Finite Group Databasebeta

← #35321778

Subgroups — #35321778

Order 21248.

506 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581731C211yesmaximal, characteristic
2C2×#19127939C211yesmaximal, characteristic
3C2×#19127505C211yesmaximal, characteristic, complemented
4#24907489C211yesmaximal, characteristic
5#24905555C211yesmaximal, characteristic, complemented
6#24905554C211yesmaximal, characteristic, complemented
7C2×#358×C83C211yesmaximal, characteristic, complemented
8C22×C8×D83C2211yescharacteristic
9C2×#14581768—21no—
10C2×#14581731—22no—
11C2×#14581731—22no—
12C2×#14581731—22no—
13C22×C83⋊C16C2211yescharacteristic
14#19127939—22no—
15#19127939C414yescomplemented
16C2×#14581726C2211yescharacteristic
17C2×#14581726C2211yescharacteristic
18C2×#14581716—21no—
19#19127558—21no—
20#19127505—22no—
21#19127463—21no—
22C22×Q16×C83C2211yescharacteristic
23C22×C16×C83C2211yescharacteristic
24C2×#107×C83C2211yescharacteristic
25#358×C83—22no—
26C22×C4×D83D411yescharacteristic
27C22×C83⋊C8D411yescharacteristic
28C2×C8×D83—21no—
29C2×C8×D83—21no—
30C2×C8×D83—21no—
31C2×C8×D83—21no—
32C2×C8×D83C2×C412yes—
33C2×C8×D83D412yes—
34#14581768—24no—
35#14581768—42no—
36C2×C4⋊Dic83—21no—
37C2×C4⋊Dic83—21no—
38C2×Dic83⋊C4—41no—
39C2×(C2×D83)⋊C4—41no—
40C2×(C2×D83)⋊C4—41no—
41#14581731—28no—
42C2×C83⋊C16—21no—
43C2×C83⋊C16—21no—
44C2×C83⋊C16D412yes—
45#14581726—21no—
46#14581726—21no—
47#14581726—21no—
48#14581726C2×C412yes—
49#14581716—24no—
50C83⋊#95—41no—
51C22×Q8×C83—21no—
52C2×Q16×C83—21no—
53C2×Q16×C83—21no—
54C2×Q16×C83—21no—
55C2×Q16×C83—21no—
56C2×Q16×C83—21no—
57C2×Q16×C83C2×C412yes—
58C22×C8×C83C2311yescharacteristic
59C2×C4⋊C4×C83—21no—
60C2×C16×C83—21no—
61C2×C16×C83—21no—
62C2×C16×C83D412yes—
63#107×C83—21no—
64#107×C83—21no—
65C23×D83D811yescharacteristic
66C22×Dic83—41no—
67C22×Dic83—41no—
68C22×Dic83D811yescharacteristic
69C2×C4×D83—21no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73C2×C4×D83C22⋊C412yes—
74C2×C4×D83SD1612yes—
75C2×C83⋊C8—21no—
76C2×C83⋊C8—21no—
77C2×C83⋊C8C22⋊C412yes—
78C8×D83—22no—
79C8×D83—22no—
80C8×D83—22no—
81C8×D83—22no—
82C4⋊Dic83—22no—
83C4⋊Dic83—41no—
84C4⋊Dic83—41no—
85C4⋊Dic83—41no—
86Dic83⋊C4—44no—
87(C2×D83)⋊C4—44no—
88(C2×D83)⋊C4—82no—
89C83⋊C16—22no—
90C2×Q8×C83—22no—
91C2×Q8×C83—41no—
92C2×Q8×C83—41no—
93C2×Q8×C83—41no—
94C2×Q8×C83—41no—
95C2×Q8×C83—41no—
96C22×C4×C83—41no—
97C22×C4×C83—41no—
98C22×C4×C83C2×D411yescharacteristic
99Q16×C83—22no—
100Q16×C83—22no—
101Q16×C83—41no—
102Q16×C83—41no—
103C2×C8×C83C22×C411yescharacteristic
104C2×C8×C83C2×D411yescharacteristic
105C2×C8×C83C2×D411yescharacteristic
106C2×C8×C83C2×D411yescharacteristic
107C2×C8×C83#4111yescharacteristic
108C2×C8×C83#4111yescharacteristic
109C4⋊C4×C83—41no—
110C4⋊C4×C83—41no—
111C16×C83—22no—
112C23×C83C2×D811yescharacteristic
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—21no—
116C22×D83—21no—
117C22×D83—21no—
118C22×D83—21no—
119C22×D83#10212yes—
120Q8×C83—42no—
121Q8×C83—42no—
122Q8×C83—81no—
123Q8×C83—81no—
124C2×C4×C83—42no—
125C2×C4×C83—81no—
126C2×C4×C83—81no—
127C2×C4×C83—81no—
128C2×C4×C83—81no—
129C2×C4×C83—81no—
130C2×C4×C83C2×C22⋊C411yescharacteristic
131C2×C4×C83#12111yescharacteristic
132C2×C4×C83#12111yescharacteristic
133C2×C4×C83C2×SD1611yescharacteristic
134C2×C4×C83#13511yescharacteristic
135C2×C4×C83#13511yescharacteristic
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—42no—
139C2×Dic83—81no—
140C2×Dic83—81no—
141C2×Dic83—81no—
142C2×Dic83—81no—
143C2×Dic83—81no—
144C2×Dic83#10212yes—
145C4×D83—22no—
146C4×D83—22no—
147C4×D83—22no—
148C4×D83—22no—
149C8×C83C4×D411yescharacteristic
150C8×C83C4×D411yescharacteristic
151C8×C83#12011yescharacteristic
152C8×C83#12311yescharacteristic
153C83⋊C8—22no—
154C22×C83C2×#10211yescharacteristic
155C22×C83#46611yescharacteristic
156C22×C83#46611yescharacteristic
157C22×C83#50911yescharacteristic
158C22×C83#51011yescharacteristic
159C22×C83#51011yescharacteristic
160C22×C83#51011yescharacteristic
161C2×D83—22no—
162C2×D83—22no—
163C2×D83—22no—
164C2×D83—41no—
165C2×D83—41no—
166C2×D83—41no—
167C2×D83—41no—
168C4×C83—82no—
169C4×C83—161no—
170C4×C83#38611yescharacteristic
171C4×C83#41611yescharacteristic
172C4×C83#46511yescharacteristic
173C4×C83#46511yescharacteristic
174Dic83—22no—
175Dic83—82no—
176Dic83—161no—
177#13062—831nomaximal, Sylow
178C2×C83#189311yescharacteristic
179C2×C83#213911yescharacteristic
180C2×C83#214111yescharacteristic
181C2×C83#221911yescharacteristic
182C2×C83#221911yescharacteristic
183C2×C83#222011yescharacteristic
184C2×C83#222011yescharacteristic
185D83—42no—
186C22×M32—831no—
187C2×#467—831no—
188C2×#358—831no—
189C2×#358D8311yescharacteristic, complemented
190#1808—831no—
191#1380—831no—
192#1380—831no—
193C83#1306211yescharacteristic, complemented, Sylow
194C22×Q16C2×D8311yescharacteristic
195C23×C8—831no—
196C2×C22⋊Q8—1661no—
197C2×M32—1662no—
198C2×M32—1662no—
199C2×M32—1662no—
200C22×C16—831no—
201C22×C16C2×D8311yescharacteristic
202#467—834no—
203#467—1662no—
204C2×#107—831no—
205C2×#107—831no—
206C2×#107C2×D8311yescharacteristic
207C2×#103—1661no—
208#380—1661no—
209#358—22no—
210#358—1662no—
211#340—1661no—
212C22×Q8—21no—
213C23×C4—831no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D8312yes—
220C22×C8—831no—
221C22×C8—832no—
222C22×C8—832no—
223C22×C8—1661no—
224C22×C8—1661no—
225C22×C8—1661no—
226C22×C8—1661no—
227C22×C8C22×D8311yescharacteristic
228C22⋊Q8—3322no—
229C22⋊Q8—1664no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—3321no—
232C2×C4⋊C4—1661no—
233C2×C4⋊C4—1661no—
234C2×C22⋊C4—3321no—
235C2×C22⋊C4—3321no—
236M32—1668no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—832no—
240C2×C16—1661no—
241C2×C16—1661no—
242C2×C16C22⋊D8312yes—
243#107—21no—
244#107—21no—
245#107—832no—
246#107—1661no—
247#107—1661no—
248#107—1661no—
249#103—1664no—
250#95—3321no—
251C24—831no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—3321no—
261C22×C4—3321no—
262C22×C4—831no—
263C22×C4—832no—
264C22×C4—832no—
265C22×C4—1661no—
266C22×C4—1661no—
267C22×C4—1661no—
268C22×C4—1661no—
269C22×C4D4×D8311yescharacteristic
270Q16—22no—
271Q16—22no—
272Q16—41no—
273Q16—41no—
274C2×C8—832no—
275C2×C8—1661no—
276C2×C8—1661no—
277C2×C8—1662no—
278C2×C8—1662no—
279C2×C8—1662no—
280C2×C8—1662no—
281C2×C8C2×C4×D8311yescharacteristic
282C2×C8D4×D8311yescharacteristic
283C2×C8D4×D8311yescharacteristic
284C2×C8C2×C22⋊D8311yescharacteristic
285C2×C8#1305058911yescharacteristic
286C2×C8#1305059011yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—3324no—
290C4⋊C4—3321no—
291C4⋊C4—3321no—
292C4⋊C4—3321no—
293C4⋊C4—1662no—
294C22⋊C4—3324no—
295C22⋊C4—6642no—
296C16—22no—
297C16—1662no—
298C23—832no—
299C23—1661no—
300C23—1661no—
301C23—1661no—
302C23—1661no—
303C23—1661no—
304C23—1661no—
305C23D8×D8311yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—81no—
309Q8—81no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—3322no—
317C2×C4—832no—
318C2×C4—6641no—
319C2×C4—6641no—
320C2×C4—6641no—
321C2×C4—6641no—
322C2×C4—6641no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4—1662no—
326C2×C4—1662no—
327C2×C4—1662no—
328C2×C4—1662no—
329C2×C4C22⋊C4×D8311yescharacteristic
330C2×C4#1458176311yescharacteristic
331C2×C4#1458176511yescharacteristic
332C2×C4SD16×D8311yescharacteristic
333C2×C4#1458181111yescharacteristic
334C2×C4#1458181311yescharacteristic
335C8—1662no—
336C8(C22⋊D83)⋊C411yescharacteristic
337C8C4×C22⋊D8311yescharacteristic
338C8#1458176011yescharacteristic
339C8#1458177411yescharacteristic
340C22—3321no—
341C22—3321no—
342C22—3321no—
343C22—3321no—
344C22—1662no—
345C22—1662no—
346C22—1662no—
347C22#102×D8311yescharacteristic
348C22#1912793411yescharacteristic
349C22#1912793611yescharacteristic
350C22#1912813811yescharacteristic
351C22#1912814011yescharacteristic
352C22#1912814311yescharacteristic
353C22#1912814311yescharacteristic
354C4—82no—
355C4—161no—
356C4—13281no—
357C4—6642no—
358C4—1662no—
359C4#1912757711yescharacteristic
360C4#1912769411yescharacteristic
361C4#1912792711yescharacteristic
362C4#1912792911yescharacteristic
363C2—3322no—
364C2#2490788811yescharacteristic
365C2#2490917411yescharacteristic
366C2#2490918111yescharacteristic
367C2#2490956811yescharacteristic
368C2#2490957011yescharacteristic
369C2#2490957311yescharacteristic
370C2#2490957911yescharacteristic
subgroupG/H
centerC23D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC2×#358×C83C2
socleC23×C83C2×D8
prime radicalC23×D83D8
cosocleC22×C8×C83C23
hypercenterC2×#358D83
hypocenterC83#13062
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#19127505C2semidirect
5. #24905555C2semidirect
6. #24905554C2semidirect
7. C2×#358×C83C2semidirect
15. #19127939C4semidirect
189. C2×#358D83semidirect
193. C83#13062semidirect

Maximal subgroups

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

subgroupindex
C22×#145817312
C2×#191279392
C2×#191275052
#249074892
#249055552
#249055542
C2×#358×C832
#1306283