Finite Group Databasebeta

← #25132159

Subgroups — #25132159

Order 11008.

1392 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583564C211yesmaximal, characteristic
2C2×#14583053C212yesmaximal, complemented
3C2×#14583004C211yesmaximal, characteristic, complemented
4#19139500C211yesmaximal, characteristic
5#19135789C212yesmaximal, complemented
6#19135602C212yesmaximal, complemented
7#19135416C211yesmaximal, characteristic, complemented
8#19135392C211yesmaximal, characteristic
9#19134611C211yesmaximal, characteristic, complemented
10#19134528C212yesmaximal, complemented
11C2×C43×#462C211yesmaximal, characteristic, complemented
12C22×Q8×D43C2212yes—
13C22×#10621885C2211yescharacteristic
14#14583564—24no—
15#14583564—24no—
16C2×#13063324—21no—
17C2×#13063303—22no—
18C2×#13063301C2212yes—
19C2×#13063301C2212yes—
20C2×C4×Dic86C2211yescharacteristic
21C2×C4×D172C2211yescharacteristic
22C2×#13063287—21no—
23#14583053—28no—
24#14583004—24no—
25C2×#13063256C2211yescharacteristic
26C2×#13063250C2212yes—
27C2×#13063249C2212yes—
28C2×#13063249C2211yescharacteristic
29C2×#13063249C2212yes—
30C2×#13063248C2211yescharacteristic
31C2×(C2×D43)⋊C8C2212yes—
32C2×(C2×D43)⋊C8C2212yes—
33C2×#13063236C2211yescharacteristic
34#14582829—21no—
35#14582816—21no—
36#14582790—22no—
37#14582765C2211yescharacteristic
38#14582756C2212yes—
39#14582662C2212yes—
40C22×Q16×C43C2212yes—
41C2×C4⋊Q8×C43C2211yescharacteristic
42C2×C4×Q8×C43C2211yescharacteristic
43C43×#462—24no—
44C2×C4⋊C8×C43C2211yescharacteristic
45C2×#103×C43C2212yes—
46C22×Dic86—22no—
47C22×Dic86C2311yescharacteristic
48C2×Q8×D43—24no—
49C2×Q8×D43—24no—
50C2×Q8×D43—24no—
51C2×Q8×D43—24no—
52C2×Q8×D43—42no—
53C2×Q8×D43—42no—
54C22×D172D411yescharacteristic
55C22×C4×D43—22no—
56C22×C4×D43C2312yes—
57C2×#10621885—22no—
58C2×#10621885—22no—
59C2×#10621885D414yescomplemented
60C22×C43⋊C8C2311yescharacteristic
61C22×C43⋊C8C2311yescharacteristic
62#13063324—28no—
63#13063303—216no—
64#13063301—24no—
65#13063301—24no—
66#13063301—24no—
67#13063301—42no—
68C4×Dic86—22no—
69C4×Dic86D414yes—
70C4×D172—22no—
71C4×D172—22no—
72#13063287—28no—
73C2×C4⋊Dic43D411yescharacteristic
74C2×C4⋊Dic43C2312yes—
75C2×C4⋊Dic43C2311yescharacteristic
76C2×Dic43⋊C4—21no—
77C2×Dic43⋊C4—22no—
78C2×Dic43⋊C4—22no—
79C2×(C2×D43)⋊C4—22no—
80C2×(C2×D43)⋊C4—22no—
81C2×(C2×D43)⋊C4—22no—
82C2×(C2×D43)⋊C4D412yes—
83C2×C4×Dic43—21no—
84#13063256—22no—
85#13063250—24no—
86#13063249—22no—
87#13063249—24no—
88#13063249—42no—
89#13063248—22no—
90(C2×D43)⋊C8—24no—
91(C2×D43)⋊C8—42no—
92#13063236—22no—
93C43⋊#95—21no—
94C43⋊#95—21no—
95C43⋊#95—22no—
96C22×Q8×C43C2311yescharacteristic
97C22×Q8×C43C2312yes—
98C2×Q16×C43—24no—
99C2×Q16×C43—24no—
100C2×Q16×C43—42no—
101C22×C8×C43C2312yes—
102C4⋊Q8×C43—22no—
103C4⋊Q8×C43D414yes—
104C4×Q8×C43—22no—
105C4×Q8×C43—22no—
106C2×C4⋊C4×C43—21no—
107C2×C4⋊C4×C43—21no—
108C2×C4⋊C4×C43—21no—
109C2×C4⋊C4×C43C2311yescharacteristic
110C2×C42×C43—21no—
111C2×C42×C43C2311yescharacteristic
112C4⋊C8×C43—22no—
113#103×C43—24no—
114C23×D43C2×D412yes—
115C2×Q8×C43—22no—
116C2×Q8×C43—21no—
117C2×Q8×C43—21no—
118C2×Q8×C43—21no—
119C2×Q8×C43—22no—
120C2×Q8×C43—21no—
121C2×Q8×C43—22no—
122C2×Q8×C43—21no—
123C2×Q8×C43—22no—
124C2×Q8×C43—22no—
125C2×Q8×C43C2×D412yes—
126C2×Q8×C43C2×D414yes—
127C22×C4×C43—21no—
128C22×C4×C43—22no—
129C22×C4×C43—21no—
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2411yescharacteristic
133C22×Dic43—21no—
134C22×Dic43—21no—
135C22×Dic43—22no—
136C22×Dic43C2×D412yes—
137Q8×D43—48no—
138Q8×D43—48no—
139Q8×D43—48no—
140Q8×D43—48no—
141C2×D172—22no—
142C2×D172—24no—
143C2×C4×D43—28no—
144C2×C4×D43—22no—
145C2×C4×D43—22no—
146C2×C4×D43—22no—
147C2×C4×D43—22no—
148C2×C4×D43—22no—
149C2×C4×D43—22no—
150C2×C4×D43—44no—
151C2×C4×D43—44no—
152C2×Dic86—21no—
153C2×Dic86—21no—
154C2×Dic86—21no—
155C2×Dic86—28no—
156C2×Dic86—44no—
157C2×Dic86—44no—
158C2×Dic86C2×D412yes—
159C2×Dic86C2×D412yes—
160C2×Dic86#4112yes—
161Q16×C43—48no—
162C2×C8×C43—22no—
163C2×C8×C43—22no—
164C2×C8×C43—22no—
165C4⋊C4×C43—24no—
166C4⋊C4×C43—24no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—42no—
170C42×C43—42no—
171C42×C43C2×D412yes—
172C42×C43C2×D412yes—
173#10621885—28no—
174(C2×D43)⋊C4—28no—
175(C2×D43)⋊C4—24no—
176(C2×D43)⋊C4—44no—
177(C2×D43)⋊C4—44no—
178C4⋊Dic43—22no—
179C4⋊Dic43—21no—
180C4⋊Dic43—22no—
181C4⋊Dic43—22no—
182C4⋊Dic43C2×D412yes—
183Dic43⋊C4—28no—
184Dic43⋊C4—24no—
185Dic43⋊C4—44no—
186C4×Dic43—24no—
187C2×C43⋊C8—21no—
188C2×C43⋊C8—21no—
189C2×C43⋊C8—21no—
190C2×C43⋊C8—21no—
191C2×C43⋊C8—21no—
192C2×C43⋊C8C2×D412yes—
193C23×C43C22×D411yescharacteristic
194C22×D43—22no—
195C22×D43—22no—
196C22×D43—22no—
197C22×D43—22no—
198C22×D43—22no—
199C22×D43—22no—
200C22×D43#13714yes—
201Q8×C43—22no—
202Q8×C43—22no—
203Q8×C43—24no—
204Q8×C43—24no—
205Q8×C43—42no—
206Q8×C43—42no—
207Q8×C43—41no—
208Q8×C43—41no—
209C2×C4×C43—22no—
210C2×C4×C43—22no—
211C2×C4×C43—24no—
212C2×C4×C43—22no—
213C2×C4×C43—41no—
214C2×C4×C43—42no—
215C2×C4×C43—42no—
216C2×C4×C43—41no—
217C2×C4×C43—41no—
218C2×C4×C43—42no—
219C2×C4×C43#12012yes—
220C2×C4×C43#12012yes—
221C2×C4×C43C22×D411yescharacteristic
222C2×C4×C43C22×D411yescharacteristic
223C2×C4×C43C22×D411yescharacteristic
224C2×C4×C43C2×#4111yescharacteristic
225C2×C4×C43H(5,𝔽2)11yescharacteristic
226C2×C4×C43H(5,𝔽2)11yescharacteristic
227C2×Dic43—22no—
228C2×Dic43—22no—
229C2×Dic43—22no—
230C2×Dic43—22no—
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—22no—
234C2×Dic43—42no—
235C2×Dic43—42no—
236C2×Dic43—42no—
237C2×Dic43—42no—
238D172—28no—
239C4×D43—416no—
240C4×D43—42no—
241C4×D43—42no—
242C4×D43—42no—
243C4×D43—42no—
244Dic86—22no—
245Dic86—22no—
246Dic86—22no—
247Dic86—416no—
248Dic86#12114yes—
249C8×C43—42no—
250C43⋊C8—22no—
251C43⋊C8—41no—
252#24255—431nomaximal, Sylow
253C22×C43C2×#12011yescharacteristic
254C22×C43D4211yescharacteristic
255C22×C43#54611yescharacteristic
256C22×C43C2×#13712yes—
257C22×C43#57812yes—
258C2×D43—24no—
259C2×D43—24no—
260C2×D43—24no—
261C2×D43—42no—
262C2×D43—42no—
263C2×D43—42no—
264C2×D43—42no—
265C4×C43—22no—
266C4×C43—22no—
267C4×C43—44no—
268C4×C43—42no—
269C4×C43—81no—
270C4×C43C2×#12111yescharacteristic
271C4×C43#53511yescharacteristic
272C4×C43D4211yescharacteristic
273C4×C43#54611yescharacteristic
274Dic43—24no—
275Dic43—42no—
276Dic43—42no—
277Dic43—82no—
278C2×D4×Q8—431no—
279C2×#462D4311yescharacteristic, complemented
280C2×#451—432no—
281C2×#441—431no—
282#2536—431no—
283#2035—432no—
284#2001—432no—
285#1960—431no—
286#1955—431no—
287#1805—431no—
288#1787—432no—
289C2×C43#240811yescharacteristic
290C2×C43#300411yescharacteristic
291C2×C43#301111yescharacteristic
292C2×C43#327012yes—
293C2×C43#331212yes—
294D43—44no—
295C23×Q8—432no—
296C22×Q16—431no—
297C22×Q16C2×D4312yes—
298D4×Q8—864no—
299D4×Q8—864no—
300C2×C4⋊Q8—861no—
301C2×C4⋊Q8C2×D4311yescharacteristic
302C2×C22⋊Q8—432no—
303C2×C22⋊Q8—432no—
304C2×C22⋊Q8—862no—
305C2×C4×Q8—431no—
306C2×C4×Q8C2×D4311yescharacteristic
307C2×C4×D4—431no—
308C2×C4×D4—861no—
309#462—24no—
310#451—868no—
311#441—864no—
312C2×#106—431no—
313C2×C4⋊C8C2×D4311yescharacteristic
314C2×#103—431no—
315C2×#103—432no—
316C2×#103—432no—
317C2×#103—431no—
318C2×#103—432no—
319C2×#103C2×D4312yes—
320C2×C22⋊C8—432no—
321C2×C22⋊C8—432no—
322C2×#97—431no—
323#398—861no—
324#394—861no—
325#388—862no—
326#382—431no—
327#380—432no—
328#340—432no—
329C43#2425511yescharacteristic, complemented, Sylow
330C22×Q8—431no—
331C22×Q8—862no—
332C22×Q8—864no—
333C22×Q8—864no—
334C22×Q8—864no—
335C22×Q8—864no—
336C22×Q8—1722no—
337C22×Q8—1722no—
338C22×Q8C22×D4311yescharacteristic
339C22×Q8C22×D4312yes—
340C22×D4—431no—
341C23×C4—432no—
342C23×C4—862no—
343C2×Q16—24no—
344C2×Q16—24no—
345C2×Q16—42no—
346C2×Q16—434no—
347C2×Q16—862no—
348C2×Q16—862no—
349C22×C8—431no—
350C22×C8—431no—
351C22×C8C22×D4312yes—
352C4⋊Q8—22no—
353C4⋊Q8—868no—
354C4⋊Q8C22⋊D4314yes—
355C22⋊Q8—8616no—
356C22⋊Q8—864no—
357C22⋊Q8—864no—
358C22⋊Q8—864no—
359C22⋊Q8—1722no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—434no—
363C4×Q8—862no—
364C4×D4—868no—
365C4×D4—862no—
366C4×D4—862no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—431no—
371C2×C4⋊C4—431no—
372C2×C4⋊C4—432no—
373C2×C4⋊C4—862no—
374C2×C4⋊C4—861no—
375C2×C4⋊C4—862no—
376C2×C4⋊C4C22×D4311yescharacteristic
377C2×C22⋊C4—432no—
378C2×C22⋊C4—862no—
379C2×C22⋊C4—862no—
380C2×C22⋊C4—862no—
381C2×C42—21no—
382C2×C42—861no—
383C2×C42C22×D4311yescharacteristic
384#106—862no—
385C4⋊C8—22no—
386#103—24no—
387#103—862no—
388#103—862no—
389#103—864no—
390#103—864no—
391#103—1722no—
392C22⋊C8—864no—
393C22⋊C8—1722no—
394#97—862no—
395#95—862no—
396#95—861no—
397#95—861no—
398C24—432no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—432no—
410C2×Q8—432no—
411C2×Q8—432no—
412C2×Q8—868no—
413C2×Q8—861no—
414C2×Q8—861no—
415C2×Q8—861no—
416C2×Q8—1724no—
417C2×Q8—1724no—
418C2×Q8—1728no—
419C2×Q8—1728no—
420C2×Q8—1728no—
421C2×Q8—1728no—
422C2×Q8D4×D4312yes—
423C2×Q8C2×C22⋊D4314yes—
424C2×D4—864no—
425C2×D4—862no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—432no—
430C22×C4—868no—
431C22×C4—861no—
432C22×C4—861no—
433C22×C4—862no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—862no—
437C22×C4—862no—
438C22×C4—862no—
439C22×C4—862no—
440C22×C4—1724no—
441C22×C4—1724no—
442C22×C4D4×D4311yescharacteristic
443C22×C4D4×D4311yescharacteristic
444C22×C4C23×D4311yescharacteristic
445Q16—48no—
446Q16—868no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—432no—
451C2×C8—861no—
452C2×C8—861no—
453C2×C8—861no—
454C2×C8—861no—
455C2×C8—861no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—432no—
462C4⋊C4—868no—
463C4⋊C4—862no—
464C4⋊C4—864no—
465C4⋊C4—861no—
466C4⋊C4—862no—
467C4⋊C4—862no—
468C4⋊C4—1724no—
469C22⋊C4—868no—
470C22⋊C4—864no—
471C22⋊C4—1724no—
472C22⋊C4—1724no—
473C42—42no—
474C42—864no—
475C42D4×D4312yes—
476C42C2×C22⋊D4312yes—
477C23—434no—
478C23—862no—
479C23—862no—
480C23—862no—
481C23—862no—
482C23—862no—
483C23—862no—
484C23C2×D4×D4311yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—24no—
488Q8—24no—
489Q8—42no—
490Q8—42no—
491Q8—41no—
492Q8—41no—
493Q8—434no—
494Q8—862no—
495Q8—862no—
496Q8—862no—
497Q8—17216no—
498D4—868no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—862no—
510C2×C4—862no—
511C2×C4—862no—
512C2×C4—862no—
513C2×C4—862no—
514C2×C4—862no—
515C2×C4—862no—
516C2×C4—17216no—
517C2×C4—1722no—
518C2×C4—1722no—
519C2×C4—1722no—
520C2×C4—1722no—
521C2×C4—1722no—
522C2×C4—1722no—
523C2×C4—1722no—
524C2×C4—1722no—
525C2×C4#1306329312yes—
526C2×C4#1306329312yes—
527C2×C4C2×D4×D4311yescharacteristic
528C2×C4C2×D4×D4311yescharacteristic
529C2×C4C22×C22⋊D4311yescharacteristic
530C2×C4#41×D4311yescharacteristic
531C2×C4#1306337011yescharacteristic
532C2×C4#1306337111yescharacteristic
533C8—42no—
534C8—862no—
535C8—1721no—
536C22—864no—
537C22—864no—
538C22—864no—
539C22—1722no—
540C22—1722no—
541C22—1722no—
542C22—1722no—
543C22C2×#1306329311yescharacteristic
544C22#1458353311yescharacteristic
545C22#1458354611yescharacteristic
546C22#137×D4312yes—
547C22#1458371212yes—
548C4—22no—
549C4—22no—
550C4—44no—
551C4—42no—
552C4—81no—
553C4—864no—
554C4—3442no—
555C4—1722no—
556C4—1722no—
557C4#121×D4311yescharacteristic
558C4#1458346511yescharacteristic
559C4D4×C22⋊D4311yescharacteristic
560C4#1458353711yescharacteristic
561C2—1724no—
562C2#1913794711yescharacteristic
563C2#1914382711yescharacteristic
564C2#1914388611yescharacteristic
565C2#1914617112yes—
566C2#1914665312yes—
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#462C2
socleC23×C43C22×D4
prime radicalC22×D172D4
hypercenterC2×#462D43
hypocenterC43#24255
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×#14583053C2semidirect
3. C2×#14583004C2semidirect
5. #19135789C2semidirect
6. #19135602C2semidirect
7. #19135416C2semidirect
9. #19134611C2semidirect
10. #19134528C2semidirect
11. C2×C43×#462C2semidirect
59. C2×#10621885D4semidirect
279. C2×#462D43semidirect
329. C43#24255semidirect

Maximal subgroups

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

subgroupindex
C2×#145835642
C2×#145830532
C2×#145830042
#191395002
#191357892
#191356022
#191354162
#191353922
#191346112
#191345282
C2×C43×#4622
#2425543