Finite Group Databasebeta

← #14727689

Subgroups — #14727689

Order 2816.

1002 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13079369C212yesmaximal
2#13074358C211yesmaximal, characteristic
3#13067624C214yesmaximal, complemented
4#13067556C214yesmaximal, complemented
5#13067188C212yesmaximal, complemented
6C2×#10622166C211yesmaximal, characteristic, complemented
7C2×C11×#544C211yesmaximal, characteristic, complemented
8C11×#544—24no—
9C11×#544—22no—
10C11×#544C2214yes—
11C2×C11×C4⋊Q8—22no—
12C2×C11×#125—22no—
13C2×C11×C22⋊Q8—21no—
14C2×C11×C22⋊Q8C2212yes—
15C2×C11×#117—21no—
16C22×C11×C4⋊C4C2211yescharacteristic
17C2×#66983C2211yescharacteristic
18C2×#66982—21no—
19#10622840—22no—
20#10622826—21no—
21#10622825C2211yescharacteristic
22C2×C22⋊(C11⋊C8)C2212yes—
23#10622661—24no—
24#10622659C2214yes—
25#10622656C2214yes—
26#10622607—22no—
27#10622604C2212yes—
28#10622596C2212yes—
29#10622581—22no—
30#10622176C2218yes—
31#10622166C2214yes—
32C22×Q8×C11—21no—
33C22×Q8×C11D412yes—
34C23×C4×C11D411yescharacteristic
35C23×C4×C11D411yescharacteristic
36C23×C4×C11C2311yescharacteristic
37C22×C22⋊D11—21no—
38C22×D44C2311yescharacteristic
39C11×C4⋊Q8—28no—
40C11×C4⋊Q8—28no—
41C11×#125—28no—
42C11×#125—28no—
43C11×C22⋊Q8—24no—
44C11×C22⋊Q8—24no—
45C11×C22⋊Q8—22no—
46C11×C22⋊Q8—22no—
47C11×C22⋊Q8D414yes—
48C11×C22⋊Q8C2314yes—
49C11×#117—24no—
50C11×#117—24no—
51C2×C11×C4⋊C4—21no—
52C2×C11×C4⋊C4—21no—
53C2×C11×C4⋊C4—21no—
54C2×C11×C4⋊C4—22no—
55C2×C11×C4⋊C4—22no—
56C2×C11×C4⋊C4—22no—
57C2×C11×C4⋊C4—22no—
58C2×C11×C4⋊C4D412yes—
59C2×C11×C4⋊C4D412yes—
60C2×C11×C4⋊C4C2312yes—
61C2×C11×C22⋊C4—22no—
62C2×C42×C11—22no—
63C2×C22⋊Dic11—21no—
64C2×#4325—24no—
65#66983—22no—
66#66983—22no—
67#66982—44no—
68C2×(C2×D11)⋊C4—22no—
69C2×(C2×D11)⋊C4—21no—
70C2×(C2×D11)⋊C4—22no—
71C2×C4⋊Dic11—21no—
72C2×C4⋊Dic11C2311yescharacteristic
73C2×Dic11⋊C4—22no—
74C22×C11⋊C8—22no—
75#66899—24no—
76C11⋊#95—22no—
77C11⋊#95C2312yes—
78C22⋊(C11⋊C8)—22no—
79C22⋊(C11⋊C8)C2314yes—
80#66873—24no—
81#66872—24no—
82#18294—111nomaximal, Sylow
83C24×C11C2×D411yescharacteristic
84C23×D11—21no—
85C2×Q8×C11—24no—
86C2×Q8×C11—22no—
87C2×Q8×C11—82no—
88C2×Q8×C11—82no—
89C2×Q8×C11C2×D414yes—
90C22×C4×C11—21no—
91C22×C4×C11—21no—
92C22×C4×C11—22no—
93C22×C4×C11—22no—
94C22×C4×C11—22no—
95C22×C4×C11—21no—
96C22×C4×C11—21no—
97C22×C4×C11—81no—
98C22×C4×C11—81no—
99C22×C4×C11SD1614yescomplemented
100C22×C4×C11C2×D412yes—
101C22×C4×C11C2×D412yes—
102C22×C4×C11C2×D411yescharacteristic
103C22×C4×C11C2×D412yes—
104C22×C4×C11C2×D412yes—
105C22×C4×C11C2411yescharacteristic
106C2×C22⋊D11—41no—
107C2×C22⋊D11—41no—
108C2×C22⋊D11—42no—
109C2×C22⋊D11—42no—
110C22×Dic11—21no—
111C22×Dic11—22no—
112C2×D44—22no—
113C2×D44—81no—
114C11×C4⋊C4—24no—
115C11×C4⋊C4—24no—
116C11×C4⋊C4—24no—
117C11×C4⋊C4—28no—
118C11×C4⋊C4—24no—
119C11×C4⋊C4—24no—
120C11×C4⋊C4—22no—
121C11×C4⋊C4—24no—
122C11×C4⋊C4—24no—
123C11×C4⋊C4C2×D414yes—
124C11×C22⋊C4—24no—
125C11×C22⋊C4—24no—
126C42×C11—28no—
127C22⋊Dic11—42no—
128#4325—84no—
129(C2×D11)⋊C4—42no—
130(C2×D11)⋊C4—44no—
131(C2×D11)⋊C4—82no—
132C4⋊Dic11—22no—
133C4⋊Dic11—81no—
134Dic11⋊C4—44no—
135C2×C11⋊C8—24no—
136C2×C11⋊C8—82no—
137C2×#544D1111yescharacteristic, complemented
138#3187—112no—
139#2618—111no—
140#1611—114no—
141#1599—114no—
142#1520—112no—
143C2×#324—111no—
144C23×C11—41no—
145C23×C11—41no—
146C23×C11#12011yescharacteristic
147C23×C11#12011yescharacteristic
148C23×C11C2×SD1612yes—
149C23×C11#13712yes—
150C23×C11C22×D411yescharacteristic
151C22×D11—42no—
152C22×D11—42no—
153C22×D11—41no—
154C22×D11—41no—
155C22×D11—41no—
156Q8×C11—44no—
157Q8×C11—82no—
158Q8×C11—84no—
159C2×C4×C11—22no—
160C2×C4×C11—22no—
161C2×C4×C11—24no—
162C2×C4×C11—24no—
163C2×C4×C11—24no—
164C2×C4×C11—24no—
165C2×C4×C11—22no—
166C2×C4×C11—42no—
167C2×C4×C11—41no—
168C2×C4×C11—41no—
169C2×C4×C11—42no—
170C2×C4×C11—44no—
171C2×C4×C11—42no—
172C2×C4×C11—44no—
173C2×C4×C11—42no—
174C2×C4×C11—81no—
175C2×C4×C11#12012yes—
176C2×C4×C11C2×SD1614yes—
177C2×C4×C11C22×D412yes—
178C22⋊D11—82no—
179C22⋊D11—82no—
180C2×Dic11—42no—
181C2×Dic11—42no—
182C2×Dic11—41no—
183C2×Dic11—41no—
184C2×Dic11—42no—
185C2×Dic11—41no—
186D44—44no—
187C11⋊C8—44no—
188#544—24no—
189#544—22no—
190#544C2×D1114yes—
191C2×C4⋊Q8—22no—
192C2×#125—22no—
193C2×#123—221no—
194C2×C22⋊Q8—21no—
195C2×C22⋊Q8C2×D1112yes—
196C2×#121—111no—
197C2×#117—21no—
198C22×C4⋊C4C2×D1111yescharacteristic
199#483—114no—
200#481—222no—
201#465—224no—
202#450—114no—
203#448—222no—
204C2×C22⋊C8—112no—
205#398—112no—
206#396—222no—
207#394—112no—
208#386—221no—
209#380—111no—
210#328—118no—
211#324—114no—
212C22×C11—21no—
213C22×C11—21no—
214C22×C11—21no—
215C22×C11—21no—
216C22×C11—21no—
217C22×C11—21no—
218C22×C11—42no—
219C22×C11—42no—
220C22×C11#44812yes—
221C22×C11#44812yes—
222C22×C11C2×#12011yescharacteristic
223C22×C11C22×SD1611yescharacteristic
224C22×C11C2×#13711yescharacteristic
225C2×D11—82no—
226C2×D11—82no—
227C2×D11—81no—
228C2×D11—81no—
229C2×D11—81no—
230C4×C11—24no—
231C4×C11—22no—
232C4×C11—22no—
233C4×C11—44no—
234C4×C11—42no—
235C4×C11—44no—
236Dic11—82no—
237Dic11—81no—
238C22×Q8—21no—
239C22×Q8C22⋊D1112yes—
240C22×D4—111no—
241C22×D4—221no—
242C23×C4C22⋊D1111yescharacteristic
243C23×C4C22⋊D1111yescharacteristic
244C23×C4C22×D1111yescharacteristic
245C2×SD16—224no—
246C22×C8—222no—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249#125—28no—
250#125—28no—
251#123—444no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8C22⋊D1114yes—
257C22⋊Q8C22×D1114yes—
258#121—222no—
259#121—222no—
260#117—24no—
261#117—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—111no—
270C2×C4⋊C4—222no—
271C2×C4⋊C4—221no—
272C2×C4⋊C4C22⋊D1112yes—
273C2×C4⋊C4C22⋊D1112yes—
274C2×C4⋊C4C22×D1112yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—221no—
277C2×C22⋊C4—221no—
278C2×C22⋊C4—222no—
279C2×C22⋊C4—222no—
280C2×C42—22no—
281#106—224no—
282#103—224no—
283#102—224no—
284C22⋊C8—114no—
285C22⋊C8—222no—
286#95—112no—
287#95—222no—
288C2×C11—21no—
289C2×C11—21no—
290C2×C11—22no—
291C2×C11—22no—
292C2×C11C2×#44811yescharacteristic
293C2×C11#302011yescharacteristic
294C2×C11#302611yescharacteristic
295D11—161no—
296C24—221no—
297C24C2×C22⋊D1111yescharacteristic
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—82no—
301C2×Q8—82no—
302C2×Q8C2×C22⋊D1114yes—
303C2×D4—222no—
304C2×D4—442no—
305C2×D4—442no—
306C2×D4—441no—
307C2×D4—441no—
308C2×D4—881no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—222no—
319C22×C4—221no—
320C22×C4#432514yescomplemented
321C22×C4C2×C22⋊D1112yes—
322C22×C4C2×C22⋊D1112yes—
323C22×C4C2×C22⋊D1111yescharacteristic
324C22×C4C2×C22⋊D1112yes—
325C22×C4C2×C22⋊D1112yes—
326C22×C4C23×D1111yescharacteristic
327SD16—884no—
328C2×C8—224no—
329C2×C8—882no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—222no—
340C4⋊C4—444no—
341C4⋊C4—881no—
342C4⋊C4C2×C22⋊D1114yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—442no—
346C22⋊C4—444no—
347C22⋊C4—442no—
348C22⋊C4—882no—
349C42—28no—
350C11#1829411yescharacteristic, complemented, Sylow
351C23—41no—
352C23—41no—
353C23—442no—
354C23—442no—
355C23—441no—
356C23—441no—
357C23—441no—
358C23C2×#432512yes—
359C23#6699512yes—
360C23#6700611yescharacteristic
361C23#6700611yescharacteristic
362C23C22×C22⋊D1111yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366D4—444no—
367D4—882no—
368D4—882no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—442no—
386C2×C4—442no—
387C2×C4—442no—
388C2×C4—441no—
389C2×C4—441no—
390C2×C4—441no—
391C2×C4C2×#432514yes—
392C2×C4#6700612yes—
393C2×C4C22×C22⋊D1112yes—
394C8—444no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—882no—
404C22—882no—
405C22—881no—
406C22—881no—
407C22—881no—
408C22#1062284012yes—
409C22#1062284012yes—
410C22C22×#432511yescharacteristic
411C22C2×#6699511yescharacteristic
412C22C2×#6700611yescharacteristic
413C4—24no—
414C4—22no—
415C4—22no—
416C4—44no—
417C4—42no—
418C4—44no—
419C4—882no—
420C4—881no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—1761no—
426C2C2×#1062284011yescharacteristic
427C2#1307798011yescharacteristic
428C2#1307801311yescharacteristic
subgroupG/H
centerC22C2×#67006
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C11×#544C2
socleC22×C11C2×#120
prime radicalC2×#66983C22
hypercenterC2×#544D11
hypocenterC11#18294
LCSG ▷ C22×C4×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #13067624C2semidirect
4. #13067556C2semidirect
5. #13067188C2semidirect
6. C2×#10622166C2semidirect
7. C2×C11×#544C2semidirect
99. C22×C4×C11SD16semidirect
137. C2×#544D11semidirect
320. C22×C4#4325semidirect
350. C11#18294semidirect

Maximal subgroups

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

subgroupindex
#130793692
#130743582
#130676242
#130675562
#130671882
C2×#106221662
C2×C11×#5442
#1829411