Finite Group Databasebeta

← #14796422

Subgroups — #14796422

Order 2816.

838 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622634C211yesmaximal, characteristic, complemented
2C2×C4×#4325C211yesmaximal, characteristic
3#13073785C211yesmaximal, characteristic
4#13070097C212yesmaximal, complemented
5#13070001C212yesmaximal
6#13069576C211yesmaximal, characteristic, complemented
7#13069335C211yesmaximal, characteristic, complemented
8#13068971C212yesmaximal
9#13068631C212yesmaximal, complemented
10#13068613C211yesmaximal, characteristic
11Q8×C11×C4⋊C4C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2212yes—
13C2×C4×Q8×C11C2211yescharacteristic
14C22×#4325C2211yescharacteristic
15C2×C4×D44C2211yescharacteristic
16C2×#66922C2211yescharacteristic
17C11×#389—22no—
18C11×#389C2212yes—
19C11×#384—21no—
20C11×#384C2211yescharacteristic
21C4×C11×C4⋊C4—21no—
22C4×C11×C4⋊C4C2211yescharacteristic
23C2×#66899C2211yescharacteristic
24C2×#66899C2212yes—
25C4×#4325—24no—
26#10622634—24no—
27#10622600—21no—
28#10622600—21no—
29C2×#66873C2211yescharacteristic
30C2×#66873C2212yes—
31C2×#66872C2211yescharacteristic
32C2×#66871C2212yes—
33#10622552C2212yes—
34#10622549C2212yes—
35C2×C4⋊(C11⋊C8)C2211yescharacteristic
36C2×C4⋊(C11⋊C8)C2212yes—
37C2×C4×C11⋊C8C2211yescharacteristic
38#10622286—22no—
39#10622167C2212yes—
40#10622167C2212yes—
41#10622167C2212yes—
42#10622162C2212yes—
43C22×Q8×C11C2311yescharacteristic
44C22×D44C2311yescharacteristic
45C22×C4×D11—21no—
46C22×Dic22—21no—
47C4×Q8×C11—22no—
48C4×Q8×C11—24no—
49C4×Q8×C11—42no—
50C4×Q8×C11D414yes—
51C2×C11×C4⋊C4—21no—
52C2×C11×C4⋊C4—21no—
53C2×C11×C4⋊C4—22no—
54C2×C11×C4⋊C4—22no—
55C2×C11×C4⋊C4—21no—
56C2×C11×C4⋊C4D411yescharacteristic
57C2×C11×C4⋊C4D411yescharacteristic
58C2×C11×C4⋊C4C2312yes—
59C2×C11×C4⋊C4C2311yescharacteristic
60C2×C42×C11—21no—
61C2×C42×C11—22no—
62C2×C42×C11C2312yes—
63C2×C42×C11C2311yescharacteristic
64C2×#4325—22no—
65C2×#4325—22no—
66C2×#4325Q814yescomplemented
67C2×(C2×D11)⋊C4—21no—
68C2×(C2×D11)⋊C4—22no—
69C2×(C2×D11)⋊C4—22no—
70C2×C4⋊Dic11—21no—
71C2×C4⋊Dic11C2312yes—
72C2×C4⋊Dic11C2311yescharacteristic
73C22×C11⋊C8C2311yescharacteristic
74C22×C11⋊C8C2312yes—
75C22×C11⋊C8C2311yescharacteristic
76C4×D44—22no—
77C4×D44—22no—
78#66922—22no—
79#66922—22no—
80C11×#95—22no—
81C11×#95D412yes—
82#66899—24no—
83#66899—22no—
84C11⋊#95—22no—
85C11⋊#95—22no—
86#66873—22no—
87#66873—42no—
88#66872—22no—
89#66871—24no—
90C4⋊(C11⋊C8)—22no—
91C4⋊(C11⋊C8)—42no—
92C4×C11⋊C8—22no—
93#24814—111nomaximal, Sylow
94C23×D11—21no—
95C2×Q8×C11—21no—
96C2×Q8×C11—21no—
97C2×Q8×C11—21no—
98C2×Q8×C11C2×D412yes—
99C2×Q8×C11C2×Q812yes—
100C2×Q8×C11#4112yes—
101C22×C4×C11—21no—
102C22×C4×C11—21no—
103C22×C4×C11—22no—
104C22×C4×C11C2×D412yes—
105C22×C4×C11C2×D412yes—
106C22×C4×C11C2×D411yescharacteristic
107C22×C4×C11C2×D411yescharacteristic
108C22×C4×C11C2411yescharacteristic
109C22×Dic11—21no—
110C22×Dic11—22no—
111C2×D44—21no—
112C2×D44—21no—
113C2×D44—21no—
114C2×D44—21no—
115C2×D44—21no—
116C2×D44C2×Q812yes—
117C2×C4×D11—42no—
118C2×C4×D11—42no—
119C2×C4×D11—42no—
120C2×Dic22—42no—
121C2×Dic22—81no—
122C2×Dic22—81no—
123C11×C4⋊C4—24no—
124C11×C4⋊C4—22no—
125C11×C4⋊C4—22no—
126C11×C4⋊C4—21no—
127C11×C4⋊C4—22no—
128C11×C4⋊C4—22no—
129C11×C4⋊C4—44no—
130C11×C4⋊C4—42no—
131C11×C4⋊C4—81no—
132C11×C4⋊C4—82no—
133C11×C4⋊C4C2×D412yes—
134C42×C11—24no—
135C42×C11—22no—
136C42×C11—22no—
137C42×C11—44no—
138C42×C11C2×D412yes—
139C42×C11#4112yes—
140#4325—28no—
141(C2×D11)⋊C4—42no—
142(C2×D11)⋊C4—82no—
143(C2×D11)⋊C4—82no—
144C4⋊Dic11—22no—
145C4⋊Dic11—21no—
146C4⋊Dic11—22no—
147C4⋊Dic11—21no—
148C4⋊Dic11—42no—
149C2×C11⋊C8—21no—
150C2×C11⋊C8—21no—
151C2×C11⋊C8—22no—
152C2×C11⋊C8—21no—
153C2×C11⋊C8—21no—
154C2×C11⋊C8—22no—
155C2×C11⋊C8—21no—
156C2×C11⋊C8—22no—
157C2×C11⋊C8C2×Q812yes—
158C2×#477—111no—
159C2×C4×SD16—111no—
160#2563—111no—
161Q8×C4⋊C4D1111yescharacteristic, complemented
162#2085—112no—
163#2067—112no—
164#1990—111no—
165#1939—111no—
166#1880—112no—
167#1805—112no—
168#1802—111no—
169C23×C11C22×D411yescharacteristic
170C22×D11—22no—
171C22×D11—41no—
172C22×D11—41no—
173C22×D11—41no—
174C22×D11—41no—
175C22×D11—41no—
176C22×D11—41no—
177Q8×C11—22no—
178Q8×C11—22no—
179Q8×C11—22no—
180Q8×C11C22⋊Q814yes—
181C2×C4×C11—22no—
182C2×C4×C11—22no—
183C2×C4×C11—22no—
184C2×C4×C11—21no—
185C2×C4×C11—21no—
186C2×C4×C11—22no—
187C2×C4×C11—22no—
188C2×C4×C11—22no—
189C2×C4×C11—22no—
190C2×C4×C11—22no—
191C2×C4×C11—22no—
192C2×C4×C11—22no—
193C2×C4×C11—41no—
194C2×C4×C11—41no—
195C2×C4×C11—42no—
196C2×C4×C11—42no—
197C2×C4×C11—42no—
198C2×C4×C11#12112yes—
199C2×C4×C11#12112yes—
200C2×C4×C11#13512yes—
201C2×C4×C11#13712yes—
202C2×C4×C11C22×D411yescharacteristic
203C2×C4×C11C22×Q811yescharacteristic
204C2×C4×C11C2×#4111yescharacteristic
205C2×C4×C11C2×#4111yescharacteristic
206C2×C4×C11H(5,𝔽2)11yescharacteristic
207C2×C4×C112−1+411yescharacteristic
208C2×Dic11—42no—
209C2×Dic11—41no—
210C2×Dic11—41no—
211C2×Dic11—41no—
212C2×Dic11—42no—
213C2×Dic11—42no—
214D44—22no—
215D44—22no—
216D44—41no—
217D44—41no—
218C4×D11—84no—
219Dic22—84no—
220C11⋊C8—22no—
221C11⋊C8—41no—
222C11⋊C8—42no—
223C22×SD16—111no—
224C2×C4⋊Q8—111no—
225C2×C4×Q8C2×D1112yes—
226C2×C4×Q8C2×D1111yescharacteristic
227C2×C4×D4—111no—
228#477—224no—
229C4×SD16—224no—
230C2×#107—112no—
231C2×#106—111no—
232C2×C4⋊C8—111no—
233C2×C4⋊C8—112no—
234C2×#103—112no—
235C2×#103—111no—
236C2×#102—111no—
237C2×#102—112no—
238C2×C4×C8—111no—
239#393—222no—
240#390—112no—
241#389—22no—
242#389C2×D1112yes—
243#384—21no—
244#384C2×D1111yescharacteristic
245#382—112no—
246#380—221no—
247#380—221no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4C2×D1111yescharacteristic
250#340—112no—
251#340—112no—
252#340—112no—
253#336—112no—
254C22×C11C2×#12111yescharacteristic
255C22×C11#54611yescharacteristic
256C22×C11D4×Q811yescharacteristic
257C22×C11C2×#13511yescharacteristic
258C22×C11C2×#13711yescharacteristic
259C22×C11#57611yescharacteristic
260C22×C11#57711yescharacteristic
261C2×D11—42no—
262C2×D11—42no—
263C2×D11—42no—
264C2×D11—81no—
265C2×D11—81no—
266C2×D11—81no—
267C2×D11—81no—
268C4×C11—24no—
269C4×C11—22no—
270C4×C11—42no—
271C4×C11—42no—
272C4×C11—42no—
273C4×C11—82no—
274C4×C11#46414yes—
275C4×C11C2×C22⋊Q811yescharacteristic
276C4×C11#53711yescharacteristic
277C4×C11#55211yescharacteristic
278C4×C11#55411yescharacteristic
279Dic11—81no—
280Dic11—82no—
281C22×Q8—221no—
282C22×Q8C22×D1111yescharacteristic
283C22×D4—111no—
284C23×C4—221no—
285C2×SD16—114no—
286C2×SD16—222no—
287C2×SD16—222no—
288C22×C8—111no—
289C22×C8—112no—
290C22×C8—111no—
291C4⋊Q8—222no—
292C4⋊Q8—222no—
293C4×Q8—22no—
294C4×Q8—24no—
295C4×Q8—42no—
296C4×Q8D4414yes—
297C4×D4—222no—
298C4×D4—222no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—111no—
305C2×C4⋊C4—112no—
306C2×C4⋊C4—221no—
307C2×C4⋊C4C22⋊D1111yescharacteristic
308C2×C4⋊C4C22⋊D1111yescharacteristic
309C2×C4⋊C4C22×D1112yes—
310C2×C4⋊C4C22×D1111yescharacteristic
311C2×C22⋊C4—221no—
312C2×C22⋊C4—222no—
313C2×C22⋊C4—222no—
314C2×C42—21no—
315C2×C42—22no—
316C2×C42C22×D1112yes—
317C2×C42C22×D1111yescharacteristic
318#107—224no—
319#106—222no—
320C4⋊C8—222no—
321C4⋊C8—442no—
322#103—222no—
323#103—224no—
324#102—222no—
325#102—442no—
326C4×C8—222no—
327#95—22no—
328#95—222no—
329#95—222no—
330#95C22⋊D1112yes—
331C2×C11#241711yescharacteristic
332C2×C11C2×#46411yescharacteristic
333C2×C11#304511yescharacteristic
334C2×C11#330511yescharacteristic
335C2×C11#332911yescharacteristic
336C2×C11#338111yescharacteristic
337C2×C11#338511yescharacteristic
338D11—82no—
339C24—221no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—442no—
344C2×Q8—881no—
345C2×Q8—881no—
346C2×Q8C2×D4412yes—
347C2×Q8#434112yes—
348C2×Q8Q8×D1112yes—
349C2×D4—112no—
350C2×D4—221no—
351C2×D4—221no—
352C2×D4—221no—
353C2×D4—221no—
354C2×D4—221no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—221no—
359C22×C4—222no—
360C22×C4—442no—
361C22×C4—442no—
362C22×C4—442no—
363C22×C4C2×C22⋊D1112yes—
364C22×C4C2×C22⋊D1112yes—
365C22×C4C2×C22⋊D1111yescharacteristic
366C22×C4C2×C22⋊D1111yescharacteristic
367C22×C4C23×D1111yescharacteristic
368SD16—228no—
369C2×C8—112no—
370C2×C8—221no—
371C2×C8—221no—
372C2×C8—221no—
373C2×C8—222no—
374C2×C8—222no—
375C2×C8—221no—
376C2×C8—222no—
377C2×C8—221no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—81no—
387C4⋊C4—82no—
388C4⋊C4—221no—
389C4⋊C4—222no—
390C4⋊C4—221no—
391C4⋊C4—222no—
392C4⋊C4—442no—
393C4⋊C4C2×D4412yes—
394C22⋊C4—442no—
395C22⋊C4—882no—
396C22⋊C4—882no—
397C42—24no—
398C42—22no—
399C42—22no—
400C42—44no—
401C42C2×D4412yes—
402C42#433912yes—
403C11#2481411yescharacteristic, complemented, Sylow
404C23—222no—
405C23—441no—
406C23—441no—
407C23—441no—
408C23—441no—
409C23—441no—
410C23—441no—
411C23C22×C22⋊D1111yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—22no—
415Q8—884no—
416Q8#6695014yes—
417D4—222no—
418D4—222no—
419D4—441no—
420D4—441no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—441no—
439C2×C4—442no—
440C2×C4—441no—
441C2×C4—441no—
442C2×C4—442no—
443C2×C4—442no—
444C2×C4—884no—
445C2×C4#6698312yes—
446C2×C4#6698312yes—
447C2×C4#6699512yes—
448C2×C4#6700312yes—
449C2×C4C22×D4411yescharacteristic
450C2×C4C2×#433911yescharacteristic
451C2×C4C2×#434111yescharacteristic
452C2×C4#6703711yescharacteristic
453C2×C4C2×Q8×D1111yescharacteristic
454C2×C4#6704011yescharacteristic
455C8—222no—
456C8—441no—
457C8—442no—
458C22—442no—
459C22—442no—
460C22—442no—
461C22—881no—
462C22—881no—
463C22—881no—
464C22—881no—
465C22C2×#6698311yescharacteristic
466C22#1062333711yescharacteristic
467C22C2×#6699511yescharacteristic
468C22Q8×C22⋊D1111yescharacteristic
469C22C2×#6700311yescharacteristic
470C22#1062336611yescharacteristic
471C22#1062336811yescharacteristic
472C4—22no—
473C4—24no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—82no—
478C4—881no—
479C4—882no—
480C4#1062264114yes—
481C4C2×#6695011yescharacteristic
482C4#1062304111yescharacteristic
483C4#1062304211yescharacteristic
484C4#1062304411yescharacteristic
485C2—882no—
486C2#1307206111yescharacteristic
487C2C2×#1062264111yescharacteristic
488C2#1307815511yescharacteristic
489C2#1308056611yescharacteristic
490C2#1308091011yescharacteristic
491C2#1308148711yescharacteristic
492C2#1308152211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingQ8×C11×C4⋊C4C2
socleC23×C11C22×D4
prime radicalC22×D44C23
hypercenterQ8×C4⋊C4D11
hypocenterC11#24814
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10622634C2semidirect
4. #13070097C2semidirect
6. #13069576C2semidirect
7. #13069335C2semidirect
9. #13068631C2semidirect
11. Q8×C11×C4⋊C4C2semidirect
66. C2×#4325Q8semidirect
161. Q8×C4⋊C4D11semidirect
403. C11#24814semidirect

Maximal subgroups

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

subgroupindex
C2×#106226342
C2×C4×#43252
#130737852
#130700972
#130700012
#130695762
#130693352
#130689712
#130686312
#130686132
Q8×C11×C4⋊C42
#2481411