Finite Group Databasebeta

← #14798898

Subgroups — #14798898

Order 2816.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622755C211yesmaximal, characteristic
2#13077469C211yesmaximal, characteristic, complemented
3#13070079C212yesmaximal, complemented
4#13069889C212yesmaximal, complemented
5#13069526C211yesmaximal, characteristic, complemented
6#13069406C211yesmaximal, characteristic, complemented
7#13069106C212yesmaximal, complemented
8#13068740C211yesmaximal, characteristic, complemented
9#13068726C212yesmaximal, complemented
10#13068670C211yesmaximal, characteristic, complemented
11C11×#1757C211yesmaximal, characteristic, complemented
12C2×C11×#131C2211yescharacteristic
13C2×#67002C2212yes—
14C22×D88C2211yescharacteristic
15#10623052C2212yes—
16C11×#413C2211yescharacteristic
17C11×#410C2211yescharacteristic
18C11×#405C2211yescharacteristic
19C11×#405C2211yescharacteristic
20#10622849C2212yes—
21#10622755—22no—
22#10622755—22no—
23#10622744C2212yes—
24C2×#66887C2212yes—
25C2×#66887C2212yes—
26#10622737C2212yes—
27C2×#66883C2211yescharacteristic
28C2×#66883C2211yescharacteristic
29#10622719C2211yescharacteristic
30#10622614C2212yes—
31#10622324—22no—
32C4×D88—22no—
33C11×#344C2211yescharacteristic
34C11×#344C2211yescharacteristic
35#10622197C2212yes—
36#10622194C2212yes—
37#10622171C2212yes—
38#10622169C2212yes—
39C2×C11×#41C2311yescharacteristic
40C22×D44C2312yes—
41C11×#131—21no—
42C11×#131—21no—
43C11×#131—21no—
44C11×#131—21no—
45C2×C11×M16—21no—
46C2×C11×M16C2311yescharacteristic
47C2×C11×M16C2311yescharacteristic
48C2×C11×M16C2311yescharacteristic
49C22×C8×C11—21no—
50C22×C8×C11C2311yescharacteristic
51C11×#117C2311yescharacteristic
52C11×#117C2311yescharacteristic
53#67002—24no—
54#67002—42no—
55C2×#4325—22no—
56C2×#4323—22no—
57#66979C2312yes—
58C2×#4318C2312yes—
59C2×#4318C2312yes—
60C2×D88—21no—
61C2×D88—21no—
62C2×D88—21no—
63C2×D88—21no—
64C2×D88D412yes—
65C2×D88D412yes—
66#66946—42no—
67C22⋊C4×D11—24no—
68C4×D44—24no—
69C11×#101—21no—
70C11×#100—21no—
71C11×#99—22no—
72C11×#97—21no—
73C11×#97—21no—
74C4×C8×C11—21no—
75C4×C8×C11—21no—
76#66901—42no—
77#66887—24no—
78#66887—42no—
79#66886—42no—
80#66885—24no—
81#66883—21no—
82#66883—21no—
83#66883—21no—
84#66883—21no—
85#66882—21no—
86#66881—21no—
87#66873—42no—
88#24988—111nomaximal, Sylow
89C23×D11—22no—
90C11×#41—21no—
91C11×#41—21no—
92C11×#41—21no—
93C11×#41—21no—
94C2×Q8×C11#4111yescharacteristic
95C2×D4×C11—21no—
96C2×D4×C11#4111yescharacteristic
97C22×C4×C11—21no—
98C22×C4×C11C2411yescharacteristic
99C2×D44—22no—
100C2×D44—22no—
101C2×D44—22no—
102C2×D44—22no—
103C2×D44—22no—
104C2×D44C2×D414yes—
105C2×C4×D11—44no—
106C11×M16—21no—
107C11×M16—21no—
108C11×M16—21no—
109C11×M16—21no—
110C11×M16—21no—
111C11×M16—21no—
112C11×M16—41no—
113C11×M16—41no—
114C2×C8×C11—21no—
115C2×C8×C11—21no—
116C2×C8×C11—21no—
117C2×C8×C11—41no—
118C2×C8×C11—41no—
119C2×C8×C11C2×D411yescharacteristic
120C2×C8×C11C2×D411yescharacteristic
121C2×C8×C11#4111yescharacteristic
122C2×C8×C11#4111yescharacteristic
123C2×C8×C11#4111yescharacteristic
124C2×C8×C11#4111yescharacteristic
125C11×C4⋊C4—21no—
126C11×C4⋊C4—21no—
127C11×C22⋊C4—21no—
128C11×C22⋊C4#4112yes—
129C42×C11—21no—
130C42×C11—21no—
131C22⋊Dic11—22no—
132#4325—44no—
133#4323—44no—
134(C2×D11)⋊C4—44no—
135C4⋊Dic11—22no—
136C4×Dic11—22no—
137#4318—22no—
138#4318—22no—
139#4318—22no—
140#4318—22no—
141C2×C11⋊C8—22no—
142C2×C11⋊C8—22no—
143D88—22no—
144D88—22no—
145D88—22no—
146D88—22no—
147C2×#472—111no—
148#2948—111no—
149#2081—112no—
150#2048—112no—
151#1978—111no—
152#1953—111no—
153#1904—112no—
154#1833—111no—
155#1829—112no—
156#1814—111no—
157#1757D1111yescharacteristic, complemented
158C23×C11—21no—
159C23×C11C2×#4111yescharacteristic
160C22×D11—24no—
161C22×D11—42no—
162C22×D11—42no—
163C22×D11—162no—
164Q8×C11—21no—
165Q8×C11—21no—
166D4×C11—21no—
167D4×C11—21no—
168D4×C11—41no—
169D4×C11—41no—
170C2×C4×C11—21no—
171C2×C4×C11—22no—
172C2×C4×C11—41no—
173C2×C4×C11—41no—
174C2×C4×C11—41no—
175C2×C4×C11C22×D411yescharacteristic
176C2×C4×C11C2×#4111yescharacteristic
177C2×C4×C11C2×#4111yescharacteristic
178C2×C4×C11C2×#4111yescharacteristic
179C2×C4×C11H(5,𝔽2)11yescharacteristic
180C2×C4×C11H(5,𝔽2)11yescharacteristic
181C2×Dic11—42no—
182D44—24no—
183D44—24no—
184D44—42no—
185D44—42no—
186C4×D11—164no—
187C8×C11—41no—
188C8×C11—41no—
189C8×C11—41no—
190C8×C11#12311yescharacteristic
191C8×C11#12311yescharacteristic
192C8×C11#12311yescharacteristic
193C8×C11#12311yescharacteristic
194C11⋊C8—42no—
195C11⋊C8—42no—
196C2×AGL(1,ℤ/8)—112no—
197C22×D8—111no—
198C2×#131C2×D1111yescharacteristic
199#518—112no—
200#472—222no—
201#472—222no—
202#442—222no—
203C4×D8—222no—
204#430—112no—
205C2×#108—111no—
206C2×#108—111no—
207#427—111no—
208#421—112no—
209#418—112no—
210#418—112no—
211#413C2×D1111yescharacteristic
212C2×#100—112no—
213C2×#100—112no—
214#410C2×D1111yescharacteristic
215#405C2×D1111yescharacteristic
216#405C2×D1111yescharacteristic
217#344—112no—
218#344—112no—
219#344—112no—
220#344C2×D1111yescharacteristic
221#344C2×D1111yescharacteristic
222H(3,ℤ/4)—112no—
223C22×C11—41no—
224C22×C11—41no—
225C22×C11—81no—
226C22×C11#52911yescharacteristic
227C22×C11#54611yescharacteristic
228C22×C11#55511yescharacteristic
229C2×D11—44no—
230C2×D11—84no—
231C2×D11—162no—
232C2×D11—162no—
233C4×C11—42no—
234C4×C11—41no—
235C4×C11—81no—
236C4×C11C2×#12311yescharacteristic
237C4×C11#54011yescharacteristic
238C4×C11#54611yescharacteristic
239C4×C11#55311yescharacteristic
240Dic11—82no—
241C2×#41C22×D1111yescharacteristic
242C22×D4—112no—
243AGL(1,ℤ/8)—224no—
244AGL(1,ℤ/8)—442no—
245C2×SD16—222no—
246C2×D8—112no—
247C2×D8—112no—
248C2×D8—222no—
249C2×D8—221no—
250C2×D8—221no—
251C2×D8—221no—
252C2×D8—221no—
253#131—21no—
254#131—21no—
255#131—21no—
256#131—21no—
257C2×M16—21no—
258C2×M16—112no—
259C2×M16—112no—
260C2×M16C22×D1111yescharacteristic
261C2×M16C22×D1111yescharacteristic
262C2×M16C22×D1111yescharacteristic
263C22×C8—21no—
264C22×C8C22×D1111yescharacteristic
265C4×D4—224no—
266C4×D4—442no—
267#117—112no—
268#117C22×D1111yescharacteristic
269#117C22×D1111yescharacteristic
270C2×C22⋊C4—224no—
271#108—221no—
272#108—221no—
273#108—221no—
274#108—221no—
275#108—442no—
276#107—221no—
277#106—221no—
278#104—442no—
279#102—224no—
280#102—442no—
281#101—21no—
282#100—21no—
283#100—224no—
284#100—442no—
285#99—22no—
286#97—21no—
287#97—21no—
288C4×C8—21no—
289C4×C8—21no—
290C2×C11—21no—
291C2×C11—21no—
292C2×C11—21no—
293C2×C11—81no—
294C2×C11#242011yescharacteristic
295D11—84no—
296C24—222no—
297#41—21no—
298#41—21no—
299#41—21no—
300#41—21no—
301C2×Q8#433911yescharacteristic
302C2×D4—21no—
303C2×D4—114no—
304C2×D4—222no—
305C2×D4—222no—
306C2×D4—222no—
307C2×D4—222no—
308C2×D4—222no—
309C2×D4#433911yescharacteristic
310C22×C4—21no—
311C22×C4—444no—
312C22×C4C23×D1111yescharacteristic
313SD16—444no—
314D8—222no—
315D8—222no—
316D8—222no—
317D8—222no—
318D8—444no—
319M16—21no—
320M16—21no—
321M16—21no—
322M16—21no—
323M16—21no—
324M16—21no—
325M16—41no—
326M16—41no—
327M16—222no—
328M16—222no—
329M16—222no—
330M16—222no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—222no—
337C2×C8—222no—
338C2×C8#433911yescharacteristic
339C2×C8#433911yescharacteristic
340C2×C8D4×D1111yescharacteristic
341C2×C8D4×D1111yescharacteristic
342C2×C8#434111yescharacteristic
343C2×C8#434111yescharacteristic
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—222no—
347C22⋊C4—21no—
348C22⋊C4—222no—
349C22⋊C4—444no—
350C22⋊C4#433912yes—
351C42—21no—
352C42—21no—
353C42—222no—
354C11#2498811yescharacteristic, complemented, Sylow
355C23—21no—
356C23—224no—
357C23—442no—
358C23—442no—
359C23—1762no—
360C23C2×#433911yescharacteristic
361Q8—21no—
362Q8—21no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—224no—
368D4—224no—
369D4—442no—
370D4—442no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—442no—
377C2×C4—1764no—
378C2×C4C2×#433911yescharacteristic
379C2×C4C2×#433911yescharacteristic
380C2×C4C2×D4×D1111yescharacteristic
381C2×C4C2×#434111yescharacteristic
382C2×C4#6704211yescharacteristic
383C2×C4#6704211yescharacteristic
384C8—41no—
385C8—41no—
386C8—41no—
387C8—442no—
388C8—442no—
389C8#6693611yescharacteristic
390C8#6693611yescharacteristic
391C8#6693611yescharacteristic
392C8#6693611yescharacteristic
393C22—41no—
394C22—41no—
395C22—81no—
396C22—444no—
397C22—884no—
398C22—1762no—
399C22—1762no—
400C22#1062301011yescharacteristic
401C22#1062308111yescharacteristic
402C22#1062309411yescharacteristic
403C4—42no—
404C4—41no—
405C4—81no—
406C4—882no—
407C4C2×#6693611yescharacteristic
408C4#1062306611yescharacteristic
409C4#1062308111yescharacteristic
410C4#1062309611yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—81no—
415C2—884no—
416C2#1307210811yescharacteristic
subgroupG/H
centerC2#13072108
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1757C2
socleC2×C11#2420
prime radicalC2×#10622755C2
hypercenter#1757D11
hypocenterC11#24988
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1757

Complements

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

NH ≅ G/Ntype
2. #13077469C2semidirect
3. #13070079C2semidirect
4. #13069889C2semidirect
5. #13069526C2semidirect
6. #13069406C2semidirect
7. #13069106C2semidirect
8. #13068740C2semidirect
9. #13068726C2semidirect
10. #13068670C2semidirect
11. C11×#1757C2semidirect
157. #1757D11semidirect
354. C11#24988semidirect

Maximal subgroups

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

subgroupindex
C2×#106227552
#130774692
#130700792
#130698892
#130695262
#130694062
#130691062
#130687402
#130687262
#130686702
C11×#17572
#2498811