Finite Group Databasebeta

← #25152719

Subgroups — #25152719

Order 11008.

1056 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583605C211yesmaximal, characteristic
2C2×#14583176C211yesmaximal, characteristic
3#19139804C211yesmaximal, characteristic, complemented
4#19135935C212yesmaximal, complemented
5#19135921C212yesmaximal, 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×#13063316C2211yescharacteristic
16C2×#13063305—21no—
17C2×#13063303—21no—
18C2×#13063303—21no—
19C2×C4×Dic86C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21C2×#13063287C2211yescharacteristic
22C2×#13063287C2211yescharacteristic
23C2×C4⋊C4×D43—21no—
24#14583176—24no—
25C2×#13063258C2212yes—
26C2×C4⋊(C43⋊C8)C2211yescharacteristic
27C2×#13063247C2212yes—
28C2×#13063245C2212yes—
29#14582818—21no—
30#14582818—21no—
31#14582804—21no—
32#14582795C2212yes—
33#14582795C2212yes—
34#14582768—21no—
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#13063316—22no—
57#13063316—22no—
58#13063305—28no—
59#13063303—28no—
60#13063303—28no—
61C4×Dic86—22no—
62C4×Dic86—22no—
63C4×D172—22no—
64C4×D172D414yes—
65#13063287—22no—
66#13063287—22no—
67#13063287—22no—
68#13063287—22no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43D411yescharacteristic
71C2×C4⋊Dic43C2311yescharacteristic
72C2×C4⋊Dic43C2312yes—
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—22no—
76C2×Dic43⋊C4—22no—
77C2×Dic43⋊C4D411yescharacteristic
78C2×Dic43⋊C4D411yescharacteristic
79C4⋊C4×D43—28no—
80C2×(C2×D43)⋊C4—21no—
81C2×(C2×D43)⋊C4—21no—
82C2×(C2×D43)⋊C4—21no—
83C2×C4×Dic43—21no—
84C2×C4×Dic43C2311yescharacteristic
85C2×C4×Dic43C2311yescharacteristic
86#13063258—24no—
87C4⋊(C43⋊C8)—22no—
88#13063247Q818yescomplemented
89#13063245—24no—
90C43⋊#95—21no—
91C43⋊#95—22no—
92C43⋊#95—22no—
93C22×Q8×C43—21no—
94C22×C8×C43C2312yes—
95C4⋊Q8×C43—22no—
96C4⋊Q8×C43—22no—
97#125×C43—22no—
98#125×C43D414yes—
99C2×C4⋊C4×C43—21no—
100C2×C4⋊C4×C43—21no—
101C2×C4⋊C4×C43—21no—
102C2×C4⋊C4×C43C2312yes—
103C2×C4⋊C4×C43C2311yescharacteristic
104C2×C4⋊C4×C43C2311yescharacteristic
105C2×C42×C43C2311yescharacteristic
106#107×C43—24no—
107#106×C43—24no—
108#97×C43—22no—
109C23×D43—21no—
110C2×Q8×C43—42no—
111C2×Q8×C43—82no—
112C22×C4×C43—21no—
113C22×C4×C43—21no—
114C22×C4×C43—22no—
115C22×C4×C43C2×D411yescharacteristic
116C22×C4×C43C2×D411yescharacteristic
117C22×C4×C43C2411yescharacteristic
118C22×Dic43—22no—
119C22×Dic43—21no—
120C22×Dic43C2×D411yescharacteristic
121C22×Dic43C2×D411yescharacteristic
122C22×Dic43C2×D411yescharacteristic
123C22×Dic43C2×D411yescharacteristic
124C2×D172—21no—
125C2×D172—22no—
126C2×D172C2×D412yes—
127C2×D172C2×Q814yes—
128C2×C4×D43—24no—
129C2×C4×D43—24no—
130C2×C4×D43—24no—
131C2×C4×D43—42no—
132C2×C4×D43—42no—
133C2×C4×D43—82no—
134C2×C4×D43—82no—
135C2×Dic86—22no—
136C2×Dic86—22no—
137C2×Dic86—24no—
138C2×Dic86—41no—
139C2×Dic86—82no—
140C2×C8×C43—22no—
141C2×C8×C43—22no—
142C2×C8×C43C2×Q814yes—
143C4⋊C4×C43—24no—
144C4⋊C4×C43—24no—
145C4⋊C4×C43—21no—
146C4⋊C4×C43—21no—
147C4⋊C4×C43—22no—
148C4⋊C4×C43—22no—
149C4⋊C4×C43—42no—
150C4⋊C4×C43C2×D412yes—
151C4⋊C4×C43C2×D412yes—
152C42×C43C2×D412yes—
153C42×C43C2×D412yes—
154(C2×D43)⋊C4—24no—
155(C2×D43)⋊C4—24no—
156(C2×D43)⋊C4—24no—
157C4⋊Dic43—24no—
158C4⋊Dic43—22no—
159C4⋊Dic43—21no—
160C4⋊Dic43—22no—
161C4⋊Dic43C2×D412yes—
162C4⋊Dic43C2×Q814yes—
163Dic43⋊C4—24no—
164Dic43⋊C4—24no—
165Dic43⋊C4—22no—
166Dic43⋊C4—22no—
167Dic43⋊C4—82no—
168Dic43⋊C4—82no—
169C4×Dic43—21no—
170C4×Dic43—21no—
171C4×Dic43—21no—
172C4×Dic43—21no—
173C4×Dic43—42no—
174C2×C43⋊C8—22no—
175C2×C43⋊C8—24no—
176C23×C43C22×D411yescharacteristic
177C22×D43—22no—
178C22×D43—24no—
179C22×D43—41no—
180C22×D43—41no—
181C22×D43—42no—
182Q8×C43—84no—
183C2×C4×C43—21no—
184C2×C4×C43—21no—
185C2×C4×C43—22no—
186C2×C4×C43—22no—
187C2×C4×C43—22no—
188C2×C4×C43—41no—
189C2×C4×C43—41no—
190C2×C4×C43—42no—
191C2×C4×C43—42no—
192C2×C4×C43—42no—
193C2×C4×C43—42no—
194C2×C4×C43#12012yes—
195C2×C4×C43#12012yes—
196C2×C4×C43C22×D411yescharacteristic
197C2×C4×C43C22×D411yescharacteristic
198C2×C4×C43C22×Q812yes—
199C2×C4×C432−1+412yes—
200C2×Dic43—21no—
201C2×Dic43—21no—
202C2×Dic43—21no—
203C2×Dic43—21no—
204C2×Dic43—21no—
205C2×Dic43—22no—
206C2×Dic43—24no—
207C2×Dic43—21no—
208C2×Dic43—21no—
209C2×Dic43—22no—
210C2×Dic43—22no—
211C2×Dic43—42no—
212C2×Dic43—42no—
213C2×Dic43—42no—
214C2×Dic43#13712yes—
215D172—22no—
216D172—24no—
217D172C4⋊Q814yes—
218C4×D43—48no—
219C4×D43—84no—
220C4×D43—84no—
221Dic86—44no—
222Dic86—84no—
223C8×C43—24no—
224C43⋊C8—42no—
225#25850—431nomaximal, Sylow
226C22×C43C2×#12011yescharacteristic
227C22×C43D4×Q812yes—
228C22×C43C2×#13711yescharacteristic
229C22×C43#57511yescharacteristic
230C22×C43#57612yes—
231C2×D43—24no—
232C2×D43—44no—
233C2×D43—44no—
234C2×D43—42no—
235C2×D43—42no—
236C4×C43—22no—
237C4×C43—22no—
238C4×C43—42no—
239C4×C43—42no—
240C4×C43—82no—
241C4×C43C2×C4⋊Q811yescharacteristic
242C4×C43#53611yescharacteristic
243C4×C43D4×Q812yes—
244Dic43—22no—
245Dic43—44no—
246Dic43—41no—
247Dic43—41no—
248Dic43—42no—
249Dic43—41no—
250C2×#554—431no—
251C2×#501D4311yescharacteristic, complemented
252C2×#474—431no—
253#2567—431no—
254#2059—432no—
255#2057—432no—
256#1989—431no—
257#1985—431no—
258#1864—432no—
259#1862—432no—
260#1762—431no—
261C2×C43#243411yescharacteristic
262C2×C43#300711yescharacteristic
263C2×C43#300811yescharacteristic
264C2×C43#338512yes—
265C2×C43#338912yes—
266D43—44no—
267#554—864no—
268#554—864no—
269C2×C4⋊Q8—431no—
270C2×C4⋊Q8C2×D4311yescharacteristic
271C2×#125—431no—
272C2×#125C2×D4311yescharacteristic
273C2×C22⋊Q8—861no—
274C2×C22⋊Q8—861no—
275C2×C22⋊Q8—861no—
276C2×C4×Q8—431no—
277C2×C4×D4—431no—
278C2×C4×D4—431no—
279C2×C4×D4—431no—
280C22×C4⋊C4—861no—
281#501—24no—
282#474—864no—
283C2×#107—432no—
284C2×#107C2×D4312yes—
285C2×#106C2×D4312yes—
286C2×C4⋊C8—431no—
287C2×#102—432no—
288C2×#102—432no—
289C2×#97C2×D4311yescharacteristic
290#395—861no—
291#395—861no—
292#391—861no—
293#389—432no—
294#389—432no—
295#384—861no—
296C4×C4⋊C4—431no—
297#340—432no—
298#340—432no—
299#340—432no—
300#336—432no—
301#336—432no—
302C43#2585011yescharacteristic, complemented, Sylow
303C22×Q8—21no—
304C22×Q8—431no—
305C22×Q8—861no—
306C22×D4—431no—
307C23×C4—861no—
308C23×C4—861no—
309C23×C4—861no—
310C22×C8—432no—
311C22×C8C22×D4312yes—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—862no—
315C4⋊Q8—862no—
316#125—22no—
317#125—862no—
318#125—862no—
319#125C22⋊D4314yes—
320C22⋊Q8—868no—
321C22⋊Q8—868no—
322C22⋊Q8—868no—
323C4×Q8—862no—
324C4×Q8—862no—
325C4×D4—434no—
326C4×D4—862no—
327C4×D4—862no—
328C4×D4—862no—
329C4×D4—862no—
330C4×D4—862no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—431no—
335C2×C4⋊C4—432no—
336C2×C4⋊C4—431no—
337C2×C4⋊C4—431no—
338C2×C4⋊C4—431no—
339C2×C4⋊C4—868no—
340C2×C4⋊C4—861no—
341C2×C4⋊C4—861no—
342C2×C4⋊C4—861no—
343C2×C4⋊C4—862no—
344C2×C4⋊C4—862no—
345C2×C4⋊C4C22×D4312yes—
346C2×C4⋊C4C22×D4311yescharacteristic
347C2×C4⋊C4C22×D4311yescharacteristic
348C2×C22⋊C4—861no—
349C2×C22⋊C4—861no—
350C2×C22⋊C4—861no—
351C2×C42—431no—
352C2×C42—431no—
353C2×C42—861no—
354C2×C42C22×D4311yescharacteristic
355#107—24no—
356#107—864no—
357#106—24no—
358C4⋊C8—862no—
359#102—438no—
360#102—864no—
361#97—22no—
362#95—862no—
363#95—862no—
364#95—861no—
365C24—861no—
366C2×Q8—42no—
367C2×Q8—82no—
368C2×Q8—862no—
369C2×Q8—864no—
370C2×Q8—862no—
371C2×Q8—3442no—
372C2×Q8—1721no—
373C2×D4—432no—
374C2×D4—434no—
375C2×D4—861no—
376C2×D4—862no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—431no—
381C22×C4—431no—
382C22×C4—431no—
383C22×C4—431no—
384C22×C4—864no—
385C22×C4—864no—
386C22×C4—862no—
387C22×C4—861no—
388C22×C4—864no—
389C22×C4—3442no—
390C22×C4—3442no—
391C22×C4—1722no—
392C22×C4—1722no—
393C22×C4D4×D4311yescharacteristic
394C22×C4D4×D4311yescharacteristic
395C22×C4C23×D4311yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—862no—
399C2×C8—864no—
400C2×C8Q8×D4314yes—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—432no—
409C4⋊C4—434no—
410C4⋊C4—864no—
411C4⋊C4—864no—
412C4⋊C4—864no—
413C4⋊C4—862no—
414C4⋊C4—862no—
415C4⋊C4—862no—
416C4⋊C4—861no—
417C4⋊C4—862no—
418C4⋊C4—3442no—
419C4⋊C4—3442no—
420C4⋊C4C2×C22⋊D4312yes—
421C4⋊C4C2×C22⋊D4312yes—
422C22⋊C4—864no—
423C22⋊C4—864no—
424C22⋊C4—864no—
425C42—861no—
426C42—861no—
427C42—861no—
428C42—861no—
429C42—1722no—
430C42D4×D4312yes—
431C42C2×C22⋊D4312yes—
432C23—864no—
433C23—862no—
434C23—1721no—
435C23—1721no—
436C23—1722no—
437C23C2×D4×D4311yescharacteristic
438Q8—84no—
439Q8—3444no—
440Q8—1724no—
441D4—434no—
442D4—864no—
443D4—862no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—432no—
456C2×C4—861no—
457C2×C4—861no—
458C2×C4—864no—
459C2×C4—861no—
460C2×C4—861no—
461C2×C4—861no—
462C2×C4—862no—
463C2×C4—861no—
464C2×C4—861no—
465C2×C4—862no—
466C2×C4—862no—
467C2×C4—3444no—
468C2×C4—3444no—
469C2×C4—1728no—
470C2×C4—1722no—
471C2×C4—1722no—
472C2×C4—1722no—
473C2×C4#1306329312yes—
474C2×C4#1306329312yes—
475C2×C4C2×D4×D4311yescharacteristic
476C2×C4C22×C22⋊D4311yescharacteristic
477C2×C4C2×Q8×D4312yes—
478C2×C4#1306337312yes—
479C8—24no—
480C8—1722no—
481C22—864no—
482C22—1724no—
483C22—1722no—
484C22—1722no—
485C22—1724no—
486C22C2×#1306329311yescharacteristic
487C22#1458356412yes—
488C22#137×D4311yescharacteristic
489C22#1458369811yescharacteristic
490C22#1458370312yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—862no—
497C4—1724no—
498C4—1721no—
499C4—1721no—
500C4—1721no—
501C4—1722no—
502C4C4⋊Q8×D4311yescharacteristic
503C4#1458347411yescharacteristic
504C4Q8×C22⋊D4312yes—
505C2—1724no—
506C2#1913823611yescharacteristic
507C2#1914385111yescharacteristic
508C2#1914386211yescharacteristic
509C2#1914750212yes—
510C2#1914755012yes—
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#501C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenterC2×#501D43
hypocenterC43#25850
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. #19139804C2semidirect
4. #19135935C2semidirect
5. #19135921C2semidirect
6. #19135553C2semidirect
7. #19135535C2semidirect
11. C2×C43×#501C2semidirect
88. #13063247Q8semidirect
251. C2×#501D43semidirect
302. C43#25850semidirect

Maximal subgroups

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

subgroupindex
C2×#145836052
C2×#145831762
#191398042
#191359352
#191359212
#191355532
#191355352
#191348532
#191348392
#191344212
C2×C43×#5012
#2585043