Finite Group Databasebeta

← #14736693

Subgroups — #14736693

Order 2816.

618 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13079010C211yesmaximal, characteristic
2#13078647C211yesmaximal, characteristic
3C2×#10622634C211yesmaximal, characteristic, complemented
4#13068198C212yesmaximal, complemented
5#13068156C212yesmaximal, complemented
6#13068074C212yesmaximal, complemented
7#13068032C212yesmaximal, complemented
8#13067368C211yesmaximal, characteristic, complemented
9#13067298C211yesmaximal, characteristic, complemented
10#13067294C211yesmaximal, characteristic
11Q8×C11×M16C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13#10623008C2211yescharacteristic
14C2×#66922C2211yescharacteristic
15C11×#446—22no—
16C11×#446C2212yes—
17C8×Q8×C11C2212yes—
18C11×#423—21no—
19C11×#423C2211yescharacteristic
20C4×C11×M16—21no—
21C4×C11×M16C2211yescharacteristic
22#10622840—21no—
23C2×#66899—21no—
24#10622746—21no—
25#10622746—21no—
26#10622640C2211yescharacteristic
27#10622640C2211yescharacteristic
28C4×#4325—21no—
29C4×#4325—21no—
30#10622634—21no—
31#10622634—21no—
32#10622634C2212yes—
33#10622634C2212yes—
34#10622633C2211yescharacteristic
35#10622633C2211yescharacteristic
36#10622581—21no—
37C2×#66871—21no—
38#10622538C2211yescharacteristic
39C2×C4⋊(C11⋊C8)C2211yescharacteristic
40#10622327—22no—
41#10622307C2212yes—
42#10622294C2212yes—
43#10622123C2212yes—
44#10622120C2212yes—
45#10622119C2212yes—
46#10622119C2212yes—
47#10622096C2212yes—
48#10622095C2212yes—
49C22×Q8×C11D411yescharacteristic
50C2×#4339—21no—
51C22×Dic22—21no—
52C2×C11×M16—21no—
53C2×C11×M16D411yescharacteristic
54C2×C11×M16D411yescharacteristic
55C4×Q8×C11—21no—
56C4×Q8×C11—21no—
57C4×Q8×C11D411yescharacteristic
58C4×Q8×C11D411yescharacteristic
59C4×Q8×C11C2311yescharacteristic
60C4×Q8×C11C2311yescharacteristic
61C2×C11×C4⋊C4—21no—
62C2×C11×C4⋊C4D411yescharacteristic
63C2×C42×C11—21no—
64C2×C42×C11C2311yescharacteristic
65C2×#4325—21no—
66C2×#4325—21no—
67#66983—21no—
68#66983—21no—
69C2×C4⋊Dic11—21no—
70C2×C4⋊Dic11—21no—
71C22×C11⋊C8—21no—
72#66928—21no—
73#66927C2311yescharacteristic
74C4×D44—21no—
75C4×D44—21no—
76#66922—21no—
77#66922—21no—
78#66922—21no—
79#66922C2312yes—
80#66922C2311yescharacteristic
81C11×C4⋊C8—22no—
82C11×C4⋊C8—22no—
83C11×C4⋊C8—21no—
84C11×C4⋊C8C2312yes—
85C11×#97—22no—
86C11×#97D412yes—
87C4×C8×C11—22no—
88C4×C8×C11C2312yes—
89#66899—22no—
90#66899—22no—
91#66899—21no—
92#66899—21no—
93C22⋊(C11⋊C8)—21no—
94#66885—22no—
95#66885—22no—
96#66880—22no—
97#66880—22no—
98#66873—21no—
99#66873—21no—
100#66872—21no—
101#66872—21no—
102#66871—22no—
103#66871—22no—
104C4⋊(C11⋊C8)—21no—
105C4⋊(C11⋊C8)C2312yes—
106C4⋊(C11⋊C8)C2311yescharacteristic
107C4⋊(C11⋊C8)C2311yescharacteristic
108C4×C11⋊C8—21no—
109#19176—111nomaximal, Sylow
110C2×Q8×C11—21no—
111C2×Q8×C11—22no—
112C2×Q8×C11—41no—
113C2×Q8×C11C2×D411yescharacteristic
114C2×Q8×C11C2×D411yescharacteristic
115C22×C4×C11—21no—
116C22×C4×C11—21no—
117C22×C4×C11C2×D411yescharacteristic
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11#4111yescharacteristic
120C2×C22⋊D11—41no—
121C22×Dic11—41no—
122#4339—42no—
123#4339—42no—
124C2×D44—21no—
125C2×D44—21no—
126C2×D44—21no—
127C2×C4×D11—41no—
128C2×Dic22—21no—
129C2×Dic22—22no—
130C2×Dic22—41no—
131C2×Dic22—82no—
132C11×M16—22no—
133C11×M16—22no—
134C11×M16—42no—
135C2×C8×C11—22no—
136C2×C8×C11C2×D412yes—
137C2×C8×C11C2×D412yes—
138C11×C4⋊C4—21no—
139C11×C4⋊C4—21no—
140C11×C4⋊C4—21no—
141C11×C4⋊C4—21no—
142C11×C4⋊C4—21no—
143C11×C4⋊C4C2×D411yescharacteristic
144C11×C4⋊C4C2×D411yescharacteristic
145C42×C11—21no—
146C42×C11—21no—
147C42×C11—21no—
148C42×C11—21no—
149C42×C11C2×D411yescharacteristic
150C42×C11C2×D411yescharacteristic
151C42×C11#4111yescharacteristic
152C42×C11C2411yescharacteristic
153#4325—42no—
154#4325—42no—
155(C2×D11)⋊C4—41no—
156(C2×D11)⋊C4—41no—
157C4⋊Dic11—22no—
158C4⋊Dic11—21no—
159C4⋊Dic11—22no—
160C4⋊Dic11—21no—
161C4⋊Dic11—41no—
162C4⋊Dic11—41no—
163C2×C11⋊C8—21no—
164C2×C11⋊C8—21no—
165C2×C11⋊C8—22no—
166C2×C11⋊C8—42no—
167#3141—111no—
168#3103—111no—
169C2×#477—111no—
170Q8×M16D1111yescharacteristic, complemented
171#1704—112no—
172#1698—112no—
173#1684—112no—
174#1678—112no—
175#1565—111no—
176#1548—111no—
177#1547—111no—
178C23×C11#12111yescharacteristic
179C22×D11—41no—
180Q8×C11—21no—
181Q8×C11—21no—
182Q8×C11—21no—
183Q8×C11—42no—
184Q8×C11#13712yes—
185C2×C4×C11—21no—
186C2×C4×C11—21no—
187C2×C4×C11—21no—
188C2×C4×C11—21no—
189C2×C4×C11—21no—
190C2×C4×C11—21no—
191C2×C4×C11—41no—
192C2×C4×C11—42no—
193C2×C4×C11—81no—
194C2×C4×C11#12111yescharacteristic
195C2×C4×C11#12111yescharacteristic
196C2×C4×C11#12111yescharacteristic
197C2×C4×C11#13512yes—
198C2×C4×C11#13712yes—
199C2×C4×C11C22×D411yescharacteristic
200C2×C4×C11C22×D411yescharacteristic
201C2×C4×C11C2×#4111yescharacteristic
202C22⋊D11—82no—
203C2×Dic11—42no—
204C2×Dic11—41no—
205C2×Dic11—82no—
206D44—41no—
207D44—41no—
208D44—41no—
209D44—41no—
210D44—81no—
211C4×D11—82no—
212Dic22—41no—
213Dic22—41no—
214Dic22—82no—
215Dic22—82no—
216C8×C11—22no—
217C8×C11—22no—
218C8×C11—42no—
219C11⋊C8—41no—
220C11⋊C8—42no—
221C11⋊C8—41no—
222#532—111no—
223C2×C4⋊Q8—111no—
224C2×C4×Q8C2×D1111yescharacteristic
225#489—222no—
226#487—112no—
227#486—112no—
228#481—221no—
229#477—112no—
230#477—112no—
231#477—221no—
232#477—221no—
233#475—111no—
234#475—111no—
235#460—111no—
236#460—111no—
237#448—221no—
238#446—22no—
239#446C2×D1112yes—
240C8×Q8C2×D1112yes—
241C4×SD16—221no—
242C4×SD16—221no—
243#431—111no—
244C2×#107—221no—
245#423—21no—
246#423C2×D1111yescharacteristic
247C2×C4⋊C8—111no—
248#417—221no—
249#417—221no—
250C2×#103—221no—
251C4×M16—21no—
252C4×M16C2×D1111yescharacteristic
253#335—112no—
254#334—112no—
255#333—112no—
256#333—112no—
257#332—112no—
258#331—112no—
259C22×C11—21no—
260C22×C11—21no—
261C22×C11#46412yes—
262C22×C11C2×#12111yescharacteristic
263C2×D11—81no—
264C2×D11—81no—
265C2×D11—81no—
266C4×C11—21no—
267C4×C11—22no—
268C4×C11—21no—
269C4×C11—21no—
270C4×C11—41no—
271C4×C11—41no—
272C4×C11—41no—
273C4×C11#46412yes—
274C4×C11C2×#13511yescharacteristic
275C4×C11C2×#13711yescharacteristic
276C4×C11C2×#13711yescharacteristic
277C4×C11#57511yescharacteristic
278Dic11—81no—
279Dic11—82no—
280C2×#41—221no—
281C22×Q8—221no—
282C22×Q8D4411yescharacteristic
283C2×SD16—221no—
284C2×SD16—221no—
285C2×M16—21no—
286C2×M16C22⋊D1111yescharacteristic
287C2×M16C22⋊D1111yescharacteristic
288C22×C8—221no—
289C4⋊Q8—112no—
290C4⋊Q8—111no—
291C4⋊Q8—221no—
292C4⋊Q8—221no—
293C4⋊Q8—221no—
294#127—111no—
295#124—221no—
296#121—221no—
297#121—221no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8D4411yescharacteristic
301C4×Q8D4411yescharacteristic
302C4×Q8C22×D1111yescharacteristic
303C4×Q8C22×D1111yescharacteristic
304C4×D4—221no—
305C4×D4—221no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—221no—
308C2×C4⋊C4—221no—
309C2×C4⋊C4D4411yescharacteristic
310C2×C42—21no—
311C2×C42C22×D1111yescharacteristic
312#107—222no—
313#107—222no—
314#106—221no—
315#106—221no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—21no—
319C4⋊C8—112no—
320C4⋊C8—111no—
321C4⋊C8—111no—
322C4⋊C8—221no—
323C4⋊C8C22×D1112yes—
324#103—222no—
325#103—222no—
326#103—221no—
327#103—221no—
328#103—222no—
329#103—222no—
330#102—222no—
331#102—222no—
332#102—221no—
333#102—221no—
334C22⋊C8—221no—
335#97—22no—
336#97C22⋊D1112yes—
337C4×C8—22no—
338C4×C8—221no—
339C4×C8C22×D1112yes—
340C2×C11—22no—
341C2×C11C2×#46411yescharacteristic
342C2×C11#305611yescharacteristic
343C2×C11#306811yescharacteristic
344D11—161no—
345#41—442no—
346#41—442no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—222no—
351C2×Q8—221no—
352C2×Q8—441no—
353C2×Q8—882no—
354C2×Q8C2×D4411yescharacteristic
355C2×Q8C2×D4411yescharacteristic
356C2×D4—221no—
357C2×D4—221no—
358C2×D4—221no—
359C2×D4—441no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—441no—
363C22×C4—441no—
364C22×C4C2×D4411yescharacteristic
365C22×C4#433911yescharacteristic
366C22×C4C2×C22⋊D1111yescharacteristic
367SD16—442no—
368SD16—442no—
369M16—22no—
370M16—22no—
371M16—42no—
372C2×C8—22no—
373C2×C8—221no—
374C2×C8—221no—
375C2×C8—222no—
376C2×C8—442no—
377C2×C8C2×C22⋊D1112yes—
378C2×C8C2×C22⋊D1112yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—221no—
385C4⋊C4—222no—
386C4⋊C4—221no—
387C4⋊C4—222no—
388C4⋊C4—441no—
389C4⋊C4—441no—
390C4⋊C4C2×D4411yescharacteristic
391C4⋊C4C2×D4411yescharacteristic
392C22⋊C4—441no—
393C22⋊C4—441no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42C2×D4411yescharacteristic
399C42#433911yescharacteristic
400C42C2×C22⋊D1111yescharacteristic
401C42C23×D1111yescharacteristic
402C11#1917611yescharacteristic, complemented, Sylow
403C23—441no—
404C23#6698311yescharacteristic
405Q8—21no—
406Q8—21no—
407Q8—21no—
408Q8—42no—
409Q8—441no—
410Q8—441no—
411Q8—882no—
412Q8—882no—
413Q8#6696212yes—
414D4—441no—
415D4—441no—
416D4—441no—
417D4—441no—
418D4—881no—
419D4—882no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4—441no—
430C2×C4—442no—
431C2×C4—882no—
432C2×C4—882no—
433C2×C4#6698311yescharacteristic
434C2×C4#6698311yescharacteristic
435C2×C4#6698311yescharacteristic
436C2×C4#6699512yes—
437C2×C4#6700312yes—
438C2×C4C22×D4411yescharacteristic
439C2×C4C2×#433911yescharacteristic
440C2×C4C22×C22⋊D1111yescharacteristic
441C8—22no—
442C8—22no—
443C8—42no—
444C8—441no—
445C8—442no—
446C8—441no—
447C22—21no—
448C22—21no—
449C22—881no—
450C22—881no—
451C22—881no—
452C22#1062264112yes—
453C22C2×#6698311yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—881no—
462C4—882no—
463C4#1062264112yes—
464C4C2×#6696211yescharacteristic
465C4#1062327911yescharacteristic
466C4C2×#6699511yescharacteristic
467C4C2×#6700311yescharacteristic
468C2—22no—
469C2—1761no—
470C2C2×#1062264111yescharacteristic
471C2#1307824611yescharacteristic
472C2#1307834411yescharacteristic
subgroupG/H
centerC22C2×#66983
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingQ8×C11×M16C2
socleC22×C11C2×#121
prime radical#10623008C22
hypercenterQ8×M16D11
hypocenterC11#19176
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C2×#10622634C2semidirect
4. #13068198C2semidirect
5. #13068156C2semidirect
6. #13068074C2semidirect
7. #13068032C2semidirect
8. #13067368C2semidirect
9. #13067298C2semidirect
11. Q8×C11×M16C2semidirect
170. Q8×M16D11semidirect
402. C11#19176semidirect

Maximal subgroups

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

subgroupindex
#130790102
#130786472
C2×#106226342
#130681982
#130681562
#130680742
#130680322
#130673682
#130672982
#130672942
Q8×C11×M162
#1917611