Finite Group Databasebeta

← #15620532

Subgroups — #15620532

Order 2816.

4732 conjugacy classes of subgroups; only the 727 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 727 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13083428C211yesmaximal, characteristic, complemented
2#13082268C212yesmaximal, complemented
3#13082264C211yesmaximal, characteristic, complemented
4#13078023C2112yesmaximal, complemented
5#13078010C2112yesmaximal, complemented
6#13077079C212yesmaximal, complemented
7C11×#2884C211yesmaximal, characteristic, complemented
8C2×C11×2−1+4C2211yescharacteristic
9#10623457—28no—
10D11×2−1+4—28no—
11#10623453—24no—
12#10623453C22112yes—
13C2×#67040—24no—
14C2×#67040C2212yes—
15C22×Q8×D11C2211yescharacteristic, complemented
16C11×#519C2212yescomplemented
17#10623362—24no—
18#10623359C2216yescomplemented
19#10623356C2216yescomplemented
20Q8×C22⋊D11—24no—
21#10623343C2211yescharacteristic, complemented
22#10623107—28no—
23Q8×D44—28no—
24#10623057C22112yescomplemented
25#10623055C22112yescomplemented
26C11×#410C22112yescomplemented
27#10622619C22124yescomplemented
28#10622355C22148yescomplemented
29#10622354C22116yescomplemented
30C11×2−1+4—28no—
31C2×C11×#41D414yescomplemented
32C2×C11×#41C2316yes—
33C22×Q8×C11D414yescomplemented
34C22×Q8×C11C2311yescharacteristic
35#67043—216no—
36#67043—224no—
37#67042—216no—
38#67042—224no—
39#41×D11—232no—
40#41×D11—248no—
41#67040—232no—
42#67040—212no—
43#67040—28no—
44#67040—28no—
45C2×#4343—216no—
46C2×#4343D418yescomplemented
47C2×Q8×D11—216no—
48C2×Q8×D11—24no—
49C2×Q8×D11—28no—
50C2×Q8×D11D418yescomplemented
51#67037—212no—
52#67037C23116yes—
53C2×#4339—224no—
54C2×#4339C23112yes—
55C22×C4×D11C2313yes—
56C22×Dic22C2313yes—
57C4×Q8×C11—28no—
58C11×#117C23112yes—
59#66999—26no—
60Q8×Dic11—24no—
61#66997—26no—
62#66991—26no—
63#66990C2314yes—
64#66989—26no—
65#66988C23112yes—
66C4×C22⋊D11—212no—
67#66979C2316yes—
68#66977—212no—
69#66950—224no—
70#66950—212no—
71#66948—212no—
72#66939C23124yes—
73#66935C2318yes—
74#66928—212no—
75C4×D44—224no—
76#66922—212no—
77C11×#99C23116yes—
78#66870C23132yes—
79#60400—111nomaximal, Sylow
80C23×D11C2411yescharacteristic
81C11×#41—216no—
82C11×#41—224no—
83C2×Q8×C11—216no—
84C2×Q8×C11C2×D416yes—
85C2×Q8×C11C2×D414yes—
86C2×Q8×C11C2×D418yes—
87C2×D4×C11C2×D416yes—
88C2×D4×C11C2414yes—
89C22×C4×C11C2×D4112yes—
90C22×C4×C11C2413yes—
91C2×C22⋊D11—216no—
92C2×C22⋊D11—24no—
93C2×C22⋊D11C2418yes—
94C22×Dic11C2413yes—
95#4343—248no—
96#4343—232no—
97Q8×D11—248no—
98Q8×D11—232no—
99Q8×D11—416no—
100#4341—248no—
101#4341—232no—
102D4×D11—248no—
103D4×D11—232no—
104#4339—248no—
105#4339—248no—
106#4339—216no—
107#4339—224no—
108#4339—224no—
109C2×D44—224no—
110C2×D44—28no—
111C2×D44C2×D4112yes—
112C2×C4×D11—248no—
113C2×C4×D11—212no—
114C2×C4×D11—26no—
115C2×C4×D11C2×D4124yes—
116C2×Dic22—224no—
117C2×Dic22—212no—
118C2×Dic22—26no—
119C2×Dic22C2×D4112yes—
120C11×C4⋊C4—212no—
121C11×C22⋊C4C2418yes—
122C42×C11—212no—
123C22⋊Dic11—212no—
124C22⋊Dic11C2414yes—
125(C2×D11)⋊C4—28no—
126(C2×D11)⋊C4—224no—
127(C2×D11)⋊C4—24no—
128C4⋊Dic11—224no—
129C4⋊Dic11—26no—
130Dic11⋊C4—212no—
131C4×Dic11—26no—
1322−1+6—111no—
133#3486—111no—
134#3486—112no—
135#3028—1112no—
136#3025—1112no—
137#2884—112no—
138#2884D1111yescharacteristic, complemented
139C23×C11C22×D414yes—
140C23×C11C2511yescharacteristic
141C22×D11—22no—
142C22×D11—21no—
143C22×D11—216no—
144C22×D11—42no—
145C22×D11C22×D418yes—
146Q8×C11—212no—
147Q8×C11#120116yes—
148D4×C11—28no—
149D4×C11—212no—
150C2×C4×C11—224no—
151C2×C4×C11—28no—
152C2×C4×C11C22×D414yes—
153C2×C4×C11C22×D416yes—
154C2×C4×C11C22×D4112yes—
155C22⋊D11—232no—
156C22⋊D11—28no—
157C22⋊D11—28no—
158C22⋊D11—48no—
159C2×Dic11—216no—
160C2×Dic11—24no—
161C2×Dic11—23no—
162C2×Dic11—26no—
163C2×Dic11C22×D418yes—
164D44—216no—
165D44—224no—
166D44—416no—
167C4×D11—232no—
168C4×D11—248no—
169C4×D11—412no—
170Dic22—216no—
171Dic22—224no—
172Dic22—412no—
173#585—1112no—
174#585—228no—
175#585—224no—
176C2×2−1+4—112no—
177C2×2−1+4—228no—
178C2×2−1+4—224no—
179C2×2−1+4C2×D1111yescharacteristic
180C23×Q8—111no—
181D4×Q8—228no—
182D4×Q8—224no—
183#548—228no—
184#548—224no—
185#536—116no—
186#536—1112no—
187#525—116no—
188#525—1112no—
189#519—111no—
190#519C2×D1112yescomplemented
191#458—1148no—
192B(4,𝔽2)—1116no—
193#410—1124no—
194#410C2×D11112yescomplemented
195C22×C11—24no—
196C22×C11—41no—
197C22×C11—44no—
198C22×C11C23×D411yescharacteristic
199C22×C11C23×D412yes—
200C2×D11—24no—
201C2×D11—22no—
202C2×D11—28no—
203C2×D11—41no—
204C2×D11—42no—
205C2×D11—41no—
206C2×D11—48no—
207C2×D11—816no—
208C4×C11—24no—
209C4×C11—48no—
210C4×C11C2×#120112yes—
211Dic11—28no—
212Dic11—44no—
213Dic11—43no—
2142−1+4—28no—
2152−1+4—2232no—
2162−1+4—2224no—
2172−1+4—2216no—
2182−1+4—228no—
2192−1+4—2212no—
2202−1+4—228no—
221H(5,𝔽2)—1116no—
222H(5,𝔽2)—2224no—
223H(5,𝔽2)—2216no—
224H(5,𝔽2)—2212no—
225C2×#41—118no—
226C2×#41—1112no—
227C2×#41—2248no—
228C2×#41—2216no—
229C2×#41—2232no—
230C2×#41—2224no—
231C2×#41D4414yescomplemented
232C2×#41C22×D1116yes—
233C22×Q8—118no—
234C22×Q8—113no—
235C22×Q8—2216no—
236C22×Q8—224no—
237C22×Q8—228no—
238C22×Q8D4414yescomplemented
239C22×Q8C22×D1111yescharacteristic
240C23×C4—113no—
241C4⋊Q8—2212no—
242C4⋊Q8—226no—
243#124—2212no—
244#124—226no—
245#123—1112no—
246#123—1124no—
247C22⋊Q8—2224no—
248C22⋊Q8—2212no—
249C22⋊Q8—226no—
250C22⋊Q8—2212no—
251#121—2212no—
252#121—226no—
253#120—114no—
254#120—118no—
255C4×Q8—28no—
256C4×Q8—224no—
257C4×D4—2224no—
258C4×D4—2212no—
259#117—116no—
260#117C22×D11112yes—
261#99—1132no—
262#99C22×D11116yes—
263C2×C11—21no—
264C2×C11—22no—
265C2×C11—44no—
266C2×C11C22×#12011yescharacteristic
267D11—42no—
268D11—48no—
269C24—111no—
270#41—216no—
271#41—224no—
272#41—2248no—
273#41—2248no—
274#41—2232no—
275#41—2248no—
276#41—2248no—
277#41—2232no—
278#41—2224no—
279#41—2216no—
280#41—2224no—
281C2×Q8—216no—
282C2×Q8—1112no—
283C2×Q8—2224no—
284C2×Q8—2248no—
285C2×Q8—2232no—
286C2×Q8—226no—
287C2×Q8—2212no—
288C2×Q8—4416no—
289C2×Q8C2×D4416yes—
290C2×Q8C2×D4414yes—
291C2×Q8D4×D1118yes—
292C2×D4—118no—
293C2×D4—1112no—
294C2×D4—2224no—
295C2×D4—228no—
296C2×D4—2232no—
297C2×D4—2248no—
298C2×D4—2216no—
299C2×D4—224no—
300C2×D4C2×D4416yes—
301C2×D4C23×D1114yes—
302C22×C4—1124no—
303C22×C4—113no—
304C22×C4—2248no—
305C22×C4—226no—
306C22×C4—2212no—
307C22×C4C2×D44112yes—
308C22×C4C23×D1113yes—
309C4⋊C4—212no—
310C4⋊C4—2224no—
311C4⋊C4—226no—
312C4⋊C4—2212no—
313C22⋊C4—114no—
314C22⋊C4—228no—
315C22⋊C4—2224no—
316C22⋊C4—2212no—
317C22⋊C4—224no—
318C22⋊C4C23×D1118yes—
319C42—212no—
320C42—226no—
321C11#6040011yescharacteristic, complemented, Sylow
322C23—118no—
323C23—2216no—
324C23—221no—
325C23—222no—
326C23—442no—
327C23C22×D4414yes—
328C23C24×D1111yescharacteristic
329Q8—212no—
330Q8—2216no—
331Q8—2224no—
332Q8—4412no—
333Q8#66935116yes—
334D4—28no—
335D4—212no—
336D4—2216no—
337D4—2232no—
338D4—228no—
339D4—228no—
340D4—2224no—
341D4—4416no—
342D4—448no—
343C2×C4—224no—
344C2×C4—28no—
345C2×C4—118no—
346C2×C4—2232no—
347C2×C4—2216no—
348C2×C4—2248no—
349C2×C4—224no—
350C2×C4—223no—
351C2×C4—226no—
352C2×C4—4412no—
353C2×C4C22×D4414yes—
354C2×C4C22×D4416yes—
355C2×C4C2×D4×D11112yes—
356C22—24no—
357C22—41no—
358C22—44no—
359C22—224no—
360C22—222no—
361C22—228no—
362C22—448no—
363C22—442no—
364C22—441no—
365C22—441no—
366C22—8816no—
367C22C23×D4411yescharacteristic
368C22C22×D4×D1112yes—
369C4—24no—
370C4—48no—
371C4—228no—
372C4—444no—
373C4—443no—
374C4C2×#66935112yes—
375C2—22no—
376C2—21no—
377C2—44no—
378C2—442no—
379C2—448no—
380C2C22×#6693511yescharacteristic
subgroupG/H
centerC2C22×#66935
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC11×#2884C2
socleC2×C11C22×#120
hypercenter#2884D11
hypocenterC11#60400
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22×Q8 ◁ #2884

Complements

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

NH ≅ G/Ntype
1. #13083428C2semidirect
2. #13082268C2semidirect
3. #13082264C2semidirect
4. #13078023C2semidirect
5. #13078010C2semidirect
6. #13077079C2semidirect
7. C11×#2884C2semidirect
15. C22×Q8×D11C22semidirect
16. C11×#519C22semidirect
18. #10623359C22semidirect
19. #10623356C22semidirect
21. #10623343C22semidirect
24. #10623057C22semidirect
25. #10623055C22semidirect
26. C11×#410C22semidirect
27. #10622619C22semidirect
28. #10622355C22semidirect
29. #10622354C22semidirect
31. C2×C11×#41D4semidirect
33. C22×Q8×C11D4semidirect
46. C2×#4343D4semidirect
50. C2×Q8×D11D4semidirect
138. #2884D11semidirect
190. #519C2×D11semidirect
194. #410C2×D11semidirect
231. C2×#41D44semidirect
238. C22×Q8D44semidirect
321. C11#60400semidirect

Maximal subgroups

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

subgroupindex
#130834282
#130822682
#130822642
#130780232
#130780102
#130770792
C11×#28842
#6040011