Finite Group Databasebeta

← #25130955

Subgroups — #25130955

Order 11008.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583454C211yesmaximal, characteristic
2C2×#14583194C211yesmaximal, characteristic, complemented
3C4×#13063301C211yesmaximal, characteristic
4#19135796C212yesmaximal
5#19135602C212yesmaximal
6#19135542C211yesmaximal, characteristic, complemented
7#19135519C211yesmaximal, characteristic, complemented
8#19134938C212yesmaximal, complemented
9#19134847C212yesmaximal, complemented
10#19134572C211yesmaximal, characteristic
11C2×C43×#462C211yesmaximal, characteristic, complemented
12#14583454—24no—
13#14583454—24no—
14C2×#13063325C2211yescharacteristic
15C2×#13063316—21no—
16C2×#13063305—21no—
17C2×#13063301C2212yes—
18C2×#13063301C2211yescharacteristic
19C2×Q8×Dic43C2212yes—
20C2×#13063290—21no—
21C2×#13063283—21no—
22C2×C42×D43C2211yescharacteristic
23#14583194—24no—
24C2×#13063258C2212yes—
25C2×C4⋊(C43⋊C8)C2211yescharacteristic
26C2×#13063249C2212yes—
27C2×#13063249C2212yes—
28C2×#13063249C2212yes—
29C2×(C2×D43)⋊C8C2212yes—
30#14582806C2211yescharacteristic
31#14582795C2211yescharacteristic
32#14582780—21no—
33#14582770—21no—
34#14582756C2211yescharacteristic
35C4×Dic43⋊C4—21no—
36C4×C4⋊Dic43C2211yescharacteristic
37C4×(C2×D43)⋊C4—21no—
38#14582662C2212yes—
39#14582662C2212yes—
40#14582653C2212yes—
41C22×Q16×C43C2212yes—
42C2×C4⋊Q8×C43C2211yescharacteristic
43C2×C4×Q8×C43C2211yescharacteristic
44C43×#462—24no—
45C2×C4⋊C8×C43C2211yescharacteristic
46C2×#103×C43C2212yes—
47C22×Dic86—21no—
48C22×C4×D43D411yescharacteristic
49C22×C4×D43D411yescharacteristic
50C22×C4×D43C2311yescharacteristic
51C22×C43⋊C8C2312yes—
52#13063325—22no—
53#13063325D414yes—
54#13063316—28no—
55#13063305—28no—
56#13063301—24no—
57#13063301—22no—
58#13063301—24no—
59#13063301—22no—
60Q8×Dic43—24no—
61Q8×Dic43—24no—
62#13063290—28no—
63#13063283—28no—
64C2×C4⋊Dic43—21no—
65C2×C4⋊Dic43—22no—
66C2×C4⋊Dic43C2311yescharacteristic
67C2×C4⋊Dic43C2312yes—
68C2×C4⋊Dic43C2311yescharacteristic
69C2×Dic43⋊C4—22no—
70C2×Dic43⋊C4—21no—
71C2×Dic43⋊C4—21no—
72C2×(C2×D43)⋊C4—22no—
73C2×(C2×D43)⋊C4—21no—
74C2×(C2×D43)⋊C4—21no—
75C2×C4×Dic43—22no—
76C2×C4×Dic43—21no—
77C2×C4×Dic43D411yescharacteristic
78C2×C4×Dic43D411yescharacteristic
79C2×C4×Dic43C2311yescharacteristic
80C42×D43—22no—
81C42×D43—22no—
82#13063258—24no—
83C4⋊(C43⋊C8)—22no—
84#13063249—24no—
85#13063249—24no—
86#13063249—42no—
87(C2×D43)⋊C8—42no—
88C43⋊#95—21no—
89C43⋊#95—21no—
90C43⋊#95—21no—
91C43⋊#95—21no—
92C43⋊#95—21no—
93C22×Q8×C43C2312yes—
94C22×Q8×C43C2311yescharacteristic
95C2×Q16×C43—24no—
96C2×Q16×C43—24no—
97C2×Q16×C43—42no—
98C22×C8×C43C2312yes—
99C4⋊Q8×C43—22no—
100C4⋊Q8×C43D414yes—
101C4×Q8×C43—22no—
102C4×Q8×C43—22no—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—21no—
105C2×C4⋊C4×C43—21no—
106C2×C4⋊C4×C43C2311yescharacteristic
107C2×C42×C43—21no—
108C2×C42×C43C2311yescharacteristic
109C4⋊C8×C43—22no—
110#103×C43—24no—
111C23×D43C2×D411yescharacteristic
112C2×Q8×C43—22no—
113C2×Q8×C43—21no—
114C2×Q8×C43—21no—
115C2×Q8×C43—21no—
116C2×Q8×C43—22no—
117C2×Q8×C43—22no—
118C2×Q8×C43—21no—
119C2×Q8×C43—22no—
120C2×Q8×C43—22no—
121C2×Q8×C43—21no—
122C2×Q8×C43C2×D414yes—
123C2×Q8×C43#4112yes—
124C22×C4×C43—21no—
125C22×C4×C43—22no—
126C22×C4×C43—21no—
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2411yescharacteristic
130C22×Dic43—21no—
131C22×Dic43—22no—
132C22×Dic43—21no—
133C22×Dic43C2×D411yescharacteristic
134C22×Dic43C2×D411yescharacteristic
135C22×Dic43C2×D411yescharacteristic
136C2×C4×D43—22no—
137C2×C4×D43—22no—
138C2×C4×D43—22no—
139C2×C4×D43—21no—
140C2×C4×D43—21no—
141C2×C4×D43—21no—
142C2×C4×D43—21no—
143C2×C4×D43—21no—
144C2×C4×D43—21no—
145C2×C4×D43—41no—
146C2×C4×D43—41no—
147C2×C4×D43Q1614yescomplemented
148C2×Dic86—24no—
149C2×Dic86—42no—
150C2×Dic86—42no—
151Q16×C43—48no—
152C2×C8×C43—22no—
153C2×C8×C43—22no—
154C2×C8×C43—22no—
155C4⋊C4×C43—24no—
156C4⋊C4×C43—24no—
157C4⋊C4×C43—21no—
158C4⋊C4×C43—21no—
159C4⋊C4×C43—42no—
160C42×C43—42no—
161C42×C43C2×D412yes—
162C42×C43C2×D412yes—
163(C2×D43)⋊C4—28no—
164(C2×D43)⋊C4—24no—
165(C2×D43)⋊C4—42no—
166C4⋊Dic43—28no—
167C4⋊Dic43—24no—
168C4⋊Dic43—22no—
169C4⋊Dic43—21no—
170C4⋊Dic43—21no—
171C4⋊Dic43—21no—
172C4⋊Dic43C2×D414yes—
173C4⋊Dic43#4112yes—
174Dic43⋊C4—28no—
175Dic43⋊C4—24no—
176Dic43⋊C4—42no—
177C4×Dic43—28no—
178C4×Dic43—22no—
179C4×Dic43—22no—
180C4×Dic43—21no—
181C4×Dic43—21no—
182C4×Dic43—42no—
183C2×C43⋊C8—22no—
184C2×C43⋊C8—22no—
185C2×C43⋊C8—22no—
186C23×C43C22×D411yescharacteristic
187C22×D43—21no—
188C22×D43—21no—
189C22×D43—21no—
190C22×D43—21no—
191C22×D43—21no—
192C22×D43—21no—
193C22×D43C2×Q1612yes—
194Q8×C43—22no—
195Q8×C43—24no—
196Q8×C43—24no—
197Q8×C43—22no—
198Q8×C43—41no—
199Q8×C43—42no—
200Q8×C43—42no—
201Q8×C43—41no—
202C2×C4×C43—21no—
203C2×C4×C43—21no—
204C2×C4×C43—24no—
205C2×C4×C43—22no—
206C2×C4×C43—41no—
207C2×C4×C43—42no—
208C2×C4×C43—41no—
209C2×C4×C43—42no—
210C2×C4×C43—41no—
211C2×C4×C43—41no—
212C2×C4×C43—41no—
213C2×C4×C43#12012yes—
214C2×C4×C43#12012yes—
215C2×C4×C43C2×Q1612yes—
216C2×C4×C43#13712yes—
217C2×C4×C43C22×D411yescharacteristic
218C2×C4×C43C22×D411yescharacteristic
219C2×C4×C43C2×#4111yescharacteristic
220C2×C4×C43C2×#4111yescharacteristic
221C2×C4×C43H(5,𝔽2)11yescharacteristic
222C2×C4×C432−1+411yescharacteristic
223C2×Dic43—21no—
224C2×Dic43—21no—
225C2×Dic43—21no—
226C2×Dic43—21no—
227C2×Dic43—24no—
228C2×Dic43—21no—
229C2×Dic43—22no—
230C2×Dic43—21no—
231C2×Dic43—21no—
232C2×Dic43—21no—
233C2×Dic43—41no—
234C2×Dic43—41no—
235C2×Dic43—42no—
236C2×Dic43—42no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43C2×Q1612yes—
241C4×D43—28no—
242C4×D43—44no—
243C4×D43—41no—
244C4×D43—41no—
245C4×D43—41no—
246C4×D43—41no—
247Dic86—48no—
248C8×C43—42no—
249C43⋊C8—42no—
250#24155—431nomaximal, Sylow
251C22×C43C2×#12011yescharacteristic
252C22×C43#52511yescharacteristic
253C22×C43#53711yescharacteristic
254C22×C43C22×Q1611yescharacteristic
255C22×C43C2×#13711yescharacteristic
256C22×C43#57511yescharacteristic
257C22×C43#57511yescharacteristic
258C2×D43—22no—
259C2×D43—22no—
260C2×D43—22no—
261C2×D43—41no—
262C2×D43—41no—
263C2×D43—41no—
264C2×D43—41no—
265C4×C43—22no—
266C4×C43—44no—
267C4×C43—42no—
268C4×C43—81no—
269C4×C43#45114yes—
270C4×C43#54611yescharacteristic
271C4×C43#54711yescharacteristic
272C4×C43#54811yescharacteristic
273C4×C43#55011yescharacteristic
274Dic43—22no—
275Dic43—42no—
276Dic43—41no—
277Dic43—44no—
278Dic43—41no—
279Dic43—81no—
280C2×#533—431no—
281C2×#477—431no—
282C2×#462D4311yescharacteristic, complemented
283C4×C22⋊Q8—431no—
284#2036—432no—
285#2001—432no—
286#1987—431no—
287#1981—431no—
288#1878—432no—
289#1863—432no—
290#1797—431no—
291C2×C43#240711yescharacteristic
292C2×C43C2×#45111yescharacteristic
293C2×C43#300511yescharacteristic
294C2×C43#310411yescharacteristic
295C2×C43#311611yescharacteristic
296C2×C43#324911yescharacteristic
297C2×C43#325211yescharacteristic
298D43—42no—
299C22×Q16C2×D4312yes—
300#533—864no—
301#533—864no—
302C2×C4⋊Q8—431no—
303C2×C4⋊Q8C2×D4311yescharacteristic
304C2×#125—861no—
305C2×C22⋊Q8—432no—
306C2×C22⋊Q8—431no—
307C2×C22⋊Q8—861no—
308C2×C4×Q8—432no—
309C2×C4×Q8—861no—
310C2×C4×Q8C2×D4311yescharacteristic
311C2×#117—861no—
312C22×C42—431no—
313#477—864no—
314#462—24no—
315C2×#107—432no—
316C2×C4⋊C8—431no—
317C2×C4⋊C8C2×D4311yescharacteristic
318C2×#103—432no—
319C2×#103—432no—
320C2×#103—432no—
321C2×#103C2×D4312yes—
322C2×C22⋊C8—432no—
323#391—431no—
324#389—431no—
325#387—861no—
326#385—861no—
327#380—431no—
328C4×C4⋊C4—431no—
329C4×C4⋊C4—861no—
330C4×C22⋊C4—861no—
331#340—432no—
332#340—432no—
333#336—432no—
334C43#2415511yescharacteristic, complemented, Sylow
335C22×Q8—861no—
336C22×Q8C22×D4311yescharacteristic
337C22×Q8C22×D4312yes—
338C23×C4—431no—
339C23×C4—431no—
340C23×C4—431no—
341C2×Q16—24no—
342C2×Q16—24no—
343C2×Q16—42no—
344C22×C8—432no—
345C22×C8C22×D4312yes—
346C4⋊Q8—22no—
347C4⋊Q8—434no—
348C4⋊Q8—862no—
349C4⋊Q8C22⋊D4314yes—
350#125—868no—
351C22⋊Q8—868no—
352C22⋊Q8—864no—
353C22⋊Q8—862no—
354C22⋊Q8—862no—
355C22⋊Q8—864no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—868no—
359C4×Q8—864no—
360C4×Q8—864no—
361#117—868no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—432no—
366C2×C4⋊C4—431no—
367C2×C4⋊C4—431no—
368C2×C4⋊C4—862no—
369C2×C4⋊C4—862no—
370C2×C4⋊C4—861no—
371C2×C4⋊C4—861no—
372C2×C4⋊C4—861no—
373C2×C4⋊C4C22×D4311yescharacteristic
374C2×C22⋊C4—862no—
375C2×C22⋊C4—861no—
376C2×C22⋊C4—861no—
377C2×C42—21no—
378C2×C42—431no—
379C2×C42—431no—
380C2×C42—431no—
381C2×C42—862no—
382C2×C42—862no—
383C2×C42—862no—
384C2×C42—861no—
385C2×C42C22×D4311yescharacteristic
386#107—864no—
387C4⋊C8—22no—
388C4⋊C8—862no—
389#103—24no—
390#103—864no—
391#103—864no—
392#103—1722no—
393C22⋊C8—1722no—
394#95—861no—
395#95—861no—
396#95—861no—
397#95—861no—
398#95—861no—
399C24—431no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—864no—
411C2×Q8—1722no—
412C2×Q8—1722no—
413C2×Q8#1062190012yes—
414C2×Q8C2×C22⋊D4314yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—431no—
419C22×C4—434no—
420C22×C4—431no—
421C22×C4—431no—
422C22×C4—862no—
423C22×C4—862no—
424C22×C4—861no—
425C22×C4—862no—
426C22×C4—862no—
427C22×C4—861no—
428C22×C4—861no—
429C22×C4—861no—
430C22×C4—861no—
431C22×C4—861no—
432C22×C4—861no—
433C22×C4—861no—
434C22×C4—1721no—
435C22×C4—1721no—
436C22×C4D4×D4311yescharacteristic
437C22×C4D4×D4311yescharacteristic
438C22×C4C23×D4311yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—862no—
444C2×C8—862no—
445C2×C8—862no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—434no—
452C4⋊C4—432no—
453C4⋊C4—868no—
454C4⋊C4—868no—
455C4⋊C4—864no—
456C4⋊C4—864no—
457C4⋊C4—862no—
458C4⋊C4—861no—
459C4⋊C4—861no—
460C4⋊C4—861no—
461C4⋊C4—1722no—
462C22⋊C4—868no—
463C22⋊C4—864no—
464C22⋊C4—1722no—
465C42—42no—
466C42—868no—
467C42—862no—
468C42—862no—
469C42—861no—
470C42—861no—
471C42—1722no—
472C42D4×D4312yes—
473C42C2×C22⋊D4312yes—
474C23—432no—
475C23—861no—
476C23—861no—
477C23—861no—
478C23—861no—
479C23—861no—
480C23—861no—
481C23C2×D4×D4311yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—24no—
485Q8—24no—
486Q8—42no—
487Q8—42no—
488Q8—41no—
489Q8—41no—
490Q8—1728no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—432no—
503C2×C4—861no—
504C2×C4—861no—
505C2×C4—868no—
506C2×C4—861no—
507C2×C4—861no—
508C2×C4—864no—
509C2×C4—862no—
510C2×C4—861no—
511C2×C4—861no—
512C2×C4—861no—
513C2×C4—861no—
514C2×C4—1724no—
515C2×C4—1721no—
516C2×C4—1721no—
517C2×C4—1721no—
518C2×C4—1721no—
519C2×C4—1721no—
520C2×C4—1722no—
521C2×C4—1722no—
522C2×C4—1721no—
523C2×C4—1721no—
524C2×C4—1721no—
525C2×C4—1721no—
526C2×C4#1306329312yes—
527C2×C4#1306329312yes—
528C2×C4Q16×D4312yes—
529C2×C4#1306335712yes—
530C2×C4C2×D4×D4311yescharacteristic
531C2×C4C22×C22⋊D4311yescharacteristic
532C2×C4#41×D4311yescharacteristic
533C2×C4C2×#1062190011yescharacteristic
534C2×C4#1306337011yescharacteristic
535C2×C4#1306337211yescharacteristic
536C8—42no—
537C8—1722no—
538C22—862no—
539C22—862no—
540C22—862no—
541C22—1721no—
542C22—1721no—
543C22—1721no—
544C22—1721no—
545C22C2×#1306329311yescharacteristic
546C22#1458340911yescharacteristic
547C22#1458347811yescharacteristic
548C22C2×Q16×D4311yescharacteristic
549C22C2×#1306335711yescharacteristic
550C22#1458369811yescharacteristic
551C22#1458369811yescharacteristic
552C4—22no—
553C4—44no—
554C4—42no—
555C4—81no—
556C4—862no—
557C4—3441no—
558C4—1722no—
559C4—1721no—
560C4—1721no—
561C4—1724no—
562C4#1458305314yes—
563C4#1458354311yescharacteristic
564C4#1458354811yescharacteristic
565C4#1458355511yescharacteristic
566C4#1458356711yescharacteristic
567C2—1722no—
568C2#1913793611yescharacteristic
569C2C2×#1458305311yescharacteristic
570C2#1914383611yescharacteristic
571C2#1914463311yescharacteristic
572C2#1914475611yescharacteristic
573C2#1914593811yescharacteristic
574C2#1914596611yescharacteristic
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#462C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenterC2×#462D43
hypocenterC43#24155
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
2. C2×#14583194C2semidirect
6. #19135542C2semidirect
7. #19135519C2semidirect
8. #19134938C2semidirect
9. #19134847C2semidirect
11. C2×C43×#462C2semidirect
147. C2×C4×D43Q16semidirect
282. C2×#462D43semidirect
334. C43#24155semidirect

Maximal subgroups

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

subgroupindex
C2×#145834542
C2×#145831942
C4×#130633012
#191357962
#191356022
#191355422
#191355192
#191349382
#191348472
#191345722
C2×C43×#4622
#2415543