Finite Group Databasebeta

← #24988385

Subgroups — #24988385

Order 11008.

858 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19134898C212yesmaximal, complemented
2#19134893C211yesmaximal, characteristic, complemented
3C4×#13063240C211yesmaximal, characteristic, complemented
4#19133086C212yesmaximal, complemented
5C43×#3460C211yesmaximal, characteristic
6C22×#10621877C2211yescharacteristic
7C2×#13063252—22no—
8C2×#13063240—21no—
9C2×C22⋊(C43⋊C8)—21no—
10C2×C22⋊(C43⋊C8)C2211yescharacteristic
11C4×C22⋊Dic43C2212yes—
12#14582666—21no—
13#14582658—22no—
14#14582654—21no—
15#14582636—22no—
16#14582633—22no—
17C43×#544C411yescharacteristic, complemented
18C43×#542—22no—
19C43×#540—22no—
20C43×#538—21no—
21C43×#538C412yescomplemented
22C43×#537C411yescharacteristic, complemented
23C43×#535—21no—
24C43×#532—22no—
25C43×#529—22no—
26C43×#525—21no—
27C43×#525C2212yes—
28C4×#41×C43—21no—
29C2×C4×D4×C43C2211yescharacteristic
30C2×#10621877—22no—
31C2×#10621877—42no—
32C22×C43⋊C8—21no—
33C22×C43⋊C8—21no—
34C2×C22⋊Dic43—22no—
35C2×C4×Dic43—22no—
36#13063252—44no—
37#13063240—24no—
38C22⋊(C43⋊C8)—22no—
39C22⋊(C43⋊C8)—42no—
40C43⋊#95—22no—
41C2×#41×C43—21no—
42C2×#41×C43—21no—
43C2×#41×C43D412yes—
44C22×D4×C43D411yescharacteristic
45C23×C4×C43D411yescharacteristic
46C23×C4×C43C2311yescharacteristic
47C4⋊Q8×C43—22no—
48#127×C43—22no—
49#126×C43—22no—
50#126×C43—22no—
51#125×C43—22no—
52#124×C43—22no—
53#124×C43—22no—
54#124×C43—22no—
55#123×C43—22no—
56#123×C43—22no—
57#123×C43—22no—
58#123×C43—22no—
59C22⋊Q8×C43—21no—
60C22⋊Q8×C43—21no—
61C22⋊Q8×C43—22no—
62C22⋊Q8×C43C2×C412yes—
63C22⋊Q8×C43D412yes—
64#121×C43—21no—
65#121×C43—22no—
66#121×C43—22no—
67#121×C43—22no—
68#121×C43—22no—
69#121×C43—22no—
70#121×C43—21no—
71#121×C43C2×C412yes—
72#121×C43D412yes—
73#120×C43—22no—
74#120×C43—22no—
75C4×Q8×C43—21no—
76C4×Q8×C43—21no—
77C4×D4×C43—21no—
78C4×D4×C43—21no—
79C4×D4×C43—22no—
80C4×D4×C43—22no—
81C4×D4×C43—22no—
82C4×D4×C43—22no—
83C4×D4×C43—22no—
84C4×D4×C43—22no—
85#117×C43—22no—
86#117×C43—21no—
87#117×C43—21no—
88#117×C43D412yes—
89C2×C4⋊C4×C43C2×C411yescharacteristic
90C2×C22⋊C4×C43C2×C411yescharacteristic
91C2×C22⋊C4×C43D411yescharacteristic
92C2×C42×C43—22no—
93C2×C42×C43D411yescharacteristic
94C24×C43C22⋊C411yescharacteristic
95C24×C43C2×D411yescharacteristic
96#41×C43—42no—
97#41×C43—42no—
98#41×C43—42no—
99#41×C43—42no—
100#41×C43—81no—
101#41×C43—81no—
102C2×Q8×C43—21no—
103C2×Q8×C43—21no—
104C2×Q8×C43C22⋊C412yes—
105C2×D4×C43—21no—
106C2×D4×C43—22no—
107C2×D4×C43—22no—
108C2×D4×C43—21no—
109C2×D4×C43—22no—
110C2×D4×C43—22no—
111C2×D4×C43—22no—
112C2×D4×C43—22no—
113C2×D4×C43—22no—
114C2×D4×C43—41no—
115C2×D4×C43C22⋊C412yes—
116C22×C4×C43—22no—
117C22×C4×C43—22no—
118C22×C4×C43—22no—
119C22×C4×C43—21no—
120C22×C4×C43—21no—
121C22×C4×C43—41no—
122C22×C4×C43—41no—
123C22×C4×C43—41no—
124C22×C4×C43—41no—
125C22×C4×C43C22⋊C411yescharacteristic
126C22×C4×C43C22⋊C411yescharacteristic
127C22×C4×C43C22⋊C411yescharacteristic
128C22×C4×C43C22×C411yescharacteristic
129C22×C4×C43C2×D411yescharacteristic
130C22×C4×C43C2×D412yes—
131C22×C4×C43C2×D412yes—
132C22×Dic43—42no—
133C4⋊C4×C43—21no—
134C4⋊C4×C43—21no—
135C4⋊C4×C43—22no—
136C4⋊C4×C43—21no—
137C4⋊C4×C43—21no—
138C4⋊C4×C43—22no—
139C4⋊C4×C43—22no—
140C4⋊C4×C43—22no—
141C4⋊C4×C43—22no—
142C4⋊C4×C43C22⋊C412yes—
143C4⋊C4×C43C22⋊C412yes—
144C22⋊C4×C43—22no—
145C22⋊C4×C43—22no—
146C22⋊C4×C43—22no—
147C22⋊C4×C43—22no—
148C22⋊C4×C43—22no—
149C22⋊C4×C43—22no—
150C22⋊C4×C43—22no—
151C22⋊C4×C43—22no—
152C22⋊C4×C43—22no—
153C22⋊C4×C43—22no—
154C22⋊C4×C43—22no—
155C42×C43—22no—
156C42×C43—22no—
157C42×C43—21no—
158C42×C43—22no—
159C42×C43—21no—
160C22⋊Dic43—24no—
161C22⋊Dic43—42no—
162C4×Dic43—42no—
163C4×Dic43—42no—
164#10621877—24no—
165#10621877—42no—
166C2×C43⋊C8—22no—
167C2×C43⋊C8—41no—
168C2×C43⋊C8—41no—
169C2×C43⋊C8—41no—
170C2×C43⋊C8—41no—
171C2×C43⋊C8—41no—
172C23×C43—21no—
173C23×C43—21no—
174C23×C43—21no—
175C23×C43—22no—
176C23×C43—21no—
177C23×C43—21no—
178C23×C43—41no—
179C23×C43—41no—
180C23×C43—41no—
181C23×C43—41no—
182C23×C43C2×C22⋊C411yescharacteristic
183C23×C43#12012yes—
184Q8×C43—42no—
185Q8×C43—42no—
186Q8×C43—81no—
187D4×C43—44no—
188D4×C43—42no—
189D4×C43—42no—
190D4×C43—42no—
191D4×C43—42no—
192D4×C43—82no—
193D4×C43—82no—
194D4×C43—82no—
195D4×C43—81no—
196D4×C43—82no—
197C2×C4×C43—21no—
198C2×C4×C43—21no—
199C2×C4×C43—21no—
200C2×C4×C43—22no—
201C2×C4×C43—21no—
202C2×C4×C43—22no—
203C2×C4×C43—21no—
204C2×C4×C43—22no—
205C2×C4×C43—22no—
206C2×C4×C43—22no—
207C2×C4×C43—21no—
208C2×C4×C43—41no—
209C2×C4×C43—42no—
210C2×C4×C43—44no—
211C2×C4×C43—41no—
212C2×C4×C43—42no—
213C2×C4×C43—41no—
214C2×C4×C43—42no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—81no—
218C2×C4×C43—82no—
219C2×C4×C43#10412yescomplemented
220C2×C4×C43C2×C22⋊C412yes—
221C2×C4×C43#12011yescharacteristic
222C2×C4×C43#12011yescharacteristic
223C2×Dic43—44no—
224C2×Dic43—82no—
225C2×Dic43—82no—
226C43⋊C8—42no—
227C43⋊C8—81no—
228#8400—431nomaximal, Sylow
229C22×C43—21no—
230C22×C43—21no—
231C22×C43—22no—
232C22×C43—22no—
233C22×C43—22no—
234C22×C43—42no—
235C22×C43—41no—
236C22×C43—42no—
237C22×C43—41no—
238C22×C43—42no—
239C22×C43—41no—
240C22×C43—42no—
241C22×C43—81no—
242C22×C43—81no—
243C22×C43—81no—
244C22×C43#37911yescharacteristic
245C22×C43C2×#10411yescharacteristic
246C22×C43#42111yescharacteristic
247C4×C43—22no—
248C4×C43—21no—
249C4×C43—22no—
250C4×C43—41no—
251C4×C43—42no—
252C4×C43—42no—
253C4×C43—41no—
254C4×C43—42no—
255C4×C43B(4,𝔽2)11yescharacteristic
256C4×C43#45811yescharacteristic
257Dic43—84no—
258#3460D4311yescharacteristic
259#1871—432no—
260#1870—431no—
261C4×#99—431no—
262#1501—432no—
263C2×C43—21no—
264C2×C43—21no—
265C2×C43—42no—
266C2×C43—82no—
267C2×C43—81no—
268C2×C43#178911yescharacteristic
269C2×C43#179011yescharacteristic
270C2×C43#179411yescharacteristic
271C22×M16—431no—
272#544Dic4311yescharacteristic, complemented
273#542—22no—
274#540—22no—
275#538—21no—
276#538Dic4312yescomplemented
277#537Dic4311yescharacteristic, complemented
278#535—21no—
279#532—22no—
280#529—22no—
281#525—21no—
282#525C2×D4312yes—
283C4×#41—21no—
284C2×C4×D4C2×D4311yescharacteristic
285C2×#104—862no—
286C2×#99—861no—
287C2×C22⋊C8—431no—
288C2×C22⋊C8—861no—
289C4×C22⋊C4—432no—
290#342—861no—
291#339—862no—
292#336—861no—
293#328—862no—
294#327—862no—
295C43#840011yescharacteristic, complemented, Sylow
296C2×#41—21no—
297C2×#41—21no—
298C2×#41C22⋊D4312yes—
299C22×D4C22⋊D4311yescharacteristic
300C23×C4C22⋊D4311yescharacteristic
301C23×C4C22×D4311yescharacteristic
302C2×M16—862no—
303C2×M16—1722no—
304C22×C8—861no—
305C22×C8—861no—
306C4⋊Q8—22no—
307#127—22no—
308#126—22no—
309#126—22no—
310#125—22no—
311#124—22no—
312#124—22no—
313#124—22no—
314#123—22no—
315#123—22no—
316#123—22no—
317#123—22no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C22⋊Q8—22no—
321C22⋊Q8C2×Dic4312yes—
322C22⋊Q8C22⋊D4312yes—
323#121—21no—
324#121—22no—
325#121—22no—
326#121—22no—
327#121—22no—
328#121—22no—
329#121—21no—
330#121C2×Dic4312yes—
331#121C22⋊D4312yes—
332#120—22no—
333#120—22no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344#117—22no—
345#117—21no—
346#117—21no—
347#117C22⋊D4312yes—
348C2×C4⋊C4C2×Dic4311yescharacteristic
349C2×C22⋊C4—862no—
350C2×C22⋊C4C2×Dic4311yescharacteristic
351C2×C22⋊C4C22⋊D4311yescharacteristic
352C2×C42—22no—
353C2×C42—862no—
354C2×C42C22⋊D4311yescharacteristic
355#104—1724no—
356#99—864no—
357C22⋊C8—862no—
358C22⋊C8—1722no—
359#95—862no—
360C24C22⋊Dic4311yescharacteristic
361C24C2×C22⋊D4311yescharacteristic
362#41—42no—
363#41—42no—
364#41—42no—
365#41—42no—
366#41—81no—
367#41—81no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8C22⋊Dic4312yes—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—41no—
381C2×D4C22⋊Dic4312yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—1722no—
392C22×C4C22⋊Dic4311yescharacteristic
393C22×C4C22⋊Dic4311yescharacteristic
394C22×C4C22⋊Dic4311yescharacteristic
395C22×C4C22×Dic4311yescharacteristic
396C22×C4C2×C22⋊D4312yes—
397C22×C4C2×C22⋊D4311yescharacteristic
398C22×C4C2×C22⋊D4312yes—
399M16—864no—
400M16—1722no—
401C2×C8—862no—
402C2×C8—1721no—
403C2×C8—1721no—
404C2×C8—1721no—
405C2×C8—1721no—
406C2×C8—1721no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4C22⋊Dic4312yes—
417C4⋊C4C22⋊Dic4312yes—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—864no—
430C22⋊C4—1722no—
431C42—22no—
432C42—22no—
433C42—21no—
434C42—21no—
435C42—22no—
436C42—1722no—
437C42—1722no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—22no—
443C23—21no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23C2×C22⋊Dic4311yescharacteristic
449C23#1306329412yes—
450Q8—42no—
451Q8—42no—
452Q8—81no—
453D4—44no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—81no—
461D4—82no—
462D4—82no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—3442no—
486C2×C4—3442no—
487C2×C4—1724no—
488C2×C4#1306325212yescomplemented
489C2×C4C2×C22⋊Dic4312yes—
490C2×C4#1306329411yescharacteristic
491C2×C4#1306329411yescharacteristic
492C8—3441no—
493C8—1722no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22#1458275511yescharacteristic
510C22C2×#1306325211yescharacteristic
511C22#1458291611yescharacteristic
512C4—22no—
513C4—21no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—3444no—
521C4#1458308611yescharacteristic
522C4#1458308911yescharacteristic
523C2—21no—
524C2—21no—
525C2—42no—
526C2—81no—
527C2—82no—
528C2#1913453911yescharacteristic
529C2#1913454311yescharacteristic
530C2#1913455911yescharacteristic
subgroupG/H
centerC2×C4#13063294
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
Fitting, prime radicalC43×#3460C2
socleC22×C43#379
cosocleC23×C4×C43C23
hypercenter#3460D43
hypocenterC43#8400
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3460

Complements

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

NH ≅ G/Ntype
1. #19134898C2semidirect
2. #19134893C2semidirect
3. C4×#13063240C2semidirect
4. #19133086C2semidirect
17. C43×#544C4semidirect
21. C43×#538C4semidirect
22. C43×#537C4semidirect
219. C2×C4×C43#104semidirect
272. #544Dic43semidirect
276. #538Dic43semidirect
277. #537Dic43semidirect
295. C43#8400semidirect
488. C2×C4#13063252semidirect

Maximal subgroups

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

subgroupindex
#191348982
#191348932
C4×#130632402
#191330862
C43×#34602
#840043