Finite Group Databasebeta

← #14719861

Subgroups — #14719861

Order 2816.

1730 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622820C212yesmaximal, complemented
2#13066976C218yesmaximal, complemented
3#13066943C214yesmaximal, complemented
4C22×C11×#121C211yesmaximal, characteristic
5C23×D4×C11C411yescharacteristic
6C23×D4×C11C412yes—
7C24×C4×C11C2211yescharacteristic
8C2×C11×#121—22no—
9C2×C11×#121—22no—
10C2×C11×#121—22no—
11C2×C11×#121—22no—
12C2×C11×#121C414yes—
13C2×C11×#121C2214yes—
14C22×C11×C4⋊C4C411yescharacteristic
15C22×C11×C22⋊C4C2212yes—
16C22×#4318—22no—
17#10622820—22no—
18#10622820—22no—
19#10622820C2214yes—
20#10622820C2214yescomplemented
21C2×C22⋊(C11⋊C8)—22no—
22C2×C22⋊(C11⋊C8)C2214yescomplemented
23#10622163C22116yescomplemented
24C25×C11C2×C411yescharacteristic
25C25×C11C2×C412yes—
26C22×D4×C11—24no—
27C22×D4×C11—22no—
28C22×D4×C11—22no—
29C22×D4×C11—21no—
30C22×D4×C11—22no—
31C22×D4×C11—21no—
32C22×D4×C11—44no—
33C22×D4×C11—48no—
34C22×D4×C11C2×C414yes—
35C22×D4×C11C2×C418yes—
36C23×C4×C11—44no—
37C23×C4×C11C2×C411yescharacteristic
38C23×C4×C11C2×C412yes—
39C23×C4×C11C2×C412yes—
40C23×C4×C11D412yes—
41C23×C4×C11D412yes—
42C23×C4×C11D412yes—
43C23×C4×C11D412yes—
44C23×C4×C11C2312yes—
45C23×C4×C11C2311yescharacteristic
46C11×#121—24no—
47C11×#121—24no—
48C11×#121—24no—
49C11×#121—24no—
50C11×#121—24no—
51C11×#121—24no—
52C11×#121—24no—
53C11×#121—24no—
54C2×C11×C4⋊C4—21no—
55C2×C11×C4⋊C4—22no—
56C2×C11×C4⋊C4—21no—
57C2×C11×C4⋊C4C2×C414yes—
58C2×C11×C22⋊C4—24no—
59C2×C11×C22⋊C4—24no—
60C2×C11×C22⋊C4C2×C414yes—
61C2×C11×C22⋊C4C2314yes—
62C2×#4318—24no—
63C2×#4318—24no—
64C2×#4318—48no—
65C22×C11⋊C8—24no—
66C22⋊(C11⋊C8)—24no—
67C22⋊(C11⋊C8)—24no—
68C22⋊(C11⋊C8)—24no—
69C22⋊(C11⋊C8)C2318yes—
70#17396—111nomaximal, Sylow
71C24×C11—24no—
72C24×C11—22no—
73C24×C11—22no—
74C24×C11—42no—
75C24×C11—42no—
76C24×C11—44no—
77C24×C11—44no—
78C24×C11—44no—
79C24×C11C22⋊C412yes—
80C24×C11C22⋊C411yescharacteristic
81C24×C11C22⋊C411yescharacteristic
82C24×C11C22×C411yescharacteristic
83C24×C11C22×C412yes—
84C24×C11C22×C414yes—
85C2×D4×C11—28no—
86C2×D4×C11—28no—
87C2×D4×C11—24no—
88C2×D4×C11—24no—
89C2×D4×C11—44no—
90C2×D4×C11—48no—
91C2×D4×C11—48no—
92C2×D4×C11—416no—
93C2×D4×C11—48no—
94C2×D4×C11—48no—
95C2×D4×C11—416no—
96C2×D4×C11—44no—
97C22×C4×C11—28no—
98C22×C4×C11—24no—
99C22×C4×C11—22no—
100C22×C4×C11—22no—
101C22×C4×C11—44no—
102C22×C4×C11—44no—
103C22×C4×C11—44no—
104C22×C4×C11—44no—
105C22×C4×C11—44no—
106C22×C4×C11—42no—
107C22×C4×C11—44no—
108C22×C4×C11C22⋊C411yescharacteristic
109C22×C4×C11C22⋊C412yes—
110C22×C4×C11C22⋊C412yes—
111C22×C4×C11C22⋊C412yes—
112C22×C4×C11C22⋊C412yes—
113C22×C4×C11C22⋊C412yes—
114C22×C4×C11C22⋊C411yescharacteristic
115C22×C4×C11C22×C412yes—
116C22×C4×C11C22×C414yes—
117C22×C4×C11C22×C411yescharacteristic
118C22×C4×C11C2×D412yes—
119C22×C4×C11C2×D412yes—
120C22×C4×C11C2×D412yes—
121C22×C4×C11C2×D412yes—
122C22×C4×C11C2×D412yes—
123C22×C4×C11C2×D412yes—
124C22×C4×C11C2411yescharacteristic
125C11×C4⋊C4—24no—
126C11×C4⋊C4—24no—
127C11×C22⋊C4—28no—
128C11×C22⋊C4—28no—
129#4318—48no—
130#4318—48no—
131C2×C11⋊C8—28no—
132C2×C11⋊C8—48no—
133C22×#121D1111yescharacteristic
134C2×#407—112no—
135#1471—118no—
136#1462—114no—
137C23×C11—22no—
138C23×C11—22no—
139C23×C11—22no—
140C23×C11—24no—
141C23×C11—24no—
142C23×C11—24no—
143C23×C11—22no—
144C23×C11—24no—
145C23×C11—24no—
146C23×C11—28no—
147C23×C11—41no—
148C23×C11—41no—
149C23×C11—42no—
150C23×C11—41no—
151C23×C11—42no—
152C23×C11—41no—
153C23×C11—416no—
154C23×C11—44no—
155C23×C11—44no—
156C23×C11—44no—
157C23×C11—44no—
158C23×C11—44no—
159C23×C11—44no—
160C23×C11—44no—
161C23×C11—44no—
162C23×C11—816no—
163C23×C11#10014yes—
164C23×C11C2×C22⋊C412yes—
165C23×C11C2×C22⋊C411yescharacteristic
166C23×C11C2×C22⋊C411yescharacteristic
167C23×C11C2×C22⋊C412yes—
168C23×C11C23×C411yescharacteristic
169D4×C11—48no—
170D4×C11—416no—
171D4×C11—416no—
172D4×C11—48no—
173C2×C4×C11—24no—
174C2×C4×C11—22no—
175C2×C4×C11—24no—
176C2×C4×C11—24no—
177C2×C4×C11—24no—
178C2×C4×C11—24no—
179C2×C4×C11—24no—
180C2×C4×C11—24no—
181C2×C4×C11—24no—
182C2×C4×C11—42no—
183C2×C4×C11—42no—
184C2×C4×C11—41no—
185C2×C4×C11—42no—
186C2×C4×C11—42no—
187C2×C4×C11—42no—
188C2×C4×C11—41no—
189C2×C4×C11—44no—
190C2×C4×C11—44no—
191C2×C4×C11—44no—
192C2×C4×C11#9914yes—
193C2×C4×C11C2×C22⋊C412yes—
194C2×C4×C11C2×C22⋊C412yes—
195C2×C4×C11C2×C22⋊C412yes—
196C2×C4×C11C22×D412yes—
197C11⋊C8—48no—
198C23×D4Dic1111yescharacteristic
199C23×D4Dic1112yes—
200C24×C4C2×D1111yescharacteristic
201C22×M16—222no—
202C2×#121—22no—
203C2×#121—22no—
204C2×#121—22no—
205C2×#121—22no—
206C2×#121Dic1114yes—
207C2×#121C2×D1114yes—
208C22×C4⋊C4Dic1111yescharacteristic
209C22×C22⋊C4C2×D1112yes—
210#407—114no—
211#407—114no—
212#407—222no—
213#407—222no—
214C2×C22⋊C8—114no—
215C2×C22⋊C8—222no—
216#323—1116no—
217C22×C11—28no—
218C22×C11—21no—
219C22×C11—21no—
220C22×C11—22no—
221C22×C11—21no—
222C22×C11—21no—
223C22×C11—22no—
224C22×C11—21no—
225C22×C11—21no—
226C22×C11—22no—
227C22×C11—22no—
228C22×C11—22no—
229C22×C11—44no—
230C22×C11—48no—
231C22×C11—48no—
232C22×C11—44no—
233C22×C11—44no—
234C22×C11—44no—
235C22×C11—44no—
236C22×C11—88no—
237C22×C11—88no—
238C22×C11—88no—
239C22×C11C2×#9912yes—
240C22×C11C2×#10012yes—
241C22×C11C2×#10014yes—
242C22×C11C22×C22⋊C411yescharacteristic
243C22×C11#56812yes—
244C4×C11—22no—
245C4×C11—22no—
246C4×C11—22no—
247C4×C11—44no—
248C4×C11C2×#9914yes—
249C25C2×Dic1111yescharacteristic
250C25C2×Dic1112yes—
251C22×D4—24no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—44no—
258C22×D4—48no—
259C22×D4C2×Dic1114yes—
260C22×D4C2×Dic1118yes—
261C23×C4—44no—
262C23×C4C2×Dic1111yescharacteristic
263C23×C4C2×Dic1112yes—
264C23×C4C2×Dic1112yes—
265C23×C4C22⋊D1112yes—
266C23×C4C22⋊D1112yes—
267C23×C4C22⋊D1112yes—
268C23×C4C22⋊D1112yes—
269C23×C4C22×D1111yescharacteristic
270C23×C4C22×D1112yes—
271C2×M16—224no—
272C2×M16—224no—
273C2×M16—448no—
274C22×C8—224no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4C2×Dic1114yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4C2×Dic1114yes—
290C2×C22⋊C4C22×D1114yes—
291C22⋊C8—118no—
292C22⋊C8—224no—
293C22⋊C8—224no—
294C22⋊C8—224no—
295C2×C11—22no—
296C2×C11—22no—
297C2×C11—22no—
298C2×C11—24no—
299C2×C11—42no—
300C2×C11—84no—
301C2×C11#287911yescharacteristic
302C2×C11C22×#9911yescharacteristic
303C2×C11C22×#10011yescharacteristic
304C24—24no—
305C24—22no—
306C24—22no—
307C24—42no—
308C24—42no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24C22⋊Dic1112yes—
313C24C22⋊Dic1111yescharacteristic
314C24C22⋊Dic1111yescharacteristic
315C24C22×Dic1111yescharacteristic
316C24C22×Dic1112yes—
317C24C22×Dic1114yes—
318C2×D4—28no—
319C2×D4—28no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—44no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—416no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—416no—
329C2×D4—44no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4C22⋊Dic1111yescharacteristic
342C22×C4C22⋊Dic1112yes—
343C22×C4C22⋊Dic1112yes—
344C22×C4C22⋊Dic1112yes—
345C22×C4C22⋊Dic1112yes—
346C22×C4C22⋊Dic1112yes—
347C22×C4C22⋊Dic1111yescharacteristic
348C22×C4C22×Dic1112yes—
349C22×C4C22×Dic1114yes—
350C22×C4C22×Dic1111yescharacteristic
351C22×C4C2×C22⋊D1112yes—
352C22×C4C2×C22⋊D1112yes—
353C22×C4C2×C22⋊D1112yes—
354C22×C4C2×C22⋊D1112yes—
355C22×C4C2×C22⋊D1112yes—
356C22×C4C2×C22⋊D1112yes—
357C22×C4C23×D1111yescharacteristic
358M16—448no—
359M16—448no—
360C2×C8—228no—
361C2×C8—448no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C22⋊C4—28no—
365C22⋊C4—28no—
366C11#1739611yescharacteristic, complemented, Sylow
367C23—22no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—22no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—28no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—41no—
381C23—42no—
382C23—41no—
383C23—416no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—816no—
393C23#6689714yes—
394C23C2×C22⋊Dic1112yes—
395C23C2×C22⋊Dic1111yescharacteristic
396C23C2×C22⋊Dic1112yes—
397C23C2×C22⋊Dic1111yescharacteristic
398C23C23×Dic1111yescharacteristic
399D4—48no—
400D4—416no—
401D4—416no—
402D4—48no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4#6689814yes—
423C2×C4C2×C22⋊Dic1112yes—
424C2×C4C2×C22⋊Dic1112yes—
425C2×C4C2×C22⋊Dic1112yes—
426C2×C4C22×C22⋊D1112yes—
427C8—448no—
428C22—28no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—44no—
441C22—48no—
442C22—48no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—88no—
448C22—88no—
449C22—88no—
450C22C2×#6689712yes—
451C22C2×#6689714yes—
452C22C2×#6689812yes—
453C22#1062335012yes—
454C22C22×C22⋊Dic1111yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—44no—
459C4C2×#6689814yes—
460C2—22no—
461C2—22no—
462C2—22no—
463C2—24no—
464C2—42no—
465C2—84no—
466C2#1307706111yescharacteristic
467C2C22×#6689811yescharacteristic
468C2C22×#6689711yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic11
commutatorC23×C11C23×C4
FrattiniC22×C4C23×D11
Fitting, prime radicalC22×C11×#121C2
socleC22×C11C22×C22⋊C4
cosocleC22×C4×C11C24
hypercenterC22×#121D11
hypocenterC11#17396
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10622820C2semidirect
2. #13066976C2semidirect
3. #13066943C2semidirect
20. #10622820C22semidirect
22. C2×C22⋊(C11⋊C8)C22semidirect
23. #10622163C22semidirect
366. C11#17396semidirect

Maximal subgroups

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

subgroupindex
C2×#106228202
#130669762
#130669432
C22×C11×#1212
#1739611