Finite Group Databasebeta

← #14726764

Subgroups — #14726764

Order 2816.

1002 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622656C212yesmaximal, complemented
2#13072924C211yesmaximal, characteristic
3#13067624C214yesmaximal, complemented
4#13067607C214yesmaximal, complemented
5#13067188C212yesmaximal, complemented
6#13066934C211yesmaximal, characteristic
7C11×#3501C211yesmaximal, characteristic
8C11×#554—24no—
9C11×#554—22no—
10D4×Q8×C11—22no—
11C11×#544C2212yes—
12C11×#543—21no—
13C11×#543C2212yes—
14C11×#533—21no—
15C2×C11×C22⋊Q8—21no—
16C2×C11×C22⋊Q8C2212yes—
17C11×#518—21no—
18C2×C4×D4×C11C2211yescharacteristic
19C2×#66988—21no—
20C22×C4⋊Dic11C2211yescharacteristic
21C2×#66899—22no—
22#10622825—21no—
23#10622825C2211yescharacteristic
24C2×C22⋊(C11⋊C8)C2212yes—
25#10622656—24no—
26#10622656C2218yes—
27#10622596C2212yes—
28C2×#66872—22no—
29#10622566C2212yes—
30#10622556—21no—
31#10622541—21no—
32#10622176C2218yes—
33#10622166C2212yes—
34#10622163C2212yes—
35C22×Q8×C11—21no—
36C22×Q8×C11D412yes—
37C22×D4×C11D411yescharacteristic
38C23×C4×C11D411yescharacteristic
39C23×C4×C11C2311yescharacteristic
40C23×Dic11—21no—
41C11×C4⋊Q8—21no—
42C11×C4⋊Q8—21no—
43C11×C4⋊Q8—24no—
44C11×#125—22no—
45C11×#125—24no—
46C11×C22⋊Q8—22no—
47C11×C22⋊Q8—22no—
48C11×C22⋊Q8—24no—
49C11×C22⋊Q8—24no—
50C11×C22⋊Q8—24no—
51C11×C22⋊Q8—22no—
52C11×C22⋊Q8—22no—
53C11×C22⋊Q8—22no—
54C11×C22⋊Q8—22no—
55C11×C22⋊Q8—24no—
56C11×C22⋊Q8D414yes—
57C11×C22⋊Q8C2314yes—
58C4×Q8×C11—22no—
59C4×Q8×C11—22no—
60C4×D4×C11—24no—
61C4×D4×C11—24no—
62C4×D4×C11—22no—
63C4×D4×C11—22no—
64C11×#117—22no—
65C2×C11×C4⋊C4—21no—
66C2×C11×C4⋊C4—22no—
67C2×C11×C4⋊C4—22no—
68C2×C11×C4⋊C4D412yes—
69C2×C11×C4⋊C4C2311yescharacteristic
70C2×C11×C22⋊C4—22no—
71C2×C11×C22⋊C4—22no—
72C2×C11×C22⋊C4D411yescharacteristic
73C2×C11×C22⋊C4C2311yescharacteristic
74C2×C42×C11D411yescharacteristic
75C2×C22⋊Dic11—21no—
76C2×C22⋊Dic11—22no—
77#66988—44no—
78C2×C4⋊Dic11—22no—
79C2×C4⋊Dic11—22no—
80C2×C4⋊Dic11—21no—
81C2×C4⋊Dic11—21no—
82C2×C4⋊Dic11C2312yes—
83C2×Dic11⋊C4—22no—
84C22×C11⋊C8—22no—
85#66899—28no—
86C11⋊#95—22no—
87C11⋊#95—22no—
88C11⋊#95C2312yes—
89C22⋊(C11⋊C8)—22no—
90C22⋊(C11⋊C8)C2314yes—
91#66872—28no—
92#18204—111nomaximal, Sylow
93C24×C11C2×D411yescharacteristic
94C24×C11C2×D411yescharacteristic
95C2×Q8×C11—22no—
96C2×Q8×C11—22no—
97C2×Q8×C11—22no—
98C2×Q8×C11—44no—
99C2×Q8×C11—82no—
100C2×Q8×C11C2×D414yes—
101C2×D4×C11—22no—
102C2×D4×C11—22no—
103C2×D4×C11SD1614yes—
104C22×C4×C11—22no—
105C22×C4×C11—22no—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—21no—
109C22×C4×C11—21no—
110C22×C4×C11—42no—
111C22×C4×C11—42no—
112C22×C4×C11C2×D411yescharacteristic
113C22×C4×C11C2×D412yes—
114C22×C4×C11C2×D411yescharacteristic
115C22×C4×C11C2×D411yescharacteristic
116C22×C4×C11C2×D411yescharacteristic
117C22×C4×C11C2×D412yes—
118C22×C4×C11C2411yescharacteristic
119C22×Dic11—22no—
120C22×Dic11—22no—
121C22×Dic11—42no—
122C22×Dic11—42no—
123C22×Dic11—42no—
124C11×C4⋊C4—22no—
125C11×C4⋊C4—24no—
126C11×C4⋊C4—22no—
127C11×C4⋊C4—22no—
128C11×C4⋊C4—24no—
129C11×C4⋊C4—22no—
130C11×C4⋊C4—24no—
131C11×C4⋊C4—22no—
132C11×C4⋊C4—24no—
133C11×C4⋊C4—22no—
134C11×C4⋊C4C2×D414yes—
135C11×C22⋊C4—24no—
136C11×C22⋊C4—24no—
137C11×C22⋊C4—22no—
138C11×C22⋊C4—24no—
139C11×C22⋊C4—22no—
140C42×C11—24no—
141C42×C11—21no—
142C42×C11—21no—
143C22⋊Dic11—44no—
144C22⋊Dic11—42no—
145C4⋊Dic11—24no—
146C4⋊Dic11—42no—
147C4⋊Dic11—41no—
148C4⋊Dic11—41no—
149Dic11⋊C4—44no—
150C2×C11⋊C8—24no—
151C2×C11⋊C8—44no—
152#3501D1111yescharacteristic
153C2×#483—112no—
154#2486—111no—
155#1611—114no—
156#1608—114no—
157#1520—112no—
158#1459—111no—
159C23×C11—21no—
160C23×C11—21no—
161C23×C11—41no—
162C23×C11—41no—
163C23×C11—41no—
164C23×C11—41no—
165C23×C11#12012yes—
166C23×C11C2×SD1612yes—
167C23×C11C2×SD1612yes—
168C23×C11#13712yes—
169C23×C11C22×D411yescharacteristic
170Q8×C11—44no—
171Q8×C11—44no—
172Q8×C11—82no—
173Q8×C11—82no—
174D4×C11—28no—
175C2×C4×C11—21no—
176C2×C4×C11—21no—
177C2×C4×C11—24no—
178C2×C4×C11—22no—
179C2×C4×C11—21no—
180C2×C4×C11—24no—
181C2×C4×C11—24no—
182C2×C4×C11—21no—
183C2×C4×C11—42no—
184C2×C4×C11—41no—
185C2×C4×C11—42no—
186C2×C4×C11—41no—
187C2×C4×C11—41no—
188C2×C4×C11—81no—
189C2×C4×C11—82no—
190C2×C4×C11—82no—
191C2×C4×C11—82no—
192C2×C4×C11—82no—
193C2×C4×C11#12011yescharacteristic
194C2×C4×C11#12011yescharacteristic
195C2×C4×C11C2×SD1612yes—
196C2×C4×C11C22×D412yes—
197C2×Dic11—42no—
198C2×Dic11—42no—
199C2×Dic11—42no—
200C2×Dic11—42no—
201C2×Dic11—42no—
202C2×Dic11—42no—
203C2×Dic11—84no—
204C11⋊C8—44no—
205#554—24no—
206#554—22no—
207D4×Q8—22no—
208#544C2×D1112yes—
209#543—21no—
210#543C2×D1112yes—
211#533—21no—
212C2×#123—221no—
213C2×C22⋊Q8—21no—
214C2×C22⋊Q8C2×D1112yes—
215#518—21no—
216C2×C4×D4C2×D1111yescharacteristic
217C22×C4⋊C4—111no—
218#483—118no—
219#483—224no—
220C2×#106—222no—
221C2×#103—222no—
222C2×C22⋊C8—112no—
223#398—112no—
224#397—112no—
225#389—221no—
226#381—221no—
227#380—111no—
228#380—221no—
229#328—118no—
230#324—112no—
231#323—112no—
232C22×C11—21no—
233C22×C11—22no—
234C22×C11—21no—
235C22×C11—22no—
236C22×C11—22no—
237C22×C11—42no—
238C22×C11—42no—
239C22×C11—42no—
240C22×C11—41no—
241C22×C11—41no—
242C22×C11—81no—
243C22×C11#44814yes—
244C22×C11C2×#12011yescharacteristic
245C22×C11C22×SD1611yescharacteristic
246C22×C11C2×#13711yescharacteristic
247C4×C11—22no—
248C4×C11—21no—
249C4×C11—22no—
250C4×C11—41no—
251C4×C11—44no—
252C4×C11—41no—
253C4×C11—44no—
254C4×C11#45812yes—
255Dic11—82no—
256Dic11—82no—
257C22×Q8—21no—
258C22×Q8C22⋊D1112yes—
259C22×D4C22⋊D1111yescharacteristic
260C23×C4—221no—
261C23×C4C22⋊D1111yescharacteristic
262C23×C4C22×D1111yescharacteristic
263C22×C8—222no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—24no—
267#125—22no—
268#125—24no—
269#123—444no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—24no—
280C22⋊Q8C22⋊D1114yes—
281C22⋊Q8C22×D1114yes—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288#117—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—112no—
293C2×C4⋊C4—222no—
294C2×C4⋊C4—222no—
295C2×C4⋊C4—221no—
296C2×C4⋊C4—222no—
297C2×C4⋊C4—221no—
298C2×C4⋊C4C22⋊D1112yes—
299C2×C4⋊C4C22×D1111yescharacteristic
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—221no—
303C2×C22⋊C4—222no—
304C2×C22⋊C4C22⋊D1111yescharacteristic
305C2×C22⋊C4C22×D1111yescharacteristic
306C2×C42C22⋊D1111yescharacteristic
307#106—228no—
308#103—228no—
309C22⋊C8—114no—
310C22⋊C8—222no—
311#95—112no—
312#95—222no—
313#95—222no—
314C2×C11—21no—
315C2×C11—21no—
316C2×C11—24no—
317C2×C11—42no—
318C2×C11C2×#44811yescharacteristic
319C2×C11#302211yescharacteristic
320C2×C11C2×#45811yescharacteristic
321C24C2×C22⋊D1111yescharacteristic
322C24C2×C22⋊D1111yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—44no—
327C2×Q8—82no—
328C2×Q8C2×C22⋊D1114yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4#432514yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—222no—
341C22×C4—222no—
342C22×C4—442no—
343C22×C4—442no—
344C22×C4—442no—
345C22×C4C2×C22⋊D1111yescharacteristic
346C22×C4C2×C22⋊D1111yescharacteristic
347C22×C4C2×C22⋊D1112yes—
348C22×C4C2×C22⋊D1111yescharacteristic
349C22×C4C2×C22⋊D1111yescharacteristic
350C22×C4C2×C22⋊D1112yes—
351C22×C4C23×D1111yescharacteristic
352C2×C8—224no—
353C2×C8—444no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—224no—
365C4⋊C4—442no—
366C4⋊C4—444no—
367C4⋊C4—441no—
368C4⋊C4—441no—
369C4⋊C4C2×C22⋊D1114yes—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C22⋊C4—22no—
375C22⋊C4—442no—
376C22⋊C4—444no—
377C42—24no—
378C42—21no—
379C42—21no—
380C11#1820411yescharacteristic, complemented, Sylow
381C23—21no—
382C23—21no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C2×#432512yes—
388C23C2×#432512yes—
389C23#6699512yes—
390C23#6700612yes—
391C23C22×C22⋊D1111yescharacteristic
392Q8—44no—
393Q8—44no—
394Q8—82no—
395Q8—82no—
396D4—28no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—442no—
416C2×C4—442no—
417C2×C4—442no—
418C2×C4—442no—
419C2×C4—442no—
420C2×C4—442no—
421C2×C4—884no—
422C2×C4C2×#432512yes—
423C2×C4#6700611yescharacteristic
424C2×C4#6700611yescharacteristic
425C2×C4C22×C22⋊D1112yes—
426C8—444no—
427C22—21no—
428C22—22no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22#1062284014yes—
439C22C22×#432511yescharacteristic
440C22C2×#6699511yescharacteristic
441C22C2×#6700611yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—41no—
446C4—44no—
447C4—41no—
448C4—44no—
449C4—882no—
450C4—882no—
451C4#1062285612yes—
452C2—21no—
453C2—21no—
454C2—24no—
455C2—42no—
456C2C2×#1062284011yescharacteristic
457C2#1307799511yescharacteristic
458C2C2×#1062285611yescharacteristic
subgroupG/H
centerC22C2×#67006
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#3501C2
socleC22×C11C2×#120
prime radicalC22×D4×C11D4
hypercenter#3501D11
hypocenterC11#18204
LCSG ▷ C22×C4×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. C2×#10622656C2semidirect
3. #13067624C2semidirect
4. #13067607C2semidirect
5. #13067188C2semidirect
380. C11#18204semidirect

Maximal subgroups

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

subgroupindex
C2×#106226562
#130729242
#130676242
#130676072
#130671882
#130669342
C11×#35012
#1820411