Finite Group Databasebeta

← #16794689

Subgroups — #16794689

Order 3328.

972 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131512C211yesmaximal, characteristic
2C2×#11719125C211yesmaximal, characteristic
3#13130362C214yesmaximal, complemented
4#13130349C212yesmaximal, complemented
5#13130345C212yesmaximal, complemented
6#13130338C214yesmaximal, complemented
7C2×#11719327C211yesmaximal, characteristic
8C2×C13×C22⋊Q8C411yescharacteristic
9C2×Q8×Dic13C411yescharacteristic
10C2×#88198C2211yescharacteristic
11C2×#88178C411yescharacteristic
12C22×C4×Dic13C2211yescharacteristic
13#11719327—21no—
14#11719327—21no—
15#11719327—21no—
16#11719327—21no—
17#11719327C412yes—
18#11719327C412yes—
19#11719327C2212yes—
20#11719327C2212yes—
21C2×#88143—21no—
22C2×#88142C2211yescharacteristic
23C2×#88141—21no—
24C2×#88141C411yescharacteristic
25C2×#88132—21no—
26C2×#88131—21no—
27#11719169—21no—
28C2×C22⋊#1057—21no—
29#11719130—21no—
30#11719129C2212yes—
31#11719128C2212yes—
32#11719125—21no—
33#11719125—21no—
34#11719125C2212yes—
35#11719125C2212yes—
36#11719125C2212yes—
37C2×#88112C2212yes—
38C4×#6122—21no—
39#11719119—21no—
40C2×C4×#1057—21no—
41#11718303C2218yescomplemented
42#11718245C2218yescomplemented
43C22×Q8×C13C2×C411yescharacteristic
44C23×C4×C13C2×C411yescharacteristic
45C23×Dic13D411yescharacteristic
46C23×Dic13D411yescharacteristic
47C22×Dic26—21no—
48C22×Dic26C2×C411yescharacteristic
49C2×#6122—22no—
50C22×#1057—22no—
51C13×C22⋊Q8—21no—
52C13×C22⋊Q8—21no—
53C13×C22⋊Q8C2×C412yes—
54C13×C22⋊Q8C2×C412yes—
55C2×C13×C4⋊C4—21no—
56C2×C13×C4⋊C4C2×C411yescharacteristic
57C2×C13×C22⋊C4—21no—
58C2×C22⋊Dic13—21no—
59Q8×Dic13—21no—
60Q8×Dic13—21no—
61Q8×Dic13C2×C412yes—
62Q8×Dic13C2×C412yes—
63#88198—21no—
64#88198—21no—
65#88198C2×C412yes—
66#88198C2312yes—
67C2×C4⋊Dic13—21no—
68C2×C4⋊Dic13C2×C411yescharacteristic
69#88178—21no—
70#88178—21no—
71#88178C2×C412yes—
72#88178C2×C412yes—
73C2×#6101—21no—
74C2×#6101—21no—
75C2×#6101—21no—
76C2×#6101—21no—
77C2×#6101—21no—
78C2×C4×Dic13—21no—
79C2×C4×Dic13—21no—
80C2×C4×Dic13C2×C411yescharacteristic
81C2×C4×Dic13C2×C411yescharacteristic
82C2×C4×Dic13C2×C411yescharacteristic
83C2×C4×Dic13D411yescharacteristic
84C2×C4×Dic13D411yescharacteristic
85C2×C4×Dic13D411yescharacteristic
86C2×C4×Dic13D411yescharacteristic
87C2×C4×Dic13D411yescharacteristic
88C2×C4×Dic13D411yescharacteristic
89C2×C4×Dic13C2311yescharacteristic
90C2×C4×Dic13C2311yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92#88143—22no—
93#88143—22no—
94#88143—22no—
95#88143—22no—
96#88142—21no—
97#88142—21no—
98#88142C2×C412yes—
99#88142C2312yes—
100#88141—22no—
101#88141—22no—
102#88141—21no—
103#88141—21no—
104#88141—22no—
105#88141—22no—
106#88141C2×C412yes—
107#88141C2×C412yes—
108#88132—22no—
109#88132—22no—
110#88132—22no—
111#88132—22no—
112#88131—22no—
113#88131—22no—
114#88131—22no—
115#88131—22no—
116C22⋊#1057—22no—
117C22⋊#1057—22no—
118C22⋊#1057—22no—
119C22⋊#1057—22no—
120#88112—22no—
121#88112C2314yes—
122#88112C2314yes—
123#88110—22no—
124C4⋊#1057—22no—
125C4×#1057—22no—
126C4×#1057—22no—
127C4×#1057—22no—
128#17718—131nomaximal, Sylow
129C24×C13C22⋊C411yescharacteristic
130C2×Q8×C13—21no—
131C2×Q8×C13—42no—
132C2×Q8×C13C22×C412yes—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—42no—
136C22×C4×C13C22⋊C411yescharacteristic
137C22×C4×C13C22⋊C411yescharacteristic
138C22×C4×C13C22⋊C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×C4×C13C22×C411yescharacteristic
141C22×C4×C13C22×C411yescharacteristic
142C22×Dic13—24no—
143C22×Dic13—24no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13C22⋊C411yescharacteristic
147C22×Dic13C22⋊C411yescharacteristic
148C22×Dic13C22⋊C411yescharacteristic
149C22×Dic13C22⋊C411yescharacteristic
150C22×Dic13C22⋊C411yescharacteristic
151C22×Dic13C22⋊C411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C22×Dic13C2×D411yescharacteristic
157C22×Dic13C2×D411yescharacteristic
158C2×Dic26—21no—
159C2×Dic26—22no—
160C2×Dic26—22no—
161C2×Dic26—44no—
162C2×Dic26—42no—
163C2×Dic26C22×C412yes—
164#6122—44no—
165C2×#1057—24no—
166C2×#1057—24no—
167C2×#1057—44no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—22no—
170C13×C4⋊C4—22no—
171C13×C4⋊C4C22×C412yes—
172C13×C22⋊C4—22no—
173C13×C22⋊C4—22no—
174C22⋊Dic13—22no—
175C22⋊Dic13—22no—
176C4⋊Dic13—21no—
177C4⋊Dic13—22no—
178C4⋊Dic13—22no—
179C4⋊Dic13C22×C412yes—
180#6101—22no—
181#6101—22no—
182#6101—22no—
183#6101—22no—
184#6101—22no—
185#6101—22no—
186#6101—22no—
187#6101—22no—
188#6101—22no—
189#6101—22no—
190C4×Dic13—22no—
191C4×Dic13—22no—
192C4×Dic13—22no—
193C4×Dic13—22no—
194C4×Dic13C22⋊C411yescharacteristic
195C4×Dic13C22⋊C411yescharacteristic
196C4×Dic13C22⋊C411yescharacteristic
197C4×Dic13C22⋊C411yescharacteristic
198C4×Dic13C22⋊C411yescharacteristic
199C4×Dic13C22⋊C411yescharacteristic
200C4×Dic13C22×C411yescharacteristic
201C4×Dic13C22×C411yescharacteristic
202C4×Dic13C22×C411yescharacteristic
203C4×Dic13C2×D411yescharacteristic
204C4×Dic13C2×D411yescharacteristic
205C4×Dic13C2×D411yescharacteristic
206C4×Dic13C2×D411yescharacteristic
207C4×Dic13C2×D411yescharacteristic
208C4×Dic13C2×D411yescharacteristic
209C4×Dic13C2411yescharacteristic
210C2×#533—131no—
211#2920—131no—
212C2×#431—131no—
213#1554—134no—
214#1548—132no—
215#1546—132no—
216#1542—134no—
217C23×C13—22no—
218C23×C13—22no—
219C23×C13—41no—
220C23×C13C2×C22⋊C411yescharacteristic
221C23×C13C2×C22⋊C411yescharacteristic
222C23×C13C2×C22⋊C411yescharacteristic
223Q8×C13—42no—
224Q8×C13—42no—
225C2×C4×C13—22no—
226C2×C4×C13—22no—
227C2×C4×C13—22no—
228C2×C4×C13—22no—
229C2×C4×C13—22no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—42no—
234C2×C4×C13—42no—
235C2×C4×C13C2×C22⋊C411yescharacteristic
236C2×C4×C13C2×C22⋊C411yescharacteristic
237C2×C4×C13C2×C22⋊C411yescharacteristic
238C2×C4×C13C23×C411yescharacteristic
239C2×Dic13—22no—
240C2×Dic13—22no—
241C2×Dic13—22no—
242C2×Dic13—22no—
243C2×Dic13—22no—
244C2×Dic13—22no—
245C2×Dic13—22no—
246C2×Dic13—22no—
247C2×Dic13—22no—
248C2×Dic13—22no—
249C2×Dic13—22no—
250C2×Dic13—22no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13C2×C22⋊C411yescharacteristic
254C2×Dic13C2×C22⋊C411yescharacteristic
255C2×Dic13C2×C22⋊C411yescharacteristic
256C2×Dic13C2×C22⋊C411yescharacteristic
257C2×Dic13C2×C22⋊C411yescharacteristic
258C2×Dic13C2×C22⋊C411yescharacteristic
259C2×Dic13#13514yes—
260C2×Dic13#13714yes—
261C2×Dic13C22×D411yescharacteristic
262C2×Dic13C22×D411yescharacteristic
263Dic26—42no—
264Dic26—44no—
265Dic26—44no—
266Dic26—42no—
267#1057—44no—
268#1057—44no—
269#533—132no—
270#533—132no—
271#533—132no—
272#533—132no—
273#533—261no—
274#533—261no—
275#533—261no—
276#533—261no—
277C2×C4⋊Q8—131no—
278C2×C4⋊Q8—131no—
279C2×#125—261no—
280C2×C22⋊Q8—131no—
281C2×C22⋊Q8—261no—
282C2×C22⋊Q8#26111yescharacteristic
283C2×C4×Q8—131no—
284C2×C4×Q8—131no—
285C2×C4×Q8—261no—
286C2×#117—261no—
287C22×C42—131no—
288#432—132no—
289#432—261no—
290#431—132no—
291#431—132no—
292#431—132no—
293#431—261no—
294#431—261no—
295#431—261no—
296#423—132no—
297C2×C4⋊C8—132no—
298#407—261no—
299C2×C22⋊C8—261no—
300C4×M16—261no—
301C2×C4×C8—261no—
302#333—138no—
303#332—138no—
304C22×C13—21no—
305C22×C13—21no—
306C22×C13—21no—
307C22×C13—21no—
308C22×C13—21no—
309C22×C13—21no—
310C22×C13—21no—
311C22×C13—44no—
312C22×C13#41612yes—
313C22×C13#41912yes—
314C22×C13C22×C22⋊C411yescharacteristic
315C4×C13—21no—
316C4×C13—21no—
317C4×C13—21no—
318C4×C13—42no—
319C4×C13—42no—
320C4×C13#41312yes—
321Dic13—21no—
322Dic13—21no—
323Dic13—21no—
324Dic13—21no—
325Dic13—42no—
326Dic13—42no—
327Dic13#41612yes—
328Dic13#41912yes—
329Dic13C2×#13512yes—
330Dic13C2×#13712yes—
331C22×Q8—131no—
332C22×Q8—261no—
333C22×Q8C2×#26111yescharacteristic
334C23×C4—131no—
335C23×C4—131no—
336C23×C4C2×#26111yescharacteristic
337C2×M16—262no—
338C22×C8—262no—
339C4⋊Q8—132no—
340C4⋊Q8—132no—
341C4⋊Q8—132no—
342C4⋊Q8—132no—
343C4⋊Q8—261no—
344C4⋊Q8—261no—
345C4⋊Q8—261no—
346C4⋊Q8—261no—
347#125—262no—
348#125—262no—
349#125—262no—
350#125—262no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8—132no—
354C22⋊Q8—132no—
355C22⋊Q8—262no—
356C22⋊Q8—262no—
357C22⋊Q8—262no—
358C22⋊Q8—262no—
359C22⋊Q8—261no—
360C22⋊Q8—261no—
361C22⋊Q8C2×#26112yes—
362C22⋊Q8C2×#26112yes—
363C4×Q8—132no—
364C4×Q8—132no—
365C4×Q8—132no—
366C4×Q8—132no—
367C4×Q8—261no—
368C4×Q8—261no—
369C4×Q8—262no—
370C4×Q8—262no—
371C4×Q8—262no—
372C4×Q8—262no—
373C4×Q8—261no—
374C4×Q8—261no—
375#117—262no—
376#117—262no—
377#117—262no—
378#117—262no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4C2×#26111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—261no—
390C2×C42—131no—
391C2×C42—131no—
392C2×C42—131no—
393C2×C42—131no—
394C2×C42—131no—
395C2×C42—131no—
396C2×C42—131no—
397C2×C42—131no—
398C2×C42—131no—
399C2×C42—131no—
400C2×C42—131no—
401C2×C42—131no—
402C2×C42—261no—
403C2×C42—261no—
404C4⋊C8—134no—
405C4⋊C8—134no—
406C4⋊C8—262no—
407C4⋊C8—262no—
408C22⋊C8—262no—
409C22⋊C8—262no—
410C22⋊C8—262no—
411C22⋊C8—262no—
412#97—262no—
413C4×C8—262no—
414C4×C8—262no—
415C4×C8—262no—
416C2×C13—22no—
417C2×C13—22no—
418C2×C13—41no—
419C2×C13C2×#41311yescharacteristic
420C2×C13C2×#41611yescharacteristic
421C2×C13C2×#41911yescharacteristic
422C24C22⋊#26111yescharacteristic
423C2×Q8—21no—
424C2×Q8—42no—
425C2×Q8—132no—
426C2×Q8—262no—
427C2×Q8—262no—
428C2×Q8—261no—
429C2×Q8—522no—
430C2×Q8—524no—
431C2×Q8C22×#26112yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—131no—
436C22×C4—131no—
437C22×C4—131no—
438C22×C4—131no—
439C22×C4—131no—
440C22×C4—131no—
441C22×C4—131no—
442C22×C4—131no—
443C22×C4—131no—
444C22×C4—131no—
445C22×C4—131no—
446C22×C4—131no—
447C22×C4—264no—
448C22×C4—264no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4C22⋊#26111yescharacteristic
452C22×C4C22⋊#26111yescharacteristic
453C22×C4C22⋊#26111yescharacteristic
454C22×C4C22×#26111yescharacteristic
455C22×C4C22×#26111yescharacteristic
456C22×C4C22×#26111yescharacteristic
457M16—524no—
458C2×C8—264no—
459C2×C8—264no—
460C2×C8—524no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—132no—
465C4⋊C4—262no—
466C4⋊C4—262no—
467C4⋊C4—262no—
468C4⋊C4—262no—
469C4⋊C4—262no—
470C4⋊C4—262no—
471C4⋊C4—262no—
472C4⋊C4—262no—
473C4⋊C4—262no—
474C4⋊C4—262no—
475C4⋊C4—261no—
476C4⋊C4—262no—
477C4⋊C4—262no—
478C4⋊C4C22×#26112yes—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—262no—
482C22⋊C4—262no—
483C42—131no—
484C42—131no—
485C42—131no—
486C42—131no—
487C42—131no—
488C42—131no—
489C42—131no—
490C42—131no—
491C42—131no—
492C42—131no—
493C42—131no—
494C42—131no—
495C42—131no—
496C42—131no—
497C42—131no—
498C42—131no—
499C42—262no—
500C42—262no—
501C42—262no—
502C42—262no—
503C13#1771811yescharacteristic, complemented, Sylow
504C23—22no—
505C23—22no—
506C23—41no—
507C23C2×C22⋊#26111yescharacteristic
508C23C2×C22⋊#26111yescharacteristic
509C23C2×C22⋊#26111yescharacteristic
510Q8—42no—
511Q8—42no—
512Q8—522no—
513Q8—524no—
514Q8—524no—
515Q8—522no—
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—42no—
525C2×C4—42no—
526C2×C4—131no—
527C2×C4—131no—
528C2×C4—131no—
529C2×C4—131no—
530C2×C4—134no—
531C2×C4—134no—
532C2×C4—131no—
533C2×C4—131no—
534C2×C4—131no—
535C2×C4—131no—
536C2×C4—262no—
537C2×C4—262no—
538C2×C4—262no—
539C2×C4—262no—
540C2×C4—262no—
541C2×C4—262no—
542C2×C4—262no—
543C2×C4—262no—
544C2×C4—262no—
545C2×C4—262no—
546C2×C4—262no—
547C2×C4—262no—
548C2×C4—522no—
549C2×C4—522no—
550C2×C4C2×C22⋊#26111yescharacteristic
551C2×C4C2×C22⋊#26111yescharacteristic
552C2×C4C2×C22⋊#26111yescharacteristic
553C2×C4C23×#26111yescharacteristic
554C8—524no—
555C8—524no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—44no—
564C22#1171915312yes—
565C22#1171916512yes—
566C22C22×C22⋊#26111yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—42no—
571C4—42no—
572C4—132no—
573C4—132no—
574C4—132no—
575C4—132no—
576C4—261no—
577C4—261no—
578C4—261no—
579C4—261no—
580C4—522no—
581C4—522no—
582C4#1171914812yes—
583C2—22no—
584C2—22no—
585C2—41no—
586C2C2×#1171914811yescharacteristic
587C2C2×#1171916511yescharacteristic
588C2C2×#1171915311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
FittingC2×C13×C22⋊Q8C4
socleC22×C13C22×C22⋊C4
prime radicalC24×C13C22⋊C4
cosocleC4×Dic13C24
hypercenterC2×C22⋊Q8#261
hypocenterC13#17718
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #13130362C2semidirect
4. #13130349C2semidirect
5. #13130345C2semidirect
6. #13130338C2semidirect
41. #11718303C22semidirect
42. #11718245C22semidirect
503. C13#17718semidirect

Maximal subgroups

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

subgroupindex
#131315122
C2×#117191252
#131303622
#131303492
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