Finite Group Databasebeta

← #35296012

Subgroups — #35296012

Order 21248.

1074 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907407C212yesmaximal, complemented
2C4×#14581707C211yesmaximal, characteristic, complemented
3#24905983C212yesmaximal, complemented
4#24905801C211yesmaximal, characteristic, complemented
5C2×#547×C83C211yesmaximal, characteristic
6C2×#14581719—24no—
7C2×#14581707—21no—
8#19127665C2212yes—
9C4×C22⋊Dic83C2212yes—
10#19127468—21no—
11#19127438—24no—
12#19127427—21no—
13#19127425—21no—
14C22×#41×C83C411yescharacteristic, complemented
15C22×#41×C83C2211yescharacteristic
16#547×C83—24no—
17#547×C83—24no—
18C2×C4⋊Q8×C83C411yescharacteristic, complemented
19C2×#123×C83—21no—
20C2×#123×C83—21no—
21C2×C22⋊Q8×C83C411yescharacteristic, complemented
22C2×C22⋊Q8×C83C2211yescharacteristic
23C2×#121×C83—21no—
24C2×C4×D4×C83C411yescharacteristic, complemented
25C2×C4×D4×C83C2211yescharacteristic
26C22×C4⋊C4×C83—21no—
27C2×#13050574—22no—
28C2×C22⋊Dic83—22no—
29C2×C4×Dic83—22no—
30#14581719—48no—
31#14581707—24no—
32C22⋊(C83⋊C8)—22no—
33C22⋊(C83⋊C8)—22no—
34C83⋊#95—22no—
35C2×#41×C83—22no—
36C2×#41×C83—22no—
37C2×#41×C83—22no—
38C2×#41×C83—44no—
39C2×#41×C83—44no—
40C2×#41×C83D414yes—
41C22×Q8×C83C2×C411yescharacteristic
42C22×Q8×C83C2×C411yescharacteristic
43C22×D4×C83—21no—
44C22×D4×C83—21no—
45C22×D4×C83C2×C411yescharacteristic
46C22×D4×C83D411yescharacteristic
47C23×C4×C83—21no—
48C23×C4×C83—21no—
49C23×C4×C83—21no—
50C23×C4×C83C2×C411yescharacteristic
51C23×C4×C83C2311yescharacteristic
52C4⋊Q8×C83—22no—
53C4⋊Q8×C83—22no—
54#123×C83—28no—
55#123×C83—28no—
56C22⋊Q8×C83—22no—
57C22⋊Q8×C83—22no—
58C22⋊Q8×C83—22no—
59C22⋊Q8×C83D414yes—
60#121×C83—28no—
61C4×D4×C83—22no—
62C4×D4×C83—22no—
63C4×D4×C83—22no—
64C4×D4×C83—22no—
65C2×C4⋊C4×C83—28no—
66C2×C4⋊C4×C83—21no—
67C2×C4⋊C4×C83—21no—
68C2×C4⋊C4×C83—21no—
69C2×C4⋊C4×C83—21no—
70C2×C4⋊C4×C83C2×C411yescharacteristic
71C2×C4⋊C4×C83D411yescharacteristic
72C2×C22⋊C4×C83—21no—
73C2×C22⋊C4×C83—21no—
74C2×C22⋊C4×C83—21no—
75C2×C22⋊C4×C83D411yescharacteristic
76C2×C22⋊C4×C83D411yescharacteristic
77C2×C42×C83C2×C411yescharacteristic
78C22×Dic83—42no—
79#13050574—44no—
80C2×C83⋊C8—42no—
81C22⋊Dic83—24no—
82C22⋊Dic83—42no—
83C4×Dic83—44no—
84C24×C83—21no—
85C24×C83C22⋊C411yescharacteristic
86C24×C83C2×D411yescharacteristic
87#41×C83—28no—
88#41×C83—44no—
89#41×C83—48no—
90#41×C83—48no—
91#41×C83—44no—
92#41×C83—44no—
93C2×Q8×C83—21no—
94C2×Q8×C83—21no—
95C2×Q8×C83—42no—
96C2×Q8×C83—42no—
97C2×Q8×C83C22⋊C412yes—
98C2×Q8×C83C22⋊C412yes—
99C2×D4×C83—24no—
100C2×D4×C83—24no—
101C2×D4×C83—22no—
102C2×D4×C83—24no—
103C2×D4×C83—22no—
104C2×D4×C83—21no—
105C2×D4×C83—41no—
106C2×D4×C83—44no—
107C2×D4×C83—44no—
108C2×D4×C83C22⋊C412yes—
109C22×C4×C83—24no—
110C22×C4×C83—24no—
111C22×C4×C83—21no—
112C22×C4×C83—21no—
113C22×C4×C83—21no—
114C22×C4×C83—21no—
115C22×C4×C83—21no—
116C22×C4×C83—24no—
117C22×C4×C83—42no—
118C22×C4×C83—42no—
119C22×C4×C83—42no—
120C22×C4×C83—44no—
121C22×C4×C83—44no—
122C22×C4×C83—42no—
123C22×C4×C83C22⋊C411yescharacteristic
124C22×C4×C83C22⋊C411yescharacteristic
125C22×C4×C83C22⋊C411yescharacteristic
126C22×C4×C83C22×C411yescharacteristic
127C22×C4×C83C2×D412yes—
128C22×C4×C83C2×D412yes—
129C22×C4×C83C2×D411yescharacteristic
130C4⋊C4×C83—24no—
131C4⋊C4×C83—24no—
132C4⋊C4×C83—24no—
133C4⋊C4×C83—21no—
134C4⋊C4×C83—22no—
135C4⋊C4×C83—24no—
136C4⋊C4×C83C22⋊C412yes—
137C22⋊C4×C83—24no—
138C22⋊C4×C83—24no—
139C22⋊C4×C83—24no—
140C22⋊C4×C83—22no—
141C22⋊C4×C83—22no—
142C42×C83—21no—
143C42×C83—21no—
144C23×C83—24no—
145C23×C83—22no—
146C23×C83—21no—
147C23×C83—21no—
148C23×C83—21no—
149C23×C83—21no—
150C23×C83—21no—
151C23×C83—41no—
152C23×C83—41no—
153C23×C83—42no—
154C23×C83—41no—
155C23×C83—41no—
156C23×C83—41no—
157C23×C83—41no—
158C23×C83C2×C22⋊C411yescharacteristic
159C23×C83#12012yes—
160Q8×C83—24no—
161Q8×C83—42no—
162Q8×C83—42no—
163Q8×C83—42no—
164D4×C83—24no—
165D4×C83—22no—
166D4×C83—44no—
167D4×C83—48no—
168D4×C83—48no—
169D4×C83#9914yescomplemented
170C2×C4×C83—21no—
171C2×C4×C83—21no—
172C2×C4×C83—22no—
173C2×C4×C83—24no—
174C2×C4×C83—21no—
175C2×C4×C83—22no—
176C2×C4×C83—21no—
177C2×C4×C83—22no—
178C2×C4×C83—22no—
179C2×C4×C83—41no—
180C2×C4×C83—42no—
181C2×C4×C83—48no—
182C2×C4×C83—48no—
183C2×C4×C83—42no—
184C2×C4×C83—42no—
185C2×C4×C83—42no—
186C2×C4×C83—48no—
187C2×C4×C83—41no—
188C2×C4×C83—41no—
189C2×C4×C83—42no—
190C2×C4×C83—41no—
191C2×C4×C83—41no—
192C2×C4×C83#10412yescomplemented
193C2×C4×C83C2×C22⋊C411yescharacteristic
194C2×C4×C83C2×C22⋊C411yescharacteristic
195C2×C4×C83#12012yes—
196C2×Dic83—44no—
197C2×Dic83—82no—
198C2×Dic83—82no—
199C83⋊C8—82no—
200C22×C83—21no—
201C22×C83—21no—
202C22×C83—22no—
203C22×C83—22no—
204C22×C83—24no—
205C22×C83—22no—
206C22×C83—44no—
207C22×C83—44no—
208C22×C83—42no—
209C22×C83—42no—
210C22×C83—41no—
211C22×C83—41no—
212C22×C83—41no—
213C22×C83—81no—
214C22×C83—81no—
215C22×C83—81no—
216C22×C83—81no—
217C22×C83#37911yescharacteristic
218C22×C83C2×#10411yescharacteristic
219C22×C83#42111yescharacteristic
220C4×C83—22no—
221C4×C83—21no—
222C4×C83—22no—
223C4×C83—41no—
224C4×C83—41no—
225C4×C83—42no—
226C4×C83—42no—
227C4×C83—42no—
228C4×C83C2×#9912yes—
229Dic83—84no—
230#8364—831nomaximal, Sylow
231C2×C83—21no—
232C2×C83—21no—
233C2×C83—42no—
234C2×C83—44no—
235C2×C83—81no—
236C2×C83#178111yescharacteristic
237C2×C83#178911yescharacteristic
238C2×C83#179611yescharacteristic
239C2×#547D8311yescharacteristic
240#1789—832no—
241C4×#99—831no—
242#1508—832no—
243#1465—831no—
244C83#836411yescharacteristic, complemented, Sylow
245C22×#41Dic8311yescharacteristic, complemented
246C22×#41C2×D8311yescharacteristic
247#547—24no—
248#547—24no—
249C2×C4⋊Q8Dic8311yescharacteristic, complemented
250C2×#123—21no—
251C2×#123—21no—
252C2×C22⋊Q8Dic8311yescharacteristic, complemented
253C2×C22⋊Q8C2×D8311yescharacteristic
254C2×#121—21no—
255C2×C4×D4Dic8311yescharacteristic, complemented
256C2×C4×D4C2×D8311yescharacteristic
257C22×C4⋊C4—21no—
258C2×#104—1664no—
259C2×#99—1661no—
260#407—832no—
261C4×C22⋊C4—832no—
262#342—1661no—
263#328—1664no—
264#324—1661no—
265#323—1661no—
266C2×#41—22no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—44no—
270C2×#41—44no—
271C2×#41C22⋊D8314yes—
272C22×Q8C2×Dic8311yescharacteristic
273C22×Q8C2×Dic8311yescharacteristic
274C22×D4—21no—
275C22×D4—21no—
276C22×D4C2×Dic8311yescharacteristic
277C22×D4C22⋊D8311yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4C2×Dic8311yescharacteristic
282C23×C4C22×D8311yescharacteristic
283C2×M16—1662no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286#123—28no—
287#123—28no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8C22⋊D8314yes—
292#121—28no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C2×Dic8311yescharacteristic
303C2×C4⋊C4C22⋊D8311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—1662no—
308C2×C22⋊C4C22⋊D8311yescharacteristic
309C2×C22⋊C4C22⋊D8311yescharacteristic
310C2×C42—1662no—
311C2×C42C2×Dic8311yescharacteristic
312#104—3328no—
313#99—1664no—
314C22⋊C8—1662no—
315C22⋊C8—1662no—
316#95—1662no—
317C24—21no—
318C24C22⋊Dic8311yescharacteristic
319C24C2×C22⋊D8311yescharacteristic
320#41—28no—
321#41—44no—
322#41—48no—
323#41—48no—
324#41—44no—
325#41—44no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8C22⋊Dic8312yes—
331C2×Q8C22⋊Dic8312yes—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4C22⋊Dic8312yes—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—3322no—
357C22×C4C22⋊Dic8311yescharacteristic
358C22×C4C22⋊Dic8311yescharacteristic
359C22×C4C22⋊Dic8311yescharacteristic
360C22×C4C22×Dic8311yescharacteristic
361C22×C4C2×C22⋊D8312yes—
362C22×C4C2×C22⋊D8311yescharacteristic
363C22×C4C2×C22⋊D8312yes—
364M16—3324no—
365C2×C8—3322no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4C22⋊Dic8312yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—3322no—
379C22⋊C4—1664no—
380C42—21no—
381C42—21no—
382C42—3324no—
383C23—24no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23C2×C22⋊Dic8311yescharacteristic
398C23#1458176112yes—
399Q8—24no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403D4—24no—
404D4—22no—
405D4—44no—
406D4—48no—
407D4—48no—
408D4#1458170714yescomplemented
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—48no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—3324no—
432C2×C4—6642no—
433C2×C4—6642no—
434C2×C4#1458171912yescomplemented
435C2×C4C2×C22⋊Dic8311yescharacteristic
436C2×C4C2×C22⋊Dic8311yescharacteristic
437C2×C4#1458176112yes—
438C8—6642no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—24no—
443C22—22no—
444C22—22no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#1912755711yescharacteristic
457C22C2×#1458171911yescharacteristic
458C22#1912771811yescharacteristic
459C4—22no—
460C4—21no—
461C4—22no—
462C4—41no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—6644no—
468C4C2×#1458170712yes—
469C2—21no—
470C2—21no—
471C2—42no—
472C2—44no—
473C2—81no—
474C2#2490737811yescharacteristic
475C2#2490740711yescharacteristic
476C2#2490743811yescharacteristic
subgroupG/H
centerC22#19127557
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
Fitting, prime radicalC2×#547×C83C2
socleC22×C83#379
cosocleC23×C4×C83C23
hypercenterC2×#547D83
hypocenterC83#8364
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #24907407C2semidirect
2. C4×#14581707C2semidirect
3. #24905983C2semidirect
4. #24905801C2semidirect
14. C22×#41×C83C4semidirect
18. C2×C4⋊Q8×C83C4semidirect
21. C2×C22⋊Q8×C83C4semidirect
24. C2×C4×D4×C83C4semidirect
169. D4×C83#99semidirect
192. C2×C4×C83#104semidirect
244. C83#8364semidirect
245. C22×#41Dic83semidirect
249. C2×C4⋊Q8Dic83semidirect
252. C2×C22⋊Q8Dic83semidirect
255. C2×C4×D4Dic83semidirect
408. D4#14581707semidirect
434. C2×C4#14581719semidirect

Maximal subgroups

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

subgroupindex
#249074072
C4×#145817072
#249059832
#249058012
C2×#547×C832
#836483