Finite Group Databasebeta

← #14656314

Subgroups — #14656314

Order 2816.

710 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622306C211yesmaximal, characteristic, complemented
2C2×#10622620C211yesmaximal, characteristic, complemented
3C2×#10622628C211yesmaximal, characteristic
4C2×#10622120C211yesmaximal, characteristic, complemented
5C2×#10622123C211yesmaximal, characteristic, complemented
6#13066451C211yesmaximal, characteristic, complemented
7C11×#1954C211yesmaximal, characteristic, complemented
8C2×C4×D4×C11C2211yescharacteristic
9C22×#4323—21no—
10C22×#4314—22no—
11C2×#66927C2211yescharacteristic
12C2×#66922C2211yescharacteristic
13C2×C11×C4⋊C8C2211yescharacteristic
14C2×C11×C22⋊C8—22no—
15C2×C4×C8×C11C2211yescharacteristic
16C2×#66896—21no—
17#10622628—22no—
18#10622628—22no—
19#10622620—22no—
20#10622620—22no—
21C2×#66873—21no—
22C2×#66871—21no—
23C2×C4⋊(C11⋊C8)C2211yescharacteristic
24C2×C4⋊(C11⋊C8)C2211yescharacteristic
25#10622306—22no—
26#10622306—22no—
27#10622123—22no—
28#10622120—22no—
29C22×D4×C11D411yescharacteristic
30C23×C4×C11—21no—
31C22×D44—21no—
32C22×D44—21no—
33C22×D44—21no—
34C22×Dic22—21no—
35C22×C8×C11—21no—
36C22×C8×C11D412yes—
37C4×D4×C11—21no—
38C4×D4×C11—21no—
39C4×D4×C11—21no—
40C4×D4×C11—21no—
41C2×C11×C4⋊C4D411yescharacteristic
42C2×C11×C22⋊C4—21no—
43C2×C42×C11C2311yescharacteristic
44C2×#4323—24no—
45C2×#4323—42no—
46C2×#4323—42no—
47C2×C4⋊Dic11—21no—
48C2×C4⋊Dic11—21no—
49C22×C11⋊C8—21no—
50C22×C11⋊C8—21no—
51C2×#4314—28no—
52C2×#4314—44no—
53C2×#4314—44no—
54#66927—21no—
55#66927—21no—
56#66927—21no—
57#66927D412yes—
58#66922—21no—
59#66922—21no—
60#66922—21no—
61#66922D412yes—
62C11×C4⋊C8—21no—
63C11×C4⋊C8—21no—
64C11×C22⋊C8—44no—
65C4×C8×C11—21no—
66C4×C8×C11—21no—
67#66896—24no—
68#66873—24no—
69#66871—24no—
70C4⋊(C11⋊C8)—21no—
71C4⋊(C11⋊C8)—21no—
72C4⋊(C11⋊C8)D412yes—
73C4⋊(C11⋊C8)D412yes—
74#10266—111nomaximal, Sylow
75C24×C11—21no—
76C23×D11—41no—
77C2×D4×C11—21no—
78C2×D4×C11—21no—
79C2×D4×C11—21no—
80C2×D4×C11—21no—
81C2×D4×C11—21no—
82C2×D4×C11SD1612yescomplemented
83C22×C4×C11—21no—
84C22×C4×C11—41no—
85C22×C4×C11—41no—
86C22×C4×C11—42no—
87C22×C4×C11—42no—
88C22×C4×C11C2×D411yescharacteristic
89C22×C4×C11C2×D411yescharacteristic
90C22×C4×C11#4111yescharacteristic
91C22×Dic11—41no—
92C2×D44—22no—
93C2×D44—22no—
94C2×D44—22no—
95C2×D44—41no—
96C2×D44—41no—
97C2×D44—41no—
98C2×D44—42no—
99C2×D44—41no—
100C2×D44—41no—
101C2×D44—42no—
102C2×D44—41no—
103C2×D44—41no—
104C2×D44—82no—
105C2×Dic22—22no—
106C2×Dic22—41no—
107C2×Dic22—42no—
108C2×Dic22—42no—
109C2×C8×C11—22no—
110C2×C8×C11—22no—
111C2×C8×C11—41no—
112C2×C8×C11—42no—
113C2×C8×C11D814yescomplemented
114C11×C4⋊C4—21no—
115C11×C4⋊C4—21no—
116C11×C22⋊C4—41no—
117C11×C22⋊C4—41no—
118C42×C11C2×D411yescharacteristic
119C42×C11C2×D411yescharacteristic
120C42×C11C2×D411yescharacteristic
121C42×C11C2×D411yescharacteristic
122#4323—48no—
123C4⋊Dic11—22no—
124C4⋊Dic11—22no—
125C4⋊Dic11—41no—
126C4⋊Dic11—41no—
127C2×C11⋊C8—22no—
128C2×C11⋊C8—22no—
129C2×C11⋊C8—41no—
130C2×C11⋊C8—41no—
131C2×C11⋊C8—41no—
132C2×C11⋊C8—41no—
133#4314—416no—
134C2×#492—111no—
135C2×#474—111no—
136C2×#459—111no—
137#1954D1111yescharacteristic, complemented
138C2×#332—111no—
139C2×#331—111no—
140#1277—111no—
141C23×C11—22no—
142C23×C11—41no—
143C23×C11—42no—
144C23×C11—41no—
145C23×C11—42no—
146C23×C11#12111yescharacteristic
147C22×D11—42no—
148C22×D11—82no—
149C22×D11—81no—
150C22×D11—81no—
151C22×D11—82no—
152D4×C11—22no—
153D4×C11—22no—
154D4×C11—41no—
155D4×C11—41no—
156C2×C4×C11—22no—
157C2×C4×C11—41no—
158C2×C4×C11—42no—
159C2×C4×C11—82no—
160C2×C4×C11—81no—
161C2×C4×C11—81no—
162C2×C4×C11#12011yescharacteristic
163C2×C4×C11#12011yescharacteristic
164C2×C4×C11#12011yescharacteristic
165C2×C4×C11#12111yescharacteristic
166C2×C4×C11#12111yescharacteristic
167C2×C4×C11#12711yescharacteristic
168C2×C4×C11C2×D812yes—
169C2×C4×C11C2×SD1611yescharacteristic
170C2×C4×C11#13511yescharacteristic
171C2×C4×C11AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C11AGL(1,ℤ/8)12yes—
173C2×C4×C11#13711yescharacteristic
174C2×Dic11—42no—
175C2×Dic11—82no—
176D44—44no—
177D44—42no—
178D44—42no—
179D44—82no—
180D44—82no—
181D44—81no—
182D44—81no—
183D44—81no—
184D44—81no—
185Dic22—44no—
186Dic22—81no—
187Dic22—81no—
188C8×C11—24no—
189C8×C11—81no—
190C11⋊C8—42no—
191C11⋊C8—42no—
192C22×SD16—222no—
193C22×D8—221no—
194C2×C4⋊Q8—111no—
195C2×#127—111no—
196C2×C4×D4C2×D1111yescharacteristic
197#492—222no—
198#492—222no—
199#474—222no—
200#474—222no—
201#459—222no—
202#459—222no—
203C2×#107—221no—
204C2×C4⋊C8—111no—
205C2×C4⋊C8—111no—
206C2×C4⋊C8C2×D1111yescharacteristic
207C2×#102—221no—
208C2×#102—221no—
209C2×C22⋊C8—22no—
210C2×C4×C8C2×D1111yescharacteristic
211#332—222no—
212#331—222no—
213C22×C11—44no—
214C22×C11—42no—
215C22×C11—82no—
216C22×C11—81no—
217C22×C11—81no—
218C22×C11#39211yescharacteristic
219C22×C11#45912yes—
220C22×C11#46511yescharacteristic
221C22×C11#46811yescharacteristic
222C22×C11#47112yes—
223C2×D11—82no—
224C2×D11—82no—
225C2×D11—82no—
226C2×D11—162no—
227C2×D11—162no—
228C4×C11—21no—
229C4×C11—21no—
230C4×C11—81no—
231C4×C11#44712yes—
232C4×C11#44811yescharacteristic
233C4×C11#44911yescharacteristic
234C4×C11#45011yescharacteristic
235C4×C11#45211yescharacteristic
236C4×C11#49311yescharacteristic
237C4×C11#49611yescharacteristic
238Dic11—82no—
239C22×Q8—221no—
240C22×D4—221no—
241C22×D4—221no—
242C22×D4—221no—
243C22×D4D4411yescharacteristic
244C23×C4—21no—
245C2×SD16—228no—
246C2×SD16—444no—
247C2×SD16—444no—
248C2×D8—224no—
249C2×D8—442no—
250C2×D8—442no—
251C22×C8—21no—
252C22×C8—221no—
253C22×C8—221no—
254C22×C8C22⋊D1112yes—
255C4⋊Q8—112no—
256C4⋊Q8—221no—
257C4⋊Q8—221no—
258C4⋊Q8—221no—
259#127—112no—
260#127—221no—
261#127—221no—
262#127—221no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C2×C4⋊C4—221no—
268C2×C4⋊C4—221no—
269C2×C4⋊C4D4411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C42C22×D1111yescharacteristic
272#107—224no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—112no—
276C4⋊C8—112no—
277C4⋊C8—221no—
278C4⋊C8—221no—
279#102—224no—
280#102—224no—
281C22⋊C8—44no—
282C4×C8—21no—
283C4×C8—21no—
284C2×C11—82no—
285C2×C11#166512yes—
286C2×C11#168512yes—
287C2×C11#200011yescharacteristic
288C2×C11#201111yescharacteristic
289C2×C11#201911yescharacteristic
290D11—162no—
291C24—21no—
292C24—441no—
293C2×Q8—222no—
294C2×Q8—441no—
295C2×Q8—442no—
296C2×Q8—442no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—222no—
303C2×D4—222no—
304C2×D4—222no—
305C2×D4—441no—
306C2×D4—441no—
307C2×D4—441no—
308C2×D4—441no—
309C2×D4—441no—
310C2×D4—441no—
311C2×D4—442no—
312C2×D4—441no—
313C2×D4—442no—
314C2×D4—882no—
315C2×D4#431412yescomplemented
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—441no—
322C22×C4C2×D4411yescharacteristic
323C22×C4#433911yescharacteristic
324C22×C4C2×C22⋊D1111yescharacteristic
325SD16—4416no—
326D8—448no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—222no—
332C2×C8—222no—
333C2×C8—441no—
334C2×C8—441no—
335C2×C8—441no—
336C2×C8—441no—
337C2×C8#432314yescomplemented
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—222no—
341C4⋊C4—222no—
342C4⋊C4—441no—
343C4⋊C4—441no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42D4×D1111yescharacteristic
347C42D4×D1111yescharacteristic
348C42D4×D1111yescharacteristic
349C42D4×D1111yescharacteristic
350C11#1026611yescharacteristic, complemented, Sylow
351C23—22no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—42no—
356C23—442no—
357C23—881no—
358C23—881no—
359C23—882no—
360C23—882no—
361C23#6698311yescharacteristic
362Q8—444no—
363Q8—881no—
364Q8—881no—
365D4—22no—
366D4—22no—
367D4—41no—
368D4—41no—
369D4—442no—
370D4—442no—
371D4—444no—
372D4—882no—
373D4—882no—
374D4—881no—
375D4—881no—
376D4—881no—
377D4—881no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—442no—
385C2×C4—882no—
386C2×C4#6693511yescharacteristic
387C2×C4#6693511yescharacteristic
388C2×C4#6693711yescharacteristic
389C2×C4#6693711yescharacteristic
390C2×C4C2×#431411yescharacteristic
391C2×C4#6695711yescharacteristic
392C2×C4#6696111yescharacteristic
393C2×C4#6696211yescharacteristic
394C2×C4C2×#432312yes—
395C2×C4#6699011yescharacteristic
396C2×C4#6699311yescharacteristic
397C2×C4#6700212yes—
398C8—24no—
399C8—81no—
400C8—442no—
401C8—442no—
402C22—42no—
403C22—44no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—882no—
408C22—882no—
409C22—882no—
410C22—1762no—
411C22—1762no—
412C22#1062257311yescharacteristic
413C22#1062262812yes—
414C22#1062272711yescharacteristic
415C22#1062274711yescharacteristic
416C22#1062275412yes—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—882no—
421C4#1062234411yescharacteristic
422C4#1062234611yescharacteristic
423C4#1062234711yescharacteristic
424C4#1062235111yescharacteristic
425C4#1062264912yes—
426C4#1062267111yescharacteristic
427C4#1062268611yescharacteristic
428C2—82no—
429C2—1762no—
430C2#1306795712yes—
431C2#1306808112yes—
432C2#1306962111yescharacteristic
433C2#1306968611yescharacteristic
434C2#1306973111yescharacteristic
subgroupG/H
centerC23#66983
commutator, cosocleC2×C42×C11C23
FrattiniC2×C42C22×D11
FittingC11×#1954C2
socleC23×C11#121
prime radicalC2×#10622628C2
hypercenter#1954D11
hypocenterC11#10266
LCSG ▷ C2×C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#10622306C2semidirect
2. C2×#10622620C2semidirect
4. C2×#10622120C2semidirect
5. C2×#10622123C2semidirect
6. #13066451C2semidirect
7. C11×#1954C2semidirect
82. C2×D4×C11SD16semidirect
113. C2×C8×C11D8semidirect
137. #1954D11semidirect
315. C2×D4#4314semidirect
337. C2×C8#4323semidirect
350. C11#10266semidirect

Maximal subgroups

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

subgroupindex
C2×#106223062
C2×#106226202
C2×#106226282
C2×#106221202
C2×#106221232
#130664512
C11×#19542
#1026611