Finite Group Databasebeta

← #35295752

Subgroups — #35295752

Order 21248.

1238 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907373C212yesmaximal, complemented
2C2×#19127427C212yesmaximal, complemented
3C2×#19127425C212yesmaximal, complemented
4C22×#123×C83C211yesmaximal, characteristic
5C2×C22⋊(C83⋊C8)—24no—
6C2×C22⋊(C83⋊C8)C2214yes—
7#19127427—24no—
8#19127425—24no—
9C23×D4×C83C411yescharacteristic
10C24×C4×C83C2211yescharacteristic
11C2×#123×C83—24no—
12C2×#123×C83—24no—
13C2×#123×C83—24no—
14C22×C4⋊C4×C83C411yescharacteristic
15C22×C4⋊C4×C83C2211yescharacteristic
16C22×C22⋊C4×C83C411yescharacteristic
17C22×C22⋊C4×C83C411yescharacteristic
18C22×C22⋊C4×C83C2211yescharacteristic
19C22×C83⋊C8—24no—
20C22⋊(C83⋊C8)—24no—
21C22⋊(C83⋊C8)—24no—
22C22⋊(C83⋊C8)—48no—
23C25×C83C2×C411yescharacteristic
24C25×C83C2×C411yescharacteristic
25C22×D4×C83—22no—
26C22×D4×C83—22no—
27C22×D4×C83—22no—
28C22×D4×C83—44no—
29C23×C4×C83—22no—
30C23×C4×C83—22no—
31C23×C4×C83—44no—
32C23×C4×C83C2×C411yescharacteristic
33C23×C4×C83C2×C411yescharacteristic
34C23×C4×C83C2×C411yescharacteristic
35C23×C4×C83C2×C411yescharacteristic
36C23×C4×C83D414yes—
37C23×C4×C83C2311yescharacteristic
38#123×C83—232no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83—22no—
41C2×C4⋊C4×C83—22no—
42C2×C4⋊C4×C83—22no—
43C2×C4⋊C4×C83—22no—
44C2×C4⋊C4×C83D414yes—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83—22no—
48C2×C22⋊C4×C83—22no—
49C2×C22⋊C4×C83—22no—
50C2×C22⋊C4×C83—22no—
51C2×C22⋊C4×C83—22no—
52C2×C22⋊C4×C83—22no—
53C2×C22⋊C4×C83D414yes—
54C2×C83⋊C8—48no—
55C2×C83⋊C8—44no—
56C24×C83—24no—
57C24×C83—21no—
58C24×C83—21no—
59C24×C83—21no—
60C24×C83—21no—
61C24×C83—24no—
62C24×C83—24no—
63C24×C83—42no—
64C24×C83—42no—
65C24×C83—42no—
66C24×C83C22⋊C412yes—
67C24×C83C22⋊C412yes—
68C24×C83C22×C411yescharacteristic
69C2×D4×C83—28no—
70C2×D4×C83—48no—
71C2×D4×C83—44no—
72C2×D4×C83—48no—
73C2×D4×C83—44no—
74C22×C4×C83—28no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—24no—
84C22×C4×C83—28no—
85C22×C4×C83—42no—
86C22×C4×C83—44no—
87C22×C4×C83—42no—
88C22×C4×C83—48no—
89C22×C4×C83—48no—
90C22×C4×C83—42no—
91C22×C4×C83—42no—
92C22×C4×C83C22⋊C412yes—
93C22×C4×C83C22⋊C412yes—
94C22×C4×C83C22⋊C412yes—
95C22×C4×C83C22⋊C412yes—
96C22×C4×C83C2×D412yes—
97C22×C4×C83C2×D412yes—
98C22×C4×C83C2×D412yes—
99C4⋊C4×C83—28no—
100C4⋊C4×C83—28no—
101C22⋊C4×C83—28no—
102C22⋊C4×C83—28no—
103C22⋊C4×C83—28no—
104C23×C83—22no—
105C23×C83—22no—
106C23×C83—24no—
107C23×C83—24no—
108C23×C83—22no—
109C23×C83—24no—
110C23×C83—22no—
111C23×C83—24no—
112C23×C83—22no—
113C23×C83—24no—
114C23×C83—416no—
115C23×C83—42no—
116C23×C83—42no—
117C23×C83—42no—
118C23×C83—42no—
119C23×C83—42no—
120C23×C83—42no—
121C23×C83—42no—
122C23×C83—42no—
123C23×C83—42no—
124C23×C83—42no—
125C23×C83—42no—
126C23×C83—42no—
127C23×C83—88no—
128C23×C83#9914yes—
129C23×C83C2×C22⋊C411yescharacteristic
130C23×C83C2×C22⋊C411yescharacteristic
131C23×C83C2×C22⋊C411yescharacteristic
132C23×C83#13114yes—
133D4×C83—416no—
134C2×C4×C83—24no—
135C2×C4×C83—22no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—24no—
140C2×C4×C83—24no—
141C2×C4×C83—42no—
142C2×C4×C83—42no—
143C2×C4×C83—416no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
146C2×C4×C83—42no—
147C2×C4×C83—42no—
148C2×C4×C83—42no—
149C2×C4×C83—42no—
150C2×C4×C83—42no—
151C2×C4×C83—42no—
152C2×C4×C83—42no—
153C2×C4×C83—42no—
154C2×C4×C83#12014yes—
155C83⋊C8—84no—
156C22×C83—28no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—28no—
162C22×C83—24no—
163C22×C83—48no—
164C22×C83—48no—
165C22×C83—44no—
166C22×C83—42no—
167C22×C83—42no—
168C22×C83—42no—
169C22×C83—42no—
170C22×C83—42no—
171C22×C83—84no—
172C22×C83—84no—
173C22×C83—84no—
174C22×C83#37911yescharacteristic
175C22×C83#40812yes—
176C22×C83#40812yes—
177C22×C83#40812yes—
178C22×C83C2×#9912yes—
179C22×C83#41312yes—
180C4×C83—24no—
181C4×C83—42no—
182C4×C83—42no—
183C4×C83—42no—
184C4×C83—42no—
185#8304—831nomaximal, Sylow
186C2×C83—22no—
187C2×C83—22no—
188C2×C83—44no—
189C2×C83—82no—
190C2×C83#146814yes—
191C2×C83#178011yescharacteristic
192C2×C83#178111yescharacteristic
193C2×C83#178511yescharacteristic
194C22×#123D8311yescharacteristic
195#1779—832no—
196C2×#324—832no—
197C2×#323—832no—
198C83#830411yescharacteristic, complemented, Sylow
199C23×D4Dic8311yescharacteristic
200C24×C4C2×D8311yescharacteristic
201C2×#123—24no—
202C2×#123—24no—
203C2×#123—24no—
204C22×C4⋊C4Dic8311yescharacteristic
205C22×C4⋊C4C2×D8311yescharacteristic
206C22×C22⋊C4Dic8311yescharacteristic
207C22×C22⋊C4Dic8311yescharacteristic
208C22×C22⋊C4C2×D8311yescharacteristic
209C2×C22⋊C8—834no—
210C2×C22⋊C8—1664no—
211#324—1664no—
212#323—1664no—
213C25C2×Dic8311yescharacteristic
214C25C2×Dic8311yescharacteristic
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—44no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4C2×Dic8311yescharacteristic
223C23×C4C2×Dic8311yescharacteristic
224C23×C4C2×Dic8311yescharacteristic
225C23×C4C2×Dic8311yescharacteristic
226C23×C4C22⋊D8314yes—
227C23×C4C22×D8311yescharacteristic
228C22×C8—1664no—
229#123—232no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C22⋊D8314yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C22⋊D8314yes—
245C22⋊C8—3328no—
246C22⋊C8—1664no—
247C22⋊C8—1664no—
248C24—24no—
249C24—21no—
250C24—21no—
251C24—24no—
252C24—21no—
253C24—21no—
254C24—24no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24C22⋊Dic8312yes—
259C24C22⋊Dic8312yes—
260C24C22×Dic8311yescharacteristic
261C2×D4—28no—
262C2×D4—48no—
263C2×D4—44no—
264C2×D4—48no—
265C2×D4—44no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—28no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C22⋊Dic8312yes—
285C22×C4C22⋊Dic8312yes—
286C22×C4C22⋊Dic8312yes—
287C22×C4C22⋊Dic8312yes—
288C22×C4C2×C22⋊D8312yes—
289C22×C4C2×C22⋊D8312yes—
290C22×C4C2×C22⋊D8312yes—
291C2×C8—3328no—
292C2×C8—3324no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C22⋊C4—28no—
296C22⋊C4—28no—
297C22⋊C4—28no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—24no—
302C23—22no—
303C23—24no—
304C23—22no—
305C23—24no—
306C23—22no—
307C23—24no—
308C23—416no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—88no—
322C23#1458170714yes—
323C23C2×C22⋊Dic8311yescharacteristic
324C23C2×C22⋊Dic8311yescharacteristic
325C23C2×C22⋊Dic8311yescharacteristic
326C23#1458179914yes—
327D4—416no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—416no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4#1458176114yes—
349C8—6644no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—28no—
356C22—24no—
357C22—48no—
358C22—48no—
359C22—44no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22#1912755711yescharacteristic
369C22#1912766812yes—
370C22#1912766812yes—
371C22#1912766812yes—
372C22C2×#1458170712yes—
373C22#1912768512yes—
374C4—24no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—42no—
379C2—22no—
380C2—22no—
381C2—44no—
382C2—82no—
383C2#2490581114yes—
384C2#2490737511yescharacteristic
385C2#2490737811yescharacteristic
386C2#2490739111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC24×C83C22×C4
FrattiniC23×C4C22×D83
FittingC22×#123×C83C2
socleC23×C83C2×C22⋊C4
prime radicalC23×D4×C83C4
cosocleC23×C4×C83C23
hypercenterC22×#123D83
hypocenterC83#8304
LCSG ▷ C24×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. #24907373C2semidirect
2. C2×#19127427C2semidirect
3. C2×#19127425C2semidirect
198. C83#8304semidirect

Maximal subgroups

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

subgroupindex
#249073732
C2×#191274272
C2×#191274252
C22×#123×C832
#830483