Finite Group Databasebeta

← #14770333

Subgroups — #14770333

Order 2816.

1198 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623258C211yesmaximal, characteristic
2M16×Dic22C211yesmaximal, characteristic, complemented
3#13077567C211yesmaximal, characteristic, complemented
4#13069606C211yesmaximal, characteristic, complemented
5#13069603C212yesmaximal, complemented
6#13069603C212yesmaximal, complemented
7#13068607C212yesmaximal
8#13068605C211yesmaximal, characteristic
9C4×#66887C211yesmaximal, characteristic
10#13068407C212yesmaximal
11C11×#1997C211yesmaximal, characteristic, complemented
12C22×Q8×D11C412yes—
13C2×C11×C4⋊Q8C2211yescharacteristic
14C2×#66999C412yes—
15#10623258—21no—
16#10623258—21no—
17#10623258—22no—
18#10623258—22no—
19#10623258—21no—
20#10623258—21no—
21C2×#66949C414yes—
22C2×#66946C2212yes—
23C2×#66941C2212yes—
24C2×C4×D44C2211yescharacteristic
25C2×C4×Dic22C2211yescharacteristic
26C11×#423C2211yescharacteristic
27C4×C11×M16C2211yescharacteristic
28C2×#66887C2212yes—
29C2×#66887C2211yescharacteristic
30C2×#66887C2211yescharacteristic
31#10622731C2212yes—
32#10622731C2212yes—
33M16×Dic11C2212yes—
34M16×Dic11C2212yes—
35#10622537C2211yescharacteristic
36C4×#4318C2211yescharacteristic
37#10622444—21no—
38#10622443—21no—
39#10622441—22no—
40#10622314—22no—
41#10622292—21no—
42C8×Dic22—21no—
43C11×#343C2212yes—
44C11×#343C2212yes—
45#10622192C2212yes—
46#10622192C2211yescharacteristic
47#10622192C2211yescharacteristic
48#10622168C2212yes—
49#10622168C2212yes—
50C22×Q8×C11C2×C412yes—
51C2×Q8×D11—22no—
52C2×Q8×D11—24no—
53C2×Q8×D11—22no—
54C2×Q8×D11—48no—
55C22×D44C2311yescharacteristic
56C22×C4×D11C2×C412yes—
57C22×C4×D11C2×C414yes—
58C22×Dic22C2×C411yescharacteristic
59C22×Dic22C2×C414yes—
60C2×C11×M16C2312yes—
61C2×C11×M16C2311yescharacteristic
62C2×C11×M16C2311yescharacteristic
63C11×C4⋊Q8—21no—
64C11×C4⋊Q8—21no—
65C11×C4⋊Q8—21no—
66C11×C4⋊Q8—21no—
67C2×C11×C4⋊C4C2×C412yes—
68C2×C11×C4⋊C4C2312yes—
69C2×C42×C11C2311yescharacteristic
70#66999—24no—
71#66999—22no—
72#66999—22no—
73C2×(C2×D11)⋊C4C2×C412yes—
74C2×(C2×D11)⋊C4C2×C414yes—
75C2×C4⋊Dic11C2×C411yescharacteristic
76C2×C4⋊Dic11C2311yescharacteristic
77C2×Dic11⋊C4C2×C414yes—
78C2×Dic11⋊C4C2×C412yes—
79C2×C4×Dic11C2312yes—
80C2×#4318C2312yes—
81C2×#4318C2311yescharacteristic
82C2×#4318C2311yescharacteristic
83#66949—24no—
84#66949—24no—
85#66949—24no—
86#66949—24no—
87#66946—22no—
88#66946—22no—
89#66946—22no—
90#66946—22no—
91#66941—22no—
92#66941—22no—
93#66941—22no—
94#66941—22no—
95C4×D44—21no—
96C4×D44—21no—
97C4×D44—21no—
98C4×D44—21no—
99C4×Dic22D411yescharacteristic
100C4×Dic22D411yescharacteristic
101C4×Dic22D412yes—
102C4×Dic22D412yes—
103C4×Dic22D411yescharacteristic
104C4×Dic22D411yescharacteristic
105C11×C4⋊C8—21no—
106C11×C4⋊C8—21no—
107C11×#97—21no—
108C4×C8×C11—21no—
109#66887—24no—
110#66887—22no—
111#66887Q814yescomplemented
112#66879—22no—
113#66879—22no—
114Dic11⋊C8—22no—
115Dic11⋊C8—22no—
116Dic11⋊C8—22no—
117Dic11⋊C8—22no—
118C8×Dic11—22no—
119C8×Dic11—22no—
120C4⋊(C11⋊C8)—21no—
121C4⋊(C11⋊C8)—21no—
122#66867—21no—
123C4×C11⋊C8—21no—
124Q8×#100—111nomaximal, Sylow
125C23×D11C22×C412yes—
126C2×Q8×C11—22no—
127C2×Q8×C11—22no—
128C2×Q8×C11—44no—
129C22×C4×C11C22×C411yescharacteristic
130C22×C4×C11C22×C414yes—
131C22×C4×C11C22×C411yescharacteristic
132C22×C4×C11C2411yescharacteristic
133C22×Dic11C22×C412yes—
134C22×Dic11C22×C412yes—
135C22×Dic11C22×C412yes—
136Q8×D11—416no—
137Q8×D11—48no—
138Q8×D11—48no—
139C2×D44—21no—
140C2×D44—21no—
141C2×D44—21no—
142C2×D44—21no—
143C2×D44C2×Q812yes—
144C2×D44#4112yes—
145C2×C4×D11—24no—
146C2×C4×D11—24no—
147C2×C4×D11—22no—
148C2×C4×D11—22no—
149C2×C4×D11—44no—
150C2×C4×D11—48no—
151C2×Dic22—24no—
152C2×Dic22—24no—
153C2×Dic22—24no—
154C2×Dic22—48no—
155C2×Dic22C22⋊C412yes—
156C2×Dic22C22⋊C411yescharacteristic
157C2×Dic22C22⋊C411yescharacteristic
158C11×M16—22no—
159C11×M16—21no—
160C11×M16—21no—
161C11×M16—21no—
162C11×M16—22no—
163C11×M16C2×Q812yescomplemented
164C2×C8×C11—21no—
165C2×C8×C11—22no—
166C2×C8×C11—21no—
167C11×C4⋊C4—22no—
168C11×C4⋊C4—22no—
169C11×C4⋊C4—22no—
170C11×C4⋊C4—22no—
171C42×C11C2×D411yescharacteristic
172C42×C11C2×D411yescharacteristic
173C42×C11C2×D411yescharacteristic
174C42×C11C2×D411yescharacteristic
175(C2×D11)⋊C4—24no—
176(C2×D11)⋊C4—24no—
177(C2×D11)⋊C4—22no—
178(C2×D11)⋊C4—22no—
179C4⋊Dic11—21no—
180C4⋊Dic11—21no—
181C4⋊Dic11C22⋊C412yes—
182C4⋊Dic11C22⋊C411yescharacteristic
183C4⋊Dic11C22⋊C411yescharacteristic
184Dic11⋊C4—24no—
185Dic11⋊C4—24no—
186Dic11⋊C4C22⋊C412yes—
187Dic11⋊C4C22⋊C412yes—
188Dic11⋊C4C22⋊C412yes—
189Dic11⋊C4C22⋊C412yes—
190C4×Dic11C2×D412yes—
191C4×Dic11C2×D412yes—
192C4×Dic11C2×D412yes—
193C4×Dic11C2×D412yes—
194#4318—22no—
195#4318—21no—
196#4318—21no—
197#4318—22no—
198#4318—21no—
199#4318C2×Q812yescomplemented
200C2×C11⋊C8—21no—
201C2×C11⋊C8—21no—
202C2×C11⋊C8—22no—
203C2×D4×Q8—111no—
204Q8×M16—111no—
205Q8×M16—111no—
206#1997—112no—
207#1997—112no—
208#1997—111no—
209#1997D1111yescharacteristic, complemented
210#1800—111no—
211#1800—112no—
212C4×#100—111no—
213C4×#100—112no—
214C23×C11C23×C411yescharacteristic
215C22×D11—22no—
216C22×D11—22no—
217C22×D11—42no—
218C22×D11—42no—
219C22×D11C4×Q814yes—
220Q8×C11—44no—
221Q8×C11—44no—
222C2×C4×C11—22no—
223C2×C4×C11—24no—
224C2×C4×C11—41no—
225C2×C4×C11—44no—
226C2×C4×C11C2×C22⋊C411yescharacteristic
227C2×C4×C11C2×C22⋊C411yescharacteristic
228C2×C4×C11C2×C22⋊C411yescharacteristic
229C2×C4×C11C2×C22⋊C411yescharacteristic
230C2×C4×C11C22×D411yescharacteristic
231C2×C4×C11C22×D411yescharacteristic
232C2×C4×C11C22×Q811yescharacteristic
233C2×C4×C11C2×#4111yescharacteristic
234C2×C4×C112−1+411yescharacteristic
235C2×C4×C112−1+411yescharacteristic
236C2×Dic11—24no—
237C2×Dic11—22no—
238C2×Dic11—42no—
239C2×Dic11—42no—
240C2×Dic11C2×C22⋊C412yes—
241C2×Dic11C2×C22⋊C412yes—
242C2×Dic11C2×C22⋊C412yes—
243C2×Dic11C2×C22⋊C412yes—
244D44—22no—
245D44—22no—
246D44—22no—
247D44—22no—
248C4×D11—48no—
249C4×D11—44no—
250C4×D11—44no—
251C4×D11—48no—
252Dic22—22no—
253Dic22—24no—
254Dic22—48no—
255Dic22—48no—
256Dic22#10014yescomplemented
257C8×C11—22no—
258C8×C11—41no—
259C8×C11—42no—
260C11⋊C8—22no—
261C11⋊C8—42no—
262C11⋊C8—41no—
263C23×Q8—112no—
264D4×Q8—221no—
265D4×Q8—221no—
266D4×Q8—222no—
267D4×Q8—222no—
268D4×Q8—221no—
269D4×Q8—221no—
270C2×C4⋊Q8—112no—
271C2×C4⋊Q8C2×D1111yescharacteristic
272C2×C22⋊Q8—112no—
273C2×C22⋊Q8—114no—
274C2×C4×Q8—111no—
275C2×C4×D4—112no—
276C2×C4×D4—111no—
277#446—221no—
278#446—222no—
279#446—222no—
280#446—221no—
281C8×Q8—221no—
282C8×Q8—221no—
283#423—112no—
284#423—112no—
285#423—111no—
286#423C2×D1111yescharacteristic
287C2×#100—111no—
288C2×#100—112no—
289C2×#100—111no—
290C4×M16—112no—
291C4×M16—112no—
292C4×M16—111no—
293C4×M16C2×D1111yescharacteristic
294#343—112no—
295#343—111no—
296#343—111no—
297#343—112no—
298#343—112no—
299#343C2×D1112yes—
300#343C2×D1112yes—
301C22×C11—21no—
302C22×C11—21no—
303C22×C11C22×C22⋊C411yescharacteristic
304C22×C11C2×C4×Q811yescharacteristic
305C22×C11#51911yescharacteristic
306C2×D11—24no—
307C2×D11—44no—
308C2×D11—42no—
309C2×D11—42no—
310C2×D11—44no—
311C4×C11—21no—
312C4×C11—21no—
313C4×C11—21no—
314C4×C11—44no—
315C4×C11C2×#10012yes—
316C4×C11#54811yescharacteristic
317C4×C11#54811yescharacteristic
318C4×C11D4×Q811yescharacteristic
319C4×C11D4×Q811yescharacteristic
320Dic11—22no—
321Dic11—22no—
322Dic11—22no—
323Dic11—42no—
324Dic11C2×#10014yes—
325C22×Q8—111no—
326C22×Q8—114no—
327C22×Q8—222no—
328C22×Q8—224no—
329C22×Q8—222no—
330C22×Q8—448no—
331C22×Q8C4×D1112yes—
332C22×D4—111no—
333C23×C4—114no—
334C23×C4—112no—
335C2×M16—111no—
336C2×M16—112no—
337C2×M16—111no—
338C2×M16C22×D1112yes—
339C2×M16C22×D1111yescharacteristic
340C2×M16C22×D1111yescharacteristic
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—21no—
345C4⋊Q8—222no—
346C4⋊Q8—222no—
347C4⋊Q8—222no—
348C4⋊Q8—222no—
349C22⋊Q8—224no—
350C22⋊Q8—224no—
351C22⋊Q8—224no—
352C22⋊Q8—224no—
353C22⋊Q8—224no—
354C22⋊Q8—222no—
355C22⋊Q8—222no—
356C4×Q8—111no—
357C4×Q8—111no—
358C4×Q8—112no—
359C4×Q8—112no—
360C4×Q8—111no—
361C4×Q8—111no—
362C4×D4—222no—
363C4×D4—222no—
364C4×D4—222no—
365C4×D4—222no—
366C4×D4—221no—
367C4×D4—221no—
368C4×D4—221no—
369C4×D4—221no—
370C2×C4⋊C4—112no—
371C2×C4⋊C4—111no—
372C2×C4⋊C4—114no—
373C2×C4⋊C4—111no—
374C2×C4⋊C4C4×D1112yes—
375C2×C4⋊C4C22×D1112yes—
376C2×C22⋊C4—114no—
377C2×C22⋊C4—112no—
378C2×C42—112no—
379C2×C42C22×D1111yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—222no—
383C4⋊C8—222no—
384C4⋊C8—222no—
385C4⋊C8—222no—
386C4⋊C8—221no—
387C4⋊C8—221no—
388#100—114no—
389#100—222no—
390#100—224no—
391#97—21no—
392#97—222no—
393#97—222no—
394#97—221no—
395C4×C8—21no—
396C4×C8—222no—
397C4×C8—222no—
398C4×C8—221no—
399C2×C11—21no—
400C2×C11—21no—
401C2×C11Q8×C22⋊C411yescharacteristic
402C2×C11C22×#10011yescharacteristic
403C2×C11#288911yescharacteristic
404D11—44no—
405C24—112no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—111no—
410C2×Q8—112no—
411C2×Q8—111no—
412C2×Q8—224no—
413C2×Q8—224no—
414C2×Q8—224no—
415C2×Q8—448no—
416C2×Q8—448no—
417C2×Q8—4416no—
418C2×Q8—448no—
419C2×D4—112no—
420C2×D4—112no—
421C2×D4—221no—
422C2×D4—221no—
423C2×D4—221no—
424C2×D4—221no—
425C22×C4—112no—
426C22×C4—112no—
427C22×C4—112no—
428C22×C4—224no—
429C22×C4—224no—
430C22×C4—222no—
431C22×C4—222no—
432C22×C4—444no—
433C22×C4—448no—
434C22×C4C2×C4×D1111yescharacteristic
435C22×C4C2×C4×D1114yes—
436C22×C4C2×C4×D1111yescharacteristic
437C22×C4C23×D1111yescharacteristic
438M16—21no—
439M16—21no—
440M16—21no—
441M16—22no—
442M16—22no—
443M16—112no—
444M16—221no—
445M16—222no—
446M16—222no—
447M16—221no—
448M16—221no—
449M16Q8×D1112yescomplemented
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—221no—
454C2×C8—222no—
455C2×C8—221no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—112no—
461C4⋊C4—112no—
462C4⋊C4—112no—
463C4⋊C4—112no—
464C4⋊C4—112no—
465C4⋊C4—111no—
466C4⋊C4—111no—
467C4⋊C4—224no—
468C4⋊C4—224no—
469C4⋊C4—221no—
470C4⋊C4—221no—
471C22⋊C4—224no—
472C22⋊C4—224no—
473C22⋊C4—222no—
474C22⋊C4—222no—
475C42—112no—
476C42—112no—
477C42—112no—
478C42—112no—
479C42D4×D1111yescharacteristic
480C42D4×D1111yescharacteristic
481C42D4×D1111yescharacteristic
482C42D4×D1111yescharacteristic
483C11Q8×#10011yescharacteristic, complemented, Sylow
484C23—114no—
485C23—222no—
486C23—222no—
487C23—442no—
488C23—442no—
489C23C22×C4×D1111yescharacteristic
490Q8—44no—
491Q8—44no—
492Q8—114no—
493Q8—224no—
494Q8—222no—
495Q8—448no—
496Q8—448no—
497D4—222no—
498D4—222no—
499D4—222no—
500D4—222no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—112no—
506C2×C4—112no—
507C2×C4—112no—
508C2×C4—112no—
509C2×C4—224no—
510C2×C4—222no—
511C2×C4—444no—
512C2×C4—448no—
513C2×C4—448no—
514C2×C4—442no—
515C2×C4—444no—
516C2×C4—442no—
517C2×C4C22⋊C4×D1111yescharacteristic
518C2×C4C22⋊C4×D1111yescharacteristic
519C2×C4C22⋊C4×D1111yescharacteristic
520C2×C4C22⋊C4×D1111yescharacteristic
521C2×C4C2×D4×D1111yescharacteristic
522C2×C4C2×D4×D1111yescharacteristic
523C2×C4C2×Q8×D1111yescharacteristic
524C2×C4#6704011yescharacteristic
525C2×C4#41×D1111yescharacteristic
526C2×C4#6704311yescharacteristic
527C8—22no—
528C8—41no—
529C8—42no—
530C8—222no—
531C8—442no—
532C8—441no—
533C22—21no—
534C22—21no—
535C22—224no—
536C22—444no—
537C22—444no—
538C22—442no—
539C22—442no—
540C22C2×C22⋊C4×D1111yescharacteristic
541C22C4×Q8×D1111yescharacteristic
542C22#1062310311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—44no—
547C4—114no—
548C4—222no—
549C4—222no—
550C4—222no—
551C4—442no—
552C4D11×#10012yes—
553C4#1062316311yescharacteristic
554C4#1062316411yescharacteristic
555C4#1062320811yescharacteristic
556C4#1062325811yescharacteristic
557C2—21no—
558C2—21no—
559C2—444no—
560C2#1307145011yescharacteristic
561C2C2×D11×#10011yescharacteristic
562C2#1307709611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D11
commutatorC23×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1997C2
socleC22×C11C22×C22⋊C4
prime radicalC22×D44C23
cosocleC22×C4×C11C24
hypercenter#1997D11
hypocenterC11Q8×#100
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. M16×Dic22C2semidirect
3. #13077567C2semidirect
4. #13069606C2semidirect
5. #13069603C2semidirect
6. #13069603C2semidirect
11. C11×#1997C2semidirect
111. #66887Q8semidirect
163. C11×M16C2×Q8semidirect
199. #4318C2×Q8semidirect
209. #1997D11semidirect
256. Dic22#100semidirect
449. M16Q8×D11semidirect
483. C11Q8×#100semidirect

Maximal subgroups

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

subgroupindex
C2×#106232582
M16×Dic222
#130775672
#130696062
#130696032
#130686072
#130686052
C4×#668872
#130684072
C11×#19972
Q8×#10011