Finite Group Databasebeta

← #14809916

Subgroups — #14809916

Order 2816.

1262 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13072261C212yesmaximal, complemented
2#13071981C211yesmaximal, characteristic
3#13069968C214yesmaximal, complemented
4#13069602C212yesmaximal, complemented
5#13068897C214yesmaximal
6C4×#66898C211yesmaximal, characteristic
7C11×#3501C211yesmaximal, characteristic
8C23×Dic22C2211yescharacteristic
9C11×#554—22no—
10C11×#554—24no—
11D4×Q8×C11—22no—
12C11×#544—21no—
13C11×#543—21no—
14C11×#543C2212yes—
15C11×#533—21no—
16C2×C11×C22⋊Q8—21no—
17C2×C11×C22⋊Q8C2212yes—
18C11×#518C2212yes—
19C2×C4×D4×C11C2211yescharacteristic
20C2×#66997—22no—
21C2×#66977—21no—
22#10622833C2212yes—
23C2×#66898C2212yes—
24#10622825C2211yescharacteristic
25C4×C22⋊Dic11C2212yes—
26#10622592—21no—
27#10622592—24no—
28#10622587—22no—
29#10622560—21no—
30#10622559C2218yes—
31#10622559C2212yes—
32#10622542—21no—
33#10622164C2218yes—
34#10622164C2212yes—
35C22×Q8×C11—21no—
36C22×Q8×C11D412yes—
37C22×D4×C11C2311yescharacteristic
38C23×C4×C11D411yescharacteristic
39C23×C4×C11C2311yescharacteristic
40C23×Dic11C2312yes—
41C22×Dic22—21no—
42C22×Dic22—22no—
43C22×Dic22—21no—
44C22×Dic22—22no—
45C22×Dic22—44no—
46C11×C4⋊Q8—21no—
47C11×C4⋊Q8—21no—
48C11×C4⋊Q8—24no—
49C11×#125—22no—
50C11×#125—24no—
51C11×C22⋊Q8—22no—
52C11×C22⋊Q8—22no—
53C11×C22⋊Q8—24no—
54C11×C22⋊Q8—24no—
55C11×C22⋊Q8—22no—
56C11×C22⋊Q8—22no—
57C11×C22⋊Q8—22no—
58C11×C22⋊Q8—22no—
59C11×C22⋊Q8—22no—
60C11×C22⋊Q8—22no—
61C11×C22⋊Q8—24no—
62C11×C22⋊Q8—24no—
63C4×Q8×C11—22no—
64C4×Q8×C11—22no—
65C4×D4×C11—22no—
66C4×D4×C11—22no—
67C4×D4×C11—24no—
68C4×D4×C11—24no—
69C11×#117—22no—
70C2×C11×C4⋊C4—21no—
71C2×C11×C4⋊C4—22no—
72C2×C11×C4⋊C4D412yes—
73C2×C11×C4⋊C4D411yescharacteristic
74C2×C11×C4⋊C4D414yes—
75C2×C11×C22⋊C4—22no—
76C2×C11×C22⋊C4D411yescharacteristic
77C2×C11×C22⋊C4C2311yescharacteristic
78C2×C11×C22⋊C4C2314yes—
79C2×C42×C11D411yescharacteristic
80C2×C22⋊Dic11—22no—
81C2×C22⋊Dic11C2312yes—
82C2×C22⋊Dic11C2314yes—
83#66997—48no—
84C2×C4⋊Dic11—21no—
85#66977—44no—
86C2×Dic11⋊C4—22no—
87C2×Dic11⋊C4—28no—
88C2×C4×Dic11—22no—
89#66898Q818yescomplemented
90C11⋊#95—22no—
91C11⋊#95—22no—
92C11⋊#95—28no—
93C11⋊#95—21no—
94C11⋊#95—22no—
95#25852—111nomaximal, Sylow
96C24×C11C2×D411yescharacteristic
97C24×C11C2411yescharacteristic
98C2×Q8×C11—22no—
99C2×Q8×C11—22no—
100C2×Q8×C11—22no—
101C2×Q8×C11—22no—
102C2×Q8×C11—44no—
103C2×Q8×C11—82no—
104C2×D4×C11—22no—
105C2×D4×C11—22no—
106C2×D4×C11C2×Q814yes—
107C22×C4×C11—22no—
108C22×C4×C11—22no—
109C22×C4×C11—21no—
110C22×C4×C11—21no—
111C22×C4×C11—42no—
112C22×C4×C11—42no—
113C22×C4×C11C2×D412yes—
114C22×C4×C11C2×D411yescharacteristic
115C22×C4×C11C2×D411yescharacteristic
116C22×C4×C11C2×D411yescharacteristic
117C22×C4×C11C2×D414yes—
118C22×C4×C11C2×D414yes—
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D412yes—
122C22×Dic11—24no—
123C22×Dic11—24no—
124C22×Dic11—22no—
125C22×Dic11—22no—
126C22×Dic11—44no—
127C2×Dic22—24no—
128C2×Dic22—28no—
129C2×Dic22—44no—
130C2×Dic22—42no—
131C2×Dic22—48no—
132C2×Dic22—44no—
133C2×Dic22—41no—
134C2×Dic22—42no—
135C2×Dic22—41no—
136C11×C4⋊C4—22no—
137C11×C4⋊C4—24no—
138C11×C4⋊C4—22no—
139C11×C4⋊C4—22no—
140C11×C4⋊C4—24no—
141C11×C4⋊C4—22no—
142C11×C4⋊C4—24no—
143C11×C4⋊C4—22no—
144C11×C4⋊C4—22no—
145C11×C4⋊C4—22no—
146C11×C4⋊C4—24no—
147C11×C22⋊C4—24no—
148C11×C22⋊C4—24no—
149C11×C22⋊C4—22no—
150C11×C22⋊C4—24no—
151C11×C22⋊C4—22no—
152C42×C11—24no—
153C42×C11—21no—
154C42×C11—21no—
155C22⋊Dic11—24no—
156C22⋊Dic11—24no—
157C22⋊Dic11—44no—
158C22⋊Dic11C2×Q818yescomplemented
159C4⋊Dic11—42no—
160Dic11⋊C4—44no—
161Dic11⋊C4—416no—
162C4×Dic11—44no—
163#3501D1111yescharacteristic
164#2434—112no—
165#2409—111no—
166#2062—114no—
167#1996—112no—
168#1867—114no—
169C4×#99—111no—
170C23×C11—21no—
171C23×C11—21no—
172C23×C11—41no—
173C23×C11—41no—
174C23×C11—41no—
175C23×C11—41no—
176C23×C11C4⋊Q812yescomplemented
177C23×C11C22×D411yescharacteristic
178C23×C11C22×D412yes—
179C23×C11C22×Q812yes—
180C23×C112−1+412yes—
181Q8×C11—44no—
182Q8×C11—82no—
183Q8×C11—82no—
184Q8×C11—82no—
185D4×C11—28no—
186C2×C4×C11—21no—
187C2×C4×C11—21no—
188C2×C4×C11—24no—
189C2×C4×C11—22no—
190C2×C4×C11—21no—
191C2×C4×C11—24no—
192C2×C4×C11—24no—
193C2×C4×C11—21no—
194C2×C4×C11—42no—
195C2×C4×C11—41no—
196C2×C4×C11—42no—
197C2×C4×C11—41no—
198C2×C4×C11—44no—
199C2×C4×C11—44no—
200C2×C4×C11—41no—
201C2×C4×C11—81no—
202C2×C4×C11—82no—
203C2×C4×C11—82no—
204C2×C4×C11#12011yescharacteristic
205C2×C4×C11#12012yes—
206C2×C4×C11#12011yescharacteristic
207C2×C4×C11C4⋊Q812yescomplemented
208C2×Dic11—28no—
209C2×Dic11—24no—
210C2×Dic11—28no—
211C2×Dic11—44no—
212C2×Dic11—44no—
213C2×Dic11—42no—
214C2×Dic11—44no—
215C2×Dic11—42no—
216C2×Dic11—42no—
217C2×Dic11—42no—
218C2×Dic11—42no—
219Dic22—44no—
220Dic22—48no—
221Dic22—44no—
222C23×Q8—111no—
223#554—22no—
224#554—24no—
225D4×Q8—22no—
226#544—21no—
227#543—21no—
228#543C2×D1112yes—
229#533—21no—
230C2×C4⋊Q8—222no—
231C2×C22⋊Q8—21no—
232C2×C22⋊Q8—221no—
233C2×C22⋊Q8C2×D1112yes—
234#518C2×D1112yes—
235C2×C4×D4C2×D1111yescharacteristic
236C2×#99—112no—
237#395—221no—
238#395—224no—
239#393—221no—
240#391—222no—
241#391—221no—
242#386—112no—
243#385—112no—
244#385—118no—
245#380—111no—
246C4×C22⋊C4—112no—
247#342—112no—
248#342—118no—
249C22×C11—22no—
250C22×C11—21no—
251C22×C11—22no—
252C22×C11—21no—
253C22×C11—22no—
254C22×C11—42no—
255C22×C11—42no—
256C22×C11—42no—
257C22×C11—41no—
258C22×C11—41no—
259C22×C11—81no—
260C22×C11C2×#12011yescharacteristic
261C22×C11C2×C4⋊Q811yescharacteristic
262C22×C11#53611yescharacteristic
263C22×C11D4×Q814yes—
264C4×C11—22no—
265C4×C11—21no—
266C4×C11—22no—
267C4×C11—41no—
268C4×C11—44no—
269C4×C11—44no—
270C4×C11—41no—
271C4×C11B(4,𝔽2)12yes—
272Dic11—48no—
273Dic11—44no—
274Dic11—82no—
275C22×Q8—21no—
276C22×Q8—221no—
277C22×Q8—222no—
278C22×Q8—221no—
279C22×Q8—222no—
280C22×Q8—444no—
281C22×Q8C22⋊D1112yes—
282C22×D4C22×D1111yescharacteristic
283C23×C4—112no—
284C23×C4C22⋊D1111yescharacteristic
285C23×C4C22×D1111yescharacteristic
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—24no—
289C4⋊Q8—448no—
290#125—22no—
291#125—24no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—444no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—24no—
310C4×D4—24no—
311#117—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—221no—
315C2×C4⋊C4—222no—
316C2×C4⋊C4—228no—
317C2×C4⋊C4C22⋊D1111yescharacteristic
318C2×C4⋊C4C22⋊D1112yes—
319C2×C4⋊C4C22⋊D1114yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—114no—
322C2×C22⋊C4—112no—
323C2×C22⋊C4—222no—
324C2×C22⋊C4C22⋊D1111yescharacteristic
325C2×C22⋊C4C22×D1111yescharacteristic
326C2×C22⋊C4C22×D1114yes—
327C2×C42—222no—
328C2×C42C22⋊D1111yescharacteristic
329#99—118no—
330#95—222no—
331#95—221no—
332#95—222no—
333#95—228no—
334#95—222no—
335C2×C11—24no—
336C2×C11—21no—
337C2×C11—21no—
338C2×C11—42no—
339C2×C11#243411yescharacteristic
340C2×C11C2×B(4,𝔽2)11yescharacteristic
341C2×C11#302811yescharacteristic
342C24C2×C22⋊D1111yescharacteristic
343C24C23×D1111yescharacteristic
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—44no—
349C2×Q8—82no—
350C2×Q8—224no—
351C2×Q8—228no—
352C2×Q8—444no—
353C2×Q8—442no—
354C2×Q8—444no—
355C2×Q8—448no—
356C2×Q8—441no—
357C2×Q8—442no—
358C2×Q8—441no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4Q8×D1114yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—224no—
369C22×C4—222no—
370C22×C4—224no—
371C22×C4—222no—
372C22×C4—444no—
373C22×C4C2×C22⋊D1111yescharacteristic
374C22×C4C2×C22⋊D1112yes—
375C22×C4C2×C22⋊D1111yescharacteristic
376C22×C4C2×C22⋊D1111yescharacteristic
377C22×C4C2×C22⋊D1114yes—
378C22×C4C2×C22⋊D1114yes—
379C22×C4C2×C22⋊D1111yescharacteristic
380C22×C4C2×C22⋊D1111yescharacteristic
381C22×C4C2×C22⋊D1112yes—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—442no—
394C4⋊C4—444no—
395C4⋊C4—4416no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—24no—
400C22⋊C4—22no—
401C22⋊C4—118no—
402C22⋊C4—224no—
403C22⋊C4—224no—
404C22⋊C4—444no—
405C42—24no—
406C42—21no—
407C42—21no—
408C42—444no—
409C11#2585211yescharacteristic, complemented, Sylow
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23#6699712yescomplemented
417C23C2×Q8×D1112yes—
418C23#6704012yes—
419C23C22×C22⋊D1111yescharacteristic
420C23C22×C22⋊D1112yes—
421Q8—44no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425Q8—444no—
426Q8—448no—
427Q8—444no—
428D4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—224no—
448C2×C4—228no—
449C2×C4—228no—
450C2×C4—444no—
451C2×C4—444no—
452C2×C4—442no—
453C2×C4—444no—
454C2×C4—442no—
455C2×C4—442no—
456C2×C4—442no—
457C2×C4—442no—
458C2×C4#6699712yescomplemented
459C2×C4#6700611yescharacteristic
460C2×C4#6700612yes—
461C2×C4#6700611yescharacteristic
462C22—22no—
463C22—21no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22C2×#6699711yescharacteristic
474C22Q8×C22⋊D1114yes—
475C22#1062334711yescharacteristic
476C22C2×#6700611yescharacteristic
477C4—22no—
478C4—21no—
479C4—22no—
480C4—44no—
481C4—41no—
482C4—41no—
483C4—44no—
484C4—448no—
485C4—444no—
486C4—882no—
487C4#1062285712yes—
488C2—24no—
489C2—21no—
490C2—21no—
491C2—42no—
492C2#1307225611yescharacteristic
493C2C2×#1062285711yescharacteristic
494C2#1307802211yescharacteristic
subgroupG/H
centerC22C2×#67006
commutator, cosocleC24×C11C24
FrattiniC24C23×D11
FittingC11×#3501C2
socleC22×C11C2×#120
prime radicalC22×D4×C11C23
hypercenter#3501D11
hypocenterC11#25852
LCSG ▷ C24×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #13072261C2semidirect
3. #13069968C2semidirect
4. #13069602C2semidirect
89. #66898Q8semidirect
158. C22⋊Dic11C2×Q8semidirect
176. C23×C11C4⋊Q8semidirect
207. C2×C4×C11C4⋊Q8semidirect
409. C11#25852semidirect
416. C23#66997semidirect
458. C2×C4#66997semidirect

Maximal subgroups

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

subgroupindex
#130722612
#130719812
#130699682
#130696022
#130688972
C4×#668982
C11×#35012
#2585211