Finite Group Databasebeta

← #14913907

Subgroups — #14913907

Order 2816.

2250 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66940C211yesmaximal, characteristic
2#13071578C218yesmaximal, complemented
3#13071570C218yesmaximal, complemented
4#13071219C211yesmaximal, characteristic
5#13071218C212yesmaximal
6#13071201C212yesmaximal
7#13071018C212yesmaximal
8#13071010C212yesmaximal
9#13071007C212yesmaximal
10#13070971C212yesmaximal
11C11×#2313C211yesmaximal, characteristic
12C23×Dic22C2211yescharacteristic
13C22×C11×C4⋊C4C2211yescharacteristic
14C22×Dic11⋊C4C2211yescharacteristic
15C22×Dic11⋊C4C2211yescharacteristic
16C22×C4×Dic11C2211yescharacteristic
17C22×C4×Dic11C2211yescharacteristic
18C22×C4×Dic11C2211yescharacteristic
19C2×#66940C4116yescomplemented
20C2×#66940C2214yes—
21C2×#66940C2214yes—
22C2×#66930C2214yes—
23C11×#387C2218yes—
24C11×#381C2212yes—
25C11×#380C2212yes—
26C4×C11×C22⋊C4C2212yes—
27#10622825C2212yes—
28#10622824C2212yes—
29C4×C22⋊Dic11C2212yes—
30#10622594C2218yes—
31#10622591C2218yes—
32#10622589C2214yes—
33C4⋊C4×Dic11C2214yes—
34#10622556C2214yes—
35#10622555C2214yes—
36C22⋊C4×Dic11C2214yes—
37#10622545C2214yes—
38C4×Dic11⋊C4C2218yes—
39C4×Dic11⋊C4C2214yes—
40#10622268C2218yes—
41#10622264C2214yes—
42#10622264C2214yes—
43#10622263C2214yes—
44#10622263C2214yes—
45#10622263C2214yes—
46#10622262C2218yes—
47#10622262C2214yes—
48#10622262C2214yes—
49#10622261C2214yes—
50#10622259C2218yes—
51#10622259C2218yes—
52C23×C4×C11C2311yescharacteristic
53C23×C4×C11C2311yescharacteristic
54C23×C4×C11C2311yescharacteristic
55C23×Dic11C2311yescharacteristic
56C23×Dic11C2311yescharacteristic
57C23×Dic11C2311yescharacteristic
58C23×Dic11C2311yescharacteristic
59C22×Dic22—28no—
60C22×Dic22C2×C418yes—
61C22×Dic22C2314yes—
62C2×C11×C4⋊C4C2×C418yes—
63C2×C11×C4⋊C4C2314yes—
64C2×C11×C4⋊C4C2314yes—
65C2×C11×C22⋊C4C2314yes—
66C2×C42×C11C2314yes—
67C2×C22⋊Dic11C2314yes—
68C2×C4⋊Dic11C2314yes—
69C2×Dic11⋊C4C2×C418yes—
70C2×Dic11⋊C4C2×C418yes—
71C2×Dic11⋊C4C2318yes—
72C2×Dic11⋊C4C2318yes—
73C2×Dic11⋊C4C2314yes—
74C2×Dic11⋊C4C2314yes—
75C2×C4×Dic11C2×C418yes—
76C2×C4×Dic11C2×C418yes—
77C2×C4×Dic11C2×C418yes—
78C2×C4×Dic11C2318yes—
79C2×C4×Dic11C2318yes—
80C2×C4×Dic11C2314yes—
81C2×C4×Dic11C2312yes—
82C2×C4×Dic11C2312yes—
83C2×C4×Dic11C2312yes—
84C2×C4×Dic11C2312yes—
85C2×C4×Dic11C2312yes—
86C2×C4×Dic11C2312yes—
87#66940C2×C4132yes—
88#66940C2×C4132yes—
89#66930—216no—
90C11×#95C2314yes—
91C11×#95C2314yes—
92C11×#95C2314yes—
93C11×#95C2314yes—
94C11⋊#95C2314yes—
95C11⋊#95C2314yes—
96C11⋊#95C2318yes—
97C11⋊#95C2318yes—
98C11⋊#95C2318yes—
99C11⋊#95C2318yes—
100C11⋊#95C2314yes—
101C11⋊#95C2314yes—
102#34187—111nomaximal, Sylow
103C24×C11C2411yescharacteristic
104C22×C4×C11—24no—
105C22×C4×C11—42no—
106C22×C4×C11—42no—
107C22×C4×C11C22×C414yes—
108C22×C4×C11C22×C414yes—
109C22×C4×C11C22×C414yes—
110C22×C4×C11C2418yes—
111C22×C4×C11C2412yes—
112C22×C4×C11C2412yes—
113C22×C4×C11C2412yes—
114C22×Dic11—24no—
115C22×Dic11—24no—
116C22×Dic11—42no—
117C22×Dic11C22×C414yes—
118C22×Dic11C22×C414yes—
119C22×Dic11C22×C414yes—
120C22×Dic11C22×C414yes—
121C22×Dic11#4118yes—
122C22×Dic11C2418yes—
123C22×Dic11C2411yescharacteristic
124C22×Dic11C2412yes—
125C22×Dic11C2412yes—
126C22×Dic11C2412yes—
127C22×Dic11C2411yescharacteristic
128C2×Dic22—232no—
129C2×Dic22—28no—
130C2×Dic22—28no—
131C2×Dic22C22×C4116yes—
132C11×C4⋊C4—44no—
133C11×C4⋊C4C22×C4116yes—
134C11×C22⋊C4—28no—
135C42×C11—44no—
136C22⋊Dic11—28no—
137C4⋊Dic11—44no—
138Dic11⋊C4—216no—
139Dic11⋊C4—216no—
140Dic11⋊C4C22×C4116yes—
141Dic11⋊C4C22×C4116yes—
142C4×Dic11—216no—
143C4×Dic11—44no—
144C4×Dic11—48no—
145C4×Dic11C22×C418yes—
146C4×Dic11C22×C418yes—
147C4×Dic11C22×C418yes—
148C4×Dic11C22×C418yes—
149C4×Dic11C22×C418yes—
150C4×Dic11C22×C418yes—
151C22×C4×Q8—111no—
152#2356—118no—
153#2355—118no—
154#2313—112no—
155#2313—111no—
156#2313D1111yescharacteristic
157#2310—112no—
158#2285—112no—
159#2283—112no—
160#2283—112no—
161#2274—112no—
162C23×C11—22no—
163C23×C11—22no—
164C23×C11C23×C412yes—
165C23×C11C2×#4114yes—
166C23×C11C2511yescharacteristic
167C2×C4×C11—28no—
168C2×C4×C11—24no—
169C2×C4×C11—24no—
170C2×C4×C11—24no—
171C2×C4×C11—28no—
172C2×C4×C11—28no—
173C2×C4×C11—28no—
174C2×C4×C11—44no—
175C2×C4×C11—44no—
176C2×C4×C11C23×C414yes—
177C2×C4×C11C23×C414yes—
178C2×C4×C11C23×C414yes—
179C2×Dic11—28no—
180C2×Dic11—24no—
181C2×Dic11—22no—
182C2×Dic11—24no—
183C2×Dic11—28no—
184C2×Dic11—28no—
185C2×Dic11—28no—
186C2×Dic11—28no—
187C2×Dic11—44no—
188C2×Dic11#117116yes—
189C2×Dic11C23×C412yes—
190C2×Dic11C23×C414yes—
191C2×Dic11C23×C414yes—
192C2×Dic11C23×C414yes—
193C2×Dic11C23×C412yes—
194C2×Dic11C2×#4114yes—
195C2×Dic11C2×#4114yes—
196C2×Dic112−1+414yes—
197Dic22—232no—
198C23×Q8—111no—
199C2×C4×Q8—114no—
200C2×C4×Q8—114no—
201C2×C4×Q8—1116no—
202C2×#117—114no—
203C22×C4⋊C4—111no—
204C22×C4⋊C4—111no—
205C22×C4⋊C4C2×D1111yescharacteristic
206C22×C42—111no—
207C22×C42—111no—
208C22×C42—111no—
209#391—118no—
210#391—118no—
211#387—118no—
212#387—118no—
213#387—118no—
214#387C2×D1118yes—
215#384—114no—
216#383—114no—
217#383—114no—
218#382—114no—
219#382—114no—
220#382—114no—
221#381—112no—
222#381—114no—
223#381C2×D1112yes—
224#380—112no—
225#380—114no—
226#380C2×D1112yes—
227C4×C4⋊C4—114no—
228C4×C4⋊C4—114no—
229C4×C4⋊C4—118no—
230C4×C4⋊C4—114no—
231C4×C4⋊C4—114no—
232C4×C4⋊C4—118no—
233C4×C22⋊C4—112no—
234C4×C22⋊C4—114no—
235C4×C22⋊C4C2×D1112yes—
236#376—114no—
237#376—114no—
238C22×C11—22no—
239C22×C11—22no—
240C22×C11—28no—
241C22×C11C2×#11714yes—
242C22×C11C24×C411yescharacteristic
243C22×C11C22×#4112yes—
244C22×C11C2×2−1+412yes—
245C22×C11#58512yes—
246C4×C11—24no—
247C4×C11—24no—
248C4×C11—24no—
249C4×C11—48no—
250Dic11—24no—
251Dic11—22no—
252Dic11—24no—
253Dic11—48no—
254Dic11C2×#11718yes—
255Dic11#51914yes—
256C22×Q8—118no—
257C22×Q8—114no—
258C22×Q8—228no—
259C23×C4—111no—
260C23×C4—111no—
261C23×C4—111no—
262C23×C4—111no—
263C23×C4C22×D1111yescharacteristic
264C23×C4C22×D1111yescharacteristic
265C23×C4C22×D1111yescharacteristic
266C4×Q8—1132no—
267C4×Q8—1132no—
268#117—2216no—
269C2×C4⋊C4—118no—
270C2×C4⋊C4—118no—
271C2×C4⋊C4—114no—
272C2×C4⋊C4—114no—
273C2×C4⋊C4—118no—
274C2×C4⋊C4—118no—
275C2×C4⋊C4—114no—
276C2×C4⋊C4C4×D1118yes—
277C2×C4⋊C4C22×D1114yes—
278C2×C4⋊C4C22×D1114yes—
279C2×C22⋊C4—114no—
280C2×C22⋊C4C22×D1114yes—
281C2×C42—118no—
282C2×C42—118no—
283C2×C42—118no—
284C2×C42—112no—
285C2×C42—112no—
286C2×C42—112no—
287C2×C42—112no—
288C2×C42—112no—
289C2×C42—112no—
290C2×C42—118no—
291C2×C42—118no—
292C2×C42—114no—
293C2×C42C22×D1114yes—
294#95—118no—
295#95—118no—
296#95—114no—
297#95—118no—
298#95—114no—
299#95—114no—
300#95—118no—
301#95—114no—
302#95C22×D1114yes—
303#95C22×D1114yes—
304#95C22×D1114yes—
305#95C22×D1114yes—
306C2×C11—24no—
307C2×C11C22×#11711yescharacteristic
308C2×C11C2×#51911yescharacteristic
309C2×C11#342811yescharacteristic
310C2×C11#350714yes—
311C24C23×D1111yescharacteristic
312C2×Q8—1116no—
313C2×Q8—2232no—
314C2×Q8—228no—
315C2×Q8—228no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—114no—
320C22×C4—118no—
321C22×C4—111no—
322C22×C4—112no—
323C22×C4—114no—
324C22×C4—114no—
325C22×C4—111no—
326C22×C4—112no—
327C22×C4—112no—
328C22×C4—114no—
329C22×C4—118no—
330C22×C4—224no—
331C22×C4—224no—
332C22×C4—442no—
333C22×C4C2×C4×D1114yes—
334C22×C4C2×C4×D1114yes—
335C22×C4C2×C4×D1114yes—
336C22×C4C23×D1118yes—
337C22×C4C23×D1112yes—
338C22×C4C23×D1112yes—
339C22×C4C23×D1112yes—
340C4⋊C4—44no—
341C4⋊C4—1116no—
342C4⋊C4—1116no—
343C4⋊C4—2216no—
344C4⋊C4—2216no—
345C4⋊C4—444no—
346C4⋊C4C2×C4×D11116yes—
347C22⋊C4—28no—
348C22⋊C4—228no—
349C42—44no—
350C42—118no—
351C42—118no—
352C42—118no—
353C42—118no—
354C42—118no—
355C42—118no—
356C42—2216no—
357C42—444no—
358C42—448no—
359C11#3418711yescharacteristic, complemented, Sylow
360C23—22no—
361C23—22no—
362C23C22×C4×D1112yes—
363C23#41×D1114yes—
364C23C24×D1111yescharacteristic
365Q8—2232no—
366C2×C4—28no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—1116no—
376C2×C4—112no—
377C2×C4—114no—
378C2×C4—114no—
379C2×C4—114no—
380C2×C4—114no—
381C2×C4—112no—
382C2×C4—114no—
383C2×C4—114no—
384C2×C4—228no—
385C2×C4—228no—
386C2×C4—224no—
387C2×C4—224no—
388C2×C4—222no—
389C2×C4—228no—
390C2×C4—228no—
391C2×C4—228no—
392C2×C4—444no—
393C2×C4C22×C4×D1114yes—
394C2×C4C22×C4×D1114yes—
395C2×C4C22×C4×D1114yes—
396C22—28no—
397C22—22no—
398C22—22no—
399C22D11×#11714yes—
400C22C23×C4×D1111yescharacteristic
401C22C2×#41×D1112yes—
402C22#1062345512yes—
403C22D11×2−1+412yes—
404C4—24no—
405C4—24no—
406C4—24no—
407C4—48no—
408C4—118no—
409C4—114no—
410C4—224no—
411C4—222no—
412C4—224no—
413C4—448no—
414C2—24no—
415C2C2×D11×#11711yescharacteristic
416C2D11×#51911yescharacteristic
417C2#1308181411yescharacteristic
418C2#1308247614yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC11×#2313C2
socle, cosocleC23×C11C25
prime radicalC24×C11C24
hypercenter#2313D11
hypocenterC11#34187
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ #2313

Complements

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

NH ≅ G/Ntype
2. #13071578C2semidirect
3. #13071570C2semidirect
19. C2×#66940C4semidirect
359. C11#34187semidirect

Maximal subgroups

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

subgroupindex
C22×#669402
#130715782
#130715702
#130712192
#130712182
#130712012
#130710182
#130710102
#130710072
#130709712
C11×#23132
#3418711