Finite Group Databasebeta

← #35299786

Subgroups — #35299786

Order 21248.

964 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C83⋊C8)C211yesmaximal, characteristic
2#24907373C211yesmaximal, characteristic
3C2×#19127427C212yesmaximal, complemented
4#24905306C212yesmaximal, complemented
5C2×#385×C83C211yesmaximal, characteristic
6C23×C83⋊C8—21no—
7C2×C22⋊(C83⋊C8)—24no—
8C2×C22⋊(C83⋊C8)—21no—
9C2×C22⋊(C83⋊C8)—21no—
10C2×C22⋊(C83⋊C8)—21no—
11C2×C22⋊(C83⋊C8)C2212yes—
12C2×C22⋊(C83⋊C8)C2212yes—
13#19127427—24no—
14C24×C4×C83C2211yescharacteristic
15C22×C4⋊C4×C83C411yescharacteristic
16C22×C4⋊C4×C83C2211yescharacteristic
17C22×C22⋊C4×C83C411yescharacteristic
18C22×C22⋊C4×C83C411yescharacteristic
19#385×C83—21no—
20#385×C83—21no—
21#385×C83—21no—
22#385×C83—21no—
23C2×#95×C83C411yescharacteristic
24C2×#95×C83C2211yescharacteristic
25C22×C83⋊C8—24no—
26C22×C83⋊C8—22no—
27C22×C83⋊C8—22no—
28C22×C83⋊C8—42no—
29C22×C83⋊C8—42no—
30C22⋊(C83⋊C8)—24no—
31C22⋊(C83⋊C8)—22no—
32C22⋊(C83⋊C8)—22no—
33C22⋊(C83⋊C8)—22no—
34C22⋊(C83⋊C8)—42no—
35C22⋊(C83⋊C8)—42no—
36C22⋊(C83⋊C8)D414yes—
37C25×C83C2×C411yescharacteristic
38C23×C4×C83—22no—
39C23×C4×C83—22no—
40C23×C4×C83—42no—
41C23×C4×C83—42no—
42C23×C4×C83C2×C411yescharacteristic
43C23×C4×C83C2×C411yescharacteristic
44C23×C4×C83C2×C411yescharacteristic
45C23×C4×C83C2×C411yescharacteristic
46C23×C4×C83C2×C411yescharacteristic
47C23×C4×C83D411yescharacteristic
48C23×C4×C83D411yescharacteristic
49C23×C4×C83D411yescharacteristic
50C23×C4×C83D411yescharacteristic
51C23×C4×C83C2311yescharacteristic
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—22no—
55C2×C4⋊C4×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83—22no—
58C2×C4⋊C4×C83—42no—
59C2×C22⋊C4×C83—21no—
60C2×C22⋊C4×C83—21no—
61C2×C22⋊C4×C83—22no—
62C2×C22⋊C4×C83—21no—
63C2×C22⋊C4×C83—21no—
64C2×C22⋊C4×C83—42no—
65C2×C22⋊C4×C83C814yescomplemented
66#95×C83—21no—
67#95×C83—21no—
68#95×C83—21no—
69#95×C83—21no—
70C2×C83⋊C8—24no—
71C2×C83⋊C8—48no—
72C2×C83⋊C8—42no—
73C2×C83⋊C8—42no—
74C2×C83⋊C8—42no—
75C2×C83⋊C8—42no—
76C2×C83⋊C8—42no—
77C24×C83—22no—
78C24×C83—22no—
79C24×C83—22no—
80C24×C83—22no—
81C24×C83—41no—
82C24×C83—41no—
83C24×C83C22⋊C411yescharacteristic
84C24×C83C22⋊C411yescharacteristic
85C24×C83C2×C812yes—
86C24×C83M1612yes—
87C24×C83C22×C411yescharacteristic
88C22×C4×C83—22no—
89C22×C4×C83—22no—
90C22×C4×C83—21no—
91C22×C4×C83—21no—
92C22×C4×C83—22no—
93C22×C4×C83—21no—
94C22×C4×C83—21no—
95C22×C4×C83—22no—
96C22×C4×C83—22no—
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83—22no—
100C22×C4×C83—22no—
101C22×C4×C83—22no—
102C22×C4×C83—21no—
103C22×C4×C83—22no—
104C22×C4×C83—22no—
105C22×C4×C83—21no—
106C22×C4×C83—21no—
107C22×C4×C83—21no—
108C22×C4×C83—22no—
109C22×C4×C83—22no—
110C22×C4×C83—41no—
111C22×C4×C83—41no—
112C22×C4×C83—41no—
113C22×C4×C83—41no—
114C22×C4×C83—41no—
115C22×C4×C83—41no—
116C22×C4×C83—41no—
117C22×C4×C83—48no—
118C22×C4×C83—48no—
119C22×C4×C83—41no—
120C22×C4×C83—41no—
121C22×C4×C83—41no—
122C22×C4×C83—41no—
123C22×C4×C83—41no—
124C22×C4×C83—41no—
125C22×C4×C83C22⋊C411yescharacteristic
126C22×C4×C83C22⋊C411yescharacteristic
127C22×C4×C83C2×C812yescomplemented
128C22×C4×C83C2×C812yescomplemented
129C22×C4×C83C2×D412yes—
130C22×C4×C83C2×D411yescharacteristic
131C22×C4×C83C2×D411yescharacteristic
132C22×C4×C83#4112yes—
133C4⋊C4×C83—42no—
134C4⋊C4×C83—42no—
135C4⋊C4×C83—41no—
136C4⋊C4×C83—42no—
137C4⋊C4×C83—41no—
138C22⋊C4×C83—24no—
139C22⋊C4×C83—24no—
140C22⋊C4×C83—42no—
141C22⋊C4×C83—42no—
142C23×C83—21no—
143C23×C83—21no—
144C23×C83—21no—
145C23×C83—21no—
146C23×C83—21no—
147C23×C83—21no—
148C23×C83—21no—
149C23×C83—21no—
150C23×C83—24no—
151C23×C83—24no—
152C23×C83—21no—
153C23×C83—21no—
154C23×C83—21no—
155C23×C83—21no—
156C23×C83—21no—
157C23×C83—21no—
158C23×C83—24no—
159C23×C83—24no—
160C23×C83—42no—
161C23×C83—42no—
162C23×C83—44no—
163C23×C83—44no—
164C23×C83—44no—
165C23×C83—44no—
166C23×C83C22⋊C812yes—
167C23×C83C22⋊C812yes—
168C23×C83#9912yes—
169C23×C83#10112yes—
170C23×C83C2×C22⋊C411yescharacteristic
171C23×C83C4×D412yes—
172C23×C83C22×C811yescharacteristic
173C23×C83C2×M1611yescharacteristic
174C23×C83#13112yes—
175C2×C4×C83—24no—
176C2×C4×C83—24no—
177C2×C4×C83—21no—
178C2×C4×C83—21no—
179C2×C4×C83—21no—
180C2×C4×C83—21no—
181C2×C4×C83—21no—
182C2×C4×C83—21no—
183C2×C4×C83—21no—
184C2×C4×C83—22no—
185C2×C4×C83—22no—
186C2×C4×C83—21no—
187C2×C4×C83—21no—
188C2×C4×C83—24no—
189C2×C4×C83—42no—
190C2×C4×C83—42no—
191C2×C4×C83—44no—
192C2×C4×C83—44no—
193C2×C4×C83—44no—
194C2×C4×C83—42no—
195C2×C4×C83—42no—
196C2×C4×C83—42no—
197C2×C4×C83—41no—
198C2×C4×C83—41no—
199C2×C4×C83—41no—
200C2×C4×C83—41no—
201C2×C4×C83—42no—
202C2×C4×C83—42no—
203C2×C4×C83—41no—
204C2×C4×C83—41no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—82no—
210C2×C4×C83—82no—
211C2×C4×C83#9912yes—
212C2×C4×C83#12011yescharacteristic
213C2×C4×C83#12112yes—
214C2×C4×C83#12311yescharacteristic
215C83⋊C8—44no—
216C83⋊C8—82no—
217C22×C83—22no—
218C22×C83—22no—
219C22×C83—22no—
220C22×C83—22no—
221C22×C83—22no—
222C22×C83—21no—
223C22×C83—21no—
224C22×C83—21no—
225C22×C83—21no—
226C22×C83—22no—
227C22×C83—22no—
228C22×C83—22no—
229C22×C83—22no—
230C22×C83—22no—
231C22×C83—22no—
232C22×C83—22no—
233C22×C83—22no—
234C22×C83—48no—
235C22×C83—48no—
236C22×C83—41no—
237C22×C83—41no—
238C22×C83—41no—
239C22×C83—41no—
240C22×C83#32414yescomplemented
241C22×C83#38611yescharacteristic
242C22×C83C2×C22⋊C811yescharacteristic
243C22×C83#40811yescharacteristic
244C22×C83C2×#9911yescharacteristic
245C22×C83C2×#9911yescharacteristic
246C22×C83C2×#10111yescharacteristic
247C22×C83#41311yescharacteristic
248C22×C83C8×D412yes—
249C22×C83#43512yes—
250C4×C83—21no—
251C4×C83—22no—
252C4×C83—22no—
253C4×C83—21no—
254C4×C83—42no—
255C4×C83—41no—
256C4×C83—42no—
257C4×C83—81no—
258C4×C83—81no—
259#9081—831nomaximal, Sylow
260C2×C83—24no—
261C2×C83—22no—
262C2×C83—22no—
263C2×C83—42no—
264C2×C83C2×#32411yescharacteristic
265C2×C83#145911yescharacteristic
266C2×C83#146811yescharacteristic
267C2×C83#147211yescharacteristic
268C2×C83#186911yescharacteristic
269C2×C83#189811yescharacteristic
270C2×C83#190111yescharacteristic
271C22×C22⋊C8—831no—
272C2×#385D8311yescharacteristic
273#1779—831no—
274C2×#324—832no—
275#1280—832no—
276C83#908111yescharacteristic, complemented, Sylow
277C24×C4C2×D8311yescharacteristic
278C23×C8—1661no—
279C22×C4⋊C4Dic8311yescharacteristic
280C22×C4⋊C4C2×D8311yescharacteristic
281C22×C22⋊C4Dic8311yescharacteristic
282C22×C22⋊C4Dic8311yescharacteristic
283C2×C22⋊C8—832no—
284C2×C22⋊C8—832no—
285C2×C22⋊C8—1661no—
286C2×C22⋊C8—1661no—
287C2×C22⋊C8—1664no—
288C2×C22⋊C8—1661no—
289#385—21no—
290#385—21no—
291#385—21no—
292#385—21no—
293C2×#95Dic8311yescharacteristic
294C2×#95C2×D8311yescharacteristic
295#324—1664no—
296C25C2×Dic8311yescharacteristic
297C23×C4—22no—
298C23×C4—22no—
299C23×C4—42no—
300C23×C4—42no—
301C23×C4C2×Dic8311yescharacteristic
302C23×C4C2×Dic8311yescharacteristic
303C23×C4C2×Dic8311yescharacteristic
304C23×C4C2×Dic8311yescharacteristic
305C23×C4C2×Dic8311yescharacteristic
306C23×C4C22⋊D8311yescharacteristic
307C23×C4C22⋊D8311yescharacteristic
308C23×C4C22⋊D8311yescharacteristic
309C23×C4C22⋊D8311yescharacteristic
310C23×C4C22×D8311yescharacteristic
311C22×C8—3322no—
312C22×C8—3322no—
313C22×C8—1662no—
314C22×C8—1664no—
315C22×C8—1662no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4C83⋊C814yescomplemented
330C22⋊C8—3322no—
331C22⋊C8—3322no—
332C22⋊C8—834no—
333C22⋊C8—1662no—
334C22⋊C8—1662no—
335C22⋊C8—1664no—
336C22⋊C8—1662no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24C22⋊Dic8311yescharacteristic
348C24C22⋊Dic8311yescharacteristic
349C24C2×C83⋊C812yes—
350C24#1305057412yes—
351C24C22×Dic8311yescharacteristic
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4C22⋊Dic8311yescharacteristic
390C22×C4C22⋊Dic8311yescharacteristic
391C22×C4C2×C83⋊C812yescomplemented
392C22×C4C2×C83⋊C812yescomplemented
393C22×C4D4×D8312yes—
394C22×C4C2×C22⋊D8311yescharacteristic
395C22×C4C2×C22⋊D8311yescharacteristic
396C22×C4#1305058912yes—
397C2×C8—3328no—
398C2×C8—3322no—
399C2×C8—3322no—
400C2×C8—3322no—
401C2×C8—3322no—
402C2×C8—3322no—
403C2×C8—1664no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—24no—
427C23—24no—
428C23—21no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23C22⋊(C83⋊C8)12yes—
438C23C22⋊(C83⋊C8)12yes—
439C23#1458170712yes—
440C23#1458171112yes—
441C23C2×C22⋊Dic8311yescharacteristic
442C23D4×Dic8312yes—
443C23C22×C83⋊C811yescharacteristic
444C23C2×#1305057411yescharacteristic
445C23#1458179912yes—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#1458170712yes—
483C2×C4#1458176011yescharacteristic
484C2×C4#1458176412yes—
485C2×C4#1458177311yescharacteristic
486C8—3324no—
487C8—6642no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—48no—
506C22—48no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22#1912742714yescomplemented
512C22#1912758111yescharacteristic
513C22C2×C22⋊(C83⋊C8)11yescharacteristic
514C22#1912766811yescharacteristic
515C22C2×#1458170711yescharacteristic
516C22C2×#1458170711yescharacteristic
517C22C2×#1458171111yescharacteristic
518C22#1912768511yescharacteristic
519C22D4×C83⋊C812yes—
520C22#1912777312yes—
521C4—21no—
522C4—22no—
523C4—21no—
524C4—22no—
525C4—42no—
526C4—41no—
527C4—42no—
528C4—81no—
529C4—81no—
530C2—24no—
531C2—22no—
532C2—22no—
533C2—42no—
534C2C2×#1912742711yescharacteristic
535C2#2490578111yescharacteristic
536C2#2490581111yescharacteristic
537C2#2490582711yescharacteristic
538C2#2490775611yescharacteristic
539C2#2490791711yescharacteristic
540C2#2490793211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC23×C83C22×C8
FrattiniC23×C4C22×D83
FittingC2×#385×C83C2
socleC23×C83C2×C22⋊C4
prime radicalC25×C83C2×C4
cosocleC23×C4×C83C23
hypercenterC2×#385D83
hypocenterC83#9081
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
3. C2×#19127427C2semidirect
4. #24905306C2semidirect
65. C2×C22⋊C4×C83C8semidirect
127. C22×C4×C83C2×C8semidirect
128. C22×C4×C83C2×C8semidirect
240. C22×C83#324semidirect
276. C83#9081semidirect
329. C2×C22⋊C4C83⋊C8semidirect
391. C22×C4C2×C83⋊C8semidirect
392. C22×C4C2×C83⋊C8semidirect
511. C22#19127427semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C83⋊C8)2
#249073732
C2×#191274272
#249053062
C2×#385×C832
#908183