Finite Group Databasebeta

← #25151031

Subgroups — #25151031

Order 11008.

968 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583510C211yesmaximal, characteristic
2C2×#14583305C211yesmaximal, characteristic, complemented
3#19140417C211yesmaximal, characteristic
4#19135796C212yesmaximal
5#19135789C212yesmaximal
6#19135560C211yesmaximal, characteristic, complemented
7#19135416C211yesmaximal, characteristic, complemented
8#19134861C212yesmaximal, complemented
9#19134847C212yesmaximal, complemented
10#19134362C211yesmaximal, characteristic
11C2×C43×#487C211yesmaximal, characteristic, complemented
12#14583510—24no—
13#14583510—24no—
14C2×#13063320—21no—
15C2×#13063320—21no—
16C2×#13063317C2211yescharacteristic
17C2×#13063315—21no—
18C2×#13063315—21no—
19C2×#13063301C2211yescharacteristic
20C2×#13063301C2211yescharacteristic
21C2×Q8×Dic43C2211yescharacteristic
22C2×Q8×Dic43C2211yescharacteristic
23C2×#13063282C2211yescharacteristic
24#14583305—24no—
25C2×#13063258C2212yes—
26C2×#13063256C2212yes—
27C2×#13063249C2212yes—
28C2×#13063249C2212yes—
29C2×(C2×D43)⋊C8C2212yes—
30C2×#13063236C2211yescharacteristic
31#14582837—21no—
32#14582834—21no—
33#14582818—21no—
34#14582810—21no—
35#14582795C2212yes—
36#14582765C2211yescharacteristic
37#14582756C2212yes—
38#14582662C2212yes—
39#14582662C2212yes—
40#14582653C2212yes—
41C2×C4⋊Q8×C43C2211yescharacteristic
42C2×C4⋊Q8×C43C2211yescharacteristic
43C43×#487—24no—
44C2×#103×C43C2212yes—
45C2×#103×C43C2212yes—
46C2×C4×C8×C43C2211yescharacteristic
47C22×C4×D43C2311yescharacteristic
48C22×C43⋊C8C2312yes—
49#13063320—28no—
50#13063320—28no—
51#13063317—22no—
52#13063317D414yes—
53#13063315—28no—
54#13063315—28no—
55#13063301—22no—
56#13063301—22no—
57#13063301—22no—
58#13063301—22no—
59Q8×Dic43—22no—
60Q8×Dic43—22no—
61Q8×Dic43—22no—
62Q8×Dic43—22no—
63#13063282—22no—
64#13063282—22no—
65C2×C4⋊Dic43—21no—
66C2×C4⋊Dic43—21no—
67C2×C4⋊Dic43C2312yes—
68C2×C4⋊Dic43C2311yescharacteristic
69C2×C4⋊Dic43C2311yescharacteristic
70C2×Dic43⋊C4—21no—
71C2×Dic43⋊C4—21no—
72C2×Dic43⋊C4—21no—
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—22no—
75C2×Dic43⋊C4D411yescharacteristic
76C2×Dic43⋊C4D411yescharacteristic
77C2×(C2×D43)⋊C4—21no—
78C2×(C2×D43)⋊C4—21no—
79C2×(C2×D43)⋊C4—22no—
80C2×(C2×D43)⋊C4D411yescharacteristic
81C2×(C2×D43)⋊C4D411yescharacteristic
82C2×C4×Dic43—21no—
83C2×C4×Dic43—21no—
84C2×C4×Dic43C2311yescharacteristic
85#13063258—24no—
86#13063256—24no—
87#13063249—24no—
88#13063249—24no—
89(C2×D43)⋊C8—24no—
90#13063236—22no—
91C43⋊#95—21no—
92C43⋊#95—21no—
93C43⋊#95—22no—
94C43⋊#95—22no—
95C22×Q8×C43—21no—
96C22×Q8×C43C2311yescharacteristic
97C22×Q8×C43C2311yescharacteristic
98C22×C8×C43C2312yes—
99C4⋊Q8×C43—22no—
100C4⋊Q8×C43—22no—
101C4⋊Q8×C43—22no—
102C4⋊Q8×C43D414yes—
103C2×C4⋊C4×C43—21no—
104C2×C4⋊C4×C43—21no—
105C2×C4⋊C4×C43—21no—
106C2×C4⋊C4×C43C2312yes—
107C2×C42×C43C2311yescharacteristic
108#103×C43—24no—
109#103×C43—24no—
110C4×C8×C43—22no—
111C23×D43C2×D411yescharacteristic
112C2×Q8×C43—21no—
113C2×Q8×C43—21no—
114C2×Q8×C43—22no—
115C2×Q8×C43—22no—
116C2×Q8×C43—22no—
117C2×Q8×C43—22no—
118C2×Q8×C43—42no—
119C2×Q8×C43—82no—
120C2×Q8×C43C2×D412yes—
121C2×Q8×C43C2×D412yes—
122C22×C4×C43—21no—
123C22×C4×C43—21no—
124C22×C4×C43—22no—
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43C2411yescharacteristic
128C22×Dic43—21no—
129C22×Dic43—21no—
130C22×Dic43—22no—
131C22×Dic43C2×D411yescharacteristic
132C22×Dic43C2×D411yescharacteristic
133C22×Dic43C2×D411yescharacteristic
134C2×C4×D43—21no—
135C2×C4×D43—21no—
136C2×C4×D43—22no—
137C2×C4×D43—22no—
138C2×C8×C43—22no—
139C2×C8×C43—22no—
140C2×C8×C43—22no—
141C4⋊C4×C43—24no—
142C4⋊C4×C43—24no—
143C4⋊C4×C43—22no—
144C4⋊C4×C43—22no—
145C4⋊C4×C43—42no—
146C42×C43C2×D412yes—
147C42×C43C2×D412yes—
148(C2×D43)⋊C4—24no—
149(C2×D43)⋊C4—24no—
150(C2×D43)⋊C4—22no—
151(C2×D43)⋊C4—22no—
152(C2×D43)⋊C4—82no—
153C4⋊Dic43—24no—
154C4⋊Dic43—24no—
155C4⋊Dic43—21no—
156C4⋊Dic43—21no—
157C4⋊Dic43—22no—
158C4⋊Dic43—22no—
159C4⋊Dic43C2×D412yes—
160C4⋊Dic43C2×D412yes—
161Dic43⋊C4—24no—
162Dic43⋊C4—24no—
163Dic43⋊C4—24no—
164Dic43⋊C4—24no—
165Dic43⋊C4—22no—
166Dic43⋊C4—22no—
167Dic43⋊C4—82no—
168C4×Dic43—24no—
169C4×Dic43—24no—
170C4×Dic43—21no—
171C4×Dic43—21no—
172C2×C43⋊C8—22no—
173C2×C43⋊C8—22no—
174C2×C43⋊C8#4114yes—
175C23×C43C22×D411yescharacteristic
176C22×D43—21no—
177C22×D43—21no—
178C22×D43—21no—
179C22×D43—22no—
180C22×D43—22no—
181Q8×C43—24no—
182Q8×C43—24no—
183Q8×C43—42no—
184Q8×C43—42no—
185Q8×C43—84no—
186C2×C4×C43—22no—
187C2×C4×C43—22no—
188C2×C4×C43—22no—
189C2×C4×C43—22no—
190C2×C4×C43—41no—
191C2×C4×C43—41no—
192C2×C4×C43—41no—
193C2×C4×C43—41no—
194C2×C4×C43—42no—
195C2×C4×C43—42no—
196C2×C4×C43—42no—
197C2×C4×C43#12012yes—
198C2×C4×C43#12012yes—
199C2×C4×C43C22×D411yescharacteristic
200C2×C4×C43C22×D411yescharacteristic
201C2×C4×C43C2×#4112yes—
202C2×C4×C432−1+412yes—
203C2×Dic43—21no—
204C2×Dic43—21no—
205C2×Dic43—22no—
206C2×Dic43—22no—
207C2×Dic43—21no—
208C2×Dic43—21no—
209C2×Dic43—21no—
210C2×Dic43—22no—
211C2×Dic43—22no—
212C2×Dic43—42no—
213C2×Dic43—42no—
214C2×Dic43—42no—
215C2×Dic43—42no—
216C2×Dic43—42no—
217C4×D43—42no—
218C4×D43—42no—
219C8×C43—42no—
220C43⋊C8—24no—
221#25686—431nomaximal, Sylow
222C22×C43C2×#12011yescharacteristic
223C22×C43#53211yescharacteristic
224C22×C43#53611yescharacteristic
225C22×C43#57712yes—
226C22×C43#57812yes—
227C2×D43—42no—
228C2×D43—41no—
229C2×D43—42no—
230C2×D43—41no—
231C2×D43—41no—
232C4×C43—22no—
233C4×C43—22no—
234C4×C43—42no—
235C4×C43—42no—
236C4×C43—82no—
237C4×C43#54712yes—
238C4×C43#54812yes—
239Dic43—41no—
240Dic43—41no—
241Dic43—42no—
242Dic43—41no—
243Dic43—42no—
244Dic43—82no—
245C2×#541—431no—
246C2×#501—431no—
247C2×#487D4311yescharacteristic, complemented
248#2628—431no—
249#2036—432no—
250#2035—432no—
251#1991—431no—
252#1960—431no—
253#1865—432no—
254#1863—432no—
255#1750—431no—
256C2×C43#243211yescharacteristic
257C2×C43#301011yescharacteristic
258C2×C43#301111yescharacteristic
259C2×C43#317812yes—
260C2×C43#324212yes—
261D43—81no—
262#541—864no—
263#541—864no—
264C2×C4⋊Q8C2×D4311yescharacteristic
265C2×C4⋊Q8C2×D4311yescharacteristic
266C2×#126—861no—
267C2×#126—861no—
268C2×#125—431no—
269C2×#125—861no—
270C2×#125—861no—
271C2×C22⋊Q8—431no—
272C2×C22⋊Q8—431no—
273C2×C4×Q8—431no—
274C2×C4×Q8—431no—
275C2×#117—431no—
276#501—864no—
277#487—24no—
278C2×#107—432no—
279C2×#106—432no—
280C2×#103—432no—
281C2×#103—432no—
282C2×#103C2×D4312yes—
283C2×#103C2×D4312yes—
284C2×C22⋊C8—432no—
285C2×#97—431no—
286C2×C4×C8C2×D4311yescharacteristic
287#400—861no—
288#399—861no—
289#395—861no—
290#393—861no—
291#389—432no—
292#382—431no—
293#380—432no—
294#340—432no—
295#340—432no—
296#336—432no—
297C43#2568611yescharacteristic, complemented, Sylow
298C22×Q8—21no—
299C22×Q8C22×D4311yescharacteristic
300C22×Q8C22×D4311yescharacteristic
301C23×C4—431no—
302C22×C8—432no—
303C22×C8C22×D4312yes—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8C22⋊D4314yes—
308#126—868no—
309#126—868no—
310#125—434no—
311#125—868no—
312#125—868no—
313#125—862no—
314C22⋊Q8—862no—
315C22⋊Q8—862no—
316C22⋊Q8—862no—
317C22⋊Q8—862no—
318C4×Q8—862no—
319C4×Q8—862no—
320C4×Q8—862no—
321C4×Q8—862no—
322#117—862no—
323#117—862no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—431no—
328C2×C4⋊C4—431no—
329C2×C4⋊C4—431no—
330C2×C4⋊C4—431no—
331C2×C4⋊C4—432no—
332C2×C4⋊C4—861no—
333C2×C4⋊C4—861no—
334C2×C4⋊C4—861no—
335C2×C4⋊C4—861no—
336C2×C4⋊C4—861no—
337C2×C4⋊C4—861no—
338C2×C4⋊C4—862no—
339C2×C4⋊C4C22×D4312yes—
340C2×C22⋊C4—431no—
341C2×C22⋊C4—431no—
342C2×C22⋊C4—861no—
343C2×C22⋊C4—861no—
344C2×C22⋊C4—862no—
345C2×C42—431no—
346C2×C42—861no—
347C2×C42—861no—
348C2×C42C22×D4311yescharacteristic
349#107—864no—
350#106—864no—
351#103—24no—
352#103—24no—
353#103—864no—
354#103—864no—
355C22⋊C8—864no—
356#97—862no—
357C4×C8—22no—
358#95—862no—
359#95—862no—
360#95—861no—
361#95—861no—
362C24—431no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—42no—
370C2×Q8—82no—
371C2×Q8C2×C22⋊D4312yes—
372C2×Q8C2×C22⋊D4312yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—431no—
377C22×C4—431no—
378C22×C4—431no—
379C22×C4—861no—
380C22×C4—861no—
381C22×C4—861no—
382C22×C4—861no—
383C22×C4—862no—
384C22×C4—862no—
385C22×C4—862no—
386C22×C4D4×D4311yescharacteristic
387C22×C4D4×D4311yescharacteristic
388C22×C4C23×D4311yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—434no—
393C2×C8—862no—
394C2×C8—862no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—432no—
401C4⋊C4—432no—
402C4⋊C4—864no—
403C4⋊C4—864no—
404C4⋊C4—864no—
405C4⋊C4—864no—
406C4⋊C4—862no—
407C4⋊C4—862no—
408C4⋊C4—864no—
409C4⋊C4—864no—
410C4⋊C4—861no—
411C4⋊C4—861no—
412C4⋊C4—862no—
413C4⋊C4—862no—
414C4⋊C4—3442no—
415C22⋊C4—864no—
416C22⋊C4—864no—
417C22⋊C4—862no—
418C22⋊C4—862no—
419C22⋊C4—3442no—
420C42—864no—
421C42—864no—
422C42—861no—
423C42—861no—
424C42D4×D4312yes—
425C42C2×C22⋊D4312yes—
426C23—861no—
427C23—861no—
428C23—862no—
429C23—862no—
430C23—861no—
431C23C2×D4×D4311yescharacteristic
432Q8—24no—
433Q8—24no—
434Q8—42no—
435Q8—42no—
436Q8—84no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—861no—
449C2×C4—861no—
450C2×C4—862no—
451C2×C4—862no—
452C2×C4—861no—
453C2×C4—861no—
454C2×C4—862no—
455C2×C4—862no—
456C2×C4—861no—
457C2×C4—1722no—
458C2×C4—1722no—
459C2×C4—1722no—
460C2×C4—1722no—
461C2×C4—1722no—
462C2×C4—1722no—
463C2×C4—1722no—
464C2×C4#1306329312yes—
465C2×C4#1306329312yes—
466C2×C4C2×D4×D4311yescharacteristic
467C2×C4C22×C22⋊D4311yescharacteristic
468C2×C4#41×D4312yes—
469C2×C4#1306337212yes—
470C8—42no—
471C8—864no—
472C22—1722no—
473C22—1722no—
474C22—1721no—
475C22—1721no—
476C22—1721no—
477C22C2×#1306329311yescharacteristic
478C22#1458344811yescharacteristic
479C22#1458346911yescharacteristic
480C22#1458370512yes—
481C22#1458371212yes—
482C4—22no—
483C4—22no—
484C4—42no—
485C4—42no—
486C4—82no—
487C4—3442no—
488C4—1721no—
489C4—1721no—
490C4—1722no—
491C4—1721no—
492C4—1722no—
493C4#1458354812yes—
494C4#1458355712yes—
495C2—3441no—
496C2#1913822611yescharacteristic
497C2#1914387711yescharacteristic
498C2#1914388611yescharacteristic
499C2#1914527012yes—
500C2#1914585612yes—
subgroupG/H
centerC23C2×D4×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#487C2
socleC23×C43C22×D4
prime radicalC23×D43C2×D4
hypercenterC2×#487D43
hypocenterC43#25686
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
2. C2×#14583305C2semidirect
6. #19135560C2semidirect
7. #19135416C2semidirect
8. #19134861C2semidirect
9. #19134847C2semidirect
11. C2×C43×#487C2semidirect
247. C2×#487D43semidirect
297. C43#25686semidirect

Maximal subgroups

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

subgroupindex
C2×#145835102
C2×#145833052
#191404172
#191357962
#191357892
#191355602
#191354162
#191348612
#191348472
#191343622
C2×C43×#4872
#2568643