Finite Group Databasebeta

← #35306955

Subgroups — #35306955

Order 21248.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128075C211yesmaximal, characteristic
2C2×#19127901C211yesmaximal, characteristic
3#24908400C211yesmaximal, characteristic, complemented
4C2×#19127447C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905297C211yesmaximal, characteristic, complemented
7C2×#475×C83C211yesmaximal, characteristic, complemented
8C22×#13050577—21no—
9C22×#13050577—22no—
10C2×#14581789C2211yescharacteristic
11#19128075—22no—
12#19128075—22no—
13#19127901—22no—
14#19127901—22no—
15C2×C4⋊(C83⋊C8)C2211yescharacteristic
16C2×C4⋊(C83⋊C8)C2211yescharacteristic
17C2×#14581712—21no—
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127456—22no—
20#19127447—22no—
21#19127444—22no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23C2×C4×Q8×C83C2211yescharacteristic
24#475×C83—22no—
25#475×C83—22no—
26C2×#106×C83—21no—
27C2×C4⋊C8×C83C2211yescharacteristic
28C2×#103×C83—21no—
29C22×D332—21no—
30C22×D332—21no—
31C22×D332—21no—
32C2×#13050577—28no—
33C2×#13050577—24no—
34C2×#13050577—42no—
35C2×#13050577—44no—
36C2×#13050577—44no—
37C2×#13050577—42no—
38C22×C83⋊C8—21no—
39C22×C83⋊C8—21no—
40C22×C83⋊C8D412yes—
41#14581789—21no—
42#14581789—21no—
43#14581789—21no—
44#14581789D412yes—
45C4⋊(C83⋊C8)—21no—
46C4⋊(C83⋊C8)—21no—
47C4⋊(C83⋊C8)—21no—
48C4⋊(C83⋊C8)D412yes—
49#14581712—24no—
50C4×C83⋊C8—21no—
51C4×C83⋊C8—21no—
52C22×Q8×C83—21no—
53C22×Q8×C83D411yescharacteristic
54C22×C8×C83—21no—
55C4⋊Q8×C83—21no—
56C4⋊Q8×C83—21no—
57C4⋊Q8×C83—21no—
58C4⋊Q8×C83D412yes—
59C4×Q8×C83—21no—
60C4×Q8×C83—21no—
61C4×Q8×C83—21no—
62C4×Q8×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C4⋊C4×C83—21no—
65C2×C4⋊C4×C83—21no—
66C2×C4⋊C4×C83D411yescharacteristic
67C2×C42×C83—21no—
68C2×C42×C83C2311yescharacteristic
69#106×C83—24no—
70C4⋊C8×C83—21no—
71C4⋊C8×C83D412yes—
72#103×C83—24no—
73C23×D83—41no—
74C2×D332—22no—
75C2×D332—22no—
76C2×D332—22no—
77C2×D332—42no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—41no—
81C2×D332—42no—
82C2×D332—41no—
83C2×D332—41no—
84C2×D332—41no—
85C2×D332—41no—
86C2×D332—82no—
87#13050577—416no—
88#13050577—48no—
89C2×C83⋊C8—22no—
90C2×C83⋊C8—22no—
91C2×C83⋊C8—22no—
92C2×C83⋊C8—41no—
93C2×C83⋊C8—41no—
94C2×C83⋊C8—41no—
95C2×C83⋊C8—42no—
96C2×C83⋊C8SD1614yescomplemented
97C2×Q8×C83—21no—
98C2×Q8×C83—21no—
99C2×Q8×C83—21no—
100C2×Q8×C83—22no—
101C2×Q8×C83—21no—
102C2×Q8×C83—21no—
103C2×Q8×C83—41no—
104C2×Q8×C83—42no—
105C2×Q8×C83—42no—
106C2×Q8×C83SD1612yescomplemented
107C22×C4×C83—21no—
108C22×C4×C83—21no—
109C22×C4×C83—41no—
110C22×C4×C83C2×D411yescharacteristic
111C22×C4×C83C2×D411yescharacteristic
112C22×C4×C83#4111yescharacteristic
113C2×C8×C83—22no—
114C2×C8×C83—41no—
115C2×C8×C83—41no—
116C4⋊C4×C83—22no—
117C4⋊C4×C83—21no—
118C4⋊C4×C83—21no—
119C4⋊C4×C83—22no—
120C4⋊C4×C83—41no—
121C4⋊C4×C83—41no—
122C4⋊C4×C83—41no—
123C4⋊C4×C83—41no—
124C42×C83—41no—
125C42×C83—41no—
126C42×C83C2×D411yescharacteristic
127C42×C83C2×D411yescharacteristic
128C42×C83C2×D411yescharacteristic
129C42×C83C2×D411yescharacteristic
130C23×C83#12111yescharacteristic
131C22×D83—42no—
132C22×D83—82no—
133C22×D83—81no—
134C22×D83—81no—
135C22×D83—82no—
136Q8×C83—22no—
137Q8×C83—22no—
138Q8×C83—41no—
139Q8×C83—41no—
140Q8×C83—44no—
141Q8×C83—81no—
142Q8×C83—81no—
143C2×C4×C83—22no—
144C2×C4×C83—22no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—42no—
149C2×C4×C83—42no—
150C2×C4×C83—82no—
151C2×C4×C83#12011yescharacteristic
152C2×C4×C83#12011yescharacteristic
153C2×C4×C83#12011yescharacteristic
154C2×C4×C83#12111yescharacteristic
155C2×C4×C83#12111yescharacteristic
156C2×C4×C83#12711yescharacteristic
157C2×C4×C83C2×SD1612yes—
158C2×C4×C83C2×SD1611yescharacteristic
159C2×C4×C83#13511yescharacteristic
160C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C83AGL(1,ℤ/8)12yes—
162C2×C4×C83#13711yescharacteristic
163D332—44no—
164D332—42no—
165D332—42no—
166D332—82no—
167D332—82no—
168D332—81no—
169D332—81no—
170D332—81no—
171D332—81no—
172C8×C83—42no—
173C83⋊C8—24no—
174C83⋊C8—42no—
175C83⋊C8—81no—
176C22×C83#39211yescharacteristic
177C22×C83#46012yes—
178C22×C83#46511yescharacteristic
179C22×C83#46811yescharacteristic
180C22×C83#47112yes—
181C2×D83—82no—
182C2×D83—82no—
183C2×D83—82no—
184C2×D83—162no—
185C2×D83—162no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—42no—
189C4×C83—82no—
190C4×C83—81no—
191C4×C83#44811yescharacteristic
192C4×C83#44911yescharacteristic
193C4×C83#45012yes—
194C4×C83#45011yescharacteristic
195C4×C83#45211yescharacteristic
196C4×C83#49211yescharacteristic
197C4×C83#49611yescharacteristic
198#10268—831nomaximal, Sylow
199C2×C83#166612yes—
200C2×C83#168612yes—
201C2×C83#200011yescharacteristic
202C2×C83#201411yescharacteristic
203C2×C83#201911yescharacteristic
204D83—162no—
205C2×#492—831no—
206C2×#475D8311yescharacteristic, complemented
207C2×#460—831no—
208#1984—831no—
209C2×#332—831no—
210C2×#331—831no—
211#1277—831no—
212C83#1026811yescharacteristic, complemented, Sylow
213C22×SD16—1662no—
214C22×SD16—1661no—
215C2×C4⋊Q8C2×D8311yescharacteristic
216C2×#127—831no—
217C2×C4×Q8C2×D8311yescharacteristic
218#492—1662no—
219#492—1662no—
220#475—22no—
221#475—22no—
222#460—1662no—
223#460—1662no—
224C2×#106—21no—
225C2×C4⋊C8—831no—
226C2×C4⋊C8—831no—
227C2×C4⋊C8C2×D8311yescharacteristic
228C2×#103—21no—
229C2×#102—1661no—
230C2×C4×C8—831no—
231#336—1662no—
232#332—1662no—
233#331—1662no—
234C22×Q8—21no—
235C22×Q8C22⋊D8311yescharacteristic
236C22×D4—1661no—
237C22×D4—1661no—
238C22×D4—1661no—
239C2×SD16—3322no—
240C2×SD16—3322no—
241C2×SD16—3324no—
242C2×SD16—3324no—
243C2×SD16—1664no—
244C2×SD16—1668no—
245C22×C8—21no—
246C22×C8—832no—
247C22×C8—1661no—
248C22×C8—1661no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8C22⋊D8312yes—
253#127—832no—
254#127—1661no—
255#127—1661no—
256#127—1661no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C22⋊D8311yescharacteristic
265C2×C42—21no—
266C2×C42C22×D8311yescharacteristic
267#106—24no—
268C4⋊C8—21no—
269C4⋊C8—832no—
270C4⋊C8—1661no—
271C4⋊C8—1661no—
272C4⋊C8—1661no—
273C4⋊C8C22⋊D8312yes—
274#103—24no—
275#102—1664no—
276C4×C8—1661no—
277C4×C8—1661no—
278C24—3321no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8#1305057712yescomplemented
289C2×D4—3321no—
290C2×D4—3321no—
291C2×D4—3321no—
292C2×D4—3322no—
293C2×D4—3321no—
294C2×D4—3321no—
295C2×D4—3321no—
296C2×D4—3322no—
297C2×D4—3321no—
298C2×D4—6642no—
299C2×D4—1662no—
300C2×D4—1662no—
301C2×D4—1662no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4D4×D8311yescharacteristic
306C22×C4C2×C22⋊D8311yescharacteristic
307C22×C4#1305058911yescharacteristic
308SD16—3328no—
309SD16—33216no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—3321no—
314C2×C8—3321no—
315C2×C8—3321no—
316C2×C8—3322no—
317C2×C8—834no—
318C2×C8—1662no—
319C2×C8—1662no—
320C2×C8—1662no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C42—41no—
330C42—41no—
331C42D4×D8311yescharacteristic
332C42D4×D8311yescharacteristic
333C42C2×C22⋊D8311yescharacteristic
334C42C2×C22⋊D8311yescharacteristic
335C23—3322no—
336C23—6642no—
337C23—6641no—
338C23—6641no—
339C23—6642no—
340C23#1458176411yescharacteristic
341Q8—22no—
342Q8—22no—
343Q8—41no—
344Q8—41no—
345Q8—44no—
346Q8—81no—
347Q8—81no—
348D4—3324no—
349D4—3322no—
350D4—3322no—
351D4—6642no—
352D4—6642no—
353D4—6641no—
354D4—6641no—
355D4—6641no—
356D4—6641no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4#1458176011yescharacteristic
366C2×C4#1458176011yescharacteristic
367C2×C4#1458176111yescharacteristic
368C2×C4#1458176311yescharacteristic
369C2×C4#1458176411yescharacteristic
370C2×C4#1458178811yescharacteristic
371C2×C4SD16×D8312yes—
372C2×C4C2×#1305057711yescharacteristic
373C2×C4#1458181211yescharacteristic
374C2×C4#1458181711yescharacteristic
375C2×C4#1458181912yes—
376C2×C4#1458182311yescharacteristic
377C8—42no—
378C8—3322no—
379C8—6641no—
380C8—1664no—
381C22—13282no—
382C22—13282no—
383C22—6642no—
384C22—6642no—
385C22—6642no—
386C22#1912760911yescharacteristic
387C22#1912789912yes—
388C22#1912793211yescharacteristic
389C22#1912794911yescharacteristic
390C22#1912796612yes—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4#1912784011yescharacteristic
397C4#1912784611yescharacteristic
398C4#1912785012yes—
399C4#1912785111yescharacteristic
400C4#1912786111yescharacteristic
401C4#1912807211yescharacteristic
402C4#1912808711yescharacteristic
403C2—13282no—
404C2#2490681312yes—
405C2#2490693712yes—
406C2#2490846411yescharacteristic
407C2#2490854911yescharacteristic
408C2#2490857411yescharacteristic
subgroupG/H
centerC23#14581764
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#475×C83C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenterC2×#475D83
hypocenterC83#10268
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #24908400C2semidirect
4. C2×#19127447C2semidirect
6. #24905297C2semidirect
7. C2×#475×C83C2semidirect
96. C2×C83⋊C8SD16semidirect
106. C2×Q8×C83SD16semidirect
206. C2×#475D83semidirect
212. C83#10268semidirect
288. C2×Q8#13050577semidirect

Maximal subgroups

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

subgroupindex
C2×#191280752
C2×#191279012
#249084002
C2×#191274472
C2×#191274442
#249052972
C2×#475×C832
#1026883