Finite Group Databasebeta

← #24988705

Subgroups — #24988705

Order 11008.

788 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583622C211yesmaximal, characteristic
2#19135556C211yesmaximal, characteristic, complemented
3#19134600C211yesmaximal, characteristic, complemented
4#19134596C211yesmaximal, characteristic
5C2×#14582646C211yesmaximal, characteristic, complemented
6C2×#14582628C211yesmaximal, characteristic
7C2×C43×Q8⋊C8C211yesmaximal, characteristic, complemented
8#14583622—22no—
9#14583622—22no—
10#14583622—22no—
11#14583622C414yescomplemented
12C2×#13063325C2211yescharacteristic
13C2×#13063320—22no—
14C2×#13063313—21no—
15C2×#13063301—21no—
16C2×C4×Dic86—21no—
17C2×C4×D172C2211yescharacteristic
18C2×#13063283—21no—
19C2×C4⋊(C43⋊C8)C2211yescharacteristic
20C2×#13063249—21no—
21C2×#13063247—21no—
22#14582769C2211yescharacteristic
23#14582662—22no—
24#14582662—21no—
25#14582661—21no—
26#14582646—22no—
27#14582628—22no—
28C2×C4×Q8×C43C2211yescharacteristic
29C2×C4⋊C8×C43C2211yescharacteristic
30C2×C4×C8×C43C2211yescharacteristic
31C43×Q8⋊C8—22no—
32C22×Dic86—21no—
33C22×D172D411yescharacteristic
34C22×C4×D43—21no—
35C22×C43⋊C8—21no—
36#13063325—21no—
37#13063325—21no—
38#13063325C2×C412yes—
39#13063325D412yes—
40#13063320—28no—
41#13063320—28no—
42#13063313—24no—
43#13063313—24no—
44#13063301—24no—
45#13063301—24no—
46C4×Dic86—24no—
47C4×Dic86—24no—
48C4×D172—21no—
49C4×D172—21no—
50C4×D172C2×C412yes—
51C4×D172D412yes—
52#13063283—24no—
53#13063283—24no—
54C2×C4⋊Dic43—21no—
55C2×C4⋊Dic43—21no—
56C2×C4⋊Dic43—21no—
57C2×C4⋊Dic43—21no—
58C2×C4⋊Dic43D411yescharacteristic
59C2×C4⋊Dic43D412yes—
60C2×Dic43⋊C4—22no—
61C2×(C2×D43)⋊C4—22no—
62C2×(C2×D43)⋊C4—22no—
63C2×(C2×D43)⋊C4—21no—
64C2×C4×Dic43—22no—
65C4⋊(C43⋊C8)—21no—
66C4⋊(C43⋊C8)—21no—
67#13063249—42no—
68#13063247—42no—
69C22×Q8×C43D411yescharacteristic
70C22×C8×C43—21no—
71C22×C8×C43—21no—
72C4×Q8×C43—21no—
73C4×Q8×C43—21no—
74C4×Q8×C43C2×C412yes—
75C4×Q8×C43D412yes—
76C2×C4⋊C4×C43—21no—
77C2×C4⋊C4×C43D411yescharacteristic
78C2×C42×C43—21no—
79C2×C42×C43C2311yescharacteristic
80C4⋊C8×C43—21no—
81C4⋊C8×C43—21no—
82C4×C8×C43—21no—
83C4×C8×C43—21no—
84C23×D43—21no—
85C2×Q8×C43—21no—
86C2×Q8×C43—21no—
87C2×Q8×C43—21no—
88C2×Q8×C43—21no—
89C2×Q8×C43—21no—
90C2×Q8×C43C22⋊C412yes—
91C22×C4×C43—21no—
92C22×C4×C43—21no—
93C22×C4×C43C2×D411yescharacteristic
94C22×C4×C43C2×D411yescharacteristic
95C22×C4×C43C2×D411yescharacteristic
96C22×Dic43—21no—
97C22×Dic43—22no—
98C22×Dic43—41no—
99C2×D172—21no—
100C2×D172—21no—
101C2×D172—21no—
102C2×D172—21no—
103C2×D172—21no—
104C2×D172C22⋊C412yes—
105C2×C4×D43—22no—
106C2×C4×D43—22no—
107C2×C4×D43—41no—
108C2×C4×D43—41no—
109C2×C4×D43—41no—
110C2×C4×D43—41no—
111C2×Dic86—22no—
112C2×Dic86—22no—
113C2×Dic86—41no—
114C2×Dic86—41no—
115C2×Dic86—41no—
116C2×Dic86—41no—
117C2×C8×C43—22no—
118C2×C8×C43—41no—
119C2×C8×C43—41no—
120C2×C8×C43—41no—
121C2×C8×C43—41no—
122C2×C8×C43—41no—
123C4⋊C4×C43—22no—
124C4⋊C4×C43—22no—
125C4⋊C4×C43—21no—
126C4⋊C4×C43C22⋊C412yes—
127C42×C43—22no—
128C42×C43—22no—
129C42×C43C22×C411yescharacteristic
130C42×C43C2×D411yescharacteristic
131C42×C43C2×D411yescharacteristic
132C42×C43C2×D411yescharacteristic
133(C2×D43)⋊C4—28no—
134(C2×D43)⋊C4—28no—
135(C2×D43)⋊C4—22no—
136(C2×D43)⋊C4—22no—
137C4⋊Dic43—22no—
138C4⋊Dic43—22no—
139C4⋊Dic43—21no—
140C4⋊Dic43—22no—
141C4⋊Dic43—42no—
142C4⋊Dic43—41no—
143C4⋊Dic43—41no—
144C4⋊Dic43—41no—
145C4⋊Dic43—41no—
146C4⋊Dic43C22⋊C412yes—
147C4⋊Dic43C22⋊C414yes—
148Dic43⋊C4—28no—
149C4×Dic43—28no—
150C2×C43⋊C8—41no—
151C2×C43⋊C8—41no—
152C2×C43⋊C8—41no—
153C23×C43#12011yescharacteristic
154C22×D43—22no—
155C22×D43—41no—
156C22×D43—41no—
157C22×D43—41no—
158C22×D43—41no—
159C22×D43—41no—
160C22×D43—41no—
161Q8×C43—22no—
162Q8×C43—22no—
163Q8×C43—41no—
164Q8×C43—41no—
165C2×C4×C43—21no—
166C2×C4×C43—21no—
167C2×C4×C43—22no—
168C2×C4×C43—22no—
169C2×C4×C43—41no—
170C2×C4×C43—41no—
171C2×C4×C43—41no—
172C2×C4×C43—41no—
173C2×C4×C43C2×C22⋊C411yescharacteristic
174C2×C4×C43C2×C22⋊C411yescharacteristic
175C2×C4×C43C2×C22⋊C411yescharacteristic
176C2×C4×C43#12011yescharacteristic
177C2×C4×C43#12011yescharacteristic
178C2×C4×C43#12011yescharacteristic
179C2×C4×C43#13511yescharacteristic
180C2×C4×C43#13511yescharacteristic
181C2×C4×C43#13511yescharacteristic
182C2×C4×C43#13511yescharacteristic
183C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C43#13711yescharacteristic
186C2×C4×C43#13711yescharacteristic
187C2×Dic43—22no—
188C2×Dic43—24no—
189C2×Dic43—42no—
190C2×Dic43—41no—
191C2×Dic43—42no—
192C2×Dic43—41no—
193C2×Dic43—81no—
194C2×Dic43—81no—
195C2×Dic43—81no—
196D172—22no—
197D172—22no—
198D172—41no—
199D172—41no—
200C4×D43—42no—
201C4×D43—42no—
202C4×D43—42no—
203C4×D43—42no—
204Dic86—42no—
205Dic86—42no—
206Dic86—42no—
207Dic86—42no—
208C8×C43—42no—
209C8×C43—81no—
210C43⋊C8—81no—
211#8449—431nomaximal, Sylow
212C22×C43#37911yescharacteristic
213C22×C43#44911yescharacteristic
214C22×C43#44911yescharacteristic
215C22×C43#45211yescharacteristic
216C22×C43#45211yescharacteristic
217C22×C43#45411yescharacteristic
218C22×C43#45611yescharacteristic
219C2×D43—42no—
220C2×D43—42no—
221C2×D43—42no—
222C2×D43—81no—
223C2×D43—81no—
224C2×D43—81no—
225C2×D43—81no—
226C4×C43—21no—
227C4×C43—21no—
228C4×C43—42no—
229C4×C43—42no—
230C4×C43#41611yescharacteristic
231C4×C43#41611yescharacteristic
232C4×C43#41711yescharacteristic
233C4×C43#41711yescharacteristic
234C4×C43#44911yescharacteristic
235C4×C43#44911yescharacteristic
236C4×C43#45211yescharacteristic
237C4×C43#45211yescharacteristic
238Dic43—44no—
239Dic43—42no—
240Dic43—161no—
241C2×#556—431no—
242#1990—431no—
243#1803—431no—
244#1802—431no—
245C2×#332—431no—
246C2×Q8⋊C8D4311yescharacteristic, complemented
247C2×D4⋊C8—431no—
248C2×C43#162811yescharacteristic
249C2×C43#163811yescharacteristic
250C2×C43#165011yescharacteristic
251C2×C43#165211yescharacteristic
252C2×C43#178811yescharacteristic
253C2×C43#178811yescharacteristic
254C2×C43#179311yescharacteristic
255D43—82no—
256#556—434no—
257#556—862no—
258#556—862no—
259#556—862no—
260C2×C4⋊Q8—431no—
261C2×#126—862no—
262C2×#124—861no—
263C2×C22⋊Q8—861no—
264C2×C4×Q8—861no—
265C2×C4×Q8C2×D4311yescharacteristic
266C2×C4×D4—431no—
267C2×#117—861no—
268C2×C4⋊C8—431no—
269C2×C4⋊C8C2×D4311yescharacteristic
270C2×#103—861no—
271C2×#102—861no—
272C2×C4×C8C2×D4311yescharacteristic
273#384—431no—
274#340—861no—
275#340—862no—
276#340—861no—
277#332—862no—
278Q8⋊C8—22no—
279D4⋊C8—862no—
280C43#844911yescharacteristic, complemented, Sylow
281C22×Q8—861no—
282C22×Q8D17211yescharacteristic
283C22×D4—431no—
284C23×C4—861no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—861no—
288C4⋊Q8—432no—
289C4⋊Q8—432no—
290C4⋊Q8—861no—
291C4⋊Q8—861no—
292#126—868no—
293#126—868no—
294#124—864no—
295#124—864no—
296C22⋊Q8—864no—
297C22⋊Q8—864no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—864no—
301C4×Q8—864no—
302C4×Q8C4×D4312yes—
303C4×Q8C22⋊D4312yes—
304C4×D4—432no—
305C4×D4—432no—
306C4×D4—861no—
307C4×D4—861no—
308#117—864no—
309#117—864no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—431no—
312C2×C4⋊C4—432no—
313C2×C4⋊C4—862no—
314C2×C4⋊C4—861no—
315C2×C4⋊C4—861no—
316C2×C4⋊C4—861no—
317C2×C4⋊C4—861no—
318C2×C4⋊C4D17211yescharacteristic
319C2×C22⋊C4—862no—
320C2×C22⋊C4—862no—
321C2×C22⋊C4—861no—
322C2×C42—21no—
323C2×C42—862no—
324C2×C42C22×D4311yescharacteristic
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—861no—
328C4⋊C8—861no—
329#103—1722no—
330#102—1722no—
331C4×C8—21no—
332C4×C8—21no—
333C24—861no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—862no—
340C2×Q8—862no—
341C2×Q8—1721no—
342C2×Q8—1721no—
343C2×Q8—1721no—
344C2×Q8—1721no—
345C2×Q8(C2×D43)⋊C412yes—
346C2×D4—432no—
347C2×D4—861no—
348C2×D4—861no—
349C2×D4—861no—
350C2×D4—861no—
351C2×D4—861no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—861no—
355C22×C4—862no—
356C22×C4—862no—
357C22×C4—862no—
358C22×C4—1721no—
359C22×C4—1721no—
360C22×C4—1721no—
361C22×C4—1721no—
362C22×C4—1721no—
363C22×C4C2×D17211yescharacteristic
364C22×C4D4×D4311yescharacteristic
365C22×C4D4×D4311yescharacteristic
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—1721no—
373C2×C8—1721no—
374C2×C8—1721no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—432no—
379C4⋊C4—434no—
380C4⋊C4—868no—
381C4⋊C4—862no—
382C4⋊C4—861no—
383C4⋊C4—862no—
384C4⋊C4—862no—
385C4⋊C4—1722no—
386C4⋊C4—1721no—
387C4⋊C4—1721no—
388C4⋊C4—1721no—
389C4⋊C4—1721no—
390C4⋊C4(C2×D43)⋊C412yes—
391C22⋊C4—868no—
392C22⋊C4—868no—
393C22⋊C4—862no—
394C22⋊C4—862no—
395C42—22no—
396C42—22no—
397C42—868no—
398C42C2×C4×D4311yescharacteristic
399C42D4×D4311yescharacteristic
400C42D4×D4311yescharacteristic
401C42C2×C22⋊D4311yescharacteristic
402C23—862no—
403C23—1721no—
404C23—1721no—
405C23—1721no—
406C23—1721no—
407C23—1721no—
408C23—1721no—
409C23#1306329511yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—41no—
413Q8—41no—
414Q8—1722no—
415Q8—1722no—
416Q8—1722no—
417Q8—1722no—
418D4—862no—
419D4—862no—
420D4—1721no—
421D4—1721no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—862no—
431C2×C4—864no—
432C2×C4—3441no—
433C2×C4—3441no—
434C2×C4—3441no—
435C2×C4—1721no—
436C2×C4—1722no—
437C2×C4—1721no—
438C2×C4—1722no—
439C2×C4—1722no—
440C2×C4—1722no—
441C2×C4—1722no—
442C2×C4—1722no—
443C2×C4C22⋊C4×D4311yescharacteristic
444C2×C4C22⋊C4×D4311yescharacteristic
445C2×C4C2×(C2×D43)⋊C411yescharacteristic
446C2×C4#1306329311yescharacteristic
447C2×C4#1306329311yescharacteristic
448C2×C4#1306329511yescharacteristic
449C2×C4#1306334611yescharacteristic
450C2×C4#1306334711yescharacteristic
451C2×C4#1306334711yescharacteristic
452C2×C4#1306334811yescharacteristic
453C2×C4#1306335211yescharacteristic
454C2×C4#1306335311yescharacteristic
455C2×C4#1306335711yescharacteristic
456C2×C4#1306335811yescharacteristic
457C8—42no—
458C8—81no—
459C8—3441no—
460C22—3441no—
461C22—3441no—
462C22—3441no—
463C22—3441no—
464C22—1722no—
465C22—1722no—
466C22—1722no—
467C22#1458275411yescharacteristic
468C22#1458304611yescharacteristic
469C22#1458304711yescharacteristic
470C22#1458306311yescharacteristic
471C22#1458306411yescharacteristic
472C22#1458307311yescharacteristic
473C22#1458308311yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—1724no—
479C4—1722no—
480C4—6881no—
481C4#1458289411yescharacteristic
482C4#1458289611yescharacteristic
483C4#1458289911yescharacteristic
484C4#1458290111yescharacteristic
485C4#1458304311yescharacteristic
486C4#1458304611yescharacteristic
487C4#1458306011yescharacteristic
488C4#1458306311yescharacteristic
489C2—3442no—
490C2#1913370811yescharacteristic
491C2#1913377011yescharacteristic
492C2#1913385411yescharacteristic
493C2#1913386711yescharacteristic
494C2#1913453411yescharacteristic
495C2#1913453611yescharacteristic
496C2#1913455511yescharacteristic
subgroupG/H
centerC23#13063295
commutatorC42×C43C22×C4
FrattiniC2×C42C22×D43
FittingC2×C43×Q8⋊C8C2
socleC23×C43#120
prime radicalC22×D172D4
cosocleC2×C42×C43C23
hypercenterC2×Q8⋊C8D43
hypocenterC43#8449
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #19135556C2semidirect
3. #19134600C2semidirect
5. C2×#14582646C2semidirect
7. C2×C43×Q8⋊C8C2semidirect
11. #14583622C4semidirect
246. C2×Q8⋊C8D43semidirect
280. C43#8449semidirect

Maximal subgroups

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

subgroupindex
C2×#145836222
#191355562
#191346002
#191345962
C2×#145826462
C2×#145826282
C2×C43×Q8⋊C82
#844943