Finite Group Databasebeta

← #14910828

Subgroups — #14910828

Order 2816.

536 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic44C211yesmaximal, characteristic
2#13070863C211yesmaximal, characteristic, complemented
3#13070824C211yesmaximal, characteristic, complemented
4#13070813C211yesmaximal, characteristic, complemented
5#13070777C211yesmaximal, characteristic, complemented
6#13070722C211yesmaximal, characteristic, complemented
7#13070705C211yesmaximal, characteristic, complemented
8#13070666C211yesmaximal, characteristic, complemented
9#13070549C211yesmaximal, characteristic, complemented
10#13070549C211yesmaximal, characteristic, complemented
11#13070485C211yesmaximal, characteristic, complemented
12#13070442C211yesmaximal, characteristic
13#13070335C211yesmaximal, characteristic, complemented
14C2×#10622130C211yesmaximal, characteristic, complemented
15C11×#3336C211yesmaximal, characteristic
16C11×#552—21no—
17C22×Dic44C2211yescharacteristic
18C22×Dic44C2211yescharacteristic
19D4×Dic22—21no—
20C11×#499C2211yescharacteristic
21C11×#485—21no—
22C11×#483—21no—
23C11×#479—21no—
24C11×#467C2211yescharacteristic
25C11×#467C2211yescharacteristic
26C4×C11×Q16C2211yescharacteristic
27C8×D4×C11C2211yescharacteristic
28C11×#427C2211yescharacteristic
29C2×C11×#107C2211yescharacteristic
30C2×#66893C2211yescharacteristic
31C2×#66893C2211yescharacteristic
32C2×#66893C2211yescharacteristic
33#10622729C2211yescharacteristic
34#10622729C2211yescharacteristic
35C2×#66876C2211yescharacteristic
36C22×C11⋊C16C2211yescharacteristic
37#10622462—21no—
38#10622352—21no—
39#10622352—21no—
40#10622312C2211yescharacteristic
41C4×Dic44C2211yescharacteristic
42#10622202C2211yescharacteristic
43C22⋊(C11⋊C16)C2211yescharacteristic
44C22⋊(C11⋊C16)C2211yescharacteristic
45#10622130C2211yescharacteristic
46#10622130C2211yescharacteristic
47#10622130C2211yescharacteristic
48#10622130C2211yescharacteristic
49#10622130C2211yescharacteristic
50#10622130C2211yescharacteristic
51#10622130C2211yescharacteristic
52#10622130C2211yescharacteristic
53#10622130C2211yescharacteristic
54#10622128C2211yescharacteristic
55#10622128C2211yescharacteristic
56#10622127C2211yescharacteristic
57C4⋊(C11⋊C16)C2211yescharacteristic
58#10622099C2211yescharacteristic
59C22×Dic22—21no—
60C22×Dic22—21no—
61C2×C11×Q16C2311yescharacteristic
62C22×C8×C11C2311yescharacteristic
63C22×C8×C11C2311yescharacteristic
64C11×#126—41no—
65C11×#125—21no—
66C11×#125—41no—
67C11×#123—41no—
68C11×C22⋊Q8—21no—
69C11×C22⋊Q8—21no—
70C4×Q8×C11—21no—
71C4×D4×C11D411yescharacteristic
72C11×#117—21no—
73C11×#117—41no—
74C2×C11×C4⋊C4—21no—
75D4×Dic11—41no—
76#66977—21no—
77#66977—21no—
78C2×Dic44—22no—
79C2×Dic44—22no—
80C2×Dic44—21no—
81C2×Dic44—21no—
82C2×Dic44—21no—
83C2×Dic44—41no—
84C2×Dic44—41no—
85C2×Dic44C2311yescharacteristic
86C2×Dic44C2311yescharacteristic
87C2×Dic44C2311yescharacteristic
88#66941—41no—
89#66931—41no—
90#66931—41no—
91#66922—21no—
92C4×Dic22—21no—
93C11×#107—21no—
94C11×#107C2311yescharacteristic
95C11×#107C2311yescharacteristic
96C11×#107C2311yescharacteristic
97C11×#106—21no—
98C11×#106—21no—
99C11×C4⋊C8D411yescharacteristic
100C11×#103—21no—
101C11×#103—21no—
102C11×#103—21no—
103C11×C22⋊C8D411yescharacteristic
104C11×C22⋊C8D411yescharacteristic
105C4×C8×C11C2311yescharacteristic
106#66893—22no—
107#66893—41no—
108#66893—41no—
109#66882C2311yescharacteristic
110#66880—21no—
111#66880—21no—
112#66880—21no—
113#66876—41no—
114C2×C11⋊C16—41no—
115C2×C11⋊C16C2311yescharacteristic
116C2×C11⋊C16C2311yescharacteristic
117C2×C11⋊C16C2311yescharacteristic
118C2×C11⋊C16C2311yescharacteristic
119#33879—111nomaximal, Sylow
120C2×Q8×C11—21no—
121C2×D4×C11D811yescharacteristic
122C22×C4×C11—41no—
123C22×C4×C11C2×D411yescharacteristic
124C22×C4×C11C2×D411yescharacteristic
125C22×Dic11—41no—
126C22×Dic11—41no—
127C2×Dic22—21no—
128C2×Dic22—21no—
129C2×Dic22—21no—
130C2×Dic22—42no—
131C2×Dic22—42no—
132C2×Dic22—41no—
133C2×Dic22—41no—
134C2×Dic22—81no—
135C2×Dic22—81no—
136C11×Q16—21no—
137C11×Q16—21no—
138C2×C8×C11—21no—
139C2×C8×C11—21no—
140C2×C8×C11—21no—
141C2×C8×C11—21no—
142C2×C8×C11C2×D411yescharacteristic
143C2×C8×C11C2×D411yescharacteristic
144C2×C8×C11C2×D411yescharacteristic
145C2×C8×C11C2411yescharacteristic
146C11×C4⋊C4—21no—
147C11×C4⋊C4—21no—
148C11×C4⋊C4—21no—
149C11×C4⋊C4—41no—
150C11×C4⋊C4—41no—
151C11×C4⋊C4—41no—
152C11×C4⋊C4—41no—
153C11×C4⋊C4—41no—
154C11×C4⋊C4—41no—
155C11×C4⋊C4D811yescharacteristic
156C11×C22⋊C4—41no—
157C11×C22⋊C4—41no—
158C11×C22⋊C4—41no—
159C11×C22⋊C4D811yescharacteristic
160C11×C22⋊C4D811yescharacteristic
161C42×C11—41no—
162C42×C11—41no—
163C42×C11C2×D411yescharacteristic
164C22⋊Dic11—41no—
165C22⋊Dic11—41no—
166C4⋊Dic11—21no—
167C4⋊Dic11—41no—
168C4⋊Dic11—41no—
169Dic11⋊C4—41no—
170Dic11⋊C4—41no—
171Dic11⋊C4—41no—
172C4×Dic11—41no—
173Dic44—21no—
174Dic44—21no—
175Dic44—21no—
176Dic44—21no—
177Dic44—21no—
178Dic44—42no—
179Dic44—42no—
180Dic44—81no—
181Dic44#4112yes—
182C11⋊C16—21no—
183C11⋊C16—21no—
184C11⋊C16—21no—
185C11⋊C16—21no—
186#3336D1111yescharacteristic
187D4×Q16—111no—
188#2245—111no—
189#2237—111no—
190#2235—111no—
191#2229—111no—
192#2219—111no—
193#2216—111no—
194#2209—111no—
195#2188—111no—
196#2188—111no—
197#2176—111no—
198#2168—111no—
199#2141—111no—
200C2×#358—111no—
201C23×C11C2×D811yescharacteristic
202C23×C11C2×D811yescharacteristic
203Q8×C11—41no—
204Q8×C11—41no—
205D4×C11—22no—
206C2×C4×C11—21no—
207C2×C4×C11—21no—
208C2×C4×C11—21no—
209C2×C4×C11—21no—
210C2×C4×C11—41no—
211C2×C4×C11—41no—
212C2×C4×C11—41no—
213C2×C4×C11—41no—
214C2×C4×C11—82no—
215C2×C4×C11C2×D811yescharacteristic
216C2×C4×C11C2×D811yescharacteristic
217C2×C4×C11C2×D811yescharacteristic
218C2×C4×C11C2×D811yescharacteristic
219C2×C4×C11C22×D411yescharacteristic
220C2×Dic11—41no—
221C2×Dic11—41no—
222C2×Dic11—41no—
223C2×Dic11—41no—
224C2×Dic11—82no—
225C2×Dic11—82no—
226Dic22—22no—
227Dic22—41no—
228Dic22—41no—
229Dic22—41no—
230Dic22—41no—
231Dic22—41no—
232Dic22—82no—
233Dic22—82no—
234C8×C11—21no—
235C8×C11—21no—
236C8×C11—21no—
237C8×C11C2×#4111yescharacteristic
238C8×C11H(5,𝔽2)11yescharacteristic
239C22×Q16—111no—
240C22×Q16—111no—
241#552—21no—
242D4×Q8—221no—
243C2×Q32—111no—
244C2×Q32—111no—
245C2×Q32—111no—
246C2×M32—111no—
247C22×C16—111no—
248#499C2×D1111yescharacteristic
249#494—111no—
250#485—21no—
251#483—21no—
252#479—21no—
253#467—111no—
254#467—111no—
255#467C2×D1111yescharacteristic
256#467C2×D1111yescharacteristic
257#462—221no—
258#451—221no—
259#451—221no—
260C4×Q16—111no—
261C4×Q16C2×D1111yescharacteristic
262C8×D4C2×D1111yescharacteristic
263#427C2×D1111yescharacteristic
264C2×#107C2×D1111yescharacteristic
265#367—111no—
266#367—111no—
267#366—111no—
268C4⋊C16—111no—
269#358—111no—
270#358—111no—
271#358—111no—
272#358—111no—
273#358—111no—
274#358—111no—
275#358—111no—
276#358—111no—
277#358—111no—
278#358—111no—
279C22⋊C16—111no—
280C22⋊C16—111no—
281#346—111no—
282C22×C11—21no—
283C22×C11—21no—
284C22×C11—21no—
285C22×C11—21no—
286C22×C11—21no—
287C22×C11#51012yes—
288C22×C11C22×D811yescharacteristic
289C4×C11—21no—
290C4×C11—21no—
291C4×C11—21no—
292C4×C11—21no—
293C4×C11—81no—
294C4×C11—81no—
295C4×C11—81no—
296C4×C11—81no—
297C4×C11#54611yescharacteristic
298C4×C11#57711yescharacteristic
299Dic11—42no—
300Dic11—81no—
301Dic11—81no—
302Dic11—81no—
303C22×Q8—221no—
304C22×Q8—221no—
305C2×Q16—111no—
306C2×Q16—111no—
307C2×Q16—111no—
308C2×Q16—222no—
309C2×Q16—222no—
310C2×Q16—221no—
311C2×Q16—221no—
312C2×Q16—221no—
313C2×Q16—441no—
314C2×Q16—441no—
315C2×Q16C22×D1111yescharacteristic
316C22×C8C22×D1111yescharacteristic
317C22×C8C22×D1111yescharacteristic
318C4⋊Q8—221no—
319C4⋊Q8—441no—
320#126—41no—
321#125—21no—
322#125—41no—
323#123—41no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—221no—
327C22⋊Q8—221no—
328C22⋊Q8—441no—
329C22⋊Q8—441no—
330C4×Q8—21no—
331C4×Q8—221no—
332C4×D4—441no—
333C4×D4C22⋊D1111yescharacteristic
334#117—21no—
335#117—41no—
336C2×C4⋊C4—21no—
337Q32—222no—
338Q32—441no—
339Q32—441no—
340M32—441no—
341C2×C16—111no—
342C2×C16—111no—
343C2×C16—111no—
344C2×C16—111no—
345C2×C16—441no—
346#107—21no—
347#107—111no—
348#107C22×D1111yescharacteristic
349#107C22×D1111yescharacteristic
350#107C22×D1111yescharacteristic
351#106—21no—
352#106—21no—
353C4⋊C8C22⋊D1111yescharacteristic
354#103—21no—
355#103—21no—
356#103—21no—
357#103—221no—
358#103—221no—
359#103—221no—
360C22⋊C8C22⋊D1111yescharacteristic
361C22⋊C8C22⋊D1111yescharacteristic
362C4×C8C22×D1111yescharacteristic
363C2×C11—22no—
364C2×C11—41no—
365C2×C11#329411yescharacteristic
366C2×C11C2×#51011yescharacteristic
367C2×C11#341711yescharacteristic
368C2×Q8—21no—
369C2×Q8—221no—
370C2×Q8—221no—
371C2×Q8—221no—
372C2×Q8—442no—
373C2×Q8—442no—
374C2×Q8—441no—
375C2×Q8—441no—
376C2×Q8—881no—
377C2×Q8—881no—
378C2×D4#432311yescharacteristic
379C22×C4—41no—
380C22×C4—441no—
381C22×C4—441no—
382C22×C4C2×C22⋊D1111yescharacteristic
383C22×C4C2×C22⋊D1111yescharacteristic
384Q16—21no—
385Q16—21no—
386Q16—112no—
387Q16—221no—
388Q16—221no—
389Q16—221no—
390Q16—221no—
391Q16—221no—
392Q16—442no—
393Q16—442no—
394Q16—881no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8C2×C22⋊D1111yescharacteristic
400C2×C8C2×C22⋊D1111yescharacteristic
401C2×C8C2×C22⋊D1111yescharacteristic
402C2×C8C23×D1111yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—221no—
413C4⋊C4—441no—
414C4⋊C4—441no—
415C4⋊C4—441no—
416C4⋊C4—441no—
417C4⋊C4—441no—
418C4⋊C4#432311yescharacteristic
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—441no—
423C22⋊C4—441no—
424C22⋊C4#432311yescharacteristic
425C22⋊C4#432311yescharacteristic
426C42—41no—
427C42—41no—
428C42—441no—
429C42C2×C22⋊D1111yescharacteristic
430C16—221no—
431C16—221no—
432C16—221no—
433C16—221no—
434C11#3387911yescharacteristic, complemented, Sylow
435C23C2×#432311yescharacteristic
436C23C2×#432311yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—222no—
440Q8—441no—
441Q8—441no—
442Q8—441no—
443Q8—441no—
444Q8—441no—
445Q8—882no—
446Q8—882no—
447D4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—441no—
458C2×C4—441no—
459C2×C4—441no—
460C2×C4—441no—
461C2×C4—882no—
462C2×C4—882no—
463C2×C4C2×#432311yescharacteristic
464C2×C4C2×#432311yescharacteristic
465C2×C4C2×#432311yescharacteristic
466C2×C4C2×#432311yescharacteristic
467C2×C4C22×C22⋊D1111yescharacteristic
468C8—21no—
469C8—21no—
470C8—21no—
471C8#41×D1111yescharacteristic
472C8#6704211yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22#1062280812yes—
479C22C22×#432311yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—442no—
489C4—881no—
490C4—881no—
491C4—881no—
492C4#1062336011yescharacteristic
493C4#1062336811yescharacteristic
494C2—22no—
495C2—41no—
496C2#1308042311yescharacteristic
497C2C2×#1062280811yescharacteristic
498C2#1308176711yescharacteristic
subgroupG/H
centerC22C22×#4323
commutator, cosocleC2×C8×C11C24
FrattiniC2×C8C23×D11
FittingC11×#3336C2
socleC22×C11C22×D8
prime radicalC2×D4×C11D8
hypercenter#3336D11
hypocenterC11#33879
LCSG ▷ C2×C8×C11 ▷ C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #13070863C2semidirect
3. #13070824C2semidirect
4. #13070813C2semidirect
5. #13070777C2semidirect
6. #13070722C2semidirect
7. #13070705C2semidirect
8. #13070666C2semidirect
9. #13070549C2semidirect
10. #13070549C2semidirect
11. #13070485C2semidirect
13. #13070335C2semidirect
14. C2×#10622130C2semidirect
434. C11#33879semidirect

Maximal subgroups

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

subgroupindex
D4×Dic442
#130708632
#130708242
#130708132
#130707772
#130707222
#130707052
#130706662
#130705492
#130704852
#130704422
#130703352
C2×#106221302
C11×#33362
#3387911