Finite Group Databasebeta

← #15896874

Subgroups — #15896874

Order 3328.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719248C211yesmaximal, characteristic
2C2×#11718682C211yesmaximal, characteristic, complemented
3C2×C4×#6107C211yesmaximal, characteristic, complemented
4#13116039C212yesmaximal, complemented
5#13115844C212yesmaximal, complemented
6#13115796C211yesmaximal, characteristic, complemented
7#13115631C211yesmaximal, characteristic
8#13115144C212yesmaximal, complemented
9#13115023C212yesmaximal, complemented
10#13114838C211yesmaximal, characteristic, complemented
11C13×#2538C211yesmaximal, characteristic
12C2×C13×C22⋊Q8C2212yes—
13C2×C4×Q8×C13C2211yescharacteristic
14C2×C4×D4×C13C2211yescharacteristic
15C22×#6107C2211yescharacteristic
16C2×D4×Dic13—21no—
17C22×C4⋊Dic13C2212yes—
18C2×#88178C2212yes—
19#11719248—24no—
20#11719248—24no—
21C2×#88144—21no—
22C2×#88132—22no—
23C2×#88123C2211yescharacteristic
24C2×C4×Dic26C2211yescharacteristic
25C13×#396—22no—
26C13×#390—21no—
27C13×#389—21no—
28C13×#384C2211yescharacteristic
29C13×#380C2212yes—
30C2×#88075C2211yescharacteristic
31C2×#88075C2212yes—
32C2×C22⋊#1055C2212yes—
33C2×C22⋊#1055C2212yes—
34C4×#6107—24no—
35#11718682—24no—
36C2×#88050C2211yescharacteristic
37C2×#88050C2212yes—
38C2×#88047C2211yescharacteristic
39C2×#88047C2212yes—
40C2×C4⋊#1055C2211yescharacteristic
41C2×C4×#1055C2211yescharacteristic
42#11718121C2212yes—
43#11718121C2212yes—
44#11718121C2212yes—
45C22×Q8×C13C2311yescharacteristic
46C22×D4×C13D411yescharacteristic
47C23×C4×C13C2312yes—
48C23×Dic13—22no—
49C22×Dic26—21no—
50C22×Dic26C2311yescharacteristic
51C13×C22⋊Q8—24no—
52C13×C22⋊Q8—42no—
53C4×Q8×C13—22no—
54C4×Q8×C13—22no—
55C4×D4×C13—22no—
56C4×D4×C13—22no—
57C2×C13×C4⋊C4—21no—
58C2×C13×C4⋊C4—22no—
59C2×C13×C4⋊C4—21no—
60C2×C13×C4⋊C4—21no—
61C2×C13×C4⋊C4D411yescharacteristic
62C2×C13×C4⋊C4C2311yescharacteristic
63C2×C13×C4⋊C4C2312yes—
64C2×C13×C22⋊C4—22no—
65C2×C13×C22⋊C4—22no—
66C2×C13×C22⋊C4D412yes—
67C2×C42×C13—21no—
68C2×C42×C13C2311yescharacteristic
69C2×C22⋊Dic13—22no—
70C2×#6107—22no—
71C2×#6107—22no—
72C2×#6107—22no—
73D4×Dic13—28no—
74C2×C4⋊Dic13—21no—
75C2×C4⋊Dic13—22no—
76C2×C4⋊Dic13—24no—
77C2×C4⋊Dic13—24no—
78C2×C4⋊Dic13C2311yescharacteristic
79C2×C4⋊Dic13C2312yes—
80#88178—24no—
81#88178—24no—
82C2×#6101—21no—
83C2×#6101—22no—
84C2×C4×Dic13—21no—
85C22×#1055C2311yescharacteristic
86C22×#1055C2312yes—
87C22×#1055C2311yescharacteristic
88#88144—28no—
89#88132—216no—
90#88123—22no—
91#88123D414yes—
92C4×Dic26—22no—
93C4×Dic26D414yes—
94C13×#95—22no—
95#88075—24no—
96#88075—22no—
97C22⋊#1055—24no—
98C22⋊#1055—42no—
99#88050—22no—
100#88050—42no—
101#88047—24no—
102#88047—22no—
103C4⋊#1055—22no—
104C4×#1055—22no—
105#24122—131nomaximal, Sylow
106C24×C13C2×D412yes—
107C2×Q8×C13—21no—
108C2×Q8×C13—21no—
109C2×Q8×C13—21no—
110C2×Q8×C13—21no—
111C2×Q8×C13—21no—
112C2×Q8×C13#4112yes—
113C2×D4×C13—22no—
114C2×D4×C13—22no—
115C2×D4×C13Q1614yescomplemented
116C22×C4×C13—21no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—21no—
124C22×C4×C13—22no—
125C22×C4×C13C2×D412yes—
126C22×C4×C13C2×D411yescharacteristic
127C22×C4×C13C2×D411yescharacteristic
128C22×C4×C13C2411yescharacteristic
129C22×Dic13—21no—
130C22×Dic13—21no—
131C22×Dic13—22no—
132C22×Dic13—28no—
133C22×Dic13—82no—
134C22×Dic13—82no—
135C2×Dic26—21no—
136C2×Dic26—21no—
137C2×Dic26—21no—
138C2×Dic26—21no—
139C2×Dic26—21no—
140C2×Dic26—24no—
141C2×Dic26—81no—
142C2×Dic26—81no—
143C2×Dic26C2×D412yes—
144C13×C4⋊C4—22no—
145C13×C4⋊C4—21no—
146C13×C4⋊C4—21no—
147C13×C4⋊C4—22no—
148C13×C4⋊C4—22no—
149C13×C4⋊C4—44no—
150C13×C4⋊C4—42no—
151C13×C4⋊C4—42no—
152C13×C4⋊C4—81no—
153C13×C22⋊C4—24no—
154C13×C22⋊C4—44no—
155C13×C22⋊C4—82no—
156C42×C13—42no—
157C42×C13C2×D412yes—
158C42×C13C2×D412yes—
159C22⋊Dic13—28no—
160#6107—28no—
161C4⋊Dic13—28no—
162C4⋊Dic13—24no—
163C4⋊Dic13—21no—
164C4⋊Dic13—22no—
165C4⋊Dic13C2×D412yes—
166C4⋊Dic13C2×D414yes—
167#6101—28no—
168#6101—24no—
169C4×Dic13—24no—
170C2×#1055—21no—
171C2×#1055—21no—
172C2×#1055—21no—
173C2×#1055—22no—
174C2×#1055—22no—
175C2×#1055—21no—
176C2×#1055—22no—
177C2×#1055—21no—
178C2×#1055#4112yes—
179C2×#554—131no—
180C2×#477—131no—
181C2×C4×Q16—131no—
182#2538D1311yescharacteristic
183#2036—132no—
184#2001—132no—
185#1990—131no—
186#1954—131no—
187#1872—132no—
188#1852—132no—
189#1803—131no—
190C23×C13—22no—
191C23×C13—22no—
192C23×C13—22no—
193C23×C13—22no—
194C23×C13—22no—
195C23×C13—22no—
196C23×C13C2×Q1614yes—
197C23×C13C22×D411yescharacteristic
198Q8×C13—22no—
199Q8×C13—22no—
200Q8×C13—41no—
201Q8×C13—41no—
202D4×C13—28no—
203C2×C4×C13—21no—
204C2×C4×C13—21no—
205C2×C4×C13—22no—
206C2×C4×C13—22no—
207C2×C4×C13—22no—
208C2×C4×C13—22no—
209C2×C4×C13—41no—
210C2×C4×C13—41no—
211C2×C4×C13—42no—
212C2×C4×C13—42no—
213C2×C4×C13—42no—
214C2×C4×C13—42no—
215C2×C4×C13—41no—
216C2×C4×C13—42no—
217C2×C4×C13—42no—
218C2×C4×C13—41no—
219C2×C4×C13—42no—
220C2×C4×C13—41no—
221C2×C4×C13#12012yes—
222C2×C4×C13#12012yes—
223C2×C4×C13C2×Q1612yes—
224C2×C4×C13#13712yes—
225C2×C4×C13C22×D411yescharacteristic
226C2×C4×C13C22×D411yescharacteristic
227C2×C4×C13C2×#4111yescharacteristic
228C2×C4×C13C2×#4111yescharacteristic
229C2×C4×C13H(5,𝔽2)11yescharacteristic
230C2×C4×C132−1+411yescharacteristic
231C2×Dic13—22no—
232C2×Dic13—22no—
233C2×Dic13—24no—
234C2×Dic13—41no—
235C2×Dic13—41no—
236C2×Dic13—42no—
237C2×Dic13—41no—
238C2×Dic13—42no—
239C2×Dic13—41no—
240C2×Dic13—88no—
241Dic26—22no—
242Dic26—22no—
243Dic26—41no—
244Dic26—41no—
245Dic26—84no—
246#1055—22no—
247#1055—41no—
248#1055—42no—
249C22×Q16—131no—
250#554—264no—
251#554—264no—
252C2×C4⋊Q8—131no—
253C2×#125—261no—
254C2×C22⋊Q8—132no—
255C2×C22⋊Q8—262no—
256C2×C22⋊Q8C2×D1312yes—
257C2×C4×Q8—131no—
258C2×C4×Q8C2×D1311yescharacteristic
259C2×C4×D4—261no—
260C2×C4×D4C2×D1311yescharacteristic
261C22×C4⋊C4—132no—
262#477—264no—
263C4×Q16—264no—
264C2×#107—132no—
265C2×#107—131no—
266C2×C4⋊C8—131no—
267C2×#103—132no—
268C2×#103—131no—
269C2×#103—131no—
270C2×#103—132no—
271C2×C22⋊C8—132no—
272C2×C22⋊C8—132no—
273C2×C4×C8—131no—
274#396—22no—
275#390—21no—
276#389—21no—
277#384C2×D1311yescharacteristic
278#380C2×D1312yes—
279#340—132no—
280#340—132no—
281#340—132no—
282C22×C13—24no—
283C22×C13—24no—
284C22×C13—24no—
285C22×C13—42no—
286C22×C13—42no—
287C22×C13—42no—
288C22×C13—42no—
289C22×C13C2×#12011yescharacteristic
290C22×C13#54611yescharacteristic
291C22×C13#54711yescharacteristic
292C22×C13C22×Q1611yescharacteristic
293C22×C13C2×#13711yescharacteristic
294C22×C13#57611yescharacteristic
295C22×C13#57711yescharacteristic
296C4×C13—22no—
297C4×C13—42no—
298C4×C13—42no—
299C4×C13—82no—
300C4×C13—81no—
301C4×C13#45114yes—
302C4×C13#52511yescharacteristic
303C4×C13#53711yescharacteristic
304C4×C13#54811yescharacteristic
305C4×C13#55011yescharacteristic
306Dic13—42no—
307Dic13—44no—
308Dic13—42no—
309C22×Q8—131no—
310C22×Q8—261no—
311C22×Q8C22×D1311yescharacteristic
312C22×D4C22⋊D1311yescharacteristic
313C23×C4—262no—
314C23×C4C22×D1312yes—
315C2×Q16—262no—
316C2×Q16—262no—
317C2×Q16—262no—
318C22×C8—131no—
319C22×C8—131no—
320C22×C8—132no—
321C4⋊Q8—134no—
322C4⋊Q8—262no—
323#125—268no—
324C22⋊Q8—24no—
325C22⋊Q8—42no—
326C22⋊Q8—2616no—
327C22⋊Q8—264no—
328C22⋊Q8—264no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—134no—
332C4×Q8—262no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—268no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—131no—
341C2×C4⋊C4—132no—
342C2×C4⋊C4—264no—
343C2×C4⋊C4—264no—
344C2×C4⋊C4—262no—
345C2×C4⋊C4—261no—
346C2×C4⋊C4—261no—
347C2×C4⋊C4—262no—
348C2×C4⋊C4C22⋊D1311yescharacteristic
349C2×C4⋊C4C22×D1311yescharacteristic
350C2×C4⋊C4C22×D1312yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—262no—
354C2×C22⋊C4C22⋊D1312yes—
355C2×C42—21no—
356C2×C42—261no—
357C2×C42C22×D1311yescharacteristic
358#107—262no—
359#107—264no—
360C4⋊C8—262no—
361#103—262no—
362#103—262no—
363#103—264no—
364#103—522no—
365C22⋊C8—264no—
366C22⋊C8—522no—
367C4×C8—262no—
368#95—22no—
369C2×C13—44no—
370C2×C13#240711yescharacteristic
371C2×C13C2×#45111yescharacteristic
372C2×C13#301211yescharacteristic
373C2×C13#329911yescharacteristic
374C2×C13#333311yescharacteristic
375C2×C13#333811yescharacteristic
376C2×C13#335611yescharacteristic
377C24C2×C22⋊D1312yes—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—132no—
384C2×Q8—264no—
385C2×Q8—261no—
386C2×Q8—261no—
387C2×Q8—261no—
388C2×Q8—261no—
389C2×Q8—261no—
390C2×Q8—1041no—
391C2×Q8—1041no—
392C2×Q8#612912yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4#610714yescomplemented
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—261no—
406C22×C4—261no—
407C22×C4—262no—
408C22×C4—268no—
409C22×C4—1042no—
410C22×C4—1042no—
411C22×C4C2×C22⋊D1312yes—
412C22×C4C2×C22⋊D1311yescharacteristic
413C22×C4C2×C22⋊D1311yescharacteristic
414C22×C4C23×D1311yescharacteristic
415Q16—268no—
416C2×C8—132no—
417C2×C8—261no—
418C2×C8—261no—
419C2×C8—261no—
420C2×C8—262no—
421C2×C8—261no—
422C2×C8—262no—
423C2×C8—262no—
424C2×C8—261no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—132no—
435C4⋊C4—134no—
436C4⋊C4—268no—
437C4⋊C4—268no—
438C4⋊C4—264no—
439C4⋊C4—264no—
440C4⋊C4—261no—
441C4⋊C4—262no—
442C22⋊C4—24no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—268no—
446C42—42no—
447C42—264no—
448C42D4×D1312yes—
449C42D4×D1312yes—
450C13#2412211yescharacteristic, complemented, Sylow
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23C2×#610714yes—
458C23C22×C22⋊D1311yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—41no—
462Q8—41no—
463Q8—262no—
464Q8—262no—
465Q8—521no—
466Q8—521no—
467Q8—1044no—
468D4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—262no—
488C2×C4—262no—
489C2×C4—264no—
490C2×C4—521no—
491C2×C4—521no—
492C2×C4—521no—
493C2×C4—522no—
494C2×C4—521no—
495C2×C4—522no—
496C2×C4—1048no—
497C2×C4#8819112yes—
498C2×C4#8819112yes—
499C2×C4C2×#610712yes—
500C2×C4#8820512yes—
501C2×C4C2×D4×D1311yescharacteristic
502C2×C4C2×D4×D1311yescharacteristic
503C2×C4C2×#612911yescharacteristic
504C2×C4#8825111yescharacteristic
505C2×C4#41×D1311yescharacteristic
506C2×C4#8825711yescharacteristic
507C8—262no—
508C8—522no—
509C8—521no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#8819111yescharacteristic
518C22#1171951111yescharacteristic
519C22C22×#610711yescharacteristic
520C22C2×#8820511yescharacteristic
521C22#1171952911yescharacteristic
522C22#1171954011yescharacteristic
523C22#1171954211yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—81no—
529C4—522no—
530C4—524no—
531C4—522no—
532C4#1171871114yes—
533C4#1171930311yescharacteristic
534C4#1171930411yescharacteristic
535C4#1171931111yescharacteristic
536C4#1171931311yescharacteristic
537C2—44no—
538C2#1311817711yescharacteristic
539C2C2×#1171871111yescharacteristic
540C2#1312413711yescharacteristic
541C2#1312671211yescharacteristic
542C2#1312717411yescharacteristic
543C2#1312722411yescharacteristic
544C2#1312744511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2538C2
socleC23×C13C22×D4
prime radicalC22×D4×C13D4
hypercenter#2538D13
hypocenterC13#24122
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#11718682C2semidirect
3. C2×C4×#6107C2semidirect
4. #13116039C2semidirect
5. #13115844C2semidirect
6. #13115796C2semidirect
8. #13115144C2semidirect
9. #13115023C2semidirect
10. #13114838C2semidirect
115. C2×D4×C13Q16semidirect
395. C2×D4#6107semidirect
450. C13#24122semidirect

Maximal subgroups

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

subgroupindex
C2×#117192482
C2×#117186822
C2×C4×#61072
#131160392
#131158442
#131157962
#131156312
#131151442
#131150232
#131148382
C13×#25382
#2412213