Finite Group Databasebeta

← #14788685

Subgroups — #14788685

Order 2816.

1084 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622696C211yesmaximal, characteristic
2C2×#10622639C211yesmaximal, characteristic
3#13073974C211yesmaximal, characteristic
4#13069859C212yesmaximal, complemented
5#13069699C212yesmaximal
6#13069557C211yesmaximal, characteristic, complemented
7#13069437C211yesmaximal, characteristic, complemented
8#13068971C212yesmaximal
9#13068870C212yesmaximal, complemented
10#13068631C211yesmaximal, characteristic, complemented
11C2×C11×#557C211yesmaximal, characteristic, complemented
12C11×#557—24no—
13C11×#557—24no—
14C2×C11×C4⋊Q8C2212yes—
15C2×C11×C4⋊Q8C2211yescharacteristic
16C2×C11×#125—22no—
17C2×C11×#125—21no—
18C2×C4×Q8×C11—21no—
19C2×C4×Q8×C11C2212yes—
20C2×C4×Q8×C11C2211yescharacteristic
21C2×C4×Q8×C11C2211yescharacteristic
22C22×#4325C2211yescharacteristic
23C22×#4325C2212yes—
24C2×C4×D44C2211yescharacteristic
25#10622844—21no—
26C2×#66899C2212yes—
27C2×#66899C2211yescharacteristic
28C2×#66899C2212yes—
29C2×#66899C2212yes—
30C2×#66899C2212yes—
31#10622696—24no—
32#10622639—24no—
33#10622605—21no—
34#10622602—22no—
35C2×#66873C2211yescharacteristic
36C2×#66871C2211yescharacteristic
37#10622552C2212yes—
38#10622549C2211yescharacteristic
39#10622541C2212yes—
40C2×C4⋊(C11⋊C8)C2211yescharacteristic
41C2×#66867C2211yescharacteristic
42#10622167C2212yes—
43#10622162C2212yes—
44#10622162C2212yes—
45C22×Q8×C11D411yescharacteristic
46C22×Q8×C11C2312yes—
47C22×Q8×C11C2311yescharacteristic
48C22×D44C2311yescharacteristic
49C22×C4×D11—21no—
50C11×C4⋊Q8—24no—
51C11×C4⋊Q8—24no—
52C11×C4⋊Q8—22no—
53C11×C4⋊Q8D414yes—
54C11×#125—216no—
55C11×#125—28no—
56C4×Q8×C11—28no—
57C4×Q8×C11—24no—
58C4×Q8×C11—24no—
59C4×Q8×C11—22no—
60C4×Q8×C11—22no—
61C4×Q8×C11—22no—
62C4×Q8×C11D414yes—
63C2×C11×C4⋊C4—22no—
64C2×C11×C4⋊C4—21no—
65C2×C11×C4⋊C4—22no—
66C2×C11×C4⋊C4—21no—
67C2×C11×C4⋊C4—22no—
68C2×C11×C4⋊C4—21no—
69C2×C11×C4⋊C4D412yes—
70C2×C11×C4⋊C4D411yescharacteristic
71C2×C11×C4⋊C4C2311yescharacteristic
72C2×C42×C11—22no—
73C2×C42×C11—21no—
74C2×C42×C11C2312yes—
75C2×C42×C11C2311yescharacteristic
76C2×#4325—22no—
77C2×#4325—22no—
78C2×#4325—24no—
79C2×#4325—24no—
80C2×#4325—22no—
81C2×#4325—42no—
82C2×(C2×D11)⋊C4—21no—
83C2×(C2×D11)⋊C4—22no—
84C2×(C2×D11)⋊C4—22no—
85C2×C4⋊Dic11C2311yescharacteristic
86C2×C4⋊Dic11C2312yes—
87C2×Dic11⋊C4—22no—
88C22×C11⋊C8C2311yescharacteristic
89C22×C11⋊C8C2312yes—
90C22×C11⋊C8C2311yescharacteristic
91C4×D44—22no—
92C4×D44—22no—
93#66899—22no—
94#66899—24no—
95#66899—24no—
96#66899—42no—
97#66899—42no—
98C11⋊#95—22no—
99C11⋊#95—21no—
100C11⋊#95—21no—
101#66873—22no—
102#66871—22no—
103C4⋊(C11⋊C8)—22no—
104#66867—22no—
105#24199—111nomaximal, Sylow
106C23×D11—21no—
107C2×Q8×C11—22no—
108C2×Q8×C11—24no—
109C2×Q8×C11—22no—
110C2×Q8×C11—21no—
111C2×Q8×C11—22no—
112C2×Q8×C11—21no—
113C2×Q8×C11—22no—
114C2×Q8×C11—21no—
115C2×Q8×C11—21no—
116C2×Q8×C11—22no—
117C2×Q8×C11—21no—
118C2×Q8×C11—22no—
119C2×Q8×C11C2×D414yes—
120C2×Q8×C11C2×D412yes—
121C22×C4×C11—21no—
122C22×C4×C11—22no—
123C22×C4×C11—21no—
124C22×C4×C11C2×D412yes—
125C22×C4×C11C2×D411yescharacteristic
126C22×C4×C11C2×D411yescharacteristic
127C22×C4×C11C2×D412yes—
128C22×C4×C11C2411yescharacteristic
129C22×Dic11—22no—
130C22×Dic11—21no—
131C2×D44—21no—
132C2×D44—21no—
133C2×D44—21no—
134C2×D44—21no—
135C2×D44—21no—
136C2×D44#4112yes—
137C2×C4×D11—42no—
138C2×C4×D11—82no—
139C11×C4⋊C4—28no—
140C11×C4⋊C4—28no—
141C11×C4⋊C4—28no—
142C11×C4⋊C4—24no—
143C11×C4⋊C4—24no—
144C11×C4⋊C4—24no—
145C11×C4⋊C4—24no—
146C11×C4⋊C4—22no—
147C11×C4⋊C4—21no—
148C11×C4⋊C4C2×D412yes—
149C42×C11—28no—
150C42×C11—24no—
151C42×C11—22no—
152C42×C11—22no—
153C42×C11C2×D412yes—
154C42×C11C2×D412yes—
155#4325—28no—
156#4325—48no—
157(C2×D11)⋊C4—42no—
158(C2×D11)⋊C4—44no—
159(C2×D11)⋊C4—82no—
160C4⋊Dic11—21no—
161C4⋊Dic11—21no—
162C4⋊Dic11—22no—
163C4⋊Dic11—22no—
164Dic11⋊C4—82no—
165C2×C11⋊C8—21no—
166C2×C11⋊C8—21no—
167C2×C11⋊C8—22no—
168C2×C11⋊C8—21no—
169C2×C11⋊C8—22no—
170C2×C11⋊C8—21no—
171C2×C11⋊C8—22no—
172C2×C11⋊C8—21no—
173C2×C11⋊C8#4112yes—
174C2×#557D1111yescharacteristic, complemented
175C2×#461—111no—
176C2×#440—111no—
177#2580—111no—
178#2043—112no—
179#2013—112no—
180#1985—111no—
181#1960—111no—
182#1880—112no—
183#1863—112no—
184#1805—111no—
185C23×C11C22×D411yescharacteristic
186C22×D11—22no—
187C22×D11—41no—
188C22×D11—41no—
189C22×D11—42no—
190C22×D11—42no—
191Q8×C11—28no—
192Q8×C11—22no—
193Q8×C11—22no—
194Q8×C11—24no—
195Q8×C11—24no—
196Q8×C11—42no—
197Q8×C11—42no—
198Q8×C11—41no—
199Q8×C11—41no—
200C2×C4×C11—22no—
201C2×C4×C11—22no—
202C2×C4×C11—22no—
203C2×C4×C11—22no—
204C2×C4×C11—22no—
205C2×C4×C11—22no—
206C2×C4×C11—24no—
207C2×C4×C11—22no—
208C2×C4×C11—22no—
209C2×C4×C11—22no—
210C2×C4×C11—41no—
211C2×C4×C11—42no—
212C2×C4×C11—42no—
213C2×C4×C11—41no—
214C2×C4×C11—42no—
215C2×C4×C11#12012yes—
216C2×C4×C11#12012yes—
217C2×C4×C11#13714yes—
218C2×C4×C11C22×D411yescharacteristic
219C2×C4×C11C22×D411yescharacteristic
220C2×C4×C11C2×#4111yescharacteristic
221C2×C4×C11C2×#4111yescharacteristic
222C2×C4×C11H(5,𝔽2)11yescharacteristic
223C2×C4×C112−1+411yescharacteristic
224C2×Dic11—41no—
225C2×Dic11—42no—
226C2×Dic11—42no—
227C2×Dic11—42no—
228C2×Dic11—42no—
229D44—22no—
230D44—22no—
231D44—41no—
232D44—41no—
233C4×D11—84no—
234C11⋊C8—22no—
235C11⋊C8—41no—
236C11⋊C8—42no—
237C22×SD16—111no—
238C22×SD16—112no—
239#557—24no—
240#557—24no—
241C2×C4⋊Q8C2×D1112yes—
242C2×C4⋊Q8C2×D1111yescharacteristic
243C2×#125—22no—
244C2×#125—21no—
245C2×C4×Q8—21no—
246C2×C4×Q8C2×D1112yes—
247C2×C4×Q8C2×D1111yescharacteristic
248C2×C4×Q8C2×D1111yescharacteristic
249C2×C4×D4—111no—
250#461—224no—
251#440—224no—
252C2×#107—111no—
253C2×C4⋊C8—111no—
254C2×#103—112no—
255C2×#103—112no—
256C2×#103—111no—
257C2×#103—112no—
258C2×#103—112no—
259C2×#102—111no—
260C2×#97—111no—
261#399—221no—
262#393—221no—
263#390—112no—
264#389—112no—
265#385—222no—
266#382—111no—
267#340—112no—
268#336—112no—
269#336—112no—
270C22×C11C2×#12011yescharacteristic
271C22×C11#54811yescharacteristic
272C22×C11#55011yescharacteristic
273C22×C11C2×#13712yes—
274C22×C11#57712yes—
275C2×D11—44no—
276C2×D11—42no—
277C2×D11—82no—
278C2×D11—81no—
279C2×D11—81no—
280C4×C11—22no—
281C4×C11—24no—
282C4×C11—22no—
283C4×C11—42no—
284C4×C11—44no—
285C4×C11—42no—
286C4×C11—42no—
287C4×C11#52511yescharacteristic
288C4×C11#53711yescharacteristic
289C4×C11#54611yescharacteristic
290C4×C11#54711yescharacteristic
291Dic11—82no—
292Dic11—81no—
293C22×Q8C22⋊D1111yescharacteristic
294C22×Q8C22×D1112yes—
295C22×Q8C22×D1111yescharacteristic
296C22×D4—111no—
297C23×C4—221no—
298C2×SD16—222no—
299C2×SD16—222no—
300C2×SD16—222no—
301C2×SD16—224no—
302C2×SD16—224no—
303C2×SD16—442no—
304C22×C8—111no—
305C22×C8—112no—
306C22×C8—111no—
307C4⋊Q8—24no—
308C4⋊Q8—24no—
309C4⋊Q8—22no—
310C4⋊Q8C22⋊D1114yes—
311#125—216no—
312#125—28no—
313C4×Q8—28no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8C22⋊D1114yes—
320C4×D4—222no—
321C4×D4—222no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—111no—
329C2×C4⋊C4—112no—
330C2×C4⋊C4—222no—
331C2×C4⋊C4C22⋊D1112yes—
332C2×C4⋊C4C22⋊D1111yescharacteristic
333C2×C4⋊C4C22×D1111yescharacteristic
334C2×C22⋊C4—221no—
335C2×C22⋊C4—222no—
336C2×C22⋊C4—222no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42C22×D1112yes—
340C2×C42C22×D1111yescharacteristic
341#107—222no—
342C4⋊C8—222no—
343#103—224no—
344#103—222no—
345#103—224no—
346#103—442no—
347#103—442no—
348#102—222no—
349#97—222no—
350#95—221no—
351#95—221no—
352#95—222no—
353C2×C11#240711yescharacteristic
354C2×C11#300411yescharacteristic
355C2×C11#301011yescharacteristic
356C2×C11#337212yes—
357C2×C11#340012yes—
358D11—82no—
359C24—221no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8C2×C22⋊D1114yes—
373C2×Q8C2×C22⋊D1112yes—
374C2×D4—112no—
375C2×D4—221no—
376C2×D4—221no—
377C2×D4—221no—
378C2×D4—221no—
379C2×D4—221no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—222no—
384C22×C4—221no—
385C22×C4—442no—
386C22×C4—882no—
387C22×C4C2×C22⋊D1112yes—
388C22×C4C2×C22⋊D1112yes—
389C22×C4C2×C22⋊D1111yescharacteristic
390C22×C4C2×C22⋊D1111yescharacteristic
391C22×C4C23×D1111yescharacteristic
392SD16—228no—
393SD16—448no—
394C2×C8—112no—
395C2×C8—221no—
396C2×C8—221no—
397C2×C8—222no—
398C2×C8—221no—
399C2×C8—222no—
400C2×C8—221no—
401C2×C8—221no—
402C2×C8—222no—
403C4⋊C4—28no—
404C4⋊C4—28no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—222no—
413C4⋊C4—222no—
414C4⋊C4—221no—
415C4⋊C4—221no—
416C4⋊C4—882no—
417C4⋊C4C2×C22⋊D1112yes—
418C22⋊C4—444no—
419C22⋊C4—442no—
420C22⋊C4—882no—
421C42—28no—
422C42—24no—
423C42—22no—
424C42—22no—
425C42C2×C22⋊D1112yes—
426C42C2×C22⋊D1112yes—
427C11#2419911yescharacteristic, complemented, Sylow
428C23—222no—
429C23—442no—
430C23—441no—
431C23—442no—
432C23—441no—
433C23C22×C22⋊D1111yescharacteristic
434Q8—28no—
435Q8—24no—
436Q8—22no—
437Q8—24no—
438Q8—22no—
439Q8—41no—
440Q8—42no—
441Q8—41no—
442Q8—42no—
443D4—222no—
444D4—222no—
445D4—441no—
446D4—441no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—442no—
463C2×C4—442no—
464C2×C4—442no—
465C2×C4—442no—
466C2×C4—441no—
467C2×C4—884no—
468C2×C4#6699514yes—
469C2×C4#6700612yes—
470C2×C4#6700612yes—
471C2×C4#41×D1111yescharacteristic
472C2×C4#41×D1111yescharacteristic
473C2×C4#6704211yescharacteristic
474C2×C4#6704311yescharacteristic
475C2×C4C22×C22⋊D1111yescharacteristic
476C2×C4C22×C22⋊D1111yescharacteristic
477C8—222no—
478C8—442no—
479C8—441no—
480C22—444no—
481C22—442no—
482C22—882no—
483C22—881no—
484C22—881no—
485C22C2×#6699512yes—
486C22#1062336211yescharacteristic
487C22#1062336511yescharacteristic
488C22#1062336812yes—
489C22C2×#6700611yescharacteristic
490C4—24no—
491C4—22no—
492C4—22no—
493C4—44no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—882no—
498C4—881no—
499C4#1062335511yescharacteristic
500C4#1062335911yescharacteristic
501C4#1062336011yescharacteristic
502C4#1062336411yescharacteristic
503C2—882no—
504C2#1307195511yescharacteristic
505C2#1307785011yescharacteristic
506C2#1307790211yescharacteristic
507C2#1308139312yes—
508C2#1308166512yes—
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C11×#557C2
socleC23×C11C22×D4
prime radicalC22×D44C23
hypercenterC2×#557D11
hypocenterC11#24199
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #13069859C2semidirect
6. #13069557C2semidirect
7. #13069437C2semidirect
9. #13068870C2semidirect
10. #13068631C2semidirect
11. C2×C11×#557C2semidirect
174. C2×#557D11semidirect
427. C11#24199semidirect

Maximal subgroups

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

subgroupindex
C2×#106226962
C2×#106226392
#130739742
#130698592
#130696992
#130695572
#130694372
#130689712
#130688702
#130686312
C2×C11×#5572
#2419911