Finite Group Databasebeta

← #25108793

Subgroups — #25108793

Order 11008.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583186C211yesmaximal, characteristic
2#19137239C211yesmaximal, characteristic
3#19135548C211yesmaximal, characteristic, complemented
4#19135361C211yesmaximal, characteristic, complemented
5#19135335C211yesmaximal, characteristic, complemented
6#19135320C211yesmaximal, characteristic, complemented
7#19135291C211yesmaximal, characteristic
8#19135277C211yesmaximal, characteristic
9#19135222C211yesmaximal, characteristic
10#19135215C211yesmaximal, characteristic
11#19135180C211yesmaximal, characteristic, complemented
12#19135173C211yesmaximal, characteristic, complemented
13#19134602C211yesmaximal, characteristic
14#19134452C211yesmaximal, characteristic, complemented
15C43×#2331C211yesmaximal, characteristic, complemented
16C2×C4×D172C2211yescharacteristic
17#14583186C418yescomplemented
18C2×#13063258C2211yescharacteristic
19C2×#13063256C2211yescharacteristic
20C2×#13063256C2211yescharacteristic
21C2×#13063256C2211yescharacteristic
22C2×C4⋊(C43⋊C8)C2211yescharacteristic
23C2×C4⋊(C43⋊C8)C2211yescharacteristic
24C2×#13063245C2211yescharacteristic
25C2×#13063245C2211yescharacteristic
26C2×#13063245C2211yescharacteristic
27C2×#13063245C2211yescharacteristic
28C2×#13063236C2211yescharacteristic
29C2×C4×C43⋊C8C2211yescharacteristic
30#14582801C2211yescharacteristic
31#14582801C2211yescharacteristic
32#14582797C2211yescharacteristic
33#14582797C2211yescharacteristic
34#14582788—21no—
35#14582788—21no—
36#14582761—21no—
37#14582758—21no—
38C4×C4⋊Dic43C2211yescharacteristic
39#14582661C2211yescharacteristic
40#14582661C2211yescharacteristic
41#14582661C2211yescharacteristic
42#14582661C2211yescharacteristic
43#14582661C2211yescharacteristic
44#14582661C2211yescharacteristic
45#14582661C2211yescharacteristic
46#14582661C2211yescharacteristic
47#14582654C2211yescharacteristic
48#14582654C2211yescharacteristic
49C2×C4⋊Q8×C43C2211yescharacteristic
50C43×#391—21no—
51C43×#391—21no—
52C43×#389C2211yescharacteristic
53C43×#389C2211yescharacteristic
54C43×#387—21no—
55C43×#387—21no—
56C43×#382C2211yescharacteristic
57C4×C4⋊C4×C43C2211yescharacteristic
58C4×C4⋊C4×C43C2211yescharacteristic
59C43×#376C2211yescharacteristic
60C22×D172C2311yescharacteristic
61C22×C4×D43—21no—
62C22×C43⋊C8C2311yescharacteristic
63C22×C43⋊C8C2311yescharacteristic
64C22×C43⋊C8C2311yescharacteristic
65C22×C43⋊C8C2311yescharacteristic
66C4×D172—22no—
67C4×D172C2×C414yes—
68C2×C4⋊Dic43C2311yescharacteristic
69C2×C4⋊Dic43C2311yescharacteristic
70C2×C4⋊Dic43C2311yescharacteristic
71C2×Dic43⋊C4—21no—
72C2×Dic43⋊C4—21no—
73C2×(C2×D43)⋊C4—21no—
74C2×(C2×D43)⋊C4—21no—
75C2×(C2×D43)⋊C4—21no—
76C2×(C2×D43)⋊C4—21no—
77C2×(C2×D43)⋊C4—21no—
78C2×C4×Dic43—21no—
79#13063258—41no—
80#13063256—41no—
81#13063256C2×C414yes—
82#13063256C2×C414yes—
83C4⋊(C43⋊C8)—22no—
84C4⋊(C43⋊C8)C2×C414yes—
85#13063245—41no—
86#13063245—41no—
87#13063245C2×C414yes—
88#13063245C2×C414yes—
89#13063236—41no—
90C4×C43⋊C8—41no—
91C43⋊#95—21no—
92C43⋊#95—21no—
93C43⋊#95—21no—
94C22×Q8×C43—21no—
95C4⋊Q8×C43—22no—
96C4⋊Q8×C43C2×C414yes—
97C2×C4⋊C4×C43—21no—
98C2×C4⋊C4×C43—21no—
99C2×C4⋊C4×C43—21no—
100C2×C4⋊C4×C43C2311yescharacteristic
101C2×C4⋊C4×C43C2311yescharacteristic
102C2×C4⋊C4×C43C2311yescharacteristic
103C2×C4⋊C4×C43C2311yescharacteristic
104C2×C42×C43—21no—
105C2×C42×C43—21no—
106C2×C42×C43C2311yescharacteristic
107C2×C42×C43C2311yescharacteristic
108C2×C42×C43C2311yescharacteristic
109#95×C43—21no—
110#95×C43—21no—
111#95×C43—21no—
112#95×C43—21no—
113#95×C43—21no—
114#95×C43—21no—
115#95×C43D411yescharacteristic
116#95×C43D411yescharacteristic
117#95×C43D411yescharacteristic
118#95×C43D411yescharacteristic
119C23×D43—21no—
120C2×Q8×C43—24no—
121C2×Q8×C43—81no—
122C2×Q8×C43—81no—
123C22×C4×C43—21no—
124C22×C4×C43—21no—
125C22×C4×C43—21no—
126C22×C4×C43—21no—
127C22×C4×C43C2×D411yescharacteristic
128C22×C4×C43C2×D411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D411yescharacteristic
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2411yescharacteristic
134C22×Dic43—21no—
135C22×Dic43—21no—
136C22×Dic43—21no—
137C2×D172—21no—
138C2×D172—21no—
139C2×D172—21no—
140C2×D172—21no—
141C2×D172—21no—
142C2×D172C22×C412yes—
143C2×C4×D43—24no—
144C2×C4×D43—81no—
145C2×C4×D43—81no—
146C4⋊C4×C43—24no—
147C4⋊C4×C43—21no—
148C4⋊C4×C43—21no—
149C4⋊C4×C43—21no—
150C4⋊C4×C43—21no—
151C4⋊C4×C43—21no—
152C4⋊C4×C43—21no—
153C4⋊C4×C43—81no—
154C4⋊C4×C43—81no—
155C4⋊C4×C43C22×C412yes—
156C4⋊C4×C43C22×C412yes—
157C42×C43—21no—
158C42×C43—21no—
159C42×C43—21no—
160C42×C43—21no—
161C42×C43—81no—
162C42×C43—81no—
163C42×C43C22×C412yes—
164C42×C43#4112yes—
165(C2×D43)⋊C4—24no—
166(C2×D43)⋊C4—81no—
167(C2×D43)⋊C4—81no—
168(C2×D43)⋊C4—81no—
169(C2×D43)⋊C4—81no—
170C4⋊Dic43—21no—
171C4⋊Dic43—21no—
172C4⋊Dic43—21no—
173C4⋊Dic43—21no—
174C4⋊Dic43—21no—
175C4⋊Dic43C22×C412yes—
176Dic43⋊C4—81no—
177Dic43⋊C4—81no—
178C4×Dic43—81no—
179C2×C43⋊C8—21no—
180C2×C43⋊C8—21no—
181C2×C43⋊C8—21no—
182C2×C43⋊C8—21no—
183C2×C43⋊C8—21no—
184C2×C43⋊C8—21no—
185C2×C43⋊C8—21no—
186C2×C43⋊C8—21no—
187C2×C43⋊C8—21no—
188C2×C43⋊C8—21no—
189C2×C43⋊C8C22×C412yes—
190C2×C43⋊C8C22×C412yes—
191C23×C43C22×D411yescharacteristic
192C22×D43—22no—
193C22×D43—41no—
194C22×D43—41no—
195C22×D43—41no—
196C22×D43—41no—
197C22×D43—41no—
198C22×D43—41no—
199Q8×C43—84no—
200C2×C4×C43—21no—
201C2×C4×C43—21no—
202C2×C4×C43—21no—
203C2×C4×C43—21no—
204C2×C4×C43—22no—
205C2×C4×C43—22no—
206C2×C4×C43—21no—
207C2×C4×C43—21no—
208C2×C4×C43—21no—
209C2×C4×C43—21no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—21no—
215C2×C4×C43—21no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43C4×D412yes—
229C2×C4×C43C4×D412yes—
230C2×C4×C43C23×C411yescharacteristic
231C2×C4×C43C2×#4111yescharacteristic
232C2×C4×C43H(5,𝔽2)11yescharacteristic
233C2×C4×C432−1+411yescharacteristic
234C2×C4×C432−1+411yescharacteristic
235C2×C4×C432−1+411yescharacteristic
236C2×Dic43—22no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43—41no—
245D172—22no—
246D172—22no—
247D172—41no—
248D172—41no—
249C4×D43—84no—
250C43⋊C8—22no—
251C43⋊C8—22no—
252C43⋊C8—41no—
253C43⋊C8—41no—
254#22123—431nomaximal, Sylow
255C22×C43C2×C4×D411yescharacteristic
256C22×C43#53611yescharacteristic
257C22×C43#53711yescharacteristic
258C22×C43#57611yescharacteristic
259C22×C43#57711yescharacteristic
260C22×C43#57711yescharacteristic
261C22×C43#57711yescharacteristic
262C2×D43—42no—
263C2×D43—42no—
264C2×D43—42no—
265C2×D43—81no—
266C2×D43—81no—
267C2×D43—81no—
268C2×D43—81no—
269C4×C43—22no—
270C4×C43—22no—
271C4×C43—42no—
272C4×C43—42no—
273C4×C43—41no—
274C4×C43—41no—
275C4×C43—41no—
276C4×C43—41no—
277C4×C43—81no—
278C4×C43—81no—
279C4×C43#51911yescharacteristic
280C4×C43#52011yescharacteristic
281C4×C43#54111yescharacteristic
282C4×C43#54211yescharacteristic
283Dic43—42no—
284Dic43—81no—
285Dic43—81no—
286C2×#476—431no—
287#2331D4311yescharacteristic, complemented
288#2319—431no—
289#1988—431no—
290#1949—431no—
291#1944—431no—
292#1941—431no—
293#1936—431no—
294#1934—431no—
295#1926—431no—
296#1925—431no—
297#1920—431no—
298#1919—431no—
299#1804—431no—
300#1768—431no—
301C2×C43#232311yescharacteristic
302C2×C43#294611yescharacteristic
303C2×C43#294911yescharacteristic
304C2×C43#324111yescharacteristic
305C2×C43#324411yescharacteristic
306C2×C43#325711yescharacteristic
307C2×C43#326311yescharacteristic
308D43—82no—
309C2×C4⋊Q8C2×D4311yescharacteristic
310C2×C4×D4—431no—
311#476—438no—
312C2×#107—431no—
313C2×#106—431no—
314C2×#106—431no—
315C2×#106—431no—
316C2×C4⋊C8—431no—
317C2×C4⋊C8—431no—
318C2×#102—431no—
319C2×#102—431no—
320C2×#102—431no—
321C2×#102—431no—
322C2×#97—431no—
323C2×C4×C8—431no—
324#391—21no—
325#391—21no—
326#390—431no—
327#390—431no—
328#389—431no—
329#389—431no—
330#389C2×D4311yescharacteristic
331#389C2×D4311yescharacteristic
332#388—861no—
333#388—861no—
334#387—21no—
335#387—21no—
336#382C2×D4311yescharacteristic
337#381—861no—
338#380—861no—
339C4×C4⋊C4—431no—
340C4×C4⋊C4C2×D4311yescharacteristic
341C4×C4⋊C4C2×D4311yescharacteristic
342#376C2×D4311yescharacteristic
343#340—431no—
344#340—431no—
345#340—431no—
346#340—431no—
347#340—431no—
348#340—431no—
349#340—431no—
350#340—431no—
351#336—431no—
352#336—431no—
353C43#2212311yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×D4—431no—
356C23×C4—861no—
357C22×C8—431no—
358C22×C8—431no—
359C22×C8—431no—
360C22×C8—431no—
361C4⋊Q8—22no—
362C4⋊Q8C4×D4314yes—
363C4×D4—434no—
364C4×D4—862no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—431no—
369C2×C4⋊C4—431no—
370C2×C4⋊C4—431no—
371C2×C4⋊C4—861no—
372C2×C4⋊C4—861no—
373C2×C4⋊C4C22×D4311yescharacteristic
374C2×C4⋊C4C22×D4311yescharacteristic
375C2×C4⋊C4C22×D4311yescharacteristic
376C2×C4⋊C4C22×D4311yescharacteristic
377C2×C22⋊C4—861no—
378C2×C22⋊C4—861no—
379C2×C22⋊C4—861no—
380C2×C22⋊C4—861no—
381C2×C22⋊C4—861no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—861no—
385C2×C42C22×D4311yescharacteristic
386C2×C42C22×D4311yescharacteristic
387C2×C42C22×D4311yescharacteristic
388#107—1721no—
389#106—434no—
390#106—434no—
391#106—1721no—
392C4⋊C8—434no—
393C4⋊C8—862no—
394#102—434no—
395#102—434no—
396#102—1721no—
397#102—1721no—
398#97—1721no—
399C4×C8—1721no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—861no—
407#95—861no—
408#95—861no—
409#95C22⋊D4311yescharacteristic
410#95C22⋊D4311yescharacteristic
411#95C22⋊D4311yescharacteristic
412#95C22⋊D4311yescharacteristic
413C24—861no—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—432no—
418C2×D4—861no—
419C2×D4—861no—
420C2×D4—861no—
421C2×D4—861no—
422C2×D4—861no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—861no—
428C22×C4—864no—
429C22×C4—861no—
430C22×C4—861no—
431C22×C4—3441no—
432C22×C4—3441no—
433C22×C4C2×C22⋊D4311yescharacteristic
434C22×C4C2×C22⋊D4311yescharacteristic
435C22×C4C2×C22⋊D4311yescharacteristic
436C22×C4C2×C22⋊D4311yescharacteristic
437C22×C4C2×C22⋊D4311yescharacteristic
438C22×C4C2×C22⋊D4311yescharacteristic
439C22×C4C23×D4311yescharacteristic
440C2×C8—432no—
441C2×C8—432no—
442C2×C8—861no—
443C2×C8—861no—
444C2×C8—861no—
445C2×C8—861no—
446C2×C8—861no—
447C2×C8—861no—
448C2×C8—861no—
449C2×C8—861no—
450C2×C8—861no—
451C2×C8—861no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—432no—
462C4⋊C4—861no—
463C4⋊C4—861no—
464C4⋊C4—861no—
465C4⋊C4—861no—
466C4⋊C4—861no—
467C4⋊C4—3441no—
468C4⋊C4—3441no—
469C4⋊C4C2×C4×D4312yes—
470C4⋊C4C2×C4×D4312yes—
471C22⋊C4—864no—
472C22⋊C4—3441no—
473C22⋊C4—3441no—
474C22⋊C4—3441no—
475C22⋊C4—3441no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—81no—
482C42—3441no—
483C42C2×C4×D4312yes—
484C42#1062189812yes—
485C23—862no—
486C23—1721no—
487C23—1721no—
488C23—1721no—
489C23—1721no—
490C23—1721no—
491C23—1721no—
492C23C22×C22⋊D4311yescharacteristic
493Q8—84no—
494D4—862no—
495D4—862no—
496D4—1721no—
497D4—1721no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—862no—
527C2×C4—3444no—
528C2×C4—1721no—
529C2×C4—1721no—
530C2×C4—1721no—
531C2×C4—1721no—
532C2×C4—1721no—
533C2×C4—1721no—
534C2×C4—1721no—
535C2×C4—1721no—
536C2×C4C4×C22⋊D4312yes—
537C2×C4C4×C22⋊D4312yes—
538C2×C4C22×C4×D4311yescharacteristic
539C2×C4C2×#1062189811yescharacteristic
540C2×C4#1306337111yescharacteristic
541C2×C4#1306337211yescharacteristic
542C2×C4#1306337311yescharacteristic
543C2×C4#1306337311yescharacteristic
544C8—862no—
545C8—862no—
546C8—1721no—
547C8—1721no—
548C22—3441no—
549C22—3441no—
550C22—3441no—
551C22—3441no—
552C22—1722no—
553C22—1722no—
554C22—1722no—
555C22C2×C4×C22⋊D4311yescharacteristic
556C22#1458347311yescharacteristic
557C22#1458347711yescharacteristic
558C22#1458370211yescharacteristic
559C22#1458370711yescharacteristic
560C22#1458370711yescharacteristic
561C22#1458371011yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—3441no—
573C4—3441no—
574C4—1722no—
575C4#1458337511yescharacteristic
576C4#1458338111yescharacteristic
577C4#1458350711yescharacteristic
578C4#1458352111yescharacteristic
579C2—3442no—
580C2#1913727111yescharacteristic
581C2#1914342711yescharacteristic
582C2#1914345411yescharacteristic
583C2#1914584811yescharacteristic
584C2#1914589311yescharacteristic
585C2#1914600611yescharacteristic
586C2#1914606011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#2331C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenter#2331D43
hypocenterC43#22123
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #19135548C2semidirect
4. #19135361C2semidirect
5. #19135335C2semidirect
6. #19135320C2semidirect
11. #19135180C2semidirect
12. #19135173C2semidirect
14. #19134452C2semidirect
15. C43×#2331C2semidirect
17. #14583186C4semidirect
287. #2331D43semidirect
353. C43#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#145831862
#191372392
#191355482
#191353612
#191353352
#191353202
#191352912
#191352772
#191352222
#191352152
#191351802
#191351732
#191346022
#191344522
C43×#23312
#2212343