Finite Group Databasebeta

← #13074913

Subgroups — #13074913

Order 1408.

676 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C11×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#67000C211yesmaximal, characteristic, complemented
3C2×#66983C211yesmaximal, characteristic, complemented
4C2×#66948C211yesmaximal, characteristic, complemented
5#10622844C211yesmaximal, characteristic, complemented
6#10622826C211yesmaximal, characteristic, complemented
7#10622607C211yesmaximal, characteristic, complemented
8#10622604C211yesmaximal, characteristic, complemented
9#10622604C211yesmaximal, characteristic, complemented
10#10622602C211yesmaximal, characteristic, complemented
11#10622594C211yesmaximal, characteristic, complemented
12#10622573C211yesmaximal, characteristic, complemented
13#10622573C211yesmaximal, characteristic, complemented
14#10622570C211yesmaximal, characteristic, complemented
15#10622569C211yesmaximal, characteristic, complemented
16C22×Q8×C11C2211yescharacteristic
17C23×C4×C11C2211yescharacteristic
18C22×C22⋊D11C2211yescharacteristic
19C22×C22⋊D11C2211yescharacteristic
20C22×C22⋊D11C2211yescharacteristic
21C22×D44C2211yescharacteristic
22C22×D44C2211yescharacteristic
23C22×D44C2211yescharacteristic
24C22×C4×D11C2211yescharacteristic
25C11×C22⋊Q8—24no—
26C2×C11×C4⋊C4C2211yescharacteristic
27C2×C11×C4⋊C4C2211yescharacteristic
28C2×C11×C4⋊C4C2211yescharacteristic
29C2×C11×C22⋊C4C2211yescharacteristic
30C2×C11×C22⋊C4C2211yescharacteristic
31C2×C22⋊Dic11C2211yescharacteristic
32#67000—24no—
33#66983—24no—
34C2×(C2×D11)⋊C4C2211yescharacteristic
35C2×(C2×D11)⋊C4C2211yescharacteristic
36C2×(C2×D11)⋊C4C2211yescharacteristic
37C2×(C2×D11)⋊C4C2211yescharacteristic
38C2×(C2×D11)⋊C4C2211yescharacteristic
39C2×(C2×D11)⋊C4C2211yescharacteristic
40C2×(C2×D11)⋊C4C2211yescharacteristic
41C2×(C2×D11)⋊C4C2211yescharacteristic
42C2×(C2×D11)⋊C4C2211yescharacteristic
43C2×(C2×D11)⋊C4C2211yescharacteristic
44C2×(C2×D11)⋊C4C2211yescharacteristic
45C2×C4⋊Dic11C2211yescharacteristic
46C2×Dic11⋊C4C2211yescharacteristic
47C2×C4×Dic11C2211yescharacteristic
48#66948—24no—
49C11⋊#95C2211yescharacteristic
50C11⋊#95C2211yescharacteristic
51C11⋊#95C2211yescharacteristic
52C11⋊#95C2211yescharacteristic
53C11⋊#95C2211yescharacteristic
54C11⋊#95C2211yescharacteristic
55C24×C11C2311yescharacteristic
56C23×D11C2311yescharacteristic
57C23×D11C2311yescharacteristic
58C23×D11C2311yescharacteristic
59C2×Q8×C11—22no—
60C2×Q8×C11—22no—
61C2×Q8×C11—41no—
62C22×C4×C11—22no—
63C22×C4×C11—22no—
64C22×C4×C11—41no—
65C22×C4×C11C2311yescharacteristic
66C22×C4×C11C2311yescharacteristic
67C22×C4×C11C2311yescharacteristic
68C22×C4×C11C2311yescharacteristic
69C22×C4×C11C2311yescharacteristic
70C22×C4×C11C2311yescharacteristic
71C2×C22⋊D11—22no—
72C2×C22⋊D11—22no—
73C2×C22⋊D11—22no—
74C2×C22⋊D11—22no—
75C2×C22⋊D11—22no—
76C2×C22⋊D11—41no—
77C2×C22⋊D11—41no—
78C2×C22⋊D11—41no—
79C2×C22⋊D11—41no—
80C22×Dic11C2311yescharacteristic
81C22×Dic11C2311yescharacteristic
82C22×Dic11C2311yescharacteristic
83C22×Dic11C2311yescharacteristic
84C22×Dic11C2311yescharacteristic
85C2×D44—22no—
86C2×D44—22no—
87C2×D44—22no—
88C2×D44—22no—
89C2×D44—22no—
90C2×D44—41no—
91C2×D44—41no—
92C2×D44—41no—
93C2×D44—41no—
94C2×C4×D11—22no—
95C2×C4×D11—41no—
96C2×C4×D11—41no—
97C11×C4⋊C4—22no—
98C11×C4⋊C4—22no—
99C11×C4⋊C4D414yescomplemented
100C11×C22⋊C4—22no—
101C11×C22⋊C4—22no—
102C22⋊Dic11—41no—
103(C2×D11)⋊C4—22no—
104(C2×D11)⋊C4—22no—
105(C2×D11)⋊C4—22no—
106(C2×D11)⋊C4—22no—
107(C2×D11)⋊C4—22no—
108(C2×D11)⋊C4—22no—
109(C2×D11)⋊C4—41no—
110(C2×D11)⋊C4—41no—
111(C2×D11)⋊C4—41no—
112(C2×D11)⋊C4—41no—
113(C2×D11)⋊C4D414yescomplemented
114C4⋊Dic11—22no—
115Dic11⋊C4—41no—
116C4×Dic11—22no—
117#2676—111nomaximal, Sylow
118C23×C11—21no—
119C23×C11—21no—
120C23×C11—21no—
121C23×C11—21no—
122C23×C11—21no—
123C23×C11—21no—
124C23×C11#4112yes—
125C23×C11C2411yescharacteristic
126C22×D11—21no—
127C22×D11—21no—
128C22×D11—21no—
129C22×D11—21no—
130C22×D11—21no—
131C22×D11—21no—
132C22×D11—21no—
133C22×D11—21no—
134C22×D11—21no—
135C22×D11—21no—
136C22×D11—21no—
137C22×D11—21no—
138C22×D11—21no—
139C22×D11—21no—
140C22×D11—21no—
141C22×D11—21no—
142C22×D11—21no—
143C22×D11—21no—
144C22×D11—21no—
145C22×D11—21no—
146C22×D11C2×D412yes—
147Q8×C11—44no—
148C2×C4×C11—21no—
149C2×C4×C11—21no—
150C2×C4×C11—21no—
151C2×C4×C11—21no—
152C2×C4×C11—21no—
153C2×C4×C11—21no—
154C2×C4×C11—21no—
155C2×C4×C11—21no—
156C2×C4×C11—21no—
157C2×C4×C11—21no—
158C2×C4×C11—21no—
159C2×C4×C11—21no—
160C2×C4×C11—21no—
161C2×C4×C11—21no—
162C2×C4×C11—44no—
163C2×C4×C11C2×D412yes—
164C2×C4×C11C2×D412yes—
165C2×C4×C11C2×D412yes—
166C2×C4×C11C2×D412yes—
167C22⋊D11—44no—
168C22⋊D11—44no—
169C22⋊D11—44no—
170C2×Dic11—21no—
171C2×Dic11—21no—
172C2×Dic11—21no—
173C2×Dic11—21no—
174C2×Dic11—21no—
175C2×Dic11—21no—
176C2×Dic11—21no—
177C2×Dic11—21no—
178C2×Dic11—21no—
179C2×Dic11—21no—
180C2×Dic11—21no—
181C2×Dic11—21no—
182C2×Dic11—21no—
183C2×Dic11—21no—
184C2×Dic11C2×D412yes—
185D44—44no—
186D44—44no—
187D44—44no—
188C4×D11—44no—
189C2×#124—111no—
190C2×C22⋊Q8D1111yescharacteristic, complemented
191C2×#121—111no—
192C2×#121—111no—
193#396—111no—
194#394—111no—
195#394—111no—
196#393—111no—
197#392—111no—
198#392—111no—
199#390—111no—
200#387—111no—
201#386—111no—
202#386—111no—
203#385—111no—
204C22×C11—22no—
205C22×C11—22no—
206C22×C11—22no—
207C22×C11—41no—
208C22×C11—41no—
209C22×C11—41no—
210C22×C11—41no—
211C22×C11C22×D411yescharacteristic
212C22×C11C22×D411yescharacteristic
213C22×C11C2×#4111yescharacteristic
214C22×C11H(5,𝔽2)11yescharacteristic
215C22×C11H(5,𝔽2)11yescharacteristic
216C22×C11H(5,𝔽2)11yescharacteristic
217C22×C112−1+411yescharacteristic
218C2×D11—22no—
219C2×D11—22no—
220C2×D11—22no—
221C2×D11—41no—
222C2×D11—41no—
223C2×D11—41no—
224C2×D11—41no—
225C2×D11—41no—
226C2×D11—41no—
227C2×D11—41no—
228C2×D11—41no—
229C2×D11—41no—
230C2×D11—41no—
231C2×D11—41no—
232C2×D11—41no—
233C2×D11—41no—
234C2×D11—41no—
235C2×D11—41no—
236C2×D11—41no—
237C2×D11—41no—
238C2×D11—41no—
239C4×C11—22no—
240C4×C11—22no—
241C4×C11—22no—
242C4×C11—22no—
243C4×C11—41no—
244C4×C11—41no—
245Dic11—22no—
246Dic11—41no—
247Dic11—41no—
248Dic11—41no—
249Dic11—41no—
250C22×Q8C2×D1111yescharacteristic
251C22×D4—111no—
252C22×D4—111no—
253C22×D4—111no—
254C22×D4—111no—
255C22×D4—111no—
256C22×D4—111no—
257C23×C4—111no—
258C23×C4C2×D1111yescharacteristic
259#124—224no—
260C22⋊Q8—24no—
261#121—224no—
262#121—224no—
263C2×C4⋊C4—111no—
264C2×C4⋊C4—111no—
265C2×C4⋊C4C2×D1111yescharacteristic
266C2×C4⋊C4C2×D1111yescharacteristic
267C2×C4⋊C4C2×D1111yescharacteristic
268C2×C22⋊C4—111no—
269C2×C22⋊C4—111no—
270C2×C22⋊C4—111no—
271C2×C22⋊C4—111no—
272C2×C22⋊C4—111no—
273C2×C22⋊C4—111no—
274C2×C22⋊C4—111no—
275C2×C22⋊C4—111no—
276C2×C22⋊C4—111no—
277C2×C22⋊C4—111no—
278C2×C22⋊C4—111no—
279C2×C22⋊C4—111no—
280C2×C22⋊C4C2×D1111yescharacteristic
281C2×C22⋊C4C2×D1111yescharacteristic
282C2×C42—111no—
283#95—111no—
284#95—111no—
285#95—111no—
286#95—111no—
287#95—111no—
288#95—111no—
289C2×C11—42no—
290C2×C11#53511yescharacteristic
291C2×C11#53711yescharacteristic
292C2×C11#53811yescharacteristic
293C2×C11D4211yescharacteristic
294C2×C11#54611yescharacteristic
295C2×C11#54811yescharacteristic
296C2×C11#56211yescharacteristic
297D11—42no—
298D11—81no—
299D11—81no—
300C24—111no—
301C24—111no—
302C24—111no—
303C24C22×D1111yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—41no—
307C2×D4—222no—
308C2×D4—222no—
309C2×D4—222no—
310C2×D4—222no—
311C2×D4—222no—
312C2×D4—222no—
313C2×D4—222no—
314C2×D4—222no—
315C2×D4—222no—
316C2×D4—222no—
317C2×D4—441no—
318C2×D4—441no—
319C2×D4—441no—
320C2×D4—441no—
321C2×D4—441no—
322C2×D4—441no—
323C2×D4—441no—
324C2×D4—441no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—111no—
329C22×C4—111no—
330C22×C4—111no—
331C22×C4—111no—
332C22×C4—111no—
333C22×C4—222no—
334C22×C4—441no—
335C22×C4—441no—
336C22×C4C22×D1111yescharacteristic
337C22×C4C22×D1111yescharacteristic
338C22×C4C22×D1111yescharacteristic
339C22×C4C22×D1111yescharacteristic
340C22×C4C22×D1111yescharacteristic
341C22×C4C22×D1111yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—222no—
345C4⋊C4—441no—
346C4⋊C4C22⋊D1114yescomplemented
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—114no—
350C22⋊C4—222no—
351C22⋊C4—222no—
352C22⋊C4—222no—
353C22⋊C4—222no—
354C22⋊C4—222no—
355C22⋊C4—222no—
356C22⋊C4—441no—
357C22⋊C4—441no—
358C22⋊C4—441no—
359C22⋊C4—441no—
360C22⋊C4—441no—
361C42—222no—
362C11#267611yescharacteristic, complemented, Sylow
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—112no—
370C23—221no—
371C23—221no—
372C23—221no—
373C23—221no—
374C23—221no—
375C23—221no—
376C23—221no—
377C23—221no—
378C23—221no—
379C23—221no—
380C23—221no—
381C23—221no—
382C23—221no—
383C23—221no—
384C23—221no—
385C23—221no—
386C23—221no—
387C23—221no—
388C23—221no—
389C23—221no—
390C23#434312yes—
391C23C23×D1111yescharacteristic
392Q8—44no—
393D4—444no—
394D4—444no—
395D4—444no—
396D4—444no—
397D4—444no—
398D4—444no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—44no—
414C2×C4—112no—
415C2×C4—221no—
416C2×C4—221no—
417C2×C4—221no—
418C2×C4—221no—
419C2×C4—221no—
420C2×C4—221no—
421C2×C4—221no—
422C2×C4—221no—
423C2×C4—221no—
424C2×C4—221no—
425C2×C4—221no—
426C2×C4—221no—
427C2×C4—221no—
428C2×C4—221no—
429C2×C4—444no—
430C2×C4D4×D1112yes—
431C2×C4C2×C22⋊D1112yes—
432C2×C4C2×C22⋊D1112yes—
433C2×C4C2×C22⋊D1112yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—222no—
442C22—222no—
443C22—222no—
444C22—441no—
445C22—441no—
446C22—441no—
447C22—441no—
448C22—441no—
449C22—441no—
450C22—441no—
451C22—441no—
452C22—441no—
453C22—441no—
454C22—441no—
455C22—441no—
456C22—441no—
457C22—441no—
458C22—441no—
459C22—441no—
460C22—441no—
461C22—441no—
462C22C2×D4×D1111yescharacteristic
463C22#6703711yescharacteristic
464C22C2×#434311yescharacteristic
465C22#6704011yescharacteristic
466C22#6704211yescharacteristic
467C22#6704211yescharacteristic
468C22C22×C22⋊D1111yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—41no—
474C4—41no—
475C4—222no—
476C4—441no—
477C4—441no—
478C4—441no—
479C4—441no—
480C2—42no—
481C2—442no—
482C2—881no—
483C2—881no—
484C2#1062316011yescharacteristic
485C2#1062316111yescharacteristic
486C2#1062317511yescharacteristic
487C2#1062317811yescharacteristic
488C2D4×C22⋊D1111yescharacteristic
489C2#1062334811yescharacteristic
490C2#1062336111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC2×C11×C22⋊Q8C2
hypercenterC2×C22⋊Q8D11
hypocenterC11#2676
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C11×C22⋊Q8C2semidirect
2. C2×#67000C2semidirect
3. C2×#66983C2semidirect
4. C2×#66948C2semidirect
5. #10622844C2semidirect
6. #10622826C2semidirect
7. #10622607C2semidirect
8. #10622604C2semidirect
9. #10622604C2semidirect
10. #10622602C2semidirect
11. #10622594C2semidirect
12. #10622573C2semidirect
13. #10622573C2semidirect
14. #10622570C2semidirect
15. #10622569C2semidirect
99. C11×C4⋊C4D4semidirect
113. (C2×D11)⋊C4D4semidirect
190. C2×C22⋊Q8D11semidirect
346. C4⋊C4C22⋊D11semidirect
362. C11#2676semidirect

Maximal subgroups

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

subgroupindex
C2×C11×C22⋊Q82
C2×#670002
C2×#669832
C2×#669482
#106228442
#106228262
#106226072
#106226042
#106226022
#106225942
#106225732
#106225702
#106225692
#267611