Finite Group Databasebeta

← #25152730

Subgroups — #25152730

Order 11008.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583605C211yesmaximal, characteristic
2C2×#14583186C211yesmaximal, characteristic
3#19139584C211yesmaximal, characteristic, complemented
4#19135931C212yesmaximal, complemented
5#19135926C212yesmaximal, complemented
6#19135553C211yesmaximal, characteristic, complemented
7#19135535C211yesmaximal, characteristic, complemented
8#19134853C212yesmaximal
9#19134839C212yesmaximal
10#19134421C211yesmaximal, characteristic
11C2×C43×#501C211yesmaximal, characteristic, complemented
12#14583605—24no—
13#14583605—24no—
14C2×#13063324C2211yescharacteristic
15C2×#13063323—21no—
16C2×#13063316C2211yescharacteristic
17C2×#13063305—21no—
18C2×#13063303—21no—
19C2×#13063303—21no—
20C2×C4×Dic86C2211yescharacteristic
21C2×C4×D172C2211yescharacteristic
22C2×#13063287C2211yescharacteristic
23C2×#13063287C2211yescharacteristic
24C2×C4⋊C4×D43—21no—
25#14583186—24no—
26C2×#13063256C2212yes—
27C2×C4⋊(C43⋊C8)C2211yescharacteristic
28C2×#13063247C2212yes—
29C2×#13063245C2212yes—
30#14582830—21no—
31#14582830—21no—
32#14582806—21no—
33#14582795C2212yes—
34#14582795C2212yes—
35C4×C4⋊Dic43C2211yescharacteristic
36#14582662C2212yes—
37#14582661C2212yes—
38#14582661C2212yes—
39#14582653C2212yes—
40#14582653C2212yes—
41C2×C4⋊Q8×C43C2211yescharacteristic
42C2×#125×C43C2211yescharacteristic
43C43×#501—24no—
44C2×#107×C43C2212yes—
45C2×#106×C43C2212yes—
46C2×#97×C43C2211yescharacteristic
47C22×Dic86—21no—
48C22×Dic86D411yescharacteristic
49C22×D172C2311yescharacteristic
50C22×C4×D43—21no—
51C22×C4×D43—21no—
52C22×C4×D43—21no—
53C22×C43⋊C8C2312yes—
54#13063324—22no—
55#13063324—22no—
56#13063323—44no—
57#13063316—22no—
58#13063316—22no—
59#13063305—28no—
60#13063303—28no—
61#13063303—28no—
62C4×Dic86—22no—
63C4×Dic86—22no—
64C4×D172—22no—
65C4×D172D414yes—
66#13063287—22no—
67#13063287—22no—
68#13063287—22no—
69#13063287—22no—
70C2×C4⋊Dic43—21no—
71C2×C4⋊Dic43D411yescharacteristic
72C2×C4⋊Dic43C2311yescharacteristic
73C2×C4⋊Dic43C2312yes—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—21no—
76C2×Dic43⋊C4—22no—
77C2×Dic43⋊C4—22no—
78C2×Dic43⋊C4D411yescharacteristic
79C2×Dic43⋊C4D411yescharacteristic
80C4⋊C4×D43—28no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×(C2×D43)⋊C4—21no—
84C2×C4×Dic43—21no—
85C2×C4×Dic43C2311yescharacteristic
86C2×C4×Dic43C2311yescharacteristic
87#13063256—24no—
88C4⋊(C43⋊C8)—22no—
89#13063247Q818yescomplemented
90#13063245—24no—
91C43⋊#95—21no—
92C43⋊#95—22no—
93C43⋊#95—22no—
94C22×Q8×C43—21no—
95C22×C8×C43C2312yes—
96C4⋊Q8×C43—22no—
97C4⋊Q8×C43—22no—
98#125×C43—22no—
99#125×C43D414yes—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43C2312yes—
104C2×C4⋊C4×C43C2311yescharacteristic
105C2×C4⋊C4×C43C2311yescharacteristic
106C2×C42×C43C2311yescharacteristic
107#107×C43—24no—
108#106×C43—24no—
109#97×C43—22no—
110C23×D43—21no—
111C2×Q8×C43—42no—
112C2×Q8×C43—42no—
113C2×Q8×C43—42no—
114C22×C4×C43—21no—
115C22×C4×C43—21no—
116C22×C4×C43—22no—
117C22×C4×C43C2×D411yescharacteristic
118C22×C4×C43C2×D411yescharacteristic
119C22×C4×C43C2411yescharacteristic
120C22×Dic43—22no—
121C22×Dic43—21no—
122C22×Dic43C2×D411yescharacteristic
123C22×Dic43C2×D411yescharacteristic
124C22×Dic43C2×D411yescharacteristic
125C22×Dic43C2×D411yescharacteristic
126C2×D172—21no—
127C2×D172—22no—
128C2×D172C2×D412yes—
129C2×D172C2×Q814yes—
130C2×C4×D43—24no—
131C2×C4×D43—24no—
132C2×C4×D43—24no—
133C2×C4×D43—42no—
134C2×C4×D43—42no—
135C2×C4×D43—82no—
136C2×C4×D43—82no—
137C2×Dic86—22no—
138C2×Dic86—22no—
139C2×Dic86—24no—
140C2×Dic86—41no—
141C2×Dic86—82no—
142C2×C8×C43—22no—
143C2×C8×C43—22no—
144C2×C8×C43C2×Q814yes—
145C4⋊C4×C43—24no—
146C4⋊C4×C43—24no—
147C4⋊C4×C43—21no—
148C4⋊C4×C43—21no—
149C4⋊C4×C43—22no—
150C4⋊C4×C43—22no—
151C4⋊C4×C43—42no—
152C4⋊C4×C43C2×D412yes—
153C4⋊C4×C43C2×D412yes—
154C42×C43C2×D412yes—
155C42×C43C2×D412yes—
156(C2×D43)⋊C4—24no—
157(C2×D43)⋊C4—24no—
158(C2×D43)⋊C4—24no—
159C4⋊Dic43—24no—
160C4⋊Dic43—22no—
161C4⋊Dic43—21no—
162C4⋊Dic43—22no—
163C4⋊Dic43C2×D412yes—
164C4⋊Dic43C2×Q814yes—
165Dic43⋊C4—24no—
166Dic43⋊C4—24no—
167Dic43⋊C4—22no—
168Dic43⋊C4—22no—
169Dic43⋊C4—44no—
170Dic43⋊C4—44no—
171C4×Dic43—21no—
172C4×Dic43—21no—
173C4×Dic43—21no—
174C4×Dic43—21no—
175C4×Dic43—42no—
176C2×C43⋊C8—22no—
177C2×C43⋊C8—24no—
178C23×C43C22×D411yescharacteristic
179C22×D43—22no—
180C22×D43—24no—
181C22×D43—41no—
182C22×D43—41no—
183C22×D43—42no—
184Q8×C43—84no—
185C2×C4×C43—21no—
186C2×C4×C43—21no—
187C2×C4×C43—22no—
188C2×C4×C43—22no—
189C2×C4×C43—22no—
190C2×C4×C43—41no—
191C2×C4×C43—41no—
192C2×C4×C43—42no—
193C2×C4×C43—42no—
194C2×C4×C43—42no—
195C2×C4×C43—42no—
196C2×C4×C43#12012yes—
197C2×C4×C43#12012yes—
198C2×C4×C43C22×D411yescharacteristic
199C2×C4×C43C22×D411yescharacteristic
200C2×C4×C43C22×Q812yes—
201C2×C4×C432−1+412yes—
202C2×Dic43—21no—
203C2×Dic43—21no—
204C2×Dic43—21no—
205C2×Dic43—21no—
206C2×Dic43—21no—
207C2×Dic43—22no—
208C2×Dic43—24no—
209C2×Dic43—21no—
210C2×Dic43—21no—
211C2×Dic43—22no—
212C2×Dic43—22no—
213C2×Dic43—42no—
214C2×Dic43—42no—
215C2×Dic43—42no—
216C2×Dic43AGL(1,ℤ/8)12yes—
217D172—22no—
218D172—24no—
219D172C4⋊Q814yescomplemented
220C4×D43—48no—
221C4×D43—84no—
222C4×D43—84no—
223Dic86—44no—
224Dic86—84no—
225C8×C43—24no—
226C43⋊C8—42no—
227#25851—431nomaximal, Sylow
228C22×C43C2×#12011yescharacteristic
229C22×C43D4×Q812yes—
230C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C43#57511yescharacteristic
232C22×C43#57712yes—
233C2×D43—24no—
234C2×D43—44no—
235C2×D43—44no—
236C2×D43—42no—
237C2×D43—42no—
238C4×C43—22no—
239C4×C43—22no—
240C4×C43—42no—
241C4×C43—42no—
242C4×C43—82no—
243C4×C43C2×C4⋊Q811yescharacteristic
244C4×C43#53611yescharacteristic
245C4×C43D4×Q812yes—
246Dic43—22no—
247Dic43—44no—
248Dic43—41no—
249Dic43—41no—
250Dic43—42no—
251Dic43—41no—
252C2×#554—431no—
253C2×#501D4311yescharacteristic, complemented
254C2×#476—431no—
255#2546—431no—
256#2059—432no—
257#2058—432no—
258#1989—431no—
259#1985—431no—
260#1864—432no—
261#1862—432no—
262#1762—431no—
263C2×C43#243411yescharacteristic
264C2×C43#300611yescharacteristic
265C2×C43#300911yescharacteristic
266C2×C43#338412yes—
267C2×C43#338812yes—
268D43—44no—
269#554—864no—
270#554—864no—
271C2×C4⋊Q8—431no—
272C2×C4⋊Q8—861no—
273C2×C4⋊Q8C2×D4311yescharacteristic
274C2×#125—431no—
275C2×#125C2×D4311yescharacteristic
276C2×C22⋊Q8—861no—
277C2×C22⋊Q8—861no—
278C2×C22⋊Q8—861no—
279C2×C4×Q8—431no—
280C2×C4×D4—431no—
281C2×C4×D4—431no—
282C2×C4×D4—431no—
283C22×C4⋊C4—861no—
284#501—24no—
285#476—864no—
286C2×#107C2×D4312yes—
287C2×#106—432no—
288C2×#106C2×D4312yes—
289C2×C4⋊C8—431no—
290C2×#102—432no—
291C2×#102—432no—
292C2×#97C2×D4311yescharacteristic
293#398—861no—
294#398—861no—
295#391—861no—
296#389—432no—
297#389—432no—
298C4×C4⋊C4—431no—
299#340—432no—
300#340—432no—
301#340—432no—
302#336—432no—
303#336—432no—
304C43#2585111yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C22×Q8—431no—
307C22×Q8—861no—
308C22×D4—431no—
309C23×C4—861no—
310C23×C4—861no—
311C23×C4—861no—
312C22×C8—432no—
313C22×C8C22×D4312yes—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—862no—
317C4⋊Q8—862no—
318C4⋊Q8—1724no—
319#125—22no—
320#125—862no—
321#125—862no—
322#125C22⋊D4314yes—
323C22⋊Q8—868no—
324C22⋊Q8—868no—
325C22⋊Q8—868no—
326C4×Q8—862no—
327C4×Q8—862no—
328C4×D4—434no—
329C4×D4—862no—
330C4×D4—862no—
331C4×D4—862no—
332C4×D4—862no—
333C4×D4—862no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—431no—
338C2×C4⋊C4—432no—
339C2×C4⋊C4—431no—
340C2×C4⋊C4—431no—
341C2×C4⋊C4—431no—
342C2×C4⋊C4—868no—
343C2×C4⋊C4—861no—
344C2×C4⋊C4—861no—
345C2×C4⋊C4—861no—
346C2×C4⋊C4—862no—
347C2×C4⋊C4—862no—
348C2×C4⋊C4C22×D4312yes—
349C2×C4⋊C4C22×D4311yescharacteristic
350C2×C4⋊C4C22×D4311yescharacteristic
351C2×C22⋊C4—861no—
352C2×C22⋊C4—861no—
353C2×C22⋊C4—861no—
354C2×C42—431no—
355C2×C42—431no—
356C2×C42—861no—
357C2×C42C22×D4311yescharacteristic
358#107—24no—
359#106—24no—
360#106—864no—
361C4⋊C8—862no—
362#102—438no—
363#102—864no—
364#97—22no—
365#95—862no—
366#95—862no—
367#95—861no—
368C24—861no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—862no—
373C2×Q8—864no—
374C2×Q8—862no—
375C2×Q8—3442no—
376C2×Q8—1721no—
377C2×D4—432no—
378C2×D4—434no—
379C2×D4—861no—
380C2×D4—862no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—431no—
385C22×C4—431no—
386C22×C4—431no—
387C22×C4—431no—
388C22×C4—864no—
389C22×C4—864no—
390C22×C4—862no—
391C22×C4—861no—
392C22×C4—864no—
393C22×C4—3442no—
394C22×C4—3442no—
395C22×C4—1722no—
396C22×C4—1722no—
397C22×C4D4×D4311yescharacteristic
398C22×C4D4×D4311yescharacteristic
399C22×C4C23×D4311yescharacteristic
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—862no—
403C2×C8—864no—
404C2×C8Q8×D4314yes—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—432no—
413C4⋊C4—434no—
414C4⋊C4—864no—
415C4⋊C4—864no—
416C4⋊C4—864no—
417C4⋊C4—862no—
418C4⋊C4—862no—
419C4⋊C4—862no—
420C4⋊C4—861no—
421C4⋊C4—862no—
422C4⋊C4—1724no—
423C4⋊C4—1724no—
424C4⋊C4C2×C22⋊D4312yes—
425C4⋊C4C2×C22⋊D4312yes—
426C22⋊C4—864no—
427C22⋊C4—864no—
428C22⋊C4—864no—
429C42—861no—
430C42—861no—
431C42—861no—
432C42—861no—
433C42—1722no—
434C42D4×D4312yes—
435C42C2×C22⋊D4312yes—
436C23—864no—
437C23—862no—
438C23—1721no—
439C23—1721no—
440C23—1722no—
441C23C2×D4×D4311yescharacteristic
442Q8—84no—
443Q8—3444no—
444Q8—1724no—
445D4—434no—
446D4—864no—
447D4—862no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—432no—
460C2×C4—861no—
461C2×C4—861no—
462C2×C4—864no—
463C2×C4—861no—
464C2×C4—861no—
465C2×C4—861no—
466C2×C4—862no—
467C2×C4—861no—
468C2×C4—861no—
469C2×C4—862no—
470C2×C4—862no—
471C2×C4—3444no—
472C2×C4—3444no—
473C2×C4—1728no—
474C2×C4—1722no—
475C2×C4—1722no—
476C2×C4—1722no—
477C2×C4#1306329312yes—
478C2×C4#1306329312yes—
479C2×C4C2×D4×D4311yescharacteristic
480C2×C4C22×C22⋊D4311yescharacteristic
481C2×C4C2×Q8×D4312yes—
482C2×C4#1306337312yes—
483C8—24no—
484C8—1722no—
485C22—864no—
486C22—1724no—
487C22—1722no—
488C22—1722no—
489C22—1724no—
490C22C2×#1306329311yescharacteristic
491C22#1458356412yes—
492C22AGL(1,ℤ/8)×D4311yescharacteristic
493C22#1458369511yescharacteristic
494C22#1458370812yes—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—862no—
501C4—1721no—
502C4—1721no—
503C4—1721no—
504C4—1722no—
505C4—1724no—
506C4C4⋊Q8×D4311yescharacteristic
507C4#1458347411yescharacteristic
508C4Q8×C22⋊D4312yes—
509C2—1724no—
510C2#1913823611yescharacteristic
511C2#1914384311yescharacteristic
512C2#1914387111yescharacteristic
513C2#1914749112yes—
514C2#1914753512yes—
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#501C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenterC2×#501D43
hypocenterC43#25851
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #19139584C2semidirect
4. #19135931C2semidirect
5. #19135926C2semidirect
6. #19135553C2semidirect
7. #19135535C2semidirect
11. C2×C43×#501C2semidirect
89. #13063247Q8semidirect
219. D172C4⋊Q8semidirect
253. C2×#501D43semidirect
304. C43#25851semidirect

Maximal subgroups

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

subgroupindex
C2×#145836052
C2×#145831862
#191395842
#191359312
#191359262
#191355532
#191355352
#191348532
#191348392
#191344212
C2×C43×#5012
#2585143