Finite Group Databasebeta

← #14672066

Subgroups — #14672066

Order 2816.

624 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#10622177C212yesmaximal, complemented
2C2×#10622174C212yesmaximal
3#13068604C211yesmaximal, characteristic, complemented
4C4×#66897C211yesmaximal, characteristic, complemented
5C11×#2534C211yesmaximal, characteristic
6C2×C11×#123C2211yescharacteristic
7C2×C11×#120C2211yescharacteristic
8C2×C4×D4×C11C2211yescharacteristic
9C22×C11×C22⋊C4C411yescharacteristic, complemented
10C11×#396—21no—
11C11×#394—21no—
12C11×#388—21no—
13C11×#386—21no—
14C11×#381C411yescharacteristic, complemented
15C11×#380C411yescharacteristic, complemented
16C11×#379C411yescharacteristic, complemented
17C2×#66898C2212yes—
18C2×#66897C2212yes—
19#10622825—21no—
20C4×#4318—21no—
21#10622177—24no—
22#10622174—24no—
23#10622168—21no—
24#10622164—21no—
25C25×C11C2×C411yescharacteristic
26C22×D4×C11—21no—
27C22×D4×C11C2311yescharacteristic
28C23×C4×C11C2×C411yescharacteristic
29C23×C4×C11C2×C411yescharacteristic
30C23×C4×C11D411yescharacteristic
31C11×#123—22no—
32C11×#123—22no—
33C11×#120—22no—
34C11×#120—22no—
35C4×D4×C11—41no—
36C4×D4×C11—41no—
37C2×C11×C4⋊C4—21no—
38C2×C11×C4⋊C4—21no—
39C2×C11×C4⋊C4D411yescharacteristic
40C2×C11×C22⋊C4—21no—
41C2×C11×C22⋊C4—21no—
42C2×C11×C22⋊C4—21no—
43C2×C11×C22⋊C4—21no—
44C2×C11×C22⋊C4—21no—
45C2×C11×C22⋊C4—21no—
46C2×C11×C22⋊C4—41no—
47C2×C11×C22⋊C4—41no—
48C2×C11×C22⋊C4C2×C411yescharacteristic
49C2×C11×C22⋊C4C2×C411yescharacteristic
50C2×C11×C22⋊C4C2×C411yescharacteristic
51C2×C11×C22⋊C4D411yescharacteristic
52C2×C42×C11D411yescharacteristic
53C2×C22⋊Dic11—22no—
54C2×C4⋊Dic11—41no—
55C2×#4318—22no—
56C11×#95—21no—
57C11×#95—21no—
58C11×#95—21no—
59#66898—22no—
60#66898—22no—
61#66897—22no—
62#66897D414yes—
63C11⋊#95—41no—
64#66867—41no—
65C4×C11⋊C8—41no—
66#13219—111nomaximal, Sylow
67C24×C11—21no—
68C24×C11—22no—
69C24×C11—41no—
70C24×C11—42no—
71C24×C11—41no—
72C24×C11—81no—
73C24×C11C22×C411yescharacteristic
74C24×C11C2×D411yescharacteristic
75C2×D4×C11—41no—
76C2×D4×C11—41no—
77C2×D4×C11—42no—
78C2×D4×C11—41no—
79C2×D4×C11—41no—
80C2×D4×C11—81no—
81C2×D4×C11C2×D412yes—
82C2×D4×C11#4112yes—
83C22×C4×C11—21no—
84C22×C4×C11—21no—
85C22×C4×C11—21no—
86C22×C4×C11—21no—
87C22×C4×C11—21no—
88C22×C4×C11—21no—
89C22×C4×C11—21no—
90C22×C4×C11—22no—
91C22×C4×C11—41no—
92C22×C4×C11—41no—
93C22×C4×C11—81no—
94C22×C4×C11—81no—
95C22×C4×C11C22⋊C411yescharacteristic
96C22×C4×C11C22⋊C411yescharacteristic
97C22×C4×C11C22⋊C411yescharacteristic
98C22×C4×C11C22⋊C411yescharacteristic
99C22×C4×C11C2×D411yescharacteristic
100C22×Dic11—42no—
101C11×C4⋊C4—41no—
102C11×C4⋊C4—42no—
103C11×C4⋊C4—81no—
104C11×C22⋊C4—22no—
105C11×C22⋊C4—22no—
106C11×C22⋊C4—41no—
107C11×C22⋊C4—41no—
108C11×C22⋊C4—42no—
109C11×C22⋊C4—42no—
110C11×C22⋊C4—41no—
111C11×C22⋊C4—41no—
112C11×C22⋊C4—41no—
113C11×C22⋊C4—41no—
114C11×C22⋊C4—42no—
115C11×C22⋊C4—81no—
116C42×C11—21no—
117C42×C11—21no—
118C22⋊Dic11—42no—
119C22⋊Dic11—42no—
120C4⋊Dic11—82no—
121#4318—24no—
122#4318—42no—
123C2×C11⋊C8—42no—
124#2534D1111yescharacteristic
125C2×#352—112no—
126C2×AΣL(1,𝔽16)—112no—
127#1799—111no—
128C4×#100—111no—
129C23×C11—21no—
130C23×C11—21no—
131C23×C11—21no—
132C23×C11—21no—
133C23×C11—22no—
134C23×C11—41no—
135C23×C11—41no—
136C23×C11—41no—
137C23×C11—41no—
138C23×C11—41no—
139C23×C11—41no—
140C23×C11—41no—
141C23×C11—42no—
142C23×C11—42no—
143C23×C11—41no—
144C23×C11—41no—
145C23×C11—41no—
146C23×C11—82no—
147C23×C11—84no—
148C23×C11—84no—
149C23×C11—82no—
150C23×C11C2×C22⋊C411yescharacteristic
151C23×C11C4×D412yes—
152C23×C11#12411yescharacteristic
153C23×C11#12711yescharacteristic
154D4×C11—42no—
155D4×C11—42no—
156D4×C11—82no—
157D4×C11—82no—
158C2×C4×C11—21no—
159C2×C4×C11—21no—
160C2×C4×C11—21no—
161C2×C4×C11—21no—
162C2×C4×C11—21no—
163C2×C4×C11—41no—
164C2×C4×C11—42no—
165C2×C4×C11—41no—
166C2×C4×C11—41no—
167C2×C4×C11—41no—
168C2×C4×C11—41no—
169C2×C4×C11—41no—
170C2×C4×C11—41no—
171C2×C4×C11—41no—
172C2×C4×C11—41no—
173C2×C4×C11—41no—
174C2×C4×C11—41no—
175C2×C4×C11—41no—
176C2×C4×C11—41no—
177C2×C4×C11—81no—
178C2×C4×C11—81no—
179C2×C4×C11—81no—
180C2×C4×C11—82no—
181C2×C4×C11#9911yescharacteristic
182C2×C4×C11#9911yescharacteristic
183C2×C4×C11#12112yes—
184C2×Dic11—82no—
185C2×Dic11—82no—
186C2×Dic11—82no—
187C11⋊C8—44no—
188C2×#123C2×D1111yescharacteristic
189C2×#120C2×D1111yescharacteristic
190C2×C4×D4C2×D1111yescharacteristic
191C22×C22⋊C4Dic1111yescharacteristic, complemented
192C2×#100—112no—
193C2×#99—112no—
194C4×M16—221no—
195#396—21no—
196#394—21no—
197#388—21no—
198#386—21no—
199#381Dic1111yescharacteristic, complemented
200#380—221no—
201#380Dic1111yescharacteristic, complemented
202#379Dic1111yescharacteristic, complemented
203#352—224no—
204AΣL(1,𝔽16)—224no—
205#343—221no—
206#342—221no—
207C22×C11—21no—
208C22×C11—21no—
209C22×C11—22no—
210C22×C11—21no—
211C22×C11—21no—
212C22×C11—41no—
213C22×C11—41no—
214C22×C11—42no—
215C22×C11—42no—
216C22×C11—42no—
217C22×C11—42no—
218C22×C11—41no—
219C22×C11—41no—
220C22×C11—41no—
221C22×C11—41no—
222C22×C11—42no—
223C22×C11—81no—
224C22×C11—81no—
225C22×C11—81no—
226C22×C11—84no—
227C22×C11—84no—
228C22×C11—81no—
229C22×C11—81no—
230C22×C11#39011yescharacteristic
231C22×C11C2×#9911yescharacteristic
232C22×C11#41011yescharacteristic
233C4×C11—21no—
234C4×C11—21no—
235C4×C11—22no—
236C4×C11—41no—
237C4×C11—41no—
238C4×C11—41no—
239C4×C11—81no—
240C4×C11—81no—
241C4×C11—81no—
242Dic11—162no—
243C25C2×Dic1111yescharacteristic
244C22×D4—21no—
245C22×D4C22×D1111yescharacteristic
246C23×C4C2×Dic1111yescharacteristic
247C23×C4C2×Dic1111yescharacteristic
248C23×C4C22⋊D1111yescharacteristic
249C2×M16—222no—
250#123—22no—
251#123—22no—
252#120—22no—
253#120—22no—
254C4×D4—41no—
255C4×D4—41no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—441no—
259C2×C4⋊C4C22⋊D1111yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—222no—
269C2×C22⋊C4C2×Dic1111yescharacteristic
270C2×C22⋊C4C2×Dic1111yescharacteristic
271C2×C22⋊C4C2×Dic1111yescharacteristic
272C2×C22⋊C4C22⋊D1111yescharacteristic
273C2×C42C22⋊D1111yescharacteristic
274#100—114no—
275#100—222no—
276#99—222no—
277#99—222no—
278#97—441no—
279C4×C8—441no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—441no—
284C2×C11—21no—
285C2×C11—21no—
286C2×C11—41no—
287C2×C11—42no—
288C2×C11—41no—
289C2×C11—82no—
290C2×C11#196411yescharacteristic
291C2×C11#212011yescharacteristic
292C2×C11#212111yescharacteristic
293C24—21no—
294C24—22no—
295C24—41no—
296C24—42no—
297C24—41no—
298C24—81no—
299C24C22×Dic1111yescharacteristic
300C24C2×C22⋊D1111yescharacteristic
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—81no—
307C2×D4D4×D1112yes—
308C2×D4#434112yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—442no—
322C22×C4C22⋊Dic1111yescharacteristic
323C22×C4C22⋊Dic1111yescharacteristic
324C22×C4C22⋊Dic1111yescharacteristic
325C22×C4C22⋊Dic1111yescharacteristic
326C22×C4C2×C22⋊D1111yescharacteristic
327M16—224no—
328M16—442no—
329C2×C8—442no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—81no—
333C4⋊C4—882no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—81no—
346C22⋊C4—442no—
347C22⋊C4—442no—
348C42—21no—
349C42—21no—
350C11#1321911yescharacteristic, complemented, Sylow
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—82no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23D4×Dic1112yes—
373C23#6698911yescharacteristic
374C23#6699311yescharacteristic
375C23C2×C22⋊Dic1111yescharacteristic
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—882no—
404C2×C4—882no—
405C2×C4—882no—
406C2×C4#6689811yescharacteristic
407C2×C4#6689811yescharacteristic
408C2×C4#6699112yes—
409C8—444no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22C2×#6689811yescharacteristic
434C22#1062283411yescharacteristic
435C22#1062285311yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—1762no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—82no—
452C2#1306945311yescharacteristic
453C2#1307025711yescharacteristic
454C2#1307026211yescharacteristic
subgroupG/H
centerC22C2×#66898
commutatorC24×C11C22×C4
FrattiniC22×D4C22×D11
FittingC11×#2534C2
socleC22×C11C2×#99
prime radicalC2×C11×#120C22
cosocleC22×D4×C11C23
hypercenter#2534D11
hypocenterC11#13219
LCSG ▷ C24×C11 ▷ C22×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. C2×#10622177C2semidirect
3. #13068604C2semidirect
4. C4×#66897C2semidirect
9. C22×C11×C22⋊C4C4semidirect
14. C11×#381C4semidirect
15. C11×#380C4semidirect
16. C11×#379C4semidirect
191. C22×C22⋊C4Dic11semidirect
199. #381Dic11semidirect
201. #380Dic11semidirect
202. #379Dic11semidirect
350. C11#13219semidirect

Maximal subgroups

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

subgroupindex
C2×#106221772
C2×#106221742
#130686042
C4×#668972
C11×#25342
#1321911