Finite Group Databasebeta

← #24991365

Subgroups — #24991365

Order 11008.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583099C211yesmaximal, characteristic
2#19135522C211yesmaximal, characteristic
3#19135400C211yesmaximal, characteristic
4C2×Q8⋊(C43⋊C8)C211yesmaximal, characteristic, complemented
5#19132478C211yesmaximal, characteristic, complemented
6#19132436C211yesmaximal, characteristic, complemented
7C43×#2355C211yesmaximal, characteristic, complemented
8C22×#10621884—21no—
9C2×#13063322C2211yescharacteristic
10#14583099—22no—
11#14583099C414yescomplemented
12C2×C4⋊(C43⋊C8)C2211yescharacteristic
13C2×C4⋊(C43⋊C8)C2211yescharacteristic
14C2×#13063245—21no—
15C2×#13063245—21no—
16C2×#13063245—21no—
17C2×#13063245—22no—
18C2×C4×C43⋊C8C2211yescharacteristic
19Q8⋊(C43⋊C8)—22no—
20C2×C4×Q8×C43C2211yescharacteristic
21C43×#391—21no—
22C43×#387—22no—
23C43×#384C2211yescharacteristic
24C43×#382—21no—
25C4×C4⋊C4×C43—21no—
26C4×C4⋊C4×C43—21no—
27C4×C4⋊C4×C43C2211yescharacteristic
28C22×D172—21no—
29C22×D172—21no—
30C22×D172—21no—
31C2×#10621884—24no—
32C2×#10621884—42no—
33C2×#10621884—42no—
34C22×C43⋊C8—21no—
35C22×C43⋊C8—21no—
36C22×C43⋊C8—21no—
37#13063322—21no—
38#13063322—21no—
39#13063322—21no—
40#13063322C2×C412yes—
41C4⋊(C43⋊C8)—21no—
42C4⋊(C43⋊C8)—21no—
43C4⋊(C43⋊C8)—21no—
44C4⋊(C43⋊C8)C2×C412yes—
45#13063245—24no—
46#13063245—44no—
47#13063245—42no—
48#13063245—42no—
49C4×C43⋊C8—21no—
50C4×C43⋊C8—21no—
51C22×Q8×C43D411yescharacteristic
52C4×Q8×C43—21no—
53C4×Q8×C43—21no—
54C4×Q8×C43C2×C412yes—
55C4×Q8×C43D412yes—
56C2×C4⋊C4×C43—21no—
57C2×C4⋊C4×C43—21no—
58C2×C4⋊C4×C43—21no—
59C2×C4⋊C4×C43D411yescharacteristic
60C2×C4⋊C4×C43D411yescharacteristic
61C2×C4⋊C4×C43D411yescharacteristic
62C2×C4⋊C4×C43D412yes—
63C2×C42×C43—21no—
64C2×C42×C43—21no—
65C2×C42×C43—22no—
66C2×C42×C43C2311yescharacteristic
67#95×C43—22no—
68#95×C43—22no—
69#95×C43—22no—
70#95×C43—21no—
71C23×D43—41no—
72C2×Q8×C43—21no—
73C2×Q8×C43—21no—
74C2×Q8×C43—21no—
75C2×Q8×C43—21no—
76C2×Q8×C43—21no—
77C2×Q8×C43C22⋊C412yes—
78C22×C4×C43—21no—
79C22×C4×C43—21no—
80C22×C4×C43—21no—
81C22×C4×C43—21no—
82C22×C4×C43—22no—
83C22×C4×C43C2×D411yescharacteristic
84C22×C4×C43C2×D411yescharacteristic
85C22×C4×C43C2×D411yescharacteristic
86C2×D172—22no—
87C2×D172—22no—
88C2×D172—22no—
89C2×D172—41no—
90C2×D172—41no—
91C2×D172—41no—
92C2×D172—42no—
93C2×D172—42no—
94C2×D172—41no—
95C2×D172—41no—
96C2×D172—41no—
97C2×D172—41no—
98C2×D172—41no—
99C2×D172—41no—
100C2×D172—41no—
101C2×D172—41no—
102C4⋊C4×C43—22no—
103C4⋊C4×C43—22no—
104C4⋊C4×C43—21no—
105C4⋊C4×C43—21no—
106C4⋊C4×C43—21no—
107C4⋊C4×C43—21no—
108C4⋊C4×C43—42no—
109C4⋊C4×C43—41no—
110C4⋊C4×C43—41no—
111C4⋊C4×C43—41no—
112C4⋊C4×C43—41no—
113C4⋊C4×C43C22⋊C412yes—
114C4⋊C4×C43SD1612yescomplemented
115C42×C43—22no—
116C42×C43—22no—
117C42×C43—41no—
118C42×C43—41no—
119C42×C43—82no—
120C42×C43C22×C411yescharacteristic
121C42×C43C2×D411yescharacteristic
122C42×C43#4111yescharacteristic
123C42×C43#4111yescharacteristic
124#10621884—48no—
125C2×C43⋊C8—22no—
126C2×C43⋊C8—22no—
127C2×C43⋊C8—41no—
128C2×C43⋊C8—41no—
129C2×C43⋊C8—41no—
130C2×C43⋊C8—41no—
131C2×C43⋊C8—41no—
132C2×C43⋊C8—41no—
133C2×C43⋊C8—41no—
134C23×C43#12011yescharacteristic
135C22×D43—42no—
136C22×D43—81no—
137C22×D43—81no—
138C22×D43—81no—
139C22×D43—81no—
140C22×D43—81no—
141C22×D43—81no—
142Q8×C43—22no—
143Q8×C43—22no—
144Q8×C43—41no—
145Q8×C43—41no—
146C2×C4×C43—21no—
147C2×C4×C43—21no—
148C2×C4×C43—22no—
149C2×C4×C43—22no—
150C2×C4×C43—22no—
151C2×C4×C43—41no—
152C2×C4×C43—41no—
153C2×C4×C43—41no—
154C2×C4×C43—41no—
155C2×C4×C43—41no—
156C2×C4×C43—41no—
157C2×C4×C43—41no—
158C2×C4×C43—41no—
159C2×C4×C43—42no—
160C2×C4×C43—42no—
161C2×C4×C43—42no—
162C2×C4×C43—41no—
163C2×C4×C43C2×C22⋊C411yescharacteristic
164C2×C4×C43C4×D411yescharacteristic
165C2×C4×C43C4×D411yescharacteristic
166C2×C4×C43#12111yescharacteristic
167C2×C4×C43#12111yescharacteristic
168C2×C4×C43#12311yescharacteristic
169C2×C4×C43C2×SD1611yescharacteristic
170C2×C4×C43#13511yescharacteristic
171C2×C4×C43#13511yescharacteristic
172C2×C4×C43#13511yescharacteristic
173C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C43#13711yescharacteristic
177D172—44no—
178D172—42no—
179D172—42no—
180D172—42no—
181D172—42no—
182D172—82no—
183D172—81no—
184D172—81no—
185D172—81no—
186D172—81no—
187C43⋊C8—42no—
188C43⋊C8—42no—
189C43⋊C8—81no—
190#8978—431nomaximal, Sylow
191C22×C43#38611yescharacteristic
192C22×C43#44811yescharacteristic
193C22×C43#44911yescharacteristic
194C22×C43#44911yescharacteristic
195C22×C43#44911yescharacteristic
196C22×C43#45311yescharacteristic
197C22×C43#45411yescharacteristic
198C2×D43—82no—
199C2×D43—82no—
200C2×D43—82no—
201C2×D43—161no—
202C2×D43—161no—
203C2×D43—161no—
204C2×D43—161no—
205C4×C43—21no—
206C4×C43—21no—
207C4×C43—42no—
208C4×C43—42no—
209C4×C43—42no—
210C4×C43—82no—
211C4×C43—81no—
212C4×C43#41611yescharacteristic
213C4×C43#41811yescharacteristic
214C4×C43C4×SD1611yescharacteristic
215C4×C43#44011yescharacteristic
216C4×C43#46011yescharacteristic
217C4×C43#46411yescharacteristic
218C4×C43#48211yescharacteristic
219C4×C43#48211yescharacteristic
220C2×#460—431no—
221#2355D4311yescharacteristic, complemented
222#1982—431no—
223#1957—431no—
224C2×Q8⋊C8—431no—
225#1296—431no—
226#1279—431no—
227C2×C43#162011yescharacteristic
228C2×C43#162211yescharacteristic
229C2×C43#162611yescharacteristic
230C2×C43#162811yescharacteristic
231C2×C43#187511yescharacteristic
232C2×C43#188011yescharacteristic
233C2×C43#188311yescharacteristic
234D43—162no—
235C22×SD16—861no—
236C2×#127—431no—
237C2×C4×Q8C2×D4311yescharacteristic
238#460—434no—
239#460—862no—
240C2×C4⋊C8—431no—
241C2×C4⋊C8—431no—
242C2×#102—861no—
243C2×#102—862no—
244C2×#102—861no—
245C2×#102—861no—
246C2×C4×C8—431no—
247#391—21no—
248#387—22no—
249#384C2×D4311yescharacteristic
250#382—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4C2×D4311yescharacteristic
254Q8⋊C8—862no—
255C43#897811yescharacteristic, complemented, Sylow
256C22×Q8C22⋊D4311yescharacteristic
257C22×D4—861no—
258C22×D4—861no—
259C22×D4—861no—
260C2×SD16—864no—
261C2×SD16—1722no—
262C2×SD16—1722no—
263C22×C8—861no—
264C22×C8—861no—
265C22×C8—861no—
266#127—432no—
267#127—861no—
268#127—861no—
269#127—861no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C4×D4312yes—
273C4×Q8D17212yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D4311yescharacteristic
278C2×C4⋊C4C22⋊D4311yescharacteristic
279C2×C4⋊C4C22⋊D4311yescharacteristic
280C2×C4⋊C4C22⋊D4312yes—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C22×D4311yescharacteristic
285C4⋊C8—432no—
286C4⋊C8—861no—
287C4⋊C8—861no—
288C4⋊C8—861no—
289#102—864no—
290#102—1722no—
291#102—1724no—
292#102—1722no—
293C4×C8—861no—
294C4×C8—861no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299C24—1721no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8(C2×D43)⋊C412yes—
306C2×D4—862no—
307C2×D4—862no—
308C2×D4—862no—
309C2×D4—1722no—
310C2×D4—1722no—
311C2×D4—1721no—
312C2×D4—1721no—
313C2×D4—1721no—
314C2×D4—1721no—
315C2×D4—1721no—
316C2×D4—1721no—
317C2×D4—1721no—
318C2×D4—1721no—
319C2×D4—1721no—
320C2×D4—1721no—
321C2×D4—1721no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4C2×C22⋊D4311yescharacteristic
328C22×C4C2×C22⋊D4311yescharacteristic
329C22×C4C2×C22⋊D4311yescharacteristic
330SD16—1728no—
331C2×C8—862no—
332C2×C8—862no—
333C2×C8—1721no—
334C2×C8—1721no—
335C2×C8—1721no—
336C2×C8—1721no—
337C2×C8—1721no—
338C2×C8—1721no—
339C2×C8—1721no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4(C2×D43)⋊C412yes—
352C4⋊C4#1062188412yescomplemented
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42C2×C4×D4311yescharacteristic
359C42C2×D17211yescharacteristic
360C42#1062189811yescharacteristic
361C42#1062189811yescharacteristic
362C23—3441no—
363C23—3441no—
364C23—3441no—
365C23—3441no—
366C23—3441no—
367C23—3441no—
368C23—1722no—
369C23#1306329411yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—3442no—
375D4—3441no—
376D4—3441no—
377D4—3441no—
378D4—3441no—
379D4—1724no—
380D4—1722no—
381D4—1722no—
382D4—1722no—
383D4—1722no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4C2×(C2×D43)⋊C411yescharacteristic
402C2×C4C4×C22⋊D4311yescharacteristic
403C2×C4C4×C22⋊D4311yescharacteristic
404C2×C4#1306329911yescharacteristic
405C2×C4#1306329911yescharacteristic
406C2×C4#1306330811yescharacteristic
407C2×C4C2×#1062188411yescharacteristic
408C2×C4#1306334511yescharacteristic
409C2×C4#1306334511yescharacteristic
410C2×C4#1306334511yescharacteristic
411C2×C4#1306335011yescharacteristic
412C2×C4#1306335011yescharacteristic
413C2×C4#1306335011yescharacteristic
414C2×C4#1306335611yescharacteristic
415C8—3441no—
416C8—1722no—
417C8—1722no—
418C22—3442no—
419C22—3442no—
420C22—3442no—
421C22—6881no—
422C22—6881no—
423C22—6881no—
424C22—6881no—
425C22#1458277811yescharacteristic
426C22#1458303811yescharacteristic
427C22#1458304411yescharacteristic
428C22#1458304411yescharacteristic
429C22#1458304411yescharacteristic
430C22#1458306711yescharacteristic
431C22#1458307211yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—82no—
439C4#1458289311yescharacteristic
440C4#1458290311yescharacteristic
441C4C4×#1062188411yescharacteristic
442C4#1458299711yescharacteristic
443C4#1458309911yescharacteristic
444C4#1458312111yescharacteristic
445C4#1458322311yescharacteristic
446C4#1458322311yescharacteristic
447C2—6882no—
448C2#1913365411yescharacteristic
449C2#1913366711yescharacteristic
450C2#1913369111yescharacteristic
451C2#1913370511yescharacteristic
452C2#1913491711yescharacteristic
453C2#1913495011yescharacteristic
454C2#1913497111yescharacteristic
subgroupG/H
centerC23#13063294
commutatorC42×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#2355C2
socleC23×C43#120
prime radicalC2×#13063322C22
cosocleC2×C42×C43C23
hypercenter#2355D43
hypocenterC43#8978
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2355

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊(C43⋊C8)C2semidirect
5. #19132478C2semidirect
6. #19132436C2semidirect
7. C43×#2355C2semidirect
11. #14583099C4semidirect
114. C4⋊C4×C43SD16semidirect
221. #2355D43semidirect
255. C43#8978semidirect
352. C4⋊C4#10621884semidirect

Maximal subgroups

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

subgroupindex
C2×#145830992
#191355222
#191354002
C2×Q8⋊(C43⋊C8)2
#191324782
#191324362
C43×#23552
#897843