Finite Group Databasebeta

← #27278718

Subgroups — #27278718

Order 13568.

566 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
1#20283826C211yesmaximal, characteristic, complemented
2#20275510C211yesmaximal, characteristic, complemented
3#20273106C211yesmaximal, characteristic, complemented
4#20273097C211yesmaximal, characteristic, complemented
5#20272566C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7C4×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic
8Q8×#88322—21no—
9D4×#88322—22no—
10D4×#88322—21no—
11#16810676—21no—
12#16810676C2211yescharacteristic
13#16810676C2211yescharacteristic
14C2×C4⋊#88322—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C22⋊#88322—22no—
17C2×C4×#88322C2211yescharacteristic
18Q8⋊#88322—21no—
19D4⋊#88322—21no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C2×AGL(1,ℤ/8)×C53C411yescharacteristic
23C4×#41×C53C2211yescharacteristic
24C2×C4×D4×C53C2211yescharacteristic
25C53×#442—21no—
26C53×#440—21no—
27C4×SD16×C53—21no—
28C4×D8×C53—21no—
29C53×#428C411yescharacteristic
30C53×#418C411yescharacteristic
31C53×#417C411yescharacteristic
32C4×M16×C53C2211yescharacteristic
33C22×#88322—22no—
34C22×#88322—21no—
35C22×#88322D412yes—
36C4⋊#88322—22no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322D411yescharacteristic
41C4⋊#88322D411yescharacteristic
42C22⋊#88322—22no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—44no—
46C4×#88322—22no—
47C4×#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C4×#88322—21no—
51C2×#41×C53C2×C411yescharacteristic
52C22×D4×C53C2×C411yescharacteristic
53C23×C4×C53—21no—
54AGL(1,ℤ/8)×C53—22no—
55AGL(1,ℤ/8)×C53C814yescomplemented
56C2×SD16×C53—21no—
57C2×D8×C53—21no—
58C2×M16×C53C2×C411yescharacteristic
59C2×M16×C53C2×C411yescharacteristic
60C4×Q8×C53D411yescharacteristic
61C4×Q8×C53D411yescharacteristic
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53D411yescharacteristic
69C4×D4×C53D411yescharacteristic
70#117×C53—21no—
71#117×C53C2×C411yescharacteristic
72C2×C4⋊C4×C53C2×C411yescharacteristic
73C2×C22⋊C4×C53—21no—
74C2×C42×C53—21no—
75C2×C42×C53C2311yescharacteristic
76#107×C53—21no—
77#106×C53—21no—
78#103×C53—21no—
79#102×C53—21no—
80#102×C53—21no—
81#102×C53—21no—
82#97×C53—21no—
83C4×C8×C53—21no—
84C24×C53—21no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53C2×C812yes—
88#41×C53M1612yes—
89C2×Q8×C53C22⋊C411yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53C22⋊C411yescharacteristic
97C2×D4×C53C2×C812yescomplemented
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4111yescharacteristic
109SD16×C53—24no—
110D8×C53—24no—
111M16×C53—21no—
112M16×C53—21no—
113M16×C53—21no—
114M16×C53C2×C812yescomplemented
115C2×C8×C53—21no—
116C2×C8×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53C22⋊C411yescharacteristic
122C4⋊C4×C53C22⋊C411yescharacteristic
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C42×C53—21no—
129C42×C53—21no—
130C42×C53—21no—
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C2×#88322—22no—
136C2×#88322—21no—
137C2×#88322—22no—
138C2×#88322—21no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—42no—
142C2×#88322—41no—
143C2×#88322—82no—
144C23×C53—21no—
145C23×C53—22no—
146C23×C53—42no—
147C23×C53—42no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53C4×D411yescharacteristic
151Q8×C53—21no—
152Q8×C53C22⋊C812yescomplemented
153D4×C53—22no—
154D4×C53—22no—
155D4×C53—22no—
156D4×C53—21no—
157D4×C53—22no—
158D4×C53—41no—
159D4×C53—41no—
160D4×C53C22⋊C812yescomplemented
161C2×C4×C53—21no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—81no—
174C2×C4×C53—82no—
175C2×C4×C53—81no—
176C2×C4×C53—81no—
177C2×C4×C53C2×C22⋊C411yescharacteristic
178C2×C4×C53C4×D411yescharacteristic
179C2×C4×C53#12011yescharacteristic
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12311yescharacteristic
183C2×C4×C53C22×C811yescharacteristic
184C2×C4×C53C2×M1611yescharacteristic
185C2×C4×C53#13111yescharacteristic
186C2×C4×C53#13111yescharacteristic
187C8×C53—22no—
188C8×C53—41no—
189#88322—42no—
190#88322—42no—
191#88322—41no—
192#88322—41no—
193#88322—81no—
194#8954—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—22no—
198C22×C53—44no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—42no—
202C22×C53—82no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53#38611yescharacteristic
206C22×C53C8×D411yescharacteristic
207C22×C53#43511yescharacteristic
208C4×C53—22no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—81no—
215C4×C53C2×C22⋊C811yescharacteristic
216C4×C53#40811yescharacteristic
217C4×C53C8×D411yescharacteristic
218C4×C53#43511yescharacteristic
219C4×C53#45311yescharacteristic
220C4×C53#45411yescharacteristic
221#2988—531no—
222C4×AGL(1,ℤ/8)D5311yescharacteristic
223#1954—531no—
224#1478—531no—
225#1476—531no—
226#1283—531no—
227#1278—531no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—42no—
231C2×C53—82no—
232C2×C53#186911yescharacteristic
233C2×C53#188111yescharacteristic
234C2×C53#188311yescharacteristic
235C2×AGL(1,ℤ/8)Dic5311yescharacteristic
236C4×#41C2×D5311yescharacteristic
237C2×C4×D4C2×D5311yescharacteristic
238C8×Q8—1061no—
239#442—21no—
240#440—21no—
241C4×SD16—21no—
242C4×D8—21no—
243C8×D4—1062no—
244C8×D4—1061no—
245#431—531no—
246#431—531no—
247#431—1061no—
248#428Dic5311yescharacteristic
249C2×C4⋊C8—531no—
250C2×C4⋊C8—1061no—
251#418Dic5311yescharacteristic
252#417Dic5311yescharacteristic
253C2×C22⋊C8—1062no—
254C4×M16C2×D5311yescharacteristic
255C2×C4×C8—531no—
256Q8⋊C8—1061no—
257D4⋊C8—1061no—
258D4⋊C8—1061no—
259D4⋊C8—1061no—
260C53#895411yescharacteristic, complemented, Sylow
261C2×#41C2×Dic5311yescharacteristic
262C22×D4C2×Dic5311yescharacteristic
263C23×C4—21no—
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)#8832214yescomplemented
266C2×SD16—21no—
267C2×D8—21no—
268C2×M16C2×Dic5311yescharacteristic
269C2×M16C2×Dic5311yescharacteristic
270C22×C8—532no—
271C22×C8—1062no—
272C22×C8—1061no—
273C4×Q8C22⋊D5311yescharacteristic
274C4×Q8C22⋊D5311yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C22⋊D5311yescharacteristic
282C4×D4C22⋊D5311yescharacteristic
283#117—21no—
284#117C2×Dic5311yescharacteristic
285C2×C4⋊C4C2×Dic5311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C22×D5311yescharacteristic
289#107—21no—
290#106—21no—
291C4⋊C8—531no—
292C4⋊C8—531no—
293C4⋊C8—1062no—
294C4⋊C8—1061no—
295C4⋊C8—1061no—
296C4⋊C8—1061no—
297#103—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—1062no—
302C22⋊C8—1061no—
303C22⋊C8—1061no—
304C22⋊C8—2124no—
305#97—21no—
306C4×C8—21no—
307C4×C8—1062no—
308C4×C8—1061no—
309C4×C8—1061no—
310C4×C8—1061no—
311C4×C8—1061no—
312C24—21no—
313#41—21no—
314#41—21no—
315#41C2×#8832212yes—
316#41#1171991712yes—
317C2×Q8C22⋊Dic5311yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4C2×#8832212yescomplemented
325C2×D4C22⋊Dic5311yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4D4×D5311yescharacteristic
335C22×C4#1171994711yescharacteristic
336C22×C4C22×Dic5311yescharacteristic
337SD16—24no—
338D8—24no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16C2×#8832212yescomplemented
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—1062no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—1062no—
349C2×C8—4242no—
350C2×C8—2121no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8—2122no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4C22⋊Dic5311yescharacteristic
359C4⋊C4C22⋊Dic5311yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42D4×D5311yescharacteristic
369C42#1171994711yescharacteristic
370C42C2×C22⋊D5311yescharacteristic
371C42C2×C22⋊D5311yescharacteristic
372C23—21no—
373C23—22no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23D4×Dic5311yescharacteristic
379Q8—21no—
380Q8C22⋊#8832212yescomplemented
381D4—22no—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—22no—
386D4—41no—
387D4—41no—
388D4C22⋊#8832212yescomplemented
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4C2×C22⋊Dic5311yescharacteristic
406C2×C4D4×Dic5311yescharacteristic
407C2×C4#1313318511yescharacteristic
408C2×C4#1313318911yescharacteristic
409C2×C4#1313318911yescharacteristic
410C2×C4#1313319811yescharacteristic
411C2×C4C22×#8832211yescharacteristic
412C2×C4C2×#1171991711yescharacteristic
413C2×C4#1313322411yescharacteristic
414C2×C4#1313322411yescharacteristic
415C8—22no—
416C8—41no—
417C8—4241no—
418C8—2122no—
419C8—2121no—
420C8—2122no—
421C8—2121no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—44no—
426C22—41no—
427C22—41no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22#1681050311yescharacteristic
433C22D4×#8832211yescharacteristic
434C22#1681069511yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4C2×C22⋊#8832211yescharacteristic
443C4#1681059011yescharacteristic
444C4D4×#8832211yescharacteristic
445C4#1681069511yescharacteristic
446C4#1681078911yescharacteristic
447C4#1681079411yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—82no—
452C2#2027500811yescharacteristic
453C2#2027507911yescharacteristic
454C2#2027509311yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC4×AGL(1,ℤ/8)×C53C2
socleC22×C53#386
prime radicalC2×AGL(1,ℤ/8)×C53C4
cosocleC2×C42×C53C23
hypercenterC4×AGL(1,ℤ/8)D53
hypocenterC53#8954
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20283826C2semidirect
2. #20275510C2semidirect
3. #20273106C2semidirect
4. #20273097C2semidirect
5. #20272566C2semidirect
6. #20272553C2semidirect
55. AGL(1,ℤ/8)×C53C8semidirect
97. C2×D4×C53C2×C8semidirect
114. M16×C53C2×C8semidirect
152. Q8×C53C22⋊C8semidirect
160. D4×C53C22⋊C8semidirect
260. C53#8954semidirect
265. AGL(1,ℤ/8)#88322semidirect
324. C2×D4C2×#88322semidirect
342. M16C2×#88322semidirect
380. Q8C22⋊#88322semidirect
388. D4C22⋊#88322semidirect

Maximal subgroups

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

subgroupindex
#202838262
#202755102
#202731062
#202730972
#202725662
#202725532
C4×AGL(1,ℤ/8)×C532
#895453