Finite Group Databasebeta

← #27280271

Subgroups — #27280271

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810818C211yesmaximal, characteristic
2C2×D4×#88322C211yesmaximal, characteristic, complemented
3#20275520C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5#20272598C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C53×#1802C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C23×#88322—21no—
10C2×#13133214C2211yescharacteristic
11#16810818—22no—
12#16810818C414yescomplemented
13D4×#88322—22no—
14D4×#88322—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137—21no—
18C2×#13133137—21no—
19C2×#13133137—21no—
20C2×C22⋊#88322—21no—
21C2×C4×#88322C2211yescharacteristic
22#16810366—22no—
23C2×C4×D4×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×#102×C53—21no—
26C53×#384C2211yescharacteristic
27C53×#340—21no—
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719922—24no—
32C2×#11719922—42no—
33C2×#11719922—42no—
34C22×#88322—24no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322—42no—
38C22×#88322—42no—
39C22×#88322D411yescharacteristic
40C22×#88322D411yescharacteristic
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44#13133214C2×C412yes—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322C2×C412yes—
48C4⋊#88322D412yes—
49#13133137—24no—
50#13133137—42no—
51#13133137—42no—
52C22⋊#88322—24no—
53C4×#88322—21no—
54C4×#88322—21no—
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—21no—
57C22×C8×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53C2×C412yes—
61C4×D4×C53D412yes—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70#102×C53—42no—
71C24×C53—21no—
72C23×D53—41no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53C22⋊C412yes—
77C2×D4×C53D812yes—
78C2×D4×C53SD1612yes—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—81no—
105C2×D212—81no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—81no—
115C4⋊C4×C53C22⋊C412yes—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122#11719922—48no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—22no—
126C2×#88322—21no—
127C2×#88322—22no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—48no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322D812yescomplemented
136C23×C53—22no—
137C23×C53—22no—
138C23×C53—22no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53#12111yescharacteristic
144C22×D53—42no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—81no—
148C22×D53—81no—
149C22×D53—81no—
150C22×D53—81no—
151D4×C53—22no—
152D4×C53—22no—
153D4×C53—22no—
154D4×C53#10214yescomplemented
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53—81no—
167C2×C4×C53C2×C22⋊C411yescharacteristic
168C2×C4×C53C4×D411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12311yescharacteristic
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53C2×SD1611yescharacteristic
176C2×C4×C53#13511yescharacteristic
177C2×C4×C53#13511yescharacteristic
178C2×C4×C53#13511yescharacteristic
179C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181D212—42no—
182D212—42no—
183D212—42no—
184D212—42no—
185D212—82no—
186D212—82no—
187D212—81no—
188D212—81no—
189D212—81no—
190D212—81no—
191C8×C53—81no—
192#88322—22no—
193#88322—42no—
194#88322—42no—
195#88322—41no—
196#9221—531nomaximal, Sylow
197C22×C53—24no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—42no—
201C22×C53—42no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53#38611yescharacteristic
205C22×C53#45911yescharacteristic
206C22×C53#46311yescharacteristic
207C22×C53#46511yescharacteristic
208C22×C53#46611yescharacteristic
209C22×C53#47111yescharacteristic
210C22×C53#47211yescharacteristic
211C2×D53—82no—
212C2×D53—82no—
213C2×D53—82no—
214C2×D53—161no—
215C2×D53—161no—
216C2×D53—161no—
217C2×D53—161no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—42no—
221C4×C53—161no—
222C4×C53C2×#10211yescharacteristic
223C4×C53#41611yescharacteristic
224C4×C53C4×D811yescharacteristic
225C4×C53#44211yescharacteristic
226C4×C53#44711yescharacteristic
227C4×C53#44911yescharacteristic
228C4×C53#46511yescharacteristic
229C4×C53#46611yescharacteristic
230C2×#459—531no—
231C2×C8×D4—531no—
232#1957—531no—
233#1802D5311yescharacteristic, complemented
234C2×#331—531no—
235#1296—531no—
236#1277—531no—
237C2×C53—44no—
238C2×C53#166511yescharacteristic
239C2×C53#166711yescharacteristic
240C2×C53#167711yescharacteristic
241C2×C53#168111yescharacteristic
242C2×C53#187911yescharacteristic
243C2×C53#189311yescharacteristic
244C2×C53#190711yescharacteristic
245D53—162no—
246C22×D8—1061no—
247C23×C8—1061no—
248C2×#127—531no—
249C2×C4×D4C2×D5311yescharacteristic
250#459—534no—
251#459—1062no—
252C8×D4—1062no—
253C8×D4—1062no—
254C2×C4⋊C8—531no—
255C2×C4⋊C8—531no—
256C2×C4⋊C8C2×D5311yescharacteristic
257C2×#102—21no—
258C2×#102—1061no—
259C2×#102—1061no—
260C2×#102—1061no—
261C2×C22⋊C8—1061no—
262C2×C4×C8—531no—
263#384C2×D5311yescharacteristic
264#340—21no—
265#331—1062no—
266C53#922111yescharacteristic, complemented, Sylow
267C22×D4—1061no—
268C22×D4—1061no—
269C22×D4—1061no—
270C22×D4C22⋊D5311yescharacteristic
271C23×C4—21no—
272C2×D8—1064no—
273C2×D8—2122no—
274C2×D8—2122no—
275C22×C8—21no—
276C22×C8—531no—
277C22×C8—531no—
278C22×C8—1064no—
279C22×C8—1061no—
280C22×C8—1061no—
281C22×C8—2122no—
282C22×C8—2122no—
283#127—532no—
284#127—1061no—
285#127—1061no—
286#127—1061no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C4×D5312yes—
290C4×D4D21212yes—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C22⋊D5311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C22×D5311yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—532no—
300C4⋊C8—532no—
301C4⋊C8—1061no—
302C4⋊C8—1061no—
303#102—42no—
304#102—1064no—
305#102—2122no—
306#102—2122no—
307C22⋊C8—1064no—
308C4×C8—1061no—
309C4×C8—1061no—
310C24—21no—
311C24—2121no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—1062no—
318C2×D4—4241no—
319C2×D4—4241no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C2×D4—2121no—
329C2×D4—2121no—
330C2×D4—2121no—
331C2×D4#1171992112yes—
332C2×D4#1171992212yes—
333C2×D4#1171992412yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4D4×D5311yescharacteristic
343C22×C4#1171994711yescharacteristic
344C22×C4C2×C22⋊D5311yescharacteristic
345D8—2128no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—532no—
350C2×C8—1061no—
351C2×C8—1061no—
352C2×C8—1062no—
353C2×C8—1061no—
354C2×C8—1062no—
355C2×C8—1061no—
356C2×C8—1061no—
357C2×C8—2128no—
358C2×C8—2121no—
359C2×C8—2121no—
360C2×C8—2121no—
361C2×C8—2121no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—81no—
368C4⋊C4#1171992112yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42C2×C4×D5311yescharacteristic
372C42C2×D21211yescharacteristic
373C42D4×D5311yescharacteristic
374C42#1171994711yescharacteristic
375C23—22no—
376C23—22no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—4241no—
383C23—4241no—
384C23—4241no—
385C23—4241no—
386C23—4241no—
387C23—4241no—
388C23—2122no—
389C23#1313318911yescharacteristic
390D4—22no—
391D4—22no—
392D4—22no—
393D4—4242no—
394D4—4242no—
395D4—4241no—
396D4—4241no—
397D4—4241no—
398D4—4241no—
399D4—2122no—
400D4—2122no—
401D4—2122no—
402D4—2122no—
403D4#1313313714yescomplemented
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4C2×#1171992111yescharacteristic
417C2×C4#1313317711yescharacteristic
418C2×C4#1313317711yescharacteristic
419C2×C4#1313318711yescharacteristic
420C2×C4#1313318911yescharacteristic
421C2×C4#1313320111yescharacteristic
422C2×C4D8×D5311yescharacteristic
423C2×C4C2×#1171992211yescharacteristic
424C2×C4C2×#1171992411yescharacteristic
425C2×C4#1313323711yescharacteristic
426C2×C4#1313323711yescharacteristic
427C2×C4#1313323811yescharacteristic
428C2×C4#1313324111yescharacteristic
429C2×C4#1313324411yescharacteristic
430C8—81no—
431C8—1062no—
432C8—2122no—
433C8—2121no—
434C8—2122no—
435C22—24no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—8481no—
443C22—8481no—
444C22—8481no—
445C22—8481no—
446C22—4242no—
447C22—4242no—
448C22—4242no—
449C22#1681050011yescharacteristic
450C22#1681081611yescharacteristic
451C22#1681084011yescharacteristic
452C22#1681085411yescharacteristic
453C22#1681085711yescharacteristic
454C22#1681088811yescharacteristic
455C22#1681089111yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—161no—
460C4C2×#1313313711yescharacteristic
461C4#1681061711yescharacteristic
462C4#1681070311yescharacteristic
463C4#1681073211yescharacteristic
464C4#1681075811yescharacteristic
465C4#1681076911yescharacteristic
466C4#1681085411yescharacteristic
467C4#1681085711yescharacteristic
468C2—44no—
469C2—8482no—
470C2#2027405811yescharacteristic
471C2#2027406811yescharacteristic
472C2#2027412611yescharacteristic
473C2#2027415411yescharacteristic
474C2#2027506111yescharacteristic
475C2#2027514111yescharacteristic
476C2#2027522111yescharacteristic
subgroupG/H
centerC23#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1802C2
socleC23×C53#121
prime radicalC2×#16810818C2
cosocleC2×C42×C53C23
hypercenter#1802D53
hypocenterC53#9221
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×D4×#88322C2semidirect
3. #20275520C2semidirect
4. C2×#16810366C2semidirect
5. #20272598C2semidirect
6. #20272549C2semidirect
7. C53×#1802C2semidirect
12. #16810818C4semidirect
135. C2×#88322D8semidirect
154. D4×C53#102semidirect
233. #1802D53semidirect
266. C53#9221semidirect
403. D4#13133137semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
C2×D4×#883222
#202755202
C2×#168103662
#202725982
#202725492
C53×#18022
#922153