Finite Group Databasebeta

← #16727933

Subgroups — #16727933

Order 3328.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#6127C211yesmaximal, characteristic
2#13123937C213yesmaximal, complemented
3#13123936C213yesmaximal, complemented
4#13123930C213yesmaximal, complemented
5#13123927C213yesmaximal, complemented
6#13123925C211yesmaximal, characteristic, complemented
7#13123924C211yesmaximal, characteristic, complemented
8#13123915C211yesmaximal, characteristic, complemented
9Q8×#6094C211yesmaximal, characteristic, complemented
10#13123778C213yesmaximal, complemented
11#13123765C213yesmaximal, complemented
12C8×Q8×D13C211yesmaximal, characteristic, complemented
13#13123497C213yesmaximal
14#13123475C213yesmaximal
15C13×#2982C211yesmaximal, characteristic, complemented
16C2×C4×Q8×C13C2211yescharacteristic
17Q8×C22⋊D13C411yescharacteristic
18Q8×C22⋊D13C411yescharacteristic
19#11719284C413yes—
20#11719283C2213yes—
21#11719283C2213yes—
22Q8×D52C411yescharacteristic
23C4×Q8×D13C2211yescharacteristic
24C4×Q8×D13C2211yescharacteristic
25#11719272C413yes—
26Q8×Dic26C411yescharacteristic
27#11719216C413yes—
28#11719216C413yes—
29C4×#6127C2213yes—
30#11719175C2213yes—
31C8×Q8×C13C2211yescharacteristic
32C8×Q8×C13C2211yescharacteristic
33C8×Q8×C13C2213yes—
34C8×Q8×C13C2213yes—
35C13×#431C2213yes—
36C13×#431C2213yes—
37#11718685C2213yes—
38#11718685C2213yes—
39Q8×#1055C2211yescharacteristic
40Q8×#1055C2211yescharacteristic
41#11718587C2213yes—
42#11718586C2213yes—
43#11718501C2213yes—
44#11718501C2213yes—
45#11718500C2213yes—
46#11718500C2213yes—
47#11718498C2213yes—
48#11718496C2213yes—
49#11718494C2213yes—
50D13×C4⋊C8C2213yes—
51#11718492C2213yes—
52#11718492C2213yes—
53#11718491C2213yes—
54#11718491C2213yes—
55#11718489C2216yes—
56#11718489C2216yes—
57#11718388C2211yescharacteristic
58#11718388C2213yes—
59#11718381C2213yes—
60#11718381C2216yes—
61#11718381C2213yes—
62#11718351C2216yes—
63#11718350C2216yes—
64C4×#6094C2213yes—
65#11718346C2213yes—
66#11718346C2213yes—
67C4×C8×D13C2213yes—
68#11718340C2213yes—
69#11718340C2216yes—
70#11718340C2213yes—
71C8×Dic26C2213yes—
72C8×Dic26C2213yes—
73C8×Dic26C2216yes—
74C22×Q8×C13C2×C411yescharacteristic
75C2×Q8×D13C2×C411yescharacteristic
76C2×Q8×D13C2×C411yescharacteristic
77C2×#6127C2311yescharacteristic
78C4×Q8×C13C2×C411yescharacteristic
79C4×Q8×C13C2×C413yes—
80C4×Q8×C13C2311yescharacteristic
81C4×Q8×C13C2313yes—
82C2×C13×C4⋊C4C2×C413yes—
83C2×C42×C13C2313yes—
84#88200C2×C413yes—
85#88200C2×C413yes—
86Q8×Dic13C2×C411yescharacteristic
87Q8×Dic13C2×C411yescharacteristic
88#88198C2×C413yes—
89#88198C2×C413yes—
90C4×C22⋊D13C2×C413yes—
91C4×C22⋊D13C2×C413yes—
92C4×C22⋊D13C2×C413yes—
93C4×C22⋊D13C2×C413yes—
94#88180C2313yes—
95#88178C2×C416yes—
96#88178C2×C413yes—
97#88178C2×C413yes—
98C2×#6094C2313yes—
99C2×#6094C2311yescharacteristic
100C2×C8×D13C2313yes—
101C2×C8×D13C2311yescharacteristic
102#88151C2×C413yes—
103#88151C2×C413yes—
104#88150C2×C416yes—
105#88150C2×C416yes—
106C4⋊C4×D13C2×C413yes—
107C4⋊C4×D13C2×C413yes—
108#88143C2×C416yes—
109#88143C2×C416yes—
110#88142C2×C413yes—
111#88142C2×C413yes—
112#88141C2×C413yes—
113#88141C2×C413yes—
114C4×D52C2×C413yes—
115C4×D52C2×C413yes—
116#88126C2313yes—
117#88126C2313yes—
118C42×D13C2313yes—
119C42×D13C2313yes—
120#88124C2×C416yes—
121#88123C2×C413yes—
122#88123C2×C413yes—
123C4×Dic26C2×C413yes—
124C4×Dic26C2×C413yes—
125C4×Dic26C2316yes—
126C4×Dic26C2313yes—
127C4×Dic26C2313yes—
128C13×C4⋊C8C2313yes—
129C13×C4⋊C8C2313yes—
130C13×C4⋊C8C2313yes—
131C13×C4⋊C8C2313yes—
132C13×C22⋊C8C2311yescharacteristic
133C13×C22⋊C8C2313yes—
134C4×C8×C13C2313yes—
135C4×C8×C13C2316yes—
136C4×C8×C13C2313yes—
137C22⋊#1055C2311yescharacteristic
138C22⋊#1055C2313yes—
139#88060C2313yes—
140#88060C2313yes—
141#88060C2311yescharacteristic
142#88060C2311yescharacteristic
143#88055C2313yes—
144#88055C2316yes—
145#88055C2313yes—
146#88054C2313yes—
147#88054C2313yes—
148#88054C2316yes—
149#88054C2316yes—
150#88054C2313yes—
151#88054C2313yes—
152C8×Dic13C2313yes—
153C8×Dic13C2316yes—
154C8×Dic13C2313yes—
155C4⋊#1055C2313yes—
156C4⋊#1055C2313yes—
157C4⋊#1055C2313yes—
158C4⋊#1055C2313yes—
159#88043C2316yes—
160C4×#1055C2313yes—
161C4×#1055C2313yes—
162#60253—131nomaximal, Sylow
163C2×Q8×C13—24no—
164C2×Q8×C13C22×C413yes—
165C2×Q8×C13C22×C411yescharacteristic
166C22×C4×C13C22×C413yes—
167C22×C4×C13C22×C413yes—
168C22×C4×C13C2411yescharacteristic
169C2×C22⋊D13C22×C411yescharacteristic
170C2×C22⋊D13C22×C411yescharacteristic
171Q8×D13—24no—
172Q8×D13—24no—
173#6127—22no—
174#6127—22no—
175C2×D52C22×C411yescharacteristic
176C2×C4×D13C22×C413yes—
177C2×C4×D13C22×C413yes—
178C2×C4×D13C22×C413yes—
179C2×C4×D13C22×C413yes—
180C2×C4×D13C2411yescharacteristic
181C2×C4×D13C2411yescharacteristic
182C2×Dic26C22×C411yescharacteristic
183C2×Dic26C22×C413yes—
184C2×Dic26C22×C413yes—
185C2×Dic26C22×C416yes—
186C2×C8×C13C2411yescharacteristic
187C2×C8×C13C2413yes—
188C2×C8×C13C2413yes—
189C2×C8×C13C2411yescharacteristic
190C13×C4⋊C4C22×C413yes—
191C13×C4⋊C4C22×C413yes—
192C13×C4⋊C4C22×C416yes—
193C42×C13C22×C413yes—
194C42×C13C22×C413yes—
195C42×C13C2413yes—
196C42×C13C2413yes—
197C22⋊Dic13C22×C413yes—
198C22⋊Dic13C22×C413yes—
199#6103C22×C413yes—
200#6103C22×C413yes—
201#6103C22×C413yes—
202#6103C22×C413yes—
203C4⋊Dic13C22×C413yes—
204C4⋊Dic13C22×C413yes—
205C4⋊Dic13C22×C413yes—
206C4⋊Dic13C22×C413yes—
207C4⋊Dic13C22×C416yes—
208#6101C22×C413yes—
209#6101C22×C413yes—
210#6101C22×C413yes—
211#6101C22×C413yes—
212#6101C22×C416yes—
213#6101C22×C416yes—
214#6101C22×C416yes—
215#6101C22×C416yes—
216C4×Dic13C22×C413yes—
217C4×Dic13C22×C413yes—
218C4×Dic13C22×C413yes—
219C4×Dic13C22×C413yes—
220C4×Dic13C2413yes—
221C4×Dic13C2413yes—
222C2×#1055C2411yescharacteristic
223C2×#1055C2413yes—
224C2×#1055C2411yescharacteristic
225C2×#1055C2413yes—
226#6094—22no—
227#6094—26no—
228C8×D13—26no—
229C8×D13—22no—
230Q8×#41—131no—
231#2986—133no—
232#2986—133no—
233#2985—133no—
234#2985—133no—
235#2984—131no—
236#2984—131no—
237#2982—131no—
238#2982D1311yescharacteristic, complemented
239Q8×M16—131no—
240#2962—133no—
241#2960—133no—
242C2×C8×Q8—131no—
243#2925—133no—
244#2922—133no—
245C23×C13C23×C411yescharacteristic
246C22×D13C23×C411yescharacteristic
247C22×D13C23×C411yescharacteristic
248Q8×C13—26no—
249Q8×C13#13114yes—
250C2×C4×C13—26no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—43no—
254C2×C4×C13C23×C413yes—
255C2×C4×C13C23×C413yes—
256C2×C4×C13C23×C413yes—
257C2×C4×C13C23×C411yescharacteristic
258C2×C4×C13C23×C413yes—
259C2×C4×C13C2511yescharacteristic
260C22⋊D13—22no—
261C22⋊D13—22no—
262C2×Dic13C23×C411yescharacteristic
263C2×Dic13C23×C413yes—
264C2×Dic13C23×C413yes—
265C2×Dic13C23×C413yes—
266C2×Dic13C23×C413yes—
267C2×Dic13C23×C411yescharacteristic
268D52—22no—
269C4×D13—26no—
270C4×D13—26no—
271C4×D13—21no—
272C4×D13—21no—
273C4×D13—43no—
274C4×D13—43no—
275C4×D132−1+412yes—
276C4×D132−1+412yes—
277Dic26—26no—
278Dic26—26no—
279Dic26—22no—
280Dic26—46no—
281C8×C13—23no—
282C8×C13—21no—
283C8×C13—21no—
284C8×C13—23no—
285#1055—21no—
286#1055—23no—
287#1055—21no—
288#1055—23no—
289Q82—131no—
290#557—133no—
291#554—133no—
292#554—133no—
293#554—133no—
294D4×Q8—131no—
295D4×Q8—131no—
296D4×Q8—131no—
297#533—133no—
298#533—133no—
299#533—133no—
300C4×#41—133no—
301C2×C4×Q8—131no—
302C2×C4×Q8—131no—
303C2×C4×Q8C2×D1311yescharacteristic
304#446—133no—
305#446—133no—
306#446—136no—
307#446—136no—
308#446—133no—
309#446—133no—
310#446—133no—
311#446—136no—
312#446—133no—
313C8×Q8—131no—
314C8×Q8—131no—
315C8×Q8—133no—
316C8×Q8—133no—
317C8×Q8—133no—
318C8×Q8—133no—
319C8×Q8—136no—
320C8×Q8C2×D1311yescharacteristic
321C8×Q8C2×D1313yes—
322C8×Q8C2×D1313yes—
323C8×Q8C2×D1311yescharacteristic
324#433—133no—
325#433—133no—
326#433—133no—
327#433—133no—
328#433—136no—
329#433—136no—
330#432—133no—
331#432—133no—
332#431—133no—
333#431—133no—
334#431—133no—
335#431—133no—
336#431C2×D1313yes—
337#431C2×D1313yes—
338#424—133no—
339#424—133no—
340#423—133no—
341C2×C4⋊C8—133no—
342#408—131no—
343#408—133no—
344#405—136no—
345C4×M16—133no—
346C2×C4×C8—133no—
347C22×C13—21no—
348C22×C13—21no—
349C22×C13—21no—
350C22×C13C24×C411yescharacteristic
351C2×D13—21no—
352C2×D13—21no—
353C2×D13—21no—
354C2×D13—21no—
355C2×D13—21no—
356C2×D13#51912yes—
357C4×C13—21no—
358C4×C13—23no—
359C4×C13—23no—
360C4×C13—23no—
361C4×C13C2×#13116yes—
362C4×C13C2×2−1+411yescharacteristic
363C4×C13C2×2−1+411yescharacteristic
364Dic13—21no—
365Dic13—23no—
366Dic13—23no—
367Dic13—23no—
368Dic13—23no—
369Dic13#51912yes—
370C2×#41—131no—
371C22×Q8—131no—
372C22×Q8—131no—
373C22×Q8C4×D1311yescharacteristic
374C2×M16—131no—
375C2×M16—133no—
376C22×C8—131no—
377C22×C8—133no—
378C4⋊Q8—133no—
379C4⋊Q8—133no—
380C4⋊Q8—133no—
381C4⋊Q8—133no—
382C4⋊Q8—133no—
383C4⋊Q8—133no—
384#125—136no—
385#125—136no—
386#125—136no—
387C22⋊Q8—133no—
388C22⋊Q8—133no—
389C22⋊Q8—133no—
390C22⋊Q8—133no—
391C22⋊Q8—136no—
392C22⋊Q8—133no—
393C22⋊Q8—133no—
394C22⋊Q8—136no—
395C22⋊Q8—136no—
396C4×Q8—131no—
397C4×Q8—131no—
398C4×Q8—133no—
399C4×Q8—133no—
400C4×Q8—133no—
401C4×Q8—133no—
402C4×Q8—133no—
403C4×Q8—133no—
404C4×Q8—136no—
405C4×Q8C4×D1311yescharacteristic
406C4×Q8C4×D1313yes—
407C4×Q8C22×D1311yescharacteristic
408C4×Q8C22×D1313yes—
409C4×D4—133no—
410C4×D4—133no—
411C4×D4—133no—
412C4×D4—133no—
413C4×D4—133no—
414C4×D4—133no—
415#117—133no—
416#117—133no—
417#117—133no—
418C2×C4⋊C4—133no—
419C2×C4⋊C4—133no—
420C2×C4⋊C4C4×D1313yes—
421C2×C42—133no—
422C2×C42—133no—
423C2×C42C22×D1313yes—
424C4⋊C8—136no—
425C4⋊C8—136no—
426C4⋊C8—133no—
427C4⋊C8—133no—
428C4⋊C8—133no—
429C4⋊C8—133no—
430C4⋊C8—133no—
431C4⋊C8—133no—
432C4⋊C8—133no—
433C4⋊C8—133no—
434C4⋊C8C22×D1313yes—
435C4⋊C8C22×D1313yes—
436C4⋊C8C22×D1313yes—
437C4⋊C8C22×D1313yes—
438C22⋊C8—131no—
439C22⋊C8—133no—
440C22⋊C8—131no—
441C22⋊C8—131no—
442C22⋊C8—133no—
443C22⋊C8—133no—
444C22⋊C8C22×D1311yescharacteristic
445C22⋊C8C22×D1313yes—
446#97—133no—
447#97—133no—
448#97—136no—
449#97—136no—
450C4×C8—133no—
451C4×C8—133no—
452C4×C8—136no—
453C4×C8—133no—
454C4×C8—133no—
455C4×C8C22×D1313yes—
456C4×C8C22×D1313yes—
457C4×C8C22×D1316yes—
458C2×C13—41no—
459C2×C13C2×#51911yescharacteristic
460C2×C13C22×#13111yescharacteristic
461C2×C13#357311yescharacteristic
462D13—22no—
463D13—41no—
464#41—262no—
465#41—262no—
466C2×Q8—24no—
467C2×Q8—131no—
468C2×Q8—133no—
469C2×Q8—133no—
470C2×Q8—136no—
471C2×Q8—264no—
472C2×Q8—264no—
473C2×Q8C2×C4×D1313yes—
474C2×Q8C2×C4×D1311yescharacteristic
475C2×D4—131no—
476C2×D4—131no—
477C2×D4—131no—
478C22×C4—131no—
479C22×C4—133no—
480C22×C4—133no—
481C22×C4—131no—
482C22×C4—133no—
483C22×C4—133no—
484C22×C4C2×C4×D1313yes—
485C22×C4C2×C4×D1313yes—
486C22×C4C23×D1311yescharacteristic
487M16—262no—
488M16—266no—
489C2×C8—131no—
490C2×C8—133no—
491C2×C8—131no—
492C2×C8—133no—
493C2×C8—266no—
494C2×C8—262no—
495C2×C8C23×D1313yes—
496C2×C8C23×D1311yescharacteristic
497C2×C8C23×D1311yescharacteristic
498C2×C8C23×D1313yes—
499C4⋊C4—136no—
500C4⋊C4—136no—
501C4⋊C4—133no—
502C4⋊C4—133no—
503C4⋊C4—133no—
504C4⋊C4—133no—
505C4⋊C4—133no—
506C4⋊C4—133no—
507C4⋊C4—133no—
508C4⋊C4—133no—
509C4⋊C4—136no—
510C4⋊C4—136no—
511C4⋊C4—136no—
512C4⋊C4C2×C4×D1313yes—
513C4⋊C4C2×C4×D1316yes—
514C4⋊C4C2×C4×D1313yes—
515C22⋊C4—133no—
516C22⋊C4—133no—
517C22⋊C4—133no—
518C22⋊C4—133no—
519C22⋊C4—133no—
520C22⋊C4—133no—
521C42—133no—
522C42—133no—
523C42—133no—
524C42—133no—
525C42—133no—
526C42—133no—
527C42C2×C4×D1313yes—
528C42C2×C4×D1313yes—
529C42C23×D1313yes—
530C42C23×D1313yes—
531C13#6025311yescharacteristic, complemented, Sylow
532C23—131no—
533C23—131no—
534C23C22×C4×D1311yescharacteristic
535Q8—26no—
536Q8—262no—
537Q8—266no—
538Q8—266no—
539Q8—526no—
540Q8#8815514yes—
541D4—262no—
542D4—262no—
543D4—262no—
544C2×C4—26no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—43no—
548C2×C4—131no—
549C2×C4—132no—
550C2×C4—132no—
551C2×C4—131no—
552C2×C4—133no—
553C2×C4—133no—
554C2×C4—133no—
555C2×C4—133no—
556C2×C4—266no—
557C2×C4—266no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—523no—
561C2×C4—523no—
562C2×C4C22×C4×D1313yes—
563C2×C4C22×C4×D1311yescharacteristic
564C2×C4C22×C4×D1313yes—
565C2×C4C22×C4×D1313yes—
566C2×C4C22×C4×D1313yes—
567C2×C4C24×D1311yescharacteristic
568C8—23no—
569C8—21no—
570C8—21no—
571C8—23no—
572C8—261no—
573C8—263no—
574C8—261no—
575C8—263no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—132no—
580C22—261no—
581C22—261no—
582C22—261no—
583C22—261no—
584C22—261no—
585C22C23×C4×D1311yescharacteristic
586C4—23no—
587C4—23no—
588C4—23no—
589C4—21no—
590C4—132no—
591C4—263no—
592C4—263no—
593C4—263no—
594C4—263no—
595C4—261no—
596C4C2×#8815516yes—
597C4D13×2−1+411yescharacteristic
598C4D13×2−1+411yescharacteristic
599C2—41no—
600C2—262no—
601C2—521no—
602C2D13×#51911yescharacteristic
603C2C22×#8815511yescharacteristic
604C2#1312954911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D13
commutator, socleC22×C13C24×C4
FittingC13×#2982C2
prime radicalC2×#6127C23
cosocleC2×C4×C13C25
hypercenter#2982D13
hypocenterC13#60253
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #13123937C2semidirect
3. #13123936C2semidirect
4. #13123930C2semidirect
5. #13123927C2semidirect
6. #13123925C2semidirect
7. #13123924C2semidirect
8. #13123915C2semidirect
9. Q8×#6094C2semidirect
10. #13123778C2semidirect
11. #13123765C2semidirect
12. C8×Q8×D13C2semidirect
15. C13×#2982C2semidirect
238. #2982D13semidirect
531. C13#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#61272
#131239372
#131239362
#131239302
#131239272
#131239252
#131239242
#131239152
Q8×#60942
#131237782
#131237652
C8×Q8×D132
#131234972
#131234752
C13×#29822
#6025313