Finite Group Databasebeta

← #27286418

Subgroups — #27286418

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic
2#20279633C211yesmaximal, characteristic
3#20275677C211yesmaximal, characteristic, complemented
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#462C211yesmaximal, characteristic, complemented
8C2×#13133193—21no—
9C2×C4×D212C2211yescharacteristic
10#16810900—22no—
11#16810900—22no—
12C2×#13133150—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15C2×C4×#88322C2211yescharacteristic
16#16810545—22no—
17#16810525—21no—
18#16810521—21no—
19#16810493C2211yescharacteristic
20#16810480—21no—
21#16810385—22no—
22Q8⋊#88322—22no—
23#16810350—22no—
24C22×Q16×C53—21no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4×Q8×C53C2211yescharacteristic
27C53×#462—22no—
28C53×#462—22no—
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×#103×C53—21no—
31C22×D212D411yescharacteristic
32C22×C4×D53—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133193—42no—
36#13133193—42no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×C4⋊Dic53D412yes—
45C2×#11719919—22no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×#11719921—21no—
49#13133150—24no—
50C4⋊#88322—21no—
51C4⋊#88322D412yes—
52#13133137—24no—
53C4×#88322—21no—
54C4×#88322—21no—
55#13133124—22no—
56C22×Q8×C53—21no—
57C22×Q8×C53D411yescharacteristic
58C2×Q16×C53—24no—
59C2×Q16×C53—42no—
60C2×Q16×C53—42no—
61C22×C8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53D412yes—
66C4×Q8×C53—21no—
67C4×Q8×C53—21no—
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C42×C53—21no—
75C2×C42×C53C2311yescharacteristic
76C4⋊C8×C53—21no—
77C4⋊C8×C53D412yes—
78#103×C53—24no—
79C23×D53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—22no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—41no—
87C2×Q8×C53—41no—
88C2×Q8×C53—41no—
89C2×Q8×C53—41no—
90C2×Q8×C53—41no—
91C2×Q8×C53D812yes—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—41no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—21no—
99C22×Dic53—22no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212Q1612yescomplemented
106C2×C4×D53—41no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C4×D53—41no—
111C2×C4×D53—41no—
112Q16×C53—48no—
113C2×C8×C53—22no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C42×C53—41no—
125C42×C53—41no—
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53#4111yescharacteristic
130#11719921—44no—
131#11719921—41no—
132#11719921—41no—
133#11719921—82no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—41no—
137C4⋊Dic53—41no—
138C4⋊Dic53—41no—
139C4⋊Dic53—41no—
140C4⋊Dic53—42no—
141#11719919—44no—
142C2×#88322—22no—
143C2×#88322—22no—
144C2×#88322—41no—
145C2×#88322—41no—
146C2×#88322—41no—
147C2×#88322—41no—
148C23×C53#12011yescharacteristic
149C22×D53—22no—
150C22×D53—41no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—41no—
155C22×D53—41no—
156Q8×C53—22no—
157Q8×C53—22no—
158Q8×C53—41no—
159Q8×C53—41no—
160Q8×C53—42no—
161Q8×C53—42no—
162Q8×C53—81no—
163Q8×C53—81no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—22no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12011yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12711yescharacteristic
181C2×C4×C53C2×D811yescharacteristic
182C2×C4×C53C2×Q1611yescharacteristic
183C2×C4×C53#13511yescharacteristic
184C2×C4×C53#13511yescharacteristic
185C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C53#13711yescharacteristic
188C2×C4×C53#13711yescharacteristic
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195D212—22no—
196D212—22no—
197D212—41no—
198D212—41no—
199C4×D53—81no—
200C4×D53—81no—
201C4×D53—81no—
202C4×D53—81no—
203C8×C53—42no—
204#88322—42no—
205#88322—42no—
206#10233—531nomaximal, Sylow
207C22×C53#39211yescharacteristic
208C22×C53#44711yescharacteristic
209C22×C53#44911yescharacteristic
210C22×C53#45111yescharacteristic
211C22×C53#45211yescharacteristic
212C22×C53#45511yescharacteristic
213C22×C53#45611yescharacteristic
214C2×D53—42no—
215C2×D53—42no—
216C2×D53—42no—
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—81no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—42no—
224C4×C53—82no—
225C4×C53—81no—
226C4×C53#44711yescharacteristic
227C4×C53#44911yescharacteristic
228C4×C53#45111yescharacteristic
229C4×C53#45211yescharacteristic
230C4×C53#46611yescharacteristic
231C4×C53#46711yescharacteristic
232C4×C53#46911yescharacteristic
233C4×C53#47011yescharacteristic
234Dic53—82no—
235Dic53—81no—
236C2×#474—531no—
237C2×#462D5311yescharacteristic, complemented
238#2538—531no—
239#1990—531no—
240C2×Q8⋊C8—531no—
241C2×D4⋊C8—531no—
242#1296—531no—
243C2×C53#164111yescharacteristic
244C2×C53#164411yescharacteristic
245C2×C53#164911yescharacteristic
246C2×C53#165211yescharacteristic
247C2×C53#199911yescharacteristic
248C2×C53#200111yescharacteristic
249C2×C53#200711yescharacteristic
250D53—82no—
251C22×Q16—21no—
252C2×C4⋊Q8C2×D5311yescharacteristic
253C2×C22⋊Q8—1061no—
254C2×C4×Q8C2×D5311yescharacteristic
255C2×C4×D4—531no—
256#474—1062no—
257#474—1062no—
258#462—22no—
259#462—22no—
260C2×#107—1061no—
261C2×C4⋊C8—531no—
262C2×C4⋊C8C2×D5311yescharacteristic
263C2×#103—21no—
264C2×#102—1061no—
265C2×C4×C8—531no—
266#396—1062no—
267#390—1061no—
268#389—1061no—
269#384—531no—
270#380—1061no—
271#340—1062no—
272Q8⋊C8—1062no—
273D4⋊C8—1062no—
274C53#1023311yescharacteristic, complemented, Sylow
275C22×Q8—21no—
276C22×Q8C22⋊D5311yescharacteristic
277C22×D4—531no—
278C23×C4—1061no—
279C2×Q16—24no—
280C2×Q16—42no—
281C2×Q16—42no—
282C22×C8—21no—
283C22×C8—1061no—
284C22×C8—1061no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8C22⋊D5312yes—
289C22⋊Q8—2122no—
290C22⋊Q8—2122no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×D4—1061no—
296C4×D4—1061no—
297C4×D4—1061no—
298C4×D4—1061no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—532no—
304C2×C4⋊C4—1062no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4C22⋊D5311yescharacteristic
308C2×C22⋊C4—1062no—
309C2×C22⋊C4—1062no—
310C2×C22⋊C4—1061no—
311C2×C42—21no—
312C2×C42C22×D5311yescharacteristic
313#107—1064no—
314C4⋊C8—21no—
315C4⋊C8—532no—
316C4⋊C8—1061no—
317C4⋊C8C22⋊D5312yes—
318#103—24no—
319#102—1064no—
320C4×C8—1061no—
321C4×C8—1061no—
322#95—1062no—
323C24—1061no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8#1171992212yes—
336C2×D4—532no—
337C2×D4—1061no—
338C2×D4—1061no—
339C2×D4—1061no—
340C2×D4—1061no—
341C2×D4—1061no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4—1061no—
346C22×C4—1062no—
347C22×C4—2121no—
348C22×C4—2121no—
349C22×C4—2121no—
350C22×C4—2121no—
351C22×C4—2121no—
352C22×C4—2121no—
353C22×C4D4×D5311yescharacteristic
354C22×C4D4×D5311yescharacteristic
355C22×C4C2×C22⋊D5311yescharacteristic
356Q16—48no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—1062no—
361C2×C8—1062no—
362C2×C8—2121no—
363C2×C8—2121no—
364C2×C8—2121no—
365C2×C8—2121no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4—2122no—
377C4⋊C4—2124no—
378C4⋊C4—2121no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4—2121no—
382C22⋊C4—4242no—
383C22⋊C4—2124no—
384C22⋊C4—2121no—
385C22⋊C4—2121no—
386C42—41no—
387C42—41no—
388C42D4×D5311yescharacteristic
389C42#1171994711yescharacteristic
390C42C2×C22⋊D5311yescharacteristic
391C42C2×C22⋊D5311yescharacteristic
392C23—1062no—
393C23—2121no—
394C23—2121no—
395C23—2121no—
396C23—2121no—
397C23—2121no—
398C23—2121no—
399C23#1313318511yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—42no—
405Q8—42no—
406Q8—81no—
407Q8—81no—
408D4—1062no—
409D4—1062no—
410D4—2121no—
411D4—2121no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—4241no—
424C2×C4—4241no—
425C2×C4—4241no—
426C2×C4—4241no—
427C2×C4—2122no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2122no—
431C2×C4—2121no—
432C2×C4—2122no—
433C2×C4#1313318511yescharacteristic
434C2×C4#1313318611yescharacteristic
435C2×C4#1313318811yescharacteristic
436C2×C4#1313318911yescharacteristic
437C2×C4#1313318911yescharacteristic
438C2×C4#1313321311yescharacteristic
439C2×C4C2×#1171992211yescharacteristic
440C2×C4Q16×D5311yescharacteristic
441C2×C4#1313323711yescharacteristic
442C2×C4#1313324011yescharacteristic
443C2×C4#1313324211yescharacteristic
444C2×C4#1313324311yescharacteristic
445C2×C4#1313324911yescharacteristic
446C2×C4#1313324911yescharacteristic
447C8—42no—
448C8—2122no—
449C8—2122no—
450C22—4241no—
451C22—4241no—
452C22—4241no—
453C22—4241no—
454C22—2122no—
455C22—2122no—
456C22—2122no—
457C22#1681053111yescharacteristic
458C22#1681075611yescharacteristic
459C22#1681076611yescharacteristic
460C22#1681077711yescharacteristic
461C22#1681078411yescharacteristic
462C22#1681079911yescharacteristic
463C22#1681080411yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—82no—
469C4—4242no—
470C4—4241no—
471C4#1681075711yescharacteristic
472C4#1681076811yescharacteristic
473C4#1681077711yescharacteristic
474C4#1681078411yescharacteristic
475C4#1681085711yescharacteristic
476C4#1681086111yescharacteristic
477C4#1681087411yescharacteristic
478C4#1681087811yescharacteristic
479C2—4242no—
480C2#2027390611yescharacteristic
481C2#2027392711yescharacteristic
482C2#2027396211yescharacteristic
483C2#2027398311yescharacteristic
484C2#2027571111yescharacteristic
485C2#2027572211yescharacteristic
486C2#2027576011yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#462C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenterC2×#462D53
hypocenterC53#10233
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #20275677C2semidirect
4. C2×Q8⋊#88322C2semidirect
6. #20272599C2semidirect
7. C2×C53×#462C2semidirect
105. C2×D212Q16semidirect
237. C2×#462D53semidirect
274. C53#10233semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
#202796332
#202756772
C2×Q8⋊#883222
C2×#168103502
#202725992
C2×C53×#4622
#1023353