Finite Group Databasebeta

← #15919446

Subgroups — #15919446

Order 3328.

1260 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718743C211yesmaximal, characteristic, complemented
2C2×#11718683C211yesmaximal, characteristic, complemented
3#13120002C211yesmaximal, characteristic
4#13116036C212yesmaximal, complemented
5#13116029C212yesmaximal, complemented
6#13115760C211yesmaximal, characteristic, complemented
7#13115635C211yesmaximal, characteristic
8#13114771C212yesmaximal, complemented
9#13114771C212yesmaximal, complemented
10#13114666C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C13C211yesmaximal, characteristic
12C23×Q8×C13C2212yes—
13D4×Q8×C13—24no—
14D4×Q8×C13—24no—
15C2×C13×C4⋊Q8—21no—
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8—22no—
18C2×C13×C22⋊Q8C2212yes—
19C2×C4×Q8×C13C2211yescharacteristic
20C2×C4×D4×C13—21no—
21C2×C4×D4×C13C2211yescharacteristic
22C2×#88124C2211yescharacteristic
23C2×#88075C2211yescharacteristic
24C2×#88075C2211yescharacteristic
25C2×#88075C2212yes—
26C2×#88075C2212yes—
27C2×#88075C2212yes—
28C2×#88075C2212yes—
29C2×#88075C2212yes—
30#11718883—21no—
31#11718873C2212yes—
32#11718873C2212yes—
33C2×C22⋊#1055C2212yes—
34C2×C22⋊#1055C2212yes—
35#11718743—24no—
36#11718683—24no—
37#11718643—21no—
38C2×#88048C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40#11718606—22no—
41C4×C4⋊Dic13C2211yescharacteristic
42C2×C4⋊#1055C2211yescharacteristic
43C2×#88043C2211yescharacteristic
44#11718121C2212yes—
45C22×Q8×C13—24no—
46C22×Q8×C13—24no—
47C22×Q8×C13—22no—
48C22×Q8×C13—24no—
49C22×Q8×C13—24no—
50C22×Q8×C13—42no—
51C22×Q8×C13—42no—
52C22×Q8×C13C2312yes—
53C22×Q8×C13C2311yescharacteristic
54C22×D4×C13D411yescharacteristic
55C23×C4×C13—22no—
56C23×C4×C13C2312yes—
57C13×C4⋊Q8—28no—
58C13×C4⋊Q8—22no—
59C13×C4⋊Q8D414yes—
60C13×C22⋊Q8—216no—
61C13×C22⋊Q8—24no—
62C13×C22⋊Q8—24no—
63C4×Q8×C13—22no—
64C4×Q8×C13D414yes—
65C4×D4×C13—28no—
66C4×D4×C13—22no—
67C4×D4×C13—22no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—22no—
72C2×C13×C4⋊C4D411yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C22⋊C4—22no—
75C2×C13×C22⋊C4D412yes—
76C2×C42×C13—21no—
77C2×C42×C13C2311yescharacteristic
78C2×C22⋊Dic13—22no—
79C2×C22⋊Dic13—22no—
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2312yes—
83C2×#6101—21no—
84C2×#6101—21no—
85C2×C4×Dic13—21no—
86C22×#1055C2312yes—
87C22×#1055C2311yescharacteristic
88C22×#1055C2311yescharacteristic
89#88124—22no—
90#88124—22no—
91#88075—22no—
92#88075—22no—
93#88075—24no—
94#88075—24no—
95#88075—24no—
96#88075—42no—
97#88075—42no—
98#88071—21no—
99#88071—22no—
100#88071—22no—
101C22⋊#1055—42no—
102C22⋊#1055—42no—
103#88048—22no—
104#88047—22no—
105C4⋊#1055—22no—
106#88043—22no—
107#25894—131nomaximal, Sylow
108C24×C13C2×D412yes—
109C2×Q8×C13—28no—
110C2×Q8×C13—21no—
111C2×Q8×C13—22no—
112C2×Q8×C13—22no—
113C2×Q8×C13—22no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13—44no—
118C2×Q8×C13—44no—
119C2×Q8×C13—48no—
120C2×Q8×C13—48no—
121C2×Q8×C13—48no—
122C2×Q8×C13—48no—
123C2×Q8×C13C2×D414yes—
124C2×Q8×C13C2×D412yes—
125C2×Q8×C13#4114yes—
126C2×D4×C13—24no—
127C2×D4×C13—22no—
128C22×C4×C13—21no—
129C22×C4×C13—28no—
130C22×C4×C13—21no—
131C22×C4×C13—22no—
132C22×C4×C13—22no—
133C22×C4×C13—22no—
134C22×C4×C13—22no—
135C22×C4×C13—22no—
136C22×C4×C13—22no—
137C22×C4×C13—22no—
138C22×C4×C13—44no—
139C22×C4×C13—44no—
140C22×C4×C13C2×D412yes—
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2411yescharacteristic
144C22×Dic13—22no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C13×C4⋊C4—28no—
148C13×C4⋊C4—24no—
149C13×C4⋊C4—24no—
150C13×C4⋊C4—24no—
151C13×C4⋊C4—22no—
152C13×C4⋊C4—21no—
153C13×C4⋊C4C2×D412yes—
154C13×C22⋊C4—28no—
155C13×C22⋊C4—24no—
156C42×C13—24no—
157C42×C13C2×D412yes—
158C42×C13C2×D412yes—
159C22⋊Dic13—44no—
160C22⋊Dic13—82no—
161C4⋊Dic13—21no—
162C4⋊Dic13—21no—
163C4⋊Dic13—21no—
164C4⋊Dic13—21no—
165C4⋊Dic13—22no—
166C4⋊Dic13—22no—
167#6101—42no—
168#6101—42no—
169C4×Dic13—81no—
170C2×#1055—22no—
171C2×#1055—21no—
172C2×#1055—21no—
173C2×#1055—22no—
174C2×#1055—22no—
175C2×#1055—22no—
176C2×#1055—22no—
177C2×D4×Q8D1311yescharacteristic
178C2×#491—131no—
179C2×#479—131no—
180#2562—131no—
181#2036—132no—
182#2035—132no—
183#1981—131no—
184#1955—131no—
185#1787—132no—
186#1787—132no—
187#1763—131no—
188C23×C13—22no—
189C23×C13—22no—
190C23×C13—22no—
191C23×C13—22no—
192C23×C13—22no—
193C23×C13—22no—
194C23×C13#13714yes—
195C23×C13C22×D411yescharacteristic
196Q8×C13—22no—
197Q8×C13—24no—
198Q8×C13—24no—
199Q8×C13—24no—
200Q8×C13—416no—
201Q8×C13—42no—
202Q8×C13—42no—
203Q8×C13#12414yes—
204D4×C13—28no—
205C2×C4×C13—22no—
206C2×C4×C13—24no—
207C2×C4×C13—22no—
208C2×C4×C13—21no—
209C2×C4×C13—21no—
210C2×C4×C13—22no—
211C2×C4×C13—24no—
212C2×C4×C13—22no—
213C2×C4×C13—22no—
214C2×C4×C13—22no—
215C2×C4×C13—416no—
216C2×C4×C13—42no—
217C2×C4×C13—42no—
218C2×C4×C13—41no—
219C2×C4×C13—41no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—42no—
224C2×C4×C13#12012yes—
225C2×C4×C13#12012yes—
226C2×C4×C13C22×D411yescharacteristic
227C2×C4×C13C22×D411yescharacteristic
228C2×C4×C13C2×#4112yes—
229C2×C4×C13H(5,𝔽2)12yes—
230C2×Dic13—41no—
231C2×Dic13—41no—
232C2×Dic13—42no—
233C2×Dic13—42no—
234C2×Dic13—42no—
235C2×Dic13—42no—
236C2×Dic13—41no—
237C2×Dic13—41no—
238#1055—41no—
239#1055—41no—
240#1055—42no—
241C23×Q8C2×D1312yes—
242D4×Q8—24no—
243D4×Q8—24no—
244C2×C4⋊Q8—21no—
245C2×C4⋊Q8C2×D1311yescharacteristic
246C2×#125—131no—
247C2×C22⋊Q8—22no—
248C2×C22⋊Q8C2×D1312yes—
249C2×C4×Q8C2×D1311yescharacteristic
250C2×C4×D4—21no—
251C2×C4×D4C2×D1311yescharacteristic
252#491—264no—
253#479—264no—
254C2×#107—131no—
255C2×#106—131no—
256C2×C4⋊C8—131no—
257C2×#103—131no—
258C2×#103—131no—
259C2×#103—132no—
260C2×#103—132no—
261C2×#103—132no—
262C2×#103—132no—
263C2×#103—132no—
264C2×C22⋊C8—132no—
265C2×C22⋊C8—132no—
266C2×#97—131no—
267#400—261no—
268#394—261no—
269#388—262no—
270#380—132no—
271#380—132no—
272C4×C4⋊C4—131no—
273#340—132no—
274C22×C13—24no—
275C22×C13—24no—
276C22×C13—24no—
277C22×C13—42no—
278C22×C13—42no—
279C22×C13—42no—
280C22×C13—42no—
281C22×C13C2×#12011yescharacteristic
282C22×C13#54612yes—
283C22×C13C2×#13712yes—
284C22×C13#57711yescharacteristic
285C22×C13#57811yescharacteristic
286C4×C13—24no—
287C4×C13—22no—
288C4×C13—22no—
289C4×C13—42no—
290C4×C13—44no—
291C4×C13—42no—
292C4×C13C2×#12411yescharacteristic
293C4×C13#53511yescharacteristic
294C4×C13#54612yes—
295Dic13—81no—
296Dic13—81no—
297Dic13—82no—
298C22×Q8—24no—
299C22×Q8—24no—
300C22×Q8—22no—
301C22×Q8—24no—
302C22×Q8—24no—
303C22×Q8—42no—
304C22×Q8—42no—
305C22×Q8C22×D1312yes—
306C22×Q8C22×D1311yescharacteristic
307C22×D4C22⋊D1311yescharacteristic
308C23×C4—22no—
309C23×C4C22×D1312yes—
310C22×C8—132no—
311C22×C8—131no—
312C22×C8—131no—
313C4⋊Q8—28no—
314C4⋊Q8—22no—
315C4⋊Q8C22⋊D1314yes—
316#125—262no—
317#125—262no—
318C22⋊Q8—216no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C4×Q8—22no—
322C4×Q8C22⋊D1314yes—
323C4×D4—28no—
324C4×D4—22no—
325C4×D4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—131no—
331C2×C4⋊C4—131no—
332C2×C4⋊C4—132no—
333C2×C4⋊C4—261no—
334C2×C4⋊C4—261no—
335C2×C4⋊C4C22⋊D1311yescharacteristic
336C2×C4⋊C4C22×D1311yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—262no—
339C2×C22⋊C4—262no—
340C2×C22⋊C4C22⋊D1312yes—
341C2×C42—21no—
342C2×C42—261no—
343C2×C42C22×D1311yescharacteristic
344#107—262no—
345#106—262no—
346C4⋊C8—262no—
347#103—264no—
348#103—262no—
349#103—262no—
350#103—264no—
351#103—264no—
352#103—522no—
353#103—522no—
354C22⋊C8—522no—
355C22⋊C8—522no—
356#97—262no—
357#95—261no—
358#95—262no—
359#95—262no—
360C2×C13—44no—
361C2×C13#243511yescharacteristic
362C2×C13#300411yescharacteristic
363C2×C13#301311yescharacteristic
364C2×C13#331212yes—
365C2×C13#332012yes—
366C24C2×C22⋊D1312yes—
367C2×Q8—28no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8#612914yes—
382C2×Q8C2×C22⋊D1314yes—
383C2×Q8C2×C22⋊D1312yes—
384C2×D4—24no—
385C2×D4—22no—
386C22×C4—21no—
387C22×C4—28no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—262no—
399C22×C4—261no—
400C22×C4—261no—
401C22×C4C2×C22⋊D1312yes—
402C22×C4C2×C22⋊D1311yescharacteristic
403C22×C4C2×C22⋊D1311yescharacteristic
404C22×C4C23×D1311yescharacteristic
405C2×C8—262no—
406C2×C8—262no—
407C2×C8—262no—
408C2×C8—262no—
409C2×C8—262no—
410C2×C8—261no—
411C2×C8—261no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—262no—
419C4⋊C4—262no—
420C4⋊C4—261no—
421C4⋊C4—261no—
422C4⋊C4—261no—
423C4⋊C4—261no—
424C4⋊C4—522no—
425C4⋊C4—522no—
426C4⋊C4C2×C22⋊D1312yes—
427C22⋊C4—28no—
428C22⋊C4—24no—
429C22⋊C4—524no—
430C22⋊C4—1042no—
431C42—24no—
432C42—1041no—
433C42C2×C22⋊D1312yes—
434C42C2×C22⋊D1312yes—
435C13#2589411yescharacteristic, complemented, Sylow
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23#8819614yes—
443C23C22×C22⋊D1311yescharacteristic
444Q8—22no—
445Q8—24no—
446Q8—24no—
447Q8—24no—
448Q8—416no—
449Q8—42no—
450Q8—42no—
451Q8#8819014yes—
452D4—28no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—416no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—522no—
473C2×C4—522no—
474C2×C4—522no—
475C2×C4—522no—
476C2×C4—521no—
477C2×C4—521no—
478C2×C4—521no—
479C2×C4—521no—
480C2×C4#8820712yes—
481C2×C4#8820712yes—
482C2×C4C2×#612912yes—
483C2×C4#8825112yes—
484C2×C4C22×C22⋊D1311yescharacteristic
485C2×C4C22×C22⋊D1311yescharacteristic
486C8—521no—
487C8—521no—
488C8—522no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22#1171951112yes—
497C22C2×#8819612yes—
498C22#1171954211yescharacteristic
499C22#1171954311yescharacteristic
500C22C2×#8820711yescharacteristic
501C4—24no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—44no—
506C4—42no—
507C4—1041no—
508C4—1042no—
509C4—1041no—
510C4C2×#8819011yescharacteristic
511C4#1171951011yescharacteristic
512C4#1171951112yes—
513C2—44no—
514C2#1311848311yescharacteristic
515C2#1312407011yescharacteristic
516C2#1312415111yescharacteristic
517C2#1312688712yes—
518C2#1312700712yes—
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×D4×Q8×C13C2
socleC23×C13C22×D4
prime radicalC22×D4×C13D4
hypercenterC2×D4×Q8D13
hypocenterC13#25894
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11718743C2semidirect
2. C2×#11718683C2semidirect
4. #13116036C2semidirect
5. #13116029C2semidirect
6. #13115760C2semidirect
8. #13114771C2semidirect
9. #13114771C2semidirect
10. #13114666C2semidirect
435. C13#25894semidirect

Maximal subgroups

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

subgroupindex
C2×#117187432
C2×#117186832
#131200022
#131160362
#131160292
#131157602
#131156352
#131147712
#131146662
C2×D4×Q8×C132
#2589413