Finite Group Databasebeta

← #15873330

Subgroups — #15873330

Order 3328.

754 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718780C211yesmaximal, characteristic
2#13117443C211yesmaximal, characteristic
3#13117443C211yesmaximal, characteristic
4#13115607C211yesmaximal, characteristic, complemented
5#13115605C211yesmaximal, characteristic, complemented
6#13115603C211yesmaximal, characteristic, complemented
7#13115601C211yesmaximal, characteristic, complemented
8#13115598C211yesmaximal, characteristic, complemented
9#13115596C211yesmaximal, characteristic, complemented
10#13115593C211yesmaximal, characteristic, complemented
11#13115591C211yesmaximal, characteristic, complemented
12#13114869C211yesmaximal, characteristic
13#13114863C211yesmaximal, characteristic
14#13114859C211yesmaximal, characteristic
15C13×#1809C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C23×#1055C2211yescharacteristic
19C2×C13×#107C2211yescharacteristic
20C2×C13×#106C2211yescharacteristic
21C13×#380C2211yescharacteristic
22C13×#380C2211yescharacteristic
23#11718873C2211yescharacteristic
24#11718873C2211yescharacteristic
25#11718780C418yescomplemented
26C2×#88058C2211yescharacteristic
27C2×#88057C2211yescharacteristic
28C2×#88054C2211yescharacteristic
29C2×#88054C2211yescharacteristic
30C2×#88054C2211yescharacteristic
31C2×#88054C2211yescharacteristic
32#11718638C2211yescharacteristic
33#11718638C2211yescharacteristic
34#11718637C2211yescharacteristic
35#11718637C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38C4⋊C4×Dic13C2211yescharacteristic
39C4⋊C4×Dic13C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44#11718606—21no—
45#11718606—21no—
46#11718605—21no—
47#11718605—21no—
48#11718315—21no—
49#11718315—21no—
50#11718314—21no—
51#11718314—21no—
52C13×#340C2211yescharacteristic
53C13×#340C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C23×C4×C13C2311yescharacteristic
61C2×C13×M16—41no—
62C2×C13×M16—41no—
63C2×C13×M16C2×C414yes—
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C2×C13×C4⋊C4C2311yescharacteristic
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C22⋊C4—21no—
71C2×C13×C22⋊C4—21no—
72C2×C22⋊Dic13—21no—
73C2×C22⋊Dic13—21no—
74C2×C22⋊Dic13D411yescharacteristic
75C2×C22⋊Dic13D411yescharacteristic
76#88180C2×C414yes—
77#88180Q814yes—
78C2×C4⋊Dic13D411yescharacteristic
79C2×C4⋊Dic13D411yescharacteristic
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×#6101—21no—
85C2×#6101—21no—
86C2×#6101—21no—
87C2×#6101—21no—
88C2×C4×Dic13—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13—21no—
92C2×C4×Dic13C2311yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C22×#1055—41no—
95C22×#1055—41no—
96C22×#1055C2×C414yes—
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C13×#107—41no—
100C13×#106—41no—
101C13×#95—21no—
102C13×#95—21no—
103#88071—21no—
104#88071—21no—
105#88071—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88058—41no—
114#88057—41no—
115#88054C2×C414yes—
116#88054C2×C414yes—
117#88054C2×C414yes—
118#88054C2×C414yes—
119#88048—41no—
120#88048—41no—
121#88047—41no—
122#88047—41no—
123#21966—131nomaximal, Sylow
124C24×C13C2×D411yescharacteristic
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13C22×C412yes—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×Q812yes—
136C22×C4×C13C2411yescharacteristic
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13C2×D411yescharacteristic
142C22×Dic13C2×D411yescharacteristic
143C22×Dic13C2×D411yescharacteristic
144C22×Dic13C2×D411yescharacteristic
145C13×M16—44no—
146C2×C8×C13—21no—
147C2×C8×C13—21no—
148C2×C8×C13—21no—
149C2×C8×C13—21no—
150C2×C8×C13C22×C412yes—
151C2×C8×C13C22×C412yes—
152C13×C4⋊C4—21no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C22⋊C4—81no—
161C13×C22⋊C4—81no—
162C22⋊Dic13—81no—
163C22⋊Dic13—81no—
164C22⋊Dic13C4⋊C414yes—
165C22⋊Dic13C4⋊C414yes—
166C4⋊Dic13—21no—
167C4⋊Dic13—21no—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13C4⋊C414yes—
175C4⋊Dic13C4⋊C414yes—
176#6101—81no—
177#6101—81no—
178#6101—81no—
179#6101—81no—
180C4×Dic13—81no—
181C4×Dic13—81no—
182C4×Dic13—81no—
183C4×Dic13—81no—
184C4×Dic13C22×C412yes—
185C4×Dic13C22×C412yes—
186C4×Dic13C2×Q812yes—
187C4×Dic13C2×Q812yes—
188C2×#1055—21no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—44no—
193C2×#1055C22×C412yes—
194C2×#1055C22×C412yes—
195C2×#424—131no—
196#2314—131no—
197#2314—131no—
198#1949—131no—
199#1949—131no—
200#1949—131no—
201#1949—131no—
202#1948—131no—
203#1948—131no—
204#1947—131no—
205#1947—131no—
206#1809—131no—
207#1809D1311yescharacteristic
208#1808—131no—
209#1807—131no—
210C23×C13—21no—
211C23×C13—21no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13—21no—
215C23×C13—21no—
216C23×C13C2×C4⋊C412yes—
217C23×C13C22×D411yescharacteristic
218C2×C4×C13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—22no—
221C2×C4×C13—22no—
222C2×C4×C13—22no—
223C2×C4×C13—22no—
224C2×C4×C13—41no—
225C2×C4×C13—41no—
226C2×C4×C13—41no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13C2×C4⋊C412yes—
237C2×C4×C13C23×C411yescharacteristic
238C2×C4×C13C22×Q811yescharacteristic
239C2×C4×C13H(5,𝔽2)11yescharacteristic
240C2×C4×C13H(5,𝔽2)11yescharacteristic
241C2×C4×C132−1+411yescharacteristic
242C2×C4×C132−1+411yescharacteristic
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—41no—
252C2×Dic13—41no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13C2×C4⋊C412yes—
264C2×Dic13C2×C4⋊C412yes—
265C2×Dic13C2×C4⋊C412yes—
266C2×Dic13C2×C4⋊C412yes—
267C8×C13—22no—
268C8×C13—22no—
269#1055—22no—
270#1055—22no—
271C22×M16C2×D1311yescharacteristic
272C23×C8—131no—
273C2×#117—131no—
274C2×#107—131no—
275C2×#107—131no—
276C2×#107—131no—
277C2×#107C2×D1311yescharacteristic
278C2×#106—131no—
279C2×#106—131no—
280C2×#106—131no—
281C2×#106C2×D1311yescharacteristic
282#424—138no—
283C2×C4⋊C8—131no—
284C2×C4⋊C8—131no—
285C2×C4⋊C8—131no—
286C2×C4⋊C8—131no—
287#388—261no—
288#388—261no—
289#388—261no—
290#388—261no—
291#387—261no—
292#387—261no—
293#387—261no—
294#387—261no—
295#384—131no—
296#384—131no—
297#382—131no—
298#382—131no—
299#380—131no—
300#380—131no—
301#380C2×D1311yescharacteristic
302#380C2×D1311yescharacteristic
303C4×C4⋊C4—131no—
304C4×C4⋊C4—131no—
305C4×C4⋊C4—131no—
306C4×C4⋊C4—131no—
307#340—131no—
308#340—131no—
309#340—131no—
310#340—131no—
311#340—131no—
312#340—131no—
313#340C2×D1311yescharacteristic
314#340C2×D1311yescharacteristic
315C22×C13—22no—
316C22×C13—22no—
317C22×C13—22no—
318C22×C13—41no—
319C22×C13—41no—
320C22×C13—41no—
321C22×C13—41no—
322C22×C13C22×C4⋊C411yescharacteristic
323C22×C13#53711yescharacteristic
324C22×C13#53711yescharacteristic
325C22×C13#57611yescharacteristic
326C22×C13#57711yescharacteristic
327C22×C13#57711yescharacteristic
328C22×C13#57811yescharacteristic
329C4×C13—22no—
330C4×C13—81no—
331C4×C13—81no—
332C4×C13—81no—
333C4×C13—81no—
334C4×C13#51811yescharacteristic
335C4×C13#51911yescharacteristic
336C4×C13#54411yescharacteristic
337C4×C13#54411yescharacteristic
338Dic13—22no—
339Dic13—22no—
340Dic13—22no—
341Dic13—22no—
342Dic13—81no—
343Dic13—81no—
344Dic13—81no—
345Dic13—81no—
346C23×C4C22×D1311yescharacteristic
347C2×M16—41no—
348C2×M16—41no—
349C2×M16C4×D1314yes—
350C22×C8—131no—
351C22×C8—131no—
352C22×C8—134no—
353C22×C8—521no—
354C22×C8—521no—
355C22×C8C22×D1311yescharacteristic
356C22×C8C22×D1311yescharacteristic
357#117—134no—
358#117—134no—
359C2×C4⋊C4—131no—
360C2×C4⋊C4—131no—
361C2×C4⋊C4—131no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—131no—
365C2×C4⋊C4—261no—
366C2×C4⋊C4—261no—
367C2×C4⋊C4—261no—
368C2×C4⋊C4—261no—
369C2×C4⋊C4C22×D1311yescharacteristic
370C2×C4⋊C4C22×D1311yescharacteristic
371C2×C4⋊C4C22×D1311yescharacteristic
372C2×C4⋊C4C22×D1311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—131no—
376C2×C22⋊C4—131no—
377C2×C22⋊C4—261no—
378C2×C22⋊C4—261no—
379C2×C42—131no—
380C2×C42—131no—
381C2×C42—261no—
382C2×C42—261no—
383C2×C42—261no—
384C2×C42—261no—
385#107—41no—
386#107—521no—
387#107—521no—
388#107—521no—
389#106—41no—
390#106—521no—
391#106—521no—
392#106—521no—
393C4⋊C8—134no—
394C4⋊C8—134no—
395C4⋊C8—134no—
396C4⋊C8—134no—
397#95—21no—
398#95—21no—
399#95—261no—
400#95—261no—
401#95—261no—
402#95—261no—
403#95—261no—
404#95—261no—
405#95—261no—
406#95—261no—
407#95—261no—
408#95—261no—
409C2×C13—42no—
410C2×C13#231611yescharacteristic
411C2×C13#291211yescharacteristic
412C2×C13#291311yescharacteristic
413C2×C13#325711yescharacteristic
414C2×C13#325711yescharacteristic
415C2×C13#325811yescharacteristic
416C2×C13#325811yescharacteristic
417C24D4×D1311yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—131no—
427C22×C4—131no—
428C22×C4—131no—
429C22×C4—131no—
430C22×C4—261no—
431C22×C4—261no—
432C22×C4—261no—
433C22×C4—261no—
434C22×C4C2×C4×D1312yes—
435C22×C4D4×D1311yescharacteristic
436C22×C4Q8×D1312yes—
437C22×C4C23×D1311yescharacteristic
438M16—44no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—132no—
444C2×C8—132no—
445C2×C8—261no—
446C2×C8—261no—
447C2×C8—261no—
448C2×C8—261no—
449C2×C8—524no—
450C2×C8C2×C4×D1312yes—
451C2×C8C2×C4×D1312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—134no—
461C4⋊C4—134no—
462C4⋊C4—261no—
463C4⋊C4—261no—
464C4⋊C4—261no—
465C4⋊C4—261no—
466C4⋊C4—261no—
467C4⋊C4—261no—
468C4⋊C4—261no—
469C4⋊C4—261no—
470C4⋊C4—1041no—
471C4⋊C4—1041no—
472C4⋊C4—1041no—
473C4⋊C4—1041no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—134no—
477C22⋊C4—134no—
478C22⋊C4—1041no—
479C22⋊C4—1041no—
480C42—132no—
481C42—132no—
482C42—132no—
483C42—132no—
484C42—1041no—
485C42—1041no—
486C42—1041no—
487C42—1041no—
488C13#2196611yescharacteristic, complemented, Sylow
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C4⋊C4×D1312yes—
496C23C2×D4×D1311yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—132no—
516C2×C4—132no—
517C2×C4—132no—
518C2×C4—132no—
519C2×C4—261no—
520C2×C4—261no—
521C2×C4—261no—
522C2×C4—261no—
523C2×C4—261no—
524C2×C4—261no—
525C2×C4—261no—
526C2×C4—261no—
527C2×C4—521no—
528C2×C4—521no—
529C2×C4—521no—
530C2×C4—521no—
531C2×C4—521no—
532C2×C4—521no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—521no—
537C2×C4—521no—
538C2×C4—521no—
539C2×C4C4⋊C4×D1312yes—
540C2×C4C22×C4×D1311yescharacteristic
541C2×C4#8825111yescharacteristic
542C2×C4#8825111yescharacteristic
543C2×C4C2×Q8×D1311yescharacteristic
544C2×C4#8825711yescharacteristic
545C2×C4#8825711yescharacteristic
546C8—22no—
547C8—22no—
548C8—262no—
549C8—262no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C4⋊C4×D1311yescharacteristic
558C22#1171930211yescharacteristic
559C22#1171930211yescharacteristic
560C22#1171947911yescharacteristic
561C22#1171948011yescharacteristic
562C22#1171948311yescharacteristic
563C22#1171948411yescharacteristic
564C4—22no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—262no—
570C4—262no—
571C4—262no—
572C4—262no—
573C4—1041no—
574C4—1041no—
575C4—1041no—
576C4—1041no—
577C4#1171919111yescharacteristic
578C4#1171922111yescharacteristic
579C4#1171932811yescharacteristic
580C4#1171932811yescharacteristic
581C2—42no—
582C2#1311745311yescharacteristic
583C2#1312343011yescharacteristic
584C2#1312343511yescharacteristic
585C2#1312624211yescharacteristic
586C2#1312624811yescharacteristic
587C2#1312625711yescharacteristic
588C2#1312626311yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1809C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1809D13
hypocenterC13#21966
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #13115607C2semidirect
5. #13115605C2semidirect
6. #13115603C2semidirect
7. #13115601C2semidirect
8. #13115598C2semidirect
9. #13115596C2semidirect
10. #13115593C2semidirect
11. #13115591C2semidirect
25. #11718780C4semidirect
488. C13#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#117187802
#131174432
#131156072
#131156052
#131156032
#131156012
#131155982
#131155962
#131155932
#131155912
#131148692
#131148632
#131148592
C13×#18092
#2196613