Finite Group Databasebeta

← #35296177

Subgroups — #35296177

Order 21248.

802 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 151 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#24907455C211yesmaximal, characteristic, complemented
2#24906148C212yesmaximal, complemented
3#24905888C212yesmaximal, complemented
4#24905784C211yesmaximal, characteristic, complemented
5C2×#555×C83C211yesmaximal, characteristic
6#19127754C2212yes—
7#19127725C2212yes—
8C2×#14581709—21no—
9#19127470—21no—
10#19127447—24no—
11D4⋊(C83⋊C8)—24no—
12#19127425—22no—
13#555×C83—24no—
14#555×C83—24no—
15C2×C4⋊Q8×C83C2211yescharacteristic
16C2×#124×C83—22no—
17C2×#121×C83—21no—
18C2×#121×C83C412yes—
19C2×C4×D4×C83C2211yescharacteristic
20C2×C4×D4×C83C2211yescharacteristic
21C2×#117×C83—21no—
22C2×#117×C83C412yes—
23C2×#13050574—22no—
24C4⋊(C83⋊C8)—22no—
25C4⋊(C83⋊C8)—22no—
26C4⋊(C83⋊C8)—22no—
27#14581709—42no—
28C22⋊(C83⋊C8)—22no—
29C4×C83⋊C8—22no—
30C22×Q8×C83—21no—
31C22×D4×C83—22no—
32C22×D4×C83C2×C411yescharacteristic
33C22×D4×C83D411yescharacteristic
34C23×C4×C83—21no—
35C23×C4×C83C2×C412yes—
36C4⋊Q8×C83—22no—
37C4⋊Q8×C83D414yes—
38#124×C83—216no—
39#121×C83—28no—
40#121×C83—24no—
41#121×C83—24no—
42C4×D4×C83—22no—
43C4×D4×C83—22no—
44C4×D4×C83—22no—
45C4×D4×C83D414yes—
46#117×C83—28no—
47#117×C83—24no—
48#117×C83—24no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C4⋊C4×C83C2×C412yes—
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C22⋊C4×C83—22no—
54C2×C22⋊C4×C83—22no—
55C2×C22⋊C4×C83—21no—
56C2×C22⋊C4×C83D412yes—
57C2×C42×C83—22no—
58C2×C42×C83C2311yescharacteristic
59#13050574—42no—
60#13050574—42no—
61C2×C83⋊C8—42no—
62C2×C83⋊C8—42no—
63C24×C83—21no—
64C24×C83C22⋊C412yes—
65C2×Q8×C83—24no—
66C2×Q8×C83—82no—
67C2×D4×C83—28no—
68C2×D4×C83—21no—
69C2×D4×C83—22no—
70C2×D4×C83—22no—
71C2×D4×C83—21no—
72C2×D4×C83—21no—
73C2×D4×C83—42no—
74C2×D4×C83—84no—
75C2×D4×C83C22⋊C412yes—
76C22×C4×C83—21no—
77C22×C4×C83—22no—
78C22×C4×C83—22no—
79C22×C4×C83—22no—
80C22×C4×C83—24no—
81C22×C4×C83—44no—
82C22×C4×C83—42no—
83C22×C4×C83—41no—
84C22×C4×C83—41no—
85C22×C4×C83C22⋊C412yes—
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C4⋊C4×C83—21no—
90C4⋊C4×C83—22no—
91C4⋊C4×C83—24no—
92C4⋊C4×C83—22no—
93C4⋊C4×C83C22⋊C412yes—
94C4⋊C4×C83C22⋊C414yes—
95C22⋊C4×C83—28no—
96C22⋊C4×C83—28no—
97C22⋊C4×C83—24no—
98C22⋊C4×C83—24no—
99C42×C83—28no—
100C42×C83C2×D412yes—
101C42×C83C2×D412yes—
102C23×C83—22no—
103C23×C83—22no—
104C23×C83—22no—
105C23×C83—22no—
106C23×C83—42no—
107C23×C83—42no—
108C23×C83—42no—
109C23×C83—42no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83C2×C22⋊C411yescharacteristic
113Q8×C83—84no—
114D4×C83—24no—
115D4×C83—42no—
116D4×C83—42no—
117D4×C83—42no—
118D4×C83—88no—
119C2×C4×C83—22no—
120C2×C4×C83—22no—
121C2×C4×C83—22no—
122C2×C4×C83—24no—
123C2×C4×C83—41no—
124C2×C4×C83—44no—
125C2×C4×C83—44no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—44no—
129C2×C4×C83—44no—
130C2×C4×C83—44no—
131C2×C4×C83—42no—
132C2×C4×C83C2×C22⋊C411yescharacteristic
133C2×C4×C83C2×C22⋊C411yescharacteristic
134C2×C4×C83#12012yes—
135C2×C4×C83#12012yes—
136C2×C4×C83#13512yes—
137C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C83#13711yescharacteristic
139C83⋊C8—82no—
140C83⋊C8—82no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—42no—
144C22×C83—42no—
145C22×C83—42no—
146C22×C83—42no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—82no—
150C22×C83—82no—
151C22×C83—82no—
152C22×C83—82no—
153C22×C83#37911yescharacteristic
154C22×C83#41611yescharacteristic
155C22×C83#41711yescharacteristic
156C4×C83—22no—
157C4×C83—21no—
158C4×C83—44no—
159C4×C83—42no—
160C4×C83—42no—
161C4×C83#41312yes—
162C4×C83#44912yes—
163C4×C83#45212yes—
164#8397—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83—82no—
168C2×C83—82no—
169C2×C83#178511yescharacteristic
170C2×C83#178811yescharacteristic
171C2×C83#179511yescharacteristic
172C2×#555D8311yescharacteristic
173#1800—831no—
174#1549—832no—
175#1486—832no—
176#1460—831no—
177C83#839711yescharacteristic, complemented, Sylow
178#555—24no—
179#555—24no—
180C2×C4⋊Q8C2×D8311yescharacteristic
181C2×#124—22no—
182C2×#121—21no—
183C2×#121Dic8312yes—
184C2×C4×D4C2×D8311yescharacteristic
185C2×C4×D4C2×D8311yescharacteristic
186C2×#117—21no—
187C2×#117Dic8312yes—
188#431—832no—
189#423—832no—
190C2×#100—1661no—
191#343—1661no—
192#332—1664no—
193D4⋊C8—1664no—
194#323—1662no—
195C22×Q8—21no—
196C22×D4—22no—
197C22×D4C2×Dic8311yescharacteristic
198C22×D4C22⋊D8311yescharacteristic
199C23×C4—21no—
200C23×C4C2×Dic8312yes—
201C2×M16—1662no—
202C4⋊Q8—22no—
203C4⋊Q8C22⋊D8314yes—
204#124—216no—
205#121—28no—
206#121—24no—
207#121—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4C22⋊D8314yes—
212#117—28no—
213#117—24no—
214#117—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×Dic8311yescharacteristic
217C2×C4⋊C4C2×Dic8312yes—
218C2×C4⋊C4C22⋊D8311yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4C22⋊D8312yes—
223C2×C42—22no—
224C2×C42C22×D8311yescharacteristic
225C4⋊C8—1662no—
226C4⋊C8—1662no—
227C4⋊C8—1662no—
228#100—3322no—
229C22⋊C8—1662no—
230C4×C8—1662no—
231C24—21no—
232C24C22⋊Dic8312yes—
233C2×Q8—24no—
234C2×Q8—82no—
235C2×D4—28no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C2×D4—84no—
243C2×D4C22⋊Dic8312yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C22⋊Dic8312yes—
254C22×C4C22×Dic8311yescharacteristic
255C22×C4C2×C22⋊D8311yescharacteristic
256C22×C4C2×C22⋊D8311yescharacteristic
257M16—3322no—
258M16—3322no—
259C2×C8—3322no—
260C2×C8—3322no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4C22⋊Dic8312yes—
266C4⋊C4C22⋊Dic8314yes—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42—28no—
272C42C2×C22⋊D8312yes—
273C42C2×C22⋊D8312yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊Dic8311yescharacteristic
285Q8—84no—
286D4—24no—
287D4—42no—
288D4—42no—
289D4—42no—
290D4—88no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4C2×C22⋊Dic8311yescharacteristic
305C2×C4C2×C22⋊Dic8311yescharacteristic
306C2×C4#1458176112yes—
307C2×C4#1458176112yes—
308C2×C4#1458181212yes—
309C2×C4#1458181711yescharacteristic
310C2×C4#1458182211yescharacteristic
311C8—6642no—
312C8—6642no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22#1912755711yescharacteristic
326C22#1912769711yescharacteristic
327C22#1912770211yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—44no—
333C4#1912768512yes—
334C4#1912784612yes—
335C4#1912786312yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—82no—
340C2#2490739111yescharacteristic
341C2#2490740311yescharacteristic
342C2#2490743311yescharacteristic
subgroupG/H
centerC22#19127557
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
Fitting, prime radicalC2×#555×C83C2
socleC22×C83#379
cosocleC2×C42×C83C23
hypercenterC2×#555D83
hypocenterC83#8397
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #24907455C2semidirect
2. #24906148C2semidirect
3. #24905888C2semidirect
4. #24905784C2semidirect
177. C83#8397semidirect

Maximal subgroups

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

subgroupindex
#249074552
#249061482
#249058882
#249057842
C2×#555×C832
#839783