Finite Group Databasebeta

← #16807947

Subgroups — #16807947

Order 3328.

2340 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131379C214yesmaximal, complemented
2#13131369C211yesmaximal, characteristic, complemented
3#13131361C214yesmaximal, complemented
4D4×C4⋊#261C212yesmaximal, complemented
5#13131346C214yesmaximal, complemented
6#13131326C214yesmaximal, complemented
7#13131324C214yesmaximal, complemented
8#13131317C214yesmaximal, complemented
9C4⋊Q8×#261C211yesmaximal, characteristic, complemented
10#13131293C212yesmaximal, complemented
11D13×#555C211yesmaximal, characteristic
12C2×Q8×#261C2212yes—
13C2×D4×#261C2214yes—
14C2×D4×#261C2214yes—
15C2×#88237C2214yes—
16C22×C4⋊#261C2214yes—
17C13×#555C411yescharacteristic, complemented
18#11719439C411yescharacteristic, complemented
19D13×C4⋊Q8C2211yescharacteristic
20D13×#124C2214yes—
21#11719375C414yescomplemented
22#11719321C414yescomplemented
23D13×#121C2214yes—
24#11719268C412yescomplemented
25C4×D4×D13C2212yes—
26#11719237C414yescomplemented
27D13×#117C2214yes—
28(C2×#261)⋊Q8C2212yes—
29(C2×#261)⋊Q8C2214yes—
30#11719151C2212yes—
31#11719151C2212yes—
32#11719151C2214yes—
33#11719150C2218yes—
34#11719136C2214yes—
35C4×C22⋊#261C2214yes—
36#11719083C2212yes—
37#11719083C2212yes—
38#11719083C2214yes—
39#11719083C2214yes—
40#11719083C2214yes—
41C4⋊C4×#261C2214yes—
42C4⋊C4×#261C2214yes—
43#11719071C2214yes—
44#11719071C2214yes—
45#11719071C2218yes—
46#11719071C2218yes—
47#11719070C2218yes—
48#11719070C2218yes—
49#11719070C2218yes—
50#11719060C2218yes—
51#11719059C2211yescharacteristic
52#11719059C2214yes—
53C4×C4⋊#261C2212yes—
54#11719057C2214yes—
55C42×#261C2211yescharacteristic
56C23×#261C2318yes—
57C2×Q8×D13C2312yes—
58C2×D4×D13C2312yes—
59C2×D4×D13C2314yes—
60C22×C4×D13C2314yes—
61C2×C22⋊#261C2314yes—
62C2×C22⋊#261C2314yes—
63C2×C22⋊#261C2314yes—
64C2×C22⋊#261C2318yes—
65C2×C22⋊#261C2314yes—
66Q8×#261—44no—
67D4×#261—48no—
68D4×#261—48no—
69#88237—48no—
70C2×C4⋊#261—48no—
71C2×C4⋊#261C2312yes—
72C2×C4⋊#261C2312yes—
73C2×C4⋊#261C2314yes—
74C2×C4⋊#261C2314yes—
75C2×C4⋊#261C2318yes—
76C2×C4⋊#261C2312yes—
77C2×C4⋊#261C2314yes—
78C2×C4⋊#261C2314yes—
79C2×C4⋊#261C2314yes—
80C2×C4×#261C2311yescharacteristic
81C2×C4×#261C2312yes—
82C2×C4×#261C2312yes—
83C2×C4×#261C2314yes—
84C2×C4×#261C2314yes—
85C2×C4×#261C2311yescharacteristic
86C2×C4×#261C2314yes—
87C13×C4⋊Q8C2×C411yescharacteristic, complemented
88C13×#124C2×C414yescomplemented
89C13×#121C2×C414yescomplemented
90C4×D4×C13C2×C412yescomplemented
91C13×#117C2×C414yescomplemented
92#88201C2×C414yescomplemented
93#88198C2×C412yescomplemented
94#88193C2×C418yescomplemented
95#88192C2×C414yescomplemented
96#88190C2×C414yescomplemented
97D4×Dic13C2×C412yescomplemented
98#88184C2×C414yescomplemented
99C4×C22⋊D13C2×C414yescomplemented
100#88180C2×C414yescomplemented
101#88149C2×C414yescomplemented
102#88147C2×C412yescomplemented
103#88146C2×C414yescomplemented
104#88146C2×C414yescomplemented
105C4⋊C4×D13C2314yes—
106C4⋊C4×D13C2312yes—
107#88142C2×C414yescomplemented
108#88139C2×C418yescomplemented
109#88139C2×C418yescomplemented
110#88138C2×C418yescomplemented
111#88135C2×C414yescomplemented
112C22⋊C4×D13C2318yes—
113C22⋊C4×D13C2314yes—
114#88131C2×C418yescomplemented
115#88129C2×C414yescomplemented
116C4×D52C2×C412yescomplemented
117#88126C2×C418yescomplemented
118C42×D13C2311yescharacteristic
119C42×D13C2314yes—
120#88123C2×C411yescharacteristic, complemented
121(C2×#261)⋊C4C2314yes—
122(C2×#261)⋊C4C2318yes—
123(C2×#261)⋊C4C2314yes—
124(C2×#261)⋊C4C2314yes—
125(C2×#261)⋊C4C2318yes—
126(C2×#261)⋊C4C2318yes—
127#34485—131nomaximal, Sylow
128C23×D13C2414yes—
129C22×#261—48no—
130C22×#261—48no—
131C22×#261—48no—
132C22×#261C2412yes—
133C22×#261C2414yes—
134C22×#261C2414yes—
135C22×#261C2414yes—
136C22×#261C2411yescharacteristic
137C22×#261C2411yescharacteristic
138C2×Q8×C13C22×C412yes—
139C2×D4×C13C22×C412yes—
140C2×D4×C13C22×C414yes—
141C22×C4×C13C22×C414yes—
142C2×C22⋊D13C22×C414yes—
143C2×C22⋊D13C22×C418yes—
144C22×Dic13C22×C414yes—
145Q8×D13—42no—
146Q8×D13—42no—
147D4×D13—24no—
148D4×D13—24no—
149D4×D13—44no—
150D4×D13—44no—
151C2×D52C22×C412yes—
152C2×D52C22×C414yes—
153C2×C4×D13—28no—
154C2×C4×D13—28no—
155C2×C4×D13C22×C414yes—
156C2×C4×D13C22×C414yes—
157C2×C4×D13C2414yes—
158C2×C4×D13C2411yescharacteristic
159C2×C4×D13C2414yes—
160C2×C4×D13C2412yes—
161C2×Dic26C22×C412yes—
162C2×Dic26C22×C414yes—
163C22⋊#261—28no—
164C22⋊#261—44no—
165C22⋊#261—44no—
166C22⋊#261—48no—
167C22⋊#261—44no—
168C4⋊#261—24no—
169C4⋊#261—28no—
170C4⋊#261—24no—
171C4⋊#261—48no—
172C4⋊#261—44no—
173C4⋊#261—44no—
174C4⋊#261—44no—
175C4⋊#261—44no—
176C4⋊#261—42no—
177C4×#261—28no—
178C4×#261—22no—
179C4×#261—24no—
180C4×#261—28no—
181C4×#261—24no—
182C4×#261—44no—
183C4×#261—41no—
184C13×C4⋊C4C22×C414yes—
185C13×C4⋊C4C22×C412yes—
186C13×C22⋊C4C22×C418yes—
187C13×C22⋊C4C22×C414yes—
188C42×C13C22×C411yescharacteristic
189C42×C13C22×C414yes—
190C22⋊Dic13C22×C418yes—
191C22⋊Dic13C22×C414yes—
192#6103C22×C418yes—
193#6103C22×C418yes—
194#6103C22×C414yes—
195#6103C22×C414yes—
196C4⋊Dic13C22×C414yes—
197C4⋊Dic13C22×C412yes—
198#6101C22×C414yes—
199#6101C22×C418yes—
200C4×Dic13C22×C418yes—
201C4×Dic13C22×C414yes—
202C4×Dic13C22×C411yescharacteristic
203C4×Dic13C22×C412yes—
204C2×#555—131no—
205#2379—134no—
206#2369—131no—
207#2361—134no—
208D4×C4⋊C4—132no—
209#2346—134no—
210#2326—134no—
211#2324—134no—
212#2317—134no—
213C4×C4⋊Q8—131no—
214#2293—132no—
215C23×C13C23×C414yes—
216C22×D13—24no—
217C22×D13—24no—
218C22×D13—24no—
219C22×D13—24no—
220C22×D13—24no—
221C22×D13—24no—
222C22×D13C23×C414yes—
223C22×D13C2511yescharacteristic
224C2×#261—22no—
225C2×#261—21no—
226C2×#261—22no—
227C2×#261—22no—
228C2×#261—24no—
229C2×#261—24no—
230C2×#261—24no—
231C2×#261—24no—
232C2×#261—28no—
233C2×#261—24no—
234C2×#261—22no—
235C2×#261—24no—
236C2×#261—24no—
237C2×#261—22no—
238C2×#261—816no—
239C2×#261H(5,𝔽2)12yes—
240Q8×C13—42no—
241D4×C13—24no—
242D4×C13—44no—
243C2×C4×C13—28no—
244C2×C4×C13C23×C414yes—
245C2×C4×C13C23×C414yes—
246C2×C4×C13C23×C411yescharacteristic
247C2×C4×C13C23×C412yes—
248C22⋊D13—28no—
249C22⋊D13—48no—
250C2×Dic13—28no—
251C2×Dic13C23×C414yes—
252C2×Dic13C23×C414yes—
253C2×Dic13C23×C411yescharacteristic
254C2×Dic13C23×C412yes—
255D52—24no—
256D52—44no—
257C4×D13—24no—
258C4×D13—21no—
259C4×D13—24no—
260C4×D13—24no—
261C4×D13—24no—
262C4×D13—21no—
263C4×D13—28no—
264C4×D13—28no—
265C4×D13H(5,𝔽2)14yes—
266C4×D132−1+414yes—
267Dic26—44no—
268Dic26—42no—
269#555—131no—
270#555—134no—
271#555—134no—
272#555—132no—
273#555—134no—
274#555#26111yescharacteristic, complemented
275C2×C4⋊Q8—131no—
276C2×#124—134no—
277C2×#121—134no—
278C2×C4×Q8—132no—
279C2×C4×D4—132no—
280C2×C4×D4—134no—
281C2×C4×D4—134no—
282C2×#117—134no—
283C2×#117—134no—
284C22×C4⋊C4—134no—
285#391—132no—
286#391—134no—
287#390—134no—
288#390—132no—
289#390—132no—
290#389—134no—
291#389—132no—
292#389—134no—
293#389—134no—
294#389—132no—
295#388—138no—
296#388—134no—
297#388—138no—
298#388—134no—
299#386—138no—
300#385—138no—
301#385—138no—
302#385—138no—
303#384—131no—
304#384—134no—
305#382—138no—
306#380—134no—
307C4×C4⋊C4—134no—
308C4×C4⋊C4—134no—
309C4×C4⋊C4—132no—
310C4×C22⋊C4—134no—
311#376—134no—
312C43—131no—
313C22×C13—24no—
314C22×C13—24no—
315C22×C13—24no—
316C22×C13C24×C411yescharacteristic
317C2×D13—24no—
318C2×D13—24no—
319C2×D13—24no—
320C2×D13—44no—
321C2×D13—44no—
322C2×D13—44no—
323C2×D13—44no—
324C2×D13C2×H(5,𝔽2)11yescharacteristic
325C2×D13C2×H(5,𝔽2)12yes—
326C2×D13C2×2−1+412yes—
327C2×D13#58511yescharacteristic
328#261—22no—
329#261—42no—
330#261—44no—
331#261—41no—
332#261—44no—
333#261—44no—
334C4×C13—24no—
335C4×C13—21no—
336C4×C13—24no—
337C4×C13#52014yes—
338Dic13—24no—
339Dic13—21no—
340Dic13—24no—
341Dic13#52014yes—
342C22×Q8—132no—
343C22×D4—132no—
344C22×D4—134no—
345C23×C4—134no—
346C23×C4—138no—
347C4⋊Q8—132no—
348C4⋊Q8—134no—
349C4⋊Q8—131no—
350C4⋊Q8C2×#26111yescharacteristic, complemented
351#124—134no—
352#124—134no—
353#124—138no—
354#124—138no—
355#124—134no—
356#124C2×#26114yescomplemented
357#121—134no—
358#121—138no—
359#121—134no—
360#121—134no—
361#121—138no—
362#121C2×#26114yescomplemented
363C4×Q8—524no—
364C4×D4—132no—
365C4×D4—134no—
366C4×D4—132no—
367C4×D4—134no—
368C4×D4—132no—
369C4×D4—528no—
370C4×D4—528no—
371C4×D4C2×#26112yescomplemented
372#117—134no—
373#117—134no—
374#117—134no—
375#117—138no—
376#117—138no—
377#117—528no—
378#117C2×#26114yescomplemented
379C2×C4⋊C4—132no—
380C2×C4⋊C4—134no—
381C2×C4⋊C4—132no—
382C2×C4⋊C4—134no—
383C2×C4⋊C4—132no—
384C2×C4⋊C4—138no—
385C2×C4⋊C4—134no—
386C2×C4⋊C4—134no—
387C2×C4⋊C4—134no—
388C2×C4⋊C4—132no—
389C2×C4⋊C4—134no—
390C2×C4⋊C4—528no—
391C2×C22⋊C4—138no—
392C2×C22⋊C4—134no—
393C2×C22⋊C4—134no—
394C2×C22⋊C4—134no—
395C2×C22⋊C4—134no—
396C2×C22⋊C4—138no—
397C2×C22⋊C4—134no—
398C2×C42—134no—
399C2×C42—131no—
400C2×C42—132no—
401C2×C42—134no—
402C2×C42—134no—
403C2×C42—131no—
404C2×C42—132no—
405C2×C42—131no—
406C2×C42—134no—
407#95—134no—
408#95—138no—
409#95—134no—
410#95—138no—
411#95—138no—
412#95—134no—
413C2×C13—44no—
414C2×C13C2×#52012yes—
415C2×C13C4×H(5,𝔽2)11yescharacteristic
416D13—44no—
417D13#354411yescharacteristic
418D13#354712yes—
419D13#354911yescharacteristic
420C24—134no—
421C2×Q8—134no—
422C2×Q8—132no—
423C2×Q8—522no—
424C2×Q8—522no—
425C2×Q8C22×#26112yes—
426C2×D4—134no—
427C2×D4—138no—
428C2×D4—132no—
429C2×D4—134no—
430C2×D4—264no—
431C2×D4—264no—
432C2×D4—524no—
433C2×D4—524no—
434C2×D4C22×#26112yes—
435C2×D4C22×#26114yes—
436C22×C4—132no—
437C22×C4—131no—
438C22×C4—134no—
439C22×C4—131no—
440C22×C4—134no—
441C22×C4—134no—
442C22×C4—134no—
443C22×C4—134no—
444C22×C4—134no—
445C22×C4—132no—
446C22×C4—134no—
447C22×C4—134no—
448C22×C4—131no—
449C22×C4—268no—
450C22×C4—268no—
451C22×C4—528no—
452C22×C4—528no—
453C22×C4—528no—
454C22×C4C22×#26114yes—
455C4⋊C4—134no—
456C4⋊C4—132no—
457C4⋊C4—138no—
458C4⋊C4—134no—
459C4⋊C4—264no—
460C4⋊C4—268no—
461C4⋊C4—264no—
462C4⋊C4—528no—
463C4⋊C4—524no—
464C4⋊C4—524no—
465C4⋊C4—524no—
466C4⋊C4—524no—
467C4⋊C4—522no—
468C4⋊C4C22×#26112yes—
469C4⋊C4C22×#26114yes—
470C22⋊C4—138no—
471C22⋊C4—134no—
472C22⋊C4—138no—
473C22⋊C4—138no—
474C22⋊C4—134no—
475C22⋊C4—134no—
476C22⋊C4—268no—
477C22⋊C4—524no—
478C22⋊C4—524no—
479C22⋊C4—528no—
480C22⋊C4—524no—
481C22⋊C4C22×#26118yes—
482C22⋊C4C22×#26114yes—
483C42—134no—
484C42—138no—
485C42—131no—
486C42—132no—
487C42—264no—
488C42—268no—
489C42—268no—
490C42—262no—
491C42—264no—
492C42—524no—
493C42—521no—
494C42C22×#26114yes—
495C42C22×#26111yescharacteristic
496C13#3448511yescharacteristic, complemented, Sylow
497C23—131no—
498C23—134no—
499C23—264no—
500C23—264no—
501C23—264no—
502C23—264no—
503C23—264no—
504C23—264no—
505C23C23×#26114yes—
506Q8—42no—
507Q8—524no—
508Q8—522no—
509D4—24no—
510D4—44no—
511D4—268no—
512D4—264no—
513D4—524no—
514D4—528no—
515C2×C4—28no—
516C2×C4—132no—
517C2×C4—134no—
518C2×C4—134no—
519C2×C4—132no—
520C2×C4—134no—
521C2×C4—131no—
522C2×C4—134no—
523C2×C4—268no—
524C2×C4—264no—
525C2×C4—261no—
526C2×C4—264no—
527C2×C4—261no—
528C2×C4—264no—
529C2×C4—264no—
530C2×C4—268no—
531C2×C4—262no—
532C2×C4—268no—
533C2×C4—264no—
534C2×C4—262no—
535C2×C4—262no—
536C2×C4—264no—
537C2×C4—261no—
538C2×C4—264no—
539C2×C4—262no—
540C2×C4—264no—
541C2×C4—262no—
542C2×C4—264no—
543C2×C4—268no—
544C2×C4—264no—
545C2×C4—264no—
546C2×C4—10416no—
547C2×C4C23×#26114yes—
548C2×C4C23×#26111yescharacteristic
549C2×C4C23×#26114yes—
550C2×C4C23×#26112yes—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—132no—
555C22—132no—
556C22—131no—
557C22—131no—
558C22—264no—
559C22—264no—
560C22—264no—
561C22—524no—
562C22—524no—
563C22—524no—
564C22—524no—
565C22C24×#26111yescharacteristic
566C4—24no—
567C4—24no—
568C4—21no—
569C4—134no—
570C4—262no—
571C4—264no—
572C4—264no—
573C4—261no—
574C4—521no—
575C4—524no—
576C4—522no—
577C4—524no—
578C4—524no—
579C4#1171963114yes—
580C2—44no—
581C2—131no—
582C2—131no—
583C2—132no—
584C2—524no—
585C2C2×#1171963112yes—
586C2H(5,𝔽2)×#26111yescharacteristic
subgroupG/H
center, FrattiniC22C24×#261
commutator, socleC22×C13C24×C4
FittingC13×#555C4
prime radicalD13×#555C2
cosocleC22×D13C25
hypercenter#555#261
hypocenterC13#34485
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #13131379C2semidirect
2. #13131369C2semidirect
3. #13131361C2semidirect
4. D4×C4⋊#261C2semidirect
5. #13131346C2semidirect
6. #13131326C2semidirect
7. #13131324C2semidirect
8. #13131317C2semidirect
9. C4⋊Q8×#261C2semidirect
10. #13131293C2semidirect
17. C13×#555C4semidirect
18. #11719439C4semidirect
21. #11719375C4semidirect
22. #11719321C4semidirect
24. #11719268C4semidirect
26. #11719237C4semidirect
87. C13×C4⋊Q8C2×C4semidirect
88. C13×#124C2×C4semidirect
89. C13×#121C2×C4semidirect
90. C4×D4×C13C2×C4semidirect
91. C13×#117C2×C4semidirect
92. #88201C2×C4semidirect
93. #88198C2×C4semidirect
94. #88193C2×C4semidirect
95. #88192C2×C4semidirect
96. #88190C2×C4semidirect
97. D4×Dic13C2×C4semidirect
98. #88184C2×C4semidirect
99. C4×C22⋊D13C2×C4semidirect
100. #88180C2×C4semidirect
101. #88149C2×C4semidirect
102. #88147C2×C4semidirect
103. #88146C2×C4semidirect
104. #88146C2×C4semidirect
107. #88142C2×C4semidirect
108. #88139C2×C4semidirect
109. #88139C2×C4semidirect
110. #88138C2×C4semidirect
111. #88135C2×C4semidirect
114. #88131C2×C4semidirect
115. #88129C2×C4semidirect
116. C4×D52C2×C4semidirect
117. #88126C2×C4semidirect
120. #88123C2×C4semidirect
274. #555#261semidirect
350. C4⋊Q8C2×#261semidirect
356. #124C2×#261semidirect
362. #121C2×#261semidirect
371. C4×D4C2×#261semidirect
378. #117C2×#261semidirect
496. C13#34485semidirect

Maximal subgroups

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

subgroupindex
#131313792
#131313692
#131313612
D4×C4⋊#2612
#131313462
#131313262
#131313242
#131313172
C4⋊Q8×#2612
#131312932
D13×#5552
#3448513