Finite Group Databasebeta

← #35306962

Subgroups — #35306962

Order 21248.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128049C211yesmaximal, characteristic
2C2×#19127901C211yesmaximal, characteristic
3#24908400C211yesmaximal, characteristic, complemented
4C2×#19127450C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905297C211yesmaximal, characteristic, complemented
7C2×#477×C83C211yesmaximal, characteristic, complemented
8C22×#13050577—21no—
9C2×#14581789C2211yescharacteristic
10#19128049—22no—
11#19128049—22no—
12#19127901—22no—
13#19127901—22no—
14C2×C4⋊(C83⋊C8)C2211yescharacteristic
15C2×C4⋊(C83⋊C8)C2211yescharacteristic
16C2×#14581712—21no—
17C2×#14581712—22no—
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127456—22no—
20#19127450—22no—
21#19127444—22no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23C2×C4×Q8×C83C2211yescharacteristic
24#477×C83—22no—
25#477×C83—22no—
26C2×#107×C83—21no—
27C2×C4⋊C8×C83C2211yescharacteristic
28C2×#103×C83—21no—
29C22×D332—21no—
30C22×D332—21no—
31C22×D332—21no—
32C2×#13050577—24no—
33C2×#13050577—42no—
34C2×#13050577—42no—
35C22×C83⋊C8—21no—
36C22×C83⋊C8—21no—
37C22×C83⋊C8D412yes—
38#14581789—21no—
39#14581789—21no—
40#14581789—21no—
41#14581789D412yes—
42C4⋊(C83⋊C8)—21no—
43C4⋊(C83⋊C8)—21no—
44C4⋊(C83⋊C8)—21no—
45C4⋊(C83⋊C8)D412yes—
46#14581712—24no—
47#14581712—28no—
48C4×C83⋊C8—21no—
49C4×C83⋊C8—21no—
50C22×Q8×C83—21no—
51C22×Q8×C83D411yescharacteristic
52C22×C8×C83—21no—
53C4⋊Q8×C83—21no—
54C4⋊Q8×C83—21no—
55C4⋊Q8×C83—21no—
56C4⋊Q8×C83D412yes—
57C4×Q8×C83—21no—
58C4×Q8×C83—21no—
59C4×Q8×C83—21no—
60C4×Q8×C83—21no—
61C2×C4⋊C4×C83—21no—
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C4⋊C4×C83D411yescharacteristic
65C2×C42×C83—21no—
66C2×C42×C83C2311yescharacteristic
67#107×C83—24no—
68C4⋊C8×C83—21no—
69C4⋊C8×C83D412yes—
70#103×C83—24no—
71C23×D83—41no—
72C2×D332—22no—
73C2×D332—22no—
74C2×D332—22no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—42no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—42no—
81C2×D332—41no—
82C2×D332—41no—
83C2×D332—41no—
84C2×D332—82no—
85#13050577—48no—
86C2×C83⋊C8—22no—
87C2×C83⋊C8—22no—
88C2×C83⋊C8—22no—
89C2×C83⋊C8—22no—
90C2×C83⋊C8—41no—
91C2×C83⋊C8—41no—
92C2×C83⋊C8—41no—
93C2×C83⋊C8—42no—
94C2×Q8×C83—21no—
95C2×Q8×C83—21no—
96C2×Q8×C83—21no—
97C2×Q8×C83—22no—
98C2×Q8×C83—21no—
99C2×Q8×C83—21no—
100C2×Q8×C83—41no—
101C2×Q8×C83—82no—
102C2×Q8×C83SD1612yes—
103C22×C4×C83—21no—
104C22×C4×C83—21no—
105C22×C4×C83—41no—
106C22×C4×C83C2×D411yescharacteristic
107C22×C4×C83C2×D411yescharacteristic
108C22×C4×C83#4111yescharacteristic
109C2×C8×C83—22no—
110C2×C8×C83—41no—
111C2×C8×C83—41no—
112C4⋊C4×C83—22no—
113C4⋊C4×C83—21no—
114C4⋊C4×C83—22no—
115C4⋊C4×C83—21no—
116C4⋊C4×C83—41no—
117C4⋊C4×C83—41no—
118C4⋊C4×C83—41no—
119C4⋊C4×C83—41no—
120C42×C83—41no—
121C42×C83—41no—
122C42×C83C2×D411yescharacteristic
123C42×C83C2×D411yescharacteristic
124C42×C83C2×D411yescharacteristic
125C42×C83C2×D411yescharacteristic
126C23×C83#12111yescharacteristic
127C22×D83—42no—
128C22×D83—82no—
129C22×D83—81no—
130C22×D83—81no—
131C22×D83—82no—
132Q8×C83—22no—
133Q8×C83—22no—
134Q8×C83—41no—
135Q8×C83—41no—
136Q8×C83—82no—
137Q8×C83—82no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—42no—
144C2×C4×C83—42no—
145C2×C4×C83—82no—
146C2×C4×C83#12011yescharacteristic
147C2×C4×C83#12011yescharacteristic
148C2×C4×C83#12011yescharacteristic
149C2×C4×C83#12111yescharacteristic
150C2×C4×C83#12111yescharacteristic
151C2×C4×C83#12711yescharacteristic
152C2×C4×C83C2×SD1611yescharacteristic
153C2×C4×C83#13511yescharacteristic
154C2×C4×C83#13512yes—
155C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C83#13711yescharacteristic
157C2×C4×C83#13712yes—
158D332—42no—
159D332—42no—
160D332—44no—
161D332—82no—
162D332—82no—
163D332—81no—
164D332—81no—
165D332—81no—
166D332—81no—
167C8×C83—42no—
168C83⋊C8—42no—
169C83⋊C8—42no—
170C83⋊C8—81no—
171C22×C83#39211yescharacteristic
172C22×C83#46412yes—
173C22×C83#46511yescharacteristic
174C22×C83#46811yescharacteristic
175C22×C83#47212yes—
176C2×D83—82no—
177C2×D83—82no—
178C2×D83—82no—
179C2×D83—162no—
180C2×D83—162no—
181C4×C83—21no—
182C4×C83—21no—
183C4×C83—42no—
184C4×C83—81no—
185C4×C83—82no—
186C4×C83#44811yescharacteristic
187C4×C83#44911yescharacteristic
188C4×C83#45011yescharacteristic
189C4×C83#45211yescharacteristic
190C4×C83#45212yes—
191C4×C83#49511yescharacteristic
192C4×C83#49711yescharacteristic
193#10269—831nomaximal, Sylow
194C2×C83#167812yes—
195C2×C83#169812yes—
196C2×C83#200011yescharacteristic
197C2×C83#201511yescharacteristic
198C2×C83#201811yescharacteristic
199D83—162no—
200C2×#486—831no—
201C2×#477D8311yescharacteristic, complemented
202C2×#460—831no—
203#1984—831no—
204C2×#333—831no—
205C2×#331—831no—
206#1277—831no—
207C83#1026911yescharacteristic, complemented, Sylow
208C22×SD16—1661no—
209C2×C4⋊Q8C2×D8311yescharacteristic
210C2×#127—831no—
211C2×C4×Q8C2×D8311yescharacteristic
212#486—1662no—
213#486—1662no—
214#477—22no—
215#477—22no—
216#460—1662no—
217#460—1662no—
218C2×#107—21no—
219C2×C4⋊C8—831no—
220C2×C4⋊C8—831no—
221C2×C4⋊C8C2×D8311yescharacteristic
222C2×#103—21no—
223C2×#102—1662no—
224C2×#102—1661no—
225C2×C4×C8—831no—
226#336—1662no—
227#333—1662no—
228#331—1662no—
229C22×Q8—21no—
230C22×Q8C22⋊D8311yescharacteristic
231C22×D4—1661no—
232C22×D4—1661no—
233C22×D4—1661no—
234C2×SD16—3322no—
235C2×SD16—3322no—
236C2×SD16—1664no—
237C22×C8—21no—
238C22×C8—832no—
239C22×C8—1661no—
240C22×C8—1661no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D8312yes—
245#127—832no—
246#127—1661no—
247#127—1661no—
248#127—1661no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D8311yescharacteristic
257C2×C42—21no—
258C2×C42C22×D8311yescharacteristic
259#107—24no—
260C4⋊C8—21no—
261C4⋊C8—832no—
262C4⋊C8—1661no—
263C4⋊C8—1661no—
264C4⋊C8—1661no—
265C4⋊C8C22⋊D8312yes—
266#103—24no—
267#102—1664no—
268#102—1668no—
269C4×C8—1661no—
270C4×C8—1661no—
271C24—3321no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8#1305057712yes—
281C2×D4—3321no—
282C2×D4—3321no—
283C2×D4—3321no—
284C2×D4—3321no—
285C2×D4—3321no—
286C2×D4—3322no—
287C2×D4—3321no—
288C2×D4—3321no—
289C2×D4—3322no—
290C2×D4—6642no—
291C2×D4—1662no—
292C2×D4—1662no—
293C2×D4—1662no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4D4×D8311yescharacteristic
298C22×C4C2×C22⋊D8311yescharacteristic
299C22×C4#1305058911yescharacteristic
300SD16—3328no—
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—3321no—
305C2×C8—3321no—
306C2×C8—3321no—
307C2×C8—3322no—
308C2×C8—1662no—
309C2×C8—1662no—
310C2×C8—1662no—
311C2×C8—1662no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C42—41no—
321C42—41no—
322C42D4×D8311yescharacteristic
323C42D4×D8311yescharacteristic
324C42C2×C22⋊D8311yescharacteristic
325C42C2×C22⋊D8311yescharacteristic
326C23—3322no—
327C23—6642no—
328C23—6641no—
329C23—6641no—
330C23—6642no—
331C23#1458176411yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—82no—
337Q8—82no—
338D4—3322no—
339D4—3322no—
340D4—3324no—
341D4—6642no—
342D4—6642no—
343D4—6641no—
344D4—6641no—
345D4—6641no—
346D4—6641no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4#1458176011yescharacteristic
356C2×C4#1458176011yescharacteristic
357C2×C4#1458176111yescharacteristic
358C2×C4#1458176311yescharacteristic
359C2×C4#1458176411yescharacteristic
360C2×C4#1458178811yescharacteristic
361C2×C4C2×#1305057711yescharacteristic
362C2×C4#1458181211yescharacteristic
363C2×C4#1458181512yes—
364C2×C4#1458181711yescharacteristic
365C2×C4#1458182311yescharacteristic
366C2×C4#1458182412yes—
367C8—42no—
368C8—3322no—
369C8—3322no—
370C8—6641no—
371C22—13282no—
372C22—13282no—
373C22—6642no—
374C22—6642no—
375C22—6642no—
376C22#1912760911yescharacteristic
377C22#1912792512yes—
378C22#1912793211yescharacteristic
379C22#1912794911yescharacteristic
380C22#1912796912yes—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4#1912784011yescharacteristic
387C4#1912784611yescharacteristic
388C4#1912785111yescharacteristic
389C4#1912786111yescharacteristic
390C4#1912786212yes—
391C4#1912808611yescharacteristic
392C4#1912809411yescharacteristic
393C2—13282no—
394C2#2490688112yes—
395C2#2490700512yes—
396C2#2490846411yescharacteristic
397C2#2490855411yescharacteristic
398C2#2490856911yescharacteristic
subgroupG/H
centerC23#14581764
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#477×C83C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenterC2×#477D83
hypocenterC83#10269
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #24908400C2semidirect
4. C2×#19127450C2semidirect
6. #24905297C2semidirect
7. C2×#477×C83C2semidirect
201. C2×#477D83semidirect
207. C83#10269semidirect

Maximal subgroups

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

subgroupindex
C2×#191280492
C2×#191279012
#249084002
C2×#191274502
C2×#191274442
#249052972
C2×#477×C832
#1026983