Finite Group Databasebeta

← #35296017

Subgroups — #35296017

Order 21248.

782 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907562C212yesmaximal, complemented
2C4×#14581707C211yesmaximal, characteristic
3#24905983C212yesmaximal, complemented
4#24905801C211yesmaximal, characteristic
5C2×#556×C83C211yesmaximal, characteristic
6C2×#14581707—21no—
7#19127665C2212yes—
8C4×C22⋊Dic83C2212yes—
9#19127468—21no—
10#19127460—24no—
11#19127438—24no—
12#19127427—21no—
13#19127425—21no—
14#556×C83—24no—
15#556×C83—24no—
16C2×C4⋊Q8×C83C411yescharacteristic, complemented
17C2×#126×C83—21no—
18C2×#126×C83—21no—
19C2×#124×C83—21no—
20C2×C22⋊Q8×C83C411yescharacteristic, complemented
21C2×C22⋊Q8×C83C2211yescharacteristic
22C2×C4×Q8×C83—21no—
23C2×C4×Q8×C83C411yescharacteristic, complemented
24C2×C4×D4×C83C2211yescharacteristic
25C2×#117×C83C411yescharacteristic, complemented
26C2×#117×C83C2211yescharacteristic
27C2×#13050574—22no—
28C2×C22⋊Dic83—22no—
29C2×C4×Dic83—22no—
30#14581707—24no—
31C22⋊(C83⋊C8)—22no—
32C22⋊(C83⋊C8)—22no—
33C83⋊#95—22no—
34C22×Q8×C83—21no—
35C22×Q8×C83C2×C411yescharacteristic
36C22×D4×C83D411yescharacteristic
37C23×C4×C83C2×C411yescharacteristic
38C23×C4×C83C2311yescharacteristic
39C4⋊Q8×C83—22no—
40C4⋊Q8×C83—22no—
41#126×C83—28no—
42#126×C83—28no—
43#124×C83—28no—
44C22⋊Q8×C83—22no—
45C22⋊Q8×C83—22no—
46C22⋊Q8×C83—22no—
47C22⋊Q8×C83D414yes—
48C4×Q8×C83—28no—
49C4×Q8×C83—22no—
50C4×Q8×C83—22no—
51C4×D4×C83—22no—
52C4×D4×C83—22no—
53#117×C83—22no—
54#117×C83—22no—
55#117×C83—22no—
56#117×C83D414yes—
57C2×C4⋊C4×C83—21no—
58C2×C4⋊C4×C83—21no—
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C4⋊C4×C83C2×C411yescharacteristic
62C2×C4⋊C4×C83C2×C411yescharacteristic
63C2×C4⋊C4×C83C2×C411yescharacteristic
64C2×C4⋊C4×C83D411yescharacteristic
65C2×C22⋊C4×C83—21no—
66C2×C22⋊C4×C83—21no—
67C2×C22⋊C4×C83—21no—
68C2×C22⋊C4×C83—21no—
69C2×C22⋊C4×C83D411yescharacteristic
70C2×C22⋊C4×C83D411yescharacteristic
71C2×C42×C83—21no—
72C2×C42×C83—21no—
73C2×C42×C83—21no—
74C2×C42×C83C2×C411yescharacteristic
75C22×Dic83—42no—
76#13050574—44no—
77C2×C83⋊C8—42no—
78C22⋊Dic83—24no—
79C22⋊Dic83—42no—
80C4×Dic83—44no—
81C24×C83C22⋊C411yescharacteristic
82C24×C83C2×D411yescharacteristic
83C2×Q8×C83—24no—
84C2×Q8×C83—21no—
85C2×Q8×C83—42no—
86C2×Q8×C83—42no—
87C2×Q8×C83—42no—
88C2×Q8×C83C22⋊C412yes—
89C2×D4×C83—22no—
90C2×D4×C83—22no—
91C2×D4×C83—41no—
92C22×C4×C83—21no—
93C22×C4×C83—21no—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83—42no—
99C22×C4×C83—42no—
100C22×C4×C83C22⋊C411yescharacteristic
101C22×C4×C83C22⋊C411yescharacteristic
102C22×C4×C83C22⋊C411yescharacteristic
103C22×C4×C83C22×C411yescharacteristic
104C22×C4×C83C2×D412yes—
105C22×C4×C83C2×D412yes—
106C22×C4×C83C2×D411yescharacteristic
107C4⋊C4×C83—24no—
108C4⋊C4×C83—24no—
109C4⋊C4×C83—24no—
110C4⋊C4×C83—21no—
111C4⋊C4×C83—21no—
112C4⋊C4×C83—21no—
113C4⋊C4×C83—24no—
114C4⋊C4×C83—22no—
115C4⋊C4×C83C22⋊C412yes—
116C4⋊C4×C83C22⋊C412yes—
117C4⋊C4×C83C22⋊C412yes—
118C22⋊C4×C83—24no—
119C22⋊C4×C83—24no—
120C22⋊C4×C83—24no—
121C22⋊C4×C83—24no—
122C22⋊C4×C83—22no—
123C22⋊C4×C83—22no—
124C42×C83—24no—
125C42×C83—24no—
126C42×C83—21no—
127C42×C83—21no—
128C42×C83—24no—
129C23×C83—21no—
130C23×C83—21no—
131C23×C83—21no—
132C23×C83—21no—
133C23×C83—21no—
134C23×C83—41no—
135C23×C83—41no—
136C23×C83—41no—
137C23×C83—41no—
138C23×C83C2×C22⋊C411yescharacteristic
139C23×C83#12012yes—
140Q8×C83—42no—
141Q8×C83—48no—
142Q8×C83—42no—
143D4×C83—44no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—22no—
147C2×C4×C83—24no—
148C2×C4×C83—22no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—22no—
152C2×C4×C83—22no—
153C2×C4×C83—22no—
154C2×C4×C83—41no—
155C2×C4×C83—42no—
156C2×C4×C83—42no—
157C2×C4×C83—42no—
158C2×C4×C83—42no—
159C2×C4×C83—41no—
160C2×C4×C83—41no—
161C2×C4×C83—41no—
162C2×C4×C83—41no—
163C2×C4×C83—42no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83#10412yes—
167C2×C4×C83C2×C22⋊C411yescharacteristic
168C2×C4×C83C2×C22⋊C411yescharacteristic
169C2×C4×C83#12012yes—
170C2×Dic83—44no—
171C2×Dic83—82no—
172C2×Dic83—82no—
173C83⋊C8—82no—
174C22×C83—21no—
175C22×C83—21no—
176C22×C83—22no—
177C22×C83—22no—
178C22×C83—22no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83—41no—
182C22×C83—81no—
183C22×C83—81no—
184C22×C83—81no—
185C22×C83—81no—
186C22×C83#37911yescharacteristic
187C22×C83C2×#10411yescharacteristic
188C22×C83#42111yescharacteristic
189C4×C83—22no—
190C4×C83—21no—
191C4×C83—22no—
192C4×C83—41no—
193C4×C83—42no—
194C4×C83—41no—
195C4×C83—42no—
196C4×C83—42no—
197C4×C83—42no—
198C4×C83#41012yes—
199Dic83—84no—
200#8365—831nomaximal, Sylow
201C2×C83—21no—
202C2×C83—21no—
203C2×C83—42no—
204C2×C83—81no—
205C2×C83#178211yescharacteristic
206C2×C83#178911yescharacteristic
207C2×C83#179611yescharacteristic
208C2×#556D8311yescharacteristic
209#1826—832no—
210C4×#99—831no—
211#1508—832no—
212#1465—831no—
213C83#836511yescharacteristic, complemented, Sylow
214#556—24no—
215#556—24no—
216C2×C4⋊Q8Dic8311yescharacteristic, complemented
217C2×#126—21no—
218C2×#126—21no—
219C2×#124—21no—
220C2×C22⋊Q8Dic8311yescharacteristic, complemented
221C2×C22⋊Q8C2×D8311yescharacteristic
222C2×C4×Q8—21no—
223C2×C4×Q8Dic8311yescharacteristic, complemented
224C2×C4×D4C2×D8311yescharacteristic
225C2×#117Dic8311yescharacteristic, complemented
226C2×#117C2×D8311yescharacteristic
227C2×#99—1661no—
228#407—832no—
229C4×C22⋊C4—832no—
230#342—1661no—
231#339—1664no—
232#328—1664no—
233#324—1661no—
234#323—1661no—
235C22×Q8—21no—
236C22×Q8C2×Dic8311yescharacteristic
237C22×D4C22⋊D8311yescharacteristic
238C23×C4C2×Dic8311yescharacteristic
239C23×C4C22×D8311yescharacteristic
240C2×M16—1662no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243#126—28no—
244#126—28no—
245#124—28no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8C22⋊D8314yes—
250C4×Q8—28no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×D4—22no—
254C4×D4—22no—
255#117—22no—
256#117—22no—
257#117—22no—
258#117C22⋊D8314yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C2×Dic8311yescharacteristic
264C2×C4⋊C4C2×Dic8311yescharacteristic
265C2×C4⋊C4C2×Dic8311yescharacteristic
266C2×C4⋊C4C22⋊D8311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1662no—
272C2×C22⋊C4C22⋊D8311yescharacteristic
273C2×C22⋊C4C22⋊D8311yescharacteristic
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—1662no—
278C2×C42C2×Dic8311yescharacteristic
279#99—1664no—
280C22⋊C8—1662no—
281C22⋊C8—1662no—
282#95—1662no—
283C24C22⋊Dic8311yescharacteristic
284C24C2×C22⋊D8311yescharacteristic
285C2×Q8—24no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C22⋊Dic8312yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—3322no—
303C22×C4C22⋊Dic8311yescharacteristic
304C22×C4C22⋊Dic8311yescharacteristic
305C22×C4C22⋊Dic8311yescharacteristic
306C22×C4C22×Dic8311yescharacteristic
307C22×C4C2×C22⋊D8312yes—
308C22×C4C2×C22⋊D8311yescharacteristic
309C22×C4C2×C22⋊D8312yes—
310M16—3324no—
311C2×C8—3322no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4C22⋊Dic8312yes—
321C4⋊C4C22⋊Dic8312yes—
322C4⋊C4C22⋊Dic8312yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—3322no—
330C22⋊C4—1664no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—21no—
335C42—21no—
336C42—3324no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C2×C22⋊Dic8311yescharacteristic
347C23#1458176112yes—
348Q8—42no—
349Q8—48no—
350Q8—42no—
351D4—44no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—3324no—
375C2×C4—6642no—
376C2×C4—6642no—
377C2×C4#1458171912yes—
378C2×C4C2×C22⋊Dic8311yescharacteristic
379C2×C4C2×C22⋊Dic8311yescharacteristic
380C2×C4#1458176112yes—
381C8—6642no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#1912755711yescharacteristic
395C22C2×#1458171911yescharacteristic
396C22#1912771811yescharacteristic
397C4—22no—
398C4—21no—
399C4—22no—
400C4—41no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—6644no—
407C4#1912767512yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—81no—
412C2#2490738111yescharacteristic
413C2#2490740711yescharacteristic
414C2#2490743811yescharacteristic
subgroupG/H
centerC22#19127557
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC2×#556×C83C2
socleC22×C83#379
prime radicalC22×D4×C83D4
cosocleC23×C4×C83C23
hypercenterC2×#556D83
hypocenterC83#8365
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #24907562C2semidirect
3. #24905983C2semidirect
16. C2×C4⋊Q8×C83C4semidirect
20. C2×C22⋊Q8×C83C4semidirect
23. C2×C4×Q8×C83C4semidirect
25. C2×#117×C83C4semidirect
213. C83#8365semidirect
216. C2×C4⋊Q8Dic83semidirect
220. C2×C22⋊Q8Dic83semidirect
223. C2×C4×Q8Dic83semidirect
225. C2×#117Dic83semidirect

Maximal subgroups

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

subgroupindex
#249075622
C4×#145817072
#249059832
#249058012
C2×#556×C832
#836583