Finite Group Databasebeta

← #35388274

Subgroups — #35388274

Order 21248.

650 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918042C211yesmaximal, characteristic
2#24917585C212yesmaximal, complemented
3#24906879C214yesmaximal, complemented
4#24906867C212yesmaximal, complemented
5#24906823C212yesmaximal, complemented
6#24906226C211yesmaximal, characteristic, complemented
7#24906156C212yesmaximal, complemented
8C2×C8×Q8×C83C211yesmaximal, characteristic, complemented
9#19128251C2212yes—
10#19128106C2211yescharacteristic
11#19128078C2211yescharacteristic
12#19128048C2212yes—
13#19128014—22no—
14#19127996C2214yes—
15#19127937—21no—
16#19127937—21no—
17#19127901C2214yes—
18#19127840—22no—
19#19127799—24no—
20C4×D664—21no—
21C4×D664—21no—
22#19127758C2212yes—
23#19127454C2212yes—
24#19127452C2212yes—
25#19127449C2214yes—
26#19127444C2214yes—
27C2×C4×Q8×C83C2211yescharacteristic
28C8×Q8×C83—21no—
29C8×Q8×C83—21no—
30C8×Q8×C83C2212yes—
31C8×Q8×C83C2212yes—
32C2×C4⋊C8×C83—21no—
33C2×C4⋊C8×C83C2211yescharacteristic
34C2×C4×C8×C83—21no—
35C2×C4×C8×C83C2211yescharacteristic
36C2×#13050590—22no—
37C2×#13050577—24no—
38C2×D664—21no—
39C2×D664—21no—
40#14581792C2312yes—
41#14581789C2312yes—
42#14581780—22no—
43#14581766—22no—
44#14581766—22no—
45C4×D332—22no—
46C4×D332—22no—
47#14581726—21no—
48#14581726—21no—
49#14581725—24no—
50C4⋊(C83⋊C8)C2314yes—
51#14581716—24no—
52#14581714—22no—
53#14581714—22no—
54#14581712—24no—
55C22⋊(C83⋊C8)—22no—
56#14581703—22no—
57C22×Q8×C83D411yescharacteristic
58C22×C8×C83—21no—
59C22×C8×C83D411yescharacteristic
60C22×C8×C83D411yescharacteristic
61C4×Q8×C83—21no—
62C4×Q8×C83—21no—
63C4×Q8×C83D412yes—
64C4×Q8×C83C2312yes—
65C2×C4⋊C4×C83—21no—
66C2×C4⋊C4×C83D411yescharacteristic
67C2×C42×C83—21no—
68C2×C42×C83C2311yescharacteristic
69C4⋊C8×C83—22no—
70C4⋊C8×C83—22no—
71C4⋊C8×C83—21no—
72C4⋊C8×C83C2312yes—
73C4×C8×C83—22no—
74C4×C8×C83—22no—
75C4×C8×C83D411yescharacteristic
76C4×C8×C83D411yescharacteristic
77C4×C8×C83C2311yescharacteristic
78C4×C8×C83C2311yescharacteristic
79#13050590—84no—
80C2×Dic166—22no—
81C2×D332—22no—
82C2×D332—22no—
83C2×D332—22no—
84C2×C22⋊D83—42no—
85C2×C4×D83—42no—
86#13050577—48no—
87D664—42no—
88D664—42no—
89C2×C83⋊C8—24no—
90C4⋊Dic83—22no—
91C4⋊Dic83—22no—
92(C2×D83)⋊C4—42no—
93(C2×D83)⋊C4—42no—
94C2×Q8×C83—24no—
95C2×Q8×C83—21no—
96C2×Q8×C83C2×D412yes—
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83C2×D411yescharacteristic
100C22×C4×C83C2×D411yescharacteristic
101C22×C4×C83#4111yescharacteristic
102C2×C8×C83—22no—
103C2×C8×C83—22no—
104C2×C8×C83—41no—
105C2×C8×C83—42no—
106C2×C8×C83C2×D411yescharacteristic
107C2×C8×C83C2×D411yescharacteristic
108C2×C8×C83C2×D411yescharacteristic
109C2×C8×C83C2×D411yescharacteristic
110C4⋊C4×C83—22no—
111C4⋊C4×C83—21no—
112C4⋊C4×C83—22no—
113C4⋊C4×C83C2×D412yes—
114C42×C83—22no—
115C42×C83—22no—
116C42×C83C2×D411yescharacteristic
117C42×C83C2×D411yescharacteristic
118C42×C83#4111yescharacteristic
119C42×C83C2411yescharacteristic
120C23×C83#12111yescharacteristic
121C22×D83—42no—
122Q8×C83—24no—
123Q8×C83—22no—
124Q8×C83#13514yes—
125C2×C4×C83—24no—
126C2×C4×C83—22no—
127C2×C4×C83—22no—
128C2×C4×C83—22no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—22no—
132C2×C4×C83—42no—
133C2×C4×C83#12111yescharacteristic
134C2×C4×C83#12111yescharacteristic
135C2×C4×C83#12111yescharacteristic
136C2×C4×C83C22×D411yescharacteristic
137C2×C4×C83C22×D411yescharacteristic
138C2×C4×C83C2×#4111yescharacteristic
139C22⋊D83—84no—
140C2×Dic83—42no—
141D332—44no—
142D332—42no—
143D332—42no—
144D332—82no—
145C4×D83—84no—
146Dic166—82no—
147C8×C83—21no—
148C8×C83—21no—
149C8×C83—21no—
150C8×C83—42no—
151C8×C83#13712yes—
152C83⋊C8—44no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83#47212yes—
156C22×C83C2×#12111yescharacteristic
157C2×D83—82no—
158C2×D83—82no—
159C2×D83—82no—
160C4×C83—21no—
161C4×C83—24no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—42no—
166C4×C83C2×#13512yes—
167C4×C83C2×#13711yescharacteristic
168C4×C83#57511yescharacteristic
169Dic83—82no—
170#19264—831nomaximal, Sylow
171C2×C83—22no—
172C2×C83#303711yescharacteristic
173C2×C83#305011yescharacteristic
174C2×C83C2×#47211yescharacteristic
175D83—162no—
176#3169—831no—
177#3112—832no—
178C2×C8×Q8D8311yescharacteristic, complemented
179#1678—834no—
180#1676—832no—
181#1668—832no—
182#1568—831no—
183#1551—832no—
184C83#1926411yescharacteristic, complemented, Sylow
185#532—832no—
186C2×C4×Q8C2×D8311yescharacteristic
187#500—831no—
188#493—831no—
189#486—832no—
190#480—1662no—
191#477—834no—
192#466—1661no—
193#466—1661no—
194#460—834no—
195#448—1662no—
196C8×Q8—21no—
197C8×Q8—21no—
198C8×Q8C2×D8312yes—
199C8×Q8C2×D8312yes—
200#440—1664no—
201C4×D8—1661no—
202C4×D8—1661no—
203#432—832no—
204C2×C4⋊C8—21no—
205C2×C4⋊C8C2×D8311yescharacteristic
206C2×C4×C8—21no—
207C2×C4×C8C2×D8311yescharacteristic
208#335—832no—
209#334—832no—
210#333—834no—
211#331—834no—
212C2×#41—1662no—
213C22×Q8D33211yescharacteristic
214C2×SD16—1664no—
215C2×D8—1661no—
216C2×D8—1661no—
217C22×C8—21no—
218C22×C8C22⋊D8311yescharacteristic
219C22×C8C22⋊D8311yescharacteristic
220C4⋊Q8—832no—
221#127—832no—
222#124—1662no—
223#121—1662no—
224#121—1662no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8D33212yes—
228C4×Q8C22×D8312yes—
229C4×D4—1662no—
230C4×D4—1662no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4D33211yescharacteristic
233C2×C42—21no—
234C2×C42C22×D8311yescharacteristic
235#107—1661no—
236#107—1661no—
237#107—1664no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—834no—
242C4⋊C8C22×D8312yes—
243#103—1664no—
244#102—1662no—
245#102—1662no—
246#102—1664no—
247C22⋊C8—1662no—
248#97—1662no—
249C4×C8—22no—
250C4×C8—22no—
251C4×C8C22⋊D8311yescharacteristic
252C4×C8C22⋊D8311yescharacteristic
253C4×C8C22×D8311yescharacteristic
254C4×C8C22×D8311yescharacteristic
255#41—6644no—
256C2×Q8—24no—
257C2×Q8—21no—
258C2×Q8—1662no—
259C2×Q8C2×D33212yes—
260C2×D4—3322no—
261C2×D4—1662no—
262C2×D4—1662no—
263C2×D4—1662no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—3322no—
267C22×C4C2×C22⋊D8311yescharacteristic
268C22×C4C2×D33211yescharacteristic
269C22×C4#1305059011yescharacteristic
270SD16—3328no—
271D8—3322no—
272D8—3322no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—1664no—
278C2×C8C2×C22⋊D8311yescharacteristic
279C2×C8C2×C22⋊D8311yescharacteristic
280C2×C8C2×C22⋊D8311yescharacteristic
281C2×C8C2×C22⋊D8311yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—1662no—
286C4⋊C4—1662no—
287C4⋊C4C2×D33212yes—
288C22⋊C4—3322no—
289C22⋊C4—3322no—
290C42—22no—
291C42—22no—
292C42C2×C22⋊D8311yescharacteristic
293C42C2×D33211yescharacteristic
294C42#1305059011yescharacteristic
295C42C23×D8311yescharacteristic
296C23—3322no—
297C23#1458176611yescharacteristic
298Q8—24no—
299Q8—22no—
300Q8—6642no—
301Q8#1458181414yes—
302D4—3324no—
303D4—3322no—
304D4—3322no—
305D4—6642no—
306D4—6644no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—3322no—
316C2×C4—6644no—
317C2×C4#1458176611yescharacteristic
318C2×C4#1458176611yescharacteristic
319C2×C4#1458176611yescharacteristic
320C2×C4C22×C22⋊D8311yescharacteristic
321C2×C4C22×D33211yescharacteristic
322C2×C4C2×#1305059011yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—42no—
327C8—3324no—
328C8#1458182312yes—
329C22—21no—
330C22—21no—
331C22—6642no—
332C22—6642no—
333C22—6642no—
334C22#1912797112yes—
335C22C2×#1458176611yescharacteristic
336C4—24no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—6642no—
343C4C2×#1458181412yes—
344C4C2×#1458182311yescharacteristic
345C4#1912849911yescharacteristic
346C2—22no—
347C2—13282no—
348C2#2491693511yescharacteristic
349C2#2491704411yescharacteristic
350C2C2×#1912797111yescharacteristic
subgroupG/H
centerC22C2×#14581766
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC2×C8×Q8×C83C2
socleC22×C83C2×#121
prime radical#24918042C2
hypercenterC2×C8×Q8D83
hypocenterC83#19264
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #24917585C2semidirect
3. #24906879C2semidirect
4. #24906867C2semidirect
5. #24906823C2semidirect
6. #24906226C2semidirect
7. #24906156C2semidirect
8. C2×C8×Q8×C83C2semidirect
178. C2×C8×Q8D83semidirect
184. C83#19264semidirect

Maximal subgroups

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

subgroupindex
#249180422
#249175852
#249068792
#249068672
#249068232
#249062262
#249061562
C2×C8×Q8×C832
#1926483