Finite Group Databasebeta

← #12129349

Subgroups — #12129349

Order 1280.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×C22⋊Q8C211yesmaximal, characteristic
2#10613791C211yesmaximal, characteristic
3#10602488C211yesmaximal, characteristic
4#10602487C211yesmaximal, characteristic
5C4×#3967C211yesmaximal, characteristic, complemented
6C4×#3967C211yesmaximal, characteristic, complemented
7C4×C22⋊#168C211yesmaximal, characteristic
8#10600479C211yesmaximal, characteristic, complemented
9#10600464C211yesmaximal, characteristic, complemented
10#10600406C211yesmaximal, characteristic, complemented
11#10600402C211yesmaximal, characteristic, complemented
12#10600313C211yesmaximal, characteristic
13#10600311C211yesmaximal, characteristic
14C4×#3939C211yesmaximal, characteristic, complemented
15C4×#3940C211yesmaximal, characteristic, complemented
16C2×C5×C22⋊Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C22×C42×C5C2211yescharacteristic
19C5×#391C2211yescharacteristic
20C5×#389C2211yescharacteristic
21C5×#387—21no—
22C5×#385—21no—
23C5×#380C2211yescharacteristic
24C4×C5×C4⋊C4—21no—
25C4×C5×C4⋊C4C2211yescharacteristic
26C4×C5×C22⋊C4—21no—
27C2×#3967C2211yescharacteristic
28C2×#3967C2211yescharacteristic
29C2×#3967C2211yescharacteristic
30C2×#3967C2211yescharacteristic
31#65501C2211yescharacteristic
32#65501C2211yescharacteristic
33#65500—21no—
34C4×C22⋊Dic5—21no—
35C2×C22⋊#168C2211yescharacteristic
36C2×C22⋊#168C2211yescharacteristic
37C2×#3940C2211yescharacteristic
38C2×#3940C2211yescharacteristic
39C2×#3939C2211yescharacteristic
40C2×#3939C2211yescharacteristic
41#65226—21no—
42#65225C2211yescharacteristic
43#65225C2211yescharacteristic
44C4×C4⋊Dic5C2211yescharacteristic
45C4×C4⋊Dic5C2211yescharacteristic
46#65221—21no—
47C2×C4×#168C2211yescharacteristic
48C2×C4×#168C2211yescharacteristic
49#64749C2211yescharacteristic
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#23759—51nomaximal, Sylow
60C22×C5×Q8C2311yescharacteristic
61C23×C4×C5D411yescharacteristic
62C23×C4×C5D411yescharacteristic
63C23×C4×C5C2311yescharacteristic
64C5×C22⋊Q8—22no—
65C5×C22⋊Q8—22no—
66C4×C5×Q8—22no—
67C4×C5×Q8—22no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C22⋊C4—21no—
75C2×C5×C22⋊C4—21no—
76C2×C42×C5—22no—
77C2×C42×C5—22no—
78C2×C42×C5—21no—
79C2×C42×C5—21no—
80C2×C42×C5D411yescharacteristic
81C2×C42×C5D411yescharacteristic
82C2×C42×C5C2311yescharacteristic
83C2×C42×C5C2311yescharacteristic
84C2×C22⋊Dic5—21no—
85C2×C22⋊Dic5—21no—
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89C2×C4⋊Dic5C2311yescharacteristic
90C2×C4×Dic5—21no—
91C2×C4×Dic5—21no—
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96C5×#95—21no—
97C5×#95—21no—
98C5×#95—21no—
99C5×#95—21no—
100C5×#95—21no—
101#3967—22no—
102#3967—22no—
103#3967—22no—
104#3967—22no—
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113C22⋊#168—22no—
114C22⋊#168—22no—
115#3940—22no—
116#3940—22no—
117#3939—22no—
118#3939—22no—
119C4×#168—22no—
120C4×#168—22no—
121C4×C22⋊Q8D511yescharacteristic
122#2283—51no—
123#2101—51no—
124#2097—51no—
125#2085—51no—
126#2077—51no—
127#2075—51no—
128#2074—51no—
129#2036—51no—
130#2035—51no—
131C4×#107—51no—
132C4×#106—51no—
133C4×#103—51no—
134C4×#103—51no—
135C4×C22⋊C8—51no—
136C24×C5C2×D411yescharacteristic
137C2×C5×Q8—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8#4112yes—
143C22×C4×C5—22no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—41no—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5—21no—
169C5×C4⋊C4—24no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—42no—
176C5×C4⋊C4—42no—
177C5×C4⋊C4#4112yes—
178C5×C22⋊C4—24no—
179C5×C22⋊C4—42no—
180C42×C5—22no—
181C42×C5—22no—
182C42×C5—21no—
183C42×C5—21no—
184C42×C5—42no—
185C42×C5—42no—
186C42×C5#4112yes—
187C42×C5#4112yes—
188C22⋊Dic5—42no—
189C22⋊Dic5—42no—
190C4⋊Dic5—21no—
191C4⋊Dic5—21no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4×Dic5—42no—
199C4×Dic5—42no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168—21no—
207C2×#168—21no—
208C2×#168—21no—
209C2×#168—21no—
210C2×#168#4112yes—
211C2×#168#4112yes—
212C2×C22⋊Q8C2×D511yescharacteristic
213C2×C4×Q8C2×D511yescharacteristic
214C22×C42C2×D511yescharacteristic
215C2×#107—51no—
216C2×#107—51no—
217C2×#106—51no—
218C2×#106—51no—
219C2×#103—51no—
220C2×#103—51no—
221C2×#103—51no—
222C2×#103—51no—
223C2×C22⋊C8—51no—
224C2×C22⋊C8—51no—
225C2×C4×C8—51no—
226C2×C4×C8—51no—
227#391C2×D511yescharacteristic
228#389C2×D511yescharacteristic
229#387—21no—
230#385—21no—
231#383—101no—
232#382—51no—
233#382—51no—
234#381—101no—
235#380—51no—
236#380—51no—
237#380C2×D511yescharacteristic
238C4×C4⋊C4—21no—
239C4×C4⋊C4—51no—
240C4×C4⋊C4—51no—
241C4×C4⋊C4C2×D511yescharacteristic
242C4×C22⋊C4—21no—
243C4×C22⋊C4—101no—
244#376—101no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#340—51no—
250#340—51no—
251#340—51no—
252#340—51no—
253#336—51no—
254#336—51no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5#13512yes—
262C23×C5C22×D411yescharacteristic
263C5×Q8—22no—
264C5×Q8—22no—
265C5×Q8—41no—
266C5×Q8—41no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—44no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—44no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5#12312yes—
297C2×C4×C5#12312yes—
298C2×C4×C5#13512yes—
299C2×C4×C5#13712yes—
300C2×C4×C5C2×#4111yescharacteristic
301C2×C4×C5C2×#4111yescharacteristic
302C2×C4×C5C2×#4111yescharacteristic
303C2×C4×C5C2×#4111yescharacteristic
304C2×C4×C5C2×#4111yescharacteristic
305C2×C4×C5C2×#4111yescharacteristic
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318#168—22no—
319#168—22no—
320#168—41no—
321#168—41no—
322C22×Q8C22×D511yescharacteristic
323C23×C4C22⋊D511yescharacteristic
324C23×C4C22⋊D511yescharacteristic
325C23×C4C22×D511yescharacteristic
326C22×C8—51no—
327C22×C8—51no—
328C22×C8—51no—
329C22×C8—51no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—51no—
338C2×C4⋊C4—51no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4C22×D511yescharacteristic
342C2×C4⋊C4C22×D511yescharacteristic
343C2×C4⋊C4C22×D511yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—101no—
347C2×C22⋊C4—101no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—101no—
353C2×C42—101no—
354C2×C42C22⋊D511yescharacteristic
355C2×C42C22⋊D511yescharacteristic
356C2×C42C22×D511yescharacteristic
357C2×C42C22×D511yescharacteristic
358#107—102no—
359#107—102no—
360#106—102no—
361#106—102no—
362#103—102no—
363#103—102no—
364#103—102no—
365#103—102no—
366C22⋊C8—102no—
367C22⋊C8—102no—
368C4×C8—102no—
369C4×C8—102no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—101no—
376#95—101no—
377#95—101no—
378#95—101no—
379#95—101no—
380#95—101no—
381#95—101no—
382#95—101no—
383C22×C5—22no—
384C22×C5—22no—
385C22×C5—22no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5—41no—
390C22×C5C2×#12311yescharacteristic
391C22×C5#52511yescharacteristic
392C22×C5#53211yescharacteristic
393C22×C5C2×#13511yescharacteristic
394C22×C5C2×#13511yescharacteristic
395C22×C5C2×#13711yescharacteristic
396C22×C5#57511yescharacteristic
397C4×C5—22no—
398C4×C5—41no—
399C4×C5—41no—
400C4×C5—42no—
401C4×C5—41no—
402C4×C5—42no—
403C4×C5—81no—
404C4×C5—81no—
405C4×C5#48514yes—
406C4×C5#52911yescharacteristic
407C4×C5#52911yescharacteristic
408C4×C5#52911yescharacteristic
409C4×C5#52911yescharacteristic
410Dic5—81no—
411Dic5—81no—
412Dic5—81no—
413Dic5—81no—
414C24C2×C22⋊D511yescharacteristic
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8#70912yes—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—41no—
437C22×C4—101no—
438C22×C4—101no—
439C22×C4—101no—
440C22×C4—101no—
441C22×C4C2×C22⋊D511yescharacteristic
442C22×C4C2×C22⋊D511yescharacteristic
443C22×C4C2×C22⋊D511yescharacteristic
444C22×C4C2×C22⋊D511yescharacteristic
445C22×C4C2×C22⋊D511yescharacteristic
446C22×C4C23×D511yescharacteristic
447C2×C8—52no—
448C2×C8—52no—
449C2×C8—101no—
450C2×C8—101no—
451C2×C8—101no—
452C2×C8—101no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—101no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—101no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—101no—
471C4⋊C4—101no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4#70912yes—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—202no—
479C22⋊C4—202no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—22no—
484C42—42no—
485C42—42no—
486C42—202no—
487C42—202no—
488C42#71112yes—
489C42#71112yes—
490C2×C5—42no—
491C2×C5#239511yescharacteristic
492C2×C5C2×#48511yescharacteristic
493C2×C5#309111yescharacteristic
494C2×C5#310211yescharacteristic
495C2×C5#310811yescharacteristic
496C2×C5#316811yescharacteristic
497C2×C5#317211yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#409612yes—
505C23C22×C22⋊D511yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—201no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—201no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—201no—
549C2×C4—201no—
550C2×C4—201no—
551C2×C4#408112yes—
552C2×C4#408112yes—
553C2×C4#408812yes—
554C2×C4#409612yes—
555C2×C4C2×#70911yescharacteristic
556C2×C4C2×#70911yescharacteristic
557C2×C4C2×#71111yescharacteristic
558C2×C4C2×#71111yescharacteristic
559C2×C4D5×#4111yescharacteristic
560C2×C4D5×#4111yescharacteristic
561C8—102no—
562C8—102no—
563C8—201no—
564C8—201no—
565C5#2375911yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#6612511yescharacteristic
574C22C2×#408111yescharacteristic
575C22C2×#408811yescharacteristic
576C22C2×#409611yescharacteristic
577C22C2×#409611yescharacteristic
578C22#6615611yescharacteristic
579C22#6616611yescharacteristic
580C4—22no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—401no—
589C4—401no—
590C4—401no—
591C4—401no—
592C4#6533414yes—
593C4#6587211yescharacteristic
594C4#6587211yescharacteristic
595C4#6589111yescharacteristic
596C4#6589111yescharacteristic
597C2—42no—
598C2#1061407711yescharacteristic
599C2#1061417411yescharacteristic
600C2#1061425911yescharacteristic
601C2C2×#6533411yescharacteristic
602C2#1061464411yescharacteristic
603C2#1061486511yescharacteristic
604C2#1061507611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC4×C5×C22⋊Q8C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenterC4×C22⋊Q8D5
hypocenterC5#23759
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
5. C4×#3967C2semidirect
6. C4×#3967C2semidirect
8. #10600479C2semidirect
9. #10600464C2semidirect
10. #10600406C2semidirect
11. #10600402C2semidirect
14. C4×#3939C2semidirect
15. C4×#3940C2semidirect
565. C5#23759semidirect

Maximal subgroups

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

subgroupindex
C4×C5×C22⋊Q82
#106137912
#106024882
#106024872
C4×#39672
C4×C22⋊#1682
#106004792
#106004642
#106004062
#106004022
#106003132
#106003112
C4×#39392
C4×#39402
#237595