×C4⋊C4—261no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4C22×D1311yescharacteristic
383C2×C4⋊C4C22×D1311yescharacteristic
384C2×C4⋊C4C22×D1311yescharacteristic
385C2×C4⋊C4C22×D1311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—261no—
388C2×C22⋊C4C22⋊D1311yescharacteristic
389C2×C22⋊C4C22⋊D1311yescharacteristic
390C2×C42—261no—
391C2×C42—261no—
392C2×C42C22×D1311yescharacteristic
393C2×C42C22×D1311yescharacteristic
394#107—262no—
395#107—262no—
396#107—521no—
397#107—521no—
398C4⋊C8—262no—
399C4⋊C8—521no—
400C22⋊C8—262no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—261no—
405#95—261no—
406#95—261no—
407#95—261no—
408#95—261no—
409#95C22⋊D1311yescharacteristic
410#95C22⋊D1311yescharacteristic
411C2×C13—24no—
412C2×C13#243811yescharacteristic
413C2×C13C2×#47711yescharacteristic
414C2×C13#308111yescharacteristic
415C2×C13#329911yescharacteristic
416C2×C13#331811yescharacteristic
417C2×C13#333811yescharacteristic
418C2×C13#334511yescharacteristic
419C24C2×C22⋊D1311yescharacteristic
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—81no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—81no—
434C22×C4—261no—
435C22×C4—261no—
436C22×C4—264no—
437C22×C4—261no—
438C22×C4—261no—
439C22×C4—1041no—
440C22×C4—1041no—
441C22×C4Q8×D1312yes—
442C22×C4#613112yes—
443C22×C4C2×C22⋊D1311yescharacteristic
444C22×C4C2×C22⋊D1311yescharacteristic
445C22×C4C2×C22⋊D1311yescharacteristic
446C22×C4C2×C22⋊D1311yescharacteristic
447C22×C4C2×C22⋊D1311yescharacteristic
448C22×C4C23×D1311yescharacteristic
449C2×C8—132no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—261no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—261no—
459C2×C8—261no—
460C2×C8—261no—
461C2×C8—524no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—132no—
476C4⋊C4—132no—
477C4⋊C4—264no—
478C4⋊C4—261no—
479C4⋊C4—261no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—261no—
483C4⋊C4—261no—
484C4⋊C4—1041no—
485C4⋊C4—1041no—
486C4⋊C4#612912yes—
487C22⋊C4—22no—
488C22⋊C4—42no—
489C22⋊C4—522no—
490C22⋊C4#610714yes—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—1041no—
496C42—1041no—
497C13#2599411yescharacteristic, complemented, Sylow
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#8818612yes—
503C23C2×#610712yes—
504C23#8820012yes—
505C23C22×C22⋊D1311yescharacteristic
506Q8—84no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—262no—
537C2×C4—262no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—521no—
544C2×C4—521no—
545C2×C4—521no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—1044no—
549C2×C4C2×#610712yes—
550C2×C4C2×#610712yes—
551C2×C4#8820012yes—
552C2×C4C2×#612911yescharacteristic
553C2×C4#8825111yescharacteristic
554C2×C4C2×Q8×D1311yescharacteristic
555C2×C4C2×#613111yescharacteristic
556C2×C4#41×D1311yescharacteristic
557C2×C4#8825711yescharacteristic
558C8—262no—
559C8—521no—
560C8—521no—
561C8—521no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#1171875214yes—
569C22C2×#8818611yescharacteristic
570C22#1171951111yescharacteristic
571C22C22×#610711yescharacteristic
572C22C2×#8820011yescharacteristic
573C22#1171952911yescharacteristic
574C22#1171954011yescharacteristic
575C22#1171954211yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—522no—
586C4—522no—
587C4—1041no—
588C4—1041no—
589C4#1171932711yescharacteristic
590C4#1171932811yescharacteristic
591C4#1171934211yescharacteristic
592C4#1171934711yescharacteristic
593C2—24no—
594C2#1311850811yescharacteristic
595C2C2×#1171875211yescharacteristic
596C2#1312466111yescharacteristic
597C2#1312671211yescharacteristic
598C2#1312697711yescharacteristic
599C2#1312722411yescharacteristic
600C2#1312730611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2440C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#2440D13
hypocenterC13#25994
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #13116398C2semidirect
3. #13116332C2semidirect
4. #13116332C2semidirect
5. #13116161C2semidirect
6. #13115785C2semidirect
7. #13115592C2semidirect
8. #13115425C2semidirect
9. #13115425C2semidirect
86. C2×C4⋊Dic13Q8semidirect
497. C13#25994semidirect

Maximal subgroups

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

subgroupindex
#131177112
#131163982
#131163322
#131161612
#131157852
#131155922
#131154252
#131150242
#131150162
#131149242
#131148632
C2×#117181212
C13×#24402
#2599413