Finite Group Databasebeta

← #35564511

Subgroups — #35564511

Order 21248.

3006 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128184C211yesmaximal, characteristic
2C2×#41×Dic83C211yesmaximal, characteristic
3#24910381C214yesmaximal, complemented
4#24910368C2112yesmaximal, complemented
5#24910087C213yesmaximal, complemented
6#24910083C213yesmaximal, complemented
7#24909866C213yesmaximal, complemented
8C2×#19127566C213yesmaximal, complemented
9C2×#520×C83C211yesmaximal, characteristic
10#19128184—28no—
11#41×Dic83C4116yescomplemented
12C2×Q8×Dic83C2212yes—
13C2×Q8×Dic83C2212yes—
14C2×D4×Dic83C2216yes—
15C2×D4×Dic83C2216yes—
16C2×#14581751C2213yes—
17C2×#14581751C2213yes—
18C22×C4⋊Dic83C2213yes—
19C22×C4×Dic83C2213yes—
20#19127599C22112yes—
21#19127597C22112yes—
22#19127574C22112yes—
23#19127573C22112yes—
24#19127570C2213yes—
25#19127570C2213yes—
26#19127566C2216yes—
27#19127566C2213yes—
28#19127566C22112yes—
29#19127566C2211yescharacteristic
30#19127559C22112yes—
31C4×Dic83⋊C4C22112yes—
32C2×C83⋊#95C22112yes—
33C22×#41×C83C2211yescharacteristic
34#520×C83—28no—
35C2×C4×Q8×C83C2212yes—
36C2×C4×D4×C83C2216yes—
37C2×#117×C83C2213yes—
38C22×C4⋊C4×C83C2213yes—
39C23×Dic83C2316yes—
40C23×Dic83C2316yes—
41Q8×Dic83—28no—
42Q8×Dic83C2×C4116yescomplemented
43D4×Dic83—224no—
44D4×Dic83C2×C4148yescomplemented
45#14581751—212no—
46#14581751C2×C4124yescomplemented
47C2×C4⋊Dic83—212no—
48C2×C4⋊Dic83C2313yes—
49C2×C4⋊Dic83C2316yes—
50C2×C4⋊Dic83C2311yescharacteristic
51C2×C4⋊Dic83C2316yes—
52C2×Dic83⋊C4C23112yes—
53C2×Dic83⋊C4C23112yes—
54C2×C22⋊Dic83C2316yes—
55C2×C22⋊Dic83C2316yes—
56C2×C4×Dic83C2×C4124yescomplemented
57C2×C4×Dic83C2316yes—
58C2×C4×Dic83C2316yes—
59C2×C4×Dic83C2311yescharacteristic
60C2×C4×Dic83C2313yes—
61C83⋊#95C2314yes—
62C83⋊#95C23112yes—
63C83⋊#95C23124yes—
64C2×#41×C83C2×C418yes—
65C2×#41×C83D418yes—
66C2×#41×C83Q818yes—
67C22×Q8×C83C2311yescharacteristic
68C22×D4×C83C2313yes—
69C23×C4×C83C2316yes—
70C23×C4×C83C2313yes—
71C4×Q8×C83—28no—
72C4×D4×C83—224no—
73#117×C83—212no—
74C2×C4⋊C4×C83—212no—
75C2×C4⋊C4×C83C2313yes—
76C2×C4⋊C4×C83C2316yes—
77C2×C4⋊C4×C83C2311yescharacteristic
78C2×C22⋊C4×C83C2316yes—
79C2×C42×C83C2316yes—
80C22×Dic83—212no—
81C22×Dic83—46no—
82C22×Dic83—46no—
83C22×Dic83—412no—
84C22×Dic83C22×C4124yes—
85C22×Dic83C2416yes—
86C22×Dic83C2412yes—
87C22×Dic83C2412yes—
88C22×Dic83C2416yes—
89C4⋊Dic83—26no—
90C4⋊Dic83—212no—
91C4⋊Dic83—22no—
92C4⋊Dic83C22×C4124yes—
93Dic83⋊C4—412no—
94Dic83⋊C4—412no—
95C22⋊Dic83—212no—
96C22⋊Dic83C22×C4124yes—
97C4×Dic83—212no—
98C4×Dic83C22×C414yes—
99C4×Dic83C22×C4112yes—
100C4×Dic83C22×C4124yes—
101C24×C83C2413yes—
102#41×C83C4⋊C4164yescomplemented
103C2×Q8×C83C22×C414yes—
104C2×Q8×C83C2×D414yes—
105C2×Q8×C83C2×Q814yes—
106C2×D4×C83C22×C4112yes—
107C2×D4×C83C2×D4112yes—
108C2×D4×C83C2×Q8112yes—
109C22×C4×C83—212no—
110C22×C4×C83—412no—
111C22×C4×C83C22×C4112yes—
112C22×C4×C83C2×D4112yes—
113C22×C4×C83C2×Q8112yes—
114C22×C4×C83C2412yes—
115C22×C4×C83C2416yes—
116C22×C4×C83C2413yes—
117C22×C4×C83C2411yescharacteristic
118C4⋊C4×C83—26no—
119C4⋊C4×C83—212no—
120C4⋊C4×C83—22no—
121C22⋊C4×C83—212no—
122C42×C83—212no—
123C23×C83—26no—
124C23×C83—26no—
125C23×C83C23×C416yes—
126C23×C83C22×D416yes—
127C23×C83C22×Q816yes—
128C23×C83C2511yescharacteristic
129Q8×C83C2×C4⋊C4116yes—
130D4×C83C2×C4⋊C4148yes—
131C2×C4×C83—22no—
132C2×C4×C83—26no—
133C2×C4×C83—212no—
134C2×C4×C83—24no—
135C2×C4×C83—812no—
136C2×C4×C83C2×C4⋊C4148yes—
137C2×C4×C83C23×C412yes—
138C2×C4×C83C23×C416yes—
139C2×C4×C83C22×D416yes—
140C2×C4×C83C22×D412yes—
141C2×C4×C83C22×Q812yes—
142C2×C4×C83C22×Q816yes—
143C2×Dic83—26no—
144C2×Dic83—26no—
145C2×Dic83—22no—
146C2×Dic83—22no—
147C2×Dic83—22no—
148C2×Dic83—26no—
149C2×Dic83—24no—
150C2×Dic83—212no—
151C2×Dic83—424no—
152C2×Dic83—812no—
153C2×Dic83C23×C414yes—
154C2×Dic83C23×C4112yes—
155C22×C83—212no—
156C22×C83—26no—
157C22×C83—26no—
158C22×C83—212no—
159C22×C83C22×C4⋊C4112yes—
160C22×C83C24×C411yescharacteristic
161C22×C83C23×D411yescharacteristic
162C22×C83C23×Q811yescharacteristic
163C22×C83#58512yes—
164C22×C83#58512yes—
165C4×C83—42no—
166C4×C83—46no—
167C4×C83C22×C4⋊C414yes—
168C4×C83C22×C4⋊C4112yes—
169Dic83—24no—
170Dic83—212no—
171Dic83—42no—
172Dic83—46no—
173#34184—831nomaximal, Sylow
174C2×C83—212no—
175C2×C83C23×C4⋊C411yescharacteristic
176C2×C83#342812yes—
177C2×C83#350011yescharacteristic
178C2×C83#350011yescharacteristic
179C2×C83#350412yes—
180C2×#520—831no—
181C2×#520D8311yescharacteristic
182C2×C4×#41—831no—
183#2351—834no—
184#2349—8312no—
185#2317—833no—
186#2317—833no—
187#2285—833no—
188C2×#382—833no—
189C83#3418411yescharacteristic, complemented, Sylow
190C22×#41C2×D8311yescharacteristic
191#520—28no—
192#520—1668no—
193C4×#41—8316no—
194C2×C4×Q8—832no—
195C2×C4×Q8—832no—
196C2×C4×Q8C2×D8312yes—
197C2×C4×D4—836no—
198C2×C4×D4—836no—
199C2×C4×D4C2×D8316yes—
200C2×#117—833no—
201C2×#117—833no—
202C2×#117C2×D8313yes—
203C22×C4⋊C4—833no—
204C22×C4⋊C4C2×D8313yes—
205C22×C42—833no—
206#389—8312no—
207#389—8312no—
208#385—8312no—
209#385—8312no—
210#384—833no—
211#384—833no—
212#382—836no—
213#382—8312no—
214#382—833no—
215#382—831no—
216#380—8312no—
217C4×C4⋊C4—8312no—
218C2×#95—8312no—
219C2×#41Dic16618yes—
220C2×#41C4×D8318yes—
221C2×#41C22⋊D8318yes—
222C22×Q8C22×D8311yescharacteristic
223C22×D4C22×D8313yes—
224C23×C4—836no—
225C23×C4—836no—
226C23×C4C22×D8316yes—
227C23×C4C22×D8313yes—
228C4×Q8—28no—
229C4×Q8—8316no—
230C4×Q8—1668no—
231C4×D4—224no—
232C4×D4—8348no—
233C4×D4—16624no—
234#117—212no—
235#117—8324no—
236#117—16612no—
237C2×C4⋊C4—212no—
238C2×C4⋊C4—8312no—
239C2×C4⋊C4—8312no—
240C2×C4⋊C4—836no—
241C2×C4⋊C4—833no—
242C2×C4⋊C4—836no—
243C2×C4⋊C4—831no—
244C2×C4⋊C4—16612no—
245C2×C4⋊C4C22×D8313yes—
246C2×C4⋊C4C22×D8316yes—
247C2×C4⋊C4C22×D8311yescharacteristic
248C2×C22⋊C4—836no—
249C2×C22⋊C4—836no—
250C2×C22⋊C4C22×D8316yes—
251C2×C42—8324no—
252C2×C42—833no—
253C2×C42—836no—
254C2×C42—831no—
255C2×C42—836no—
256C2×C42C22×D8316yes—
257#95—8312no—
258#95—8324no—
259#95—834no—
260C24C23×D8313yes—
261#41Dic83⋊C4164yescomplemented
262C2×Q8C2×C4×D8314yes—
263C2×Q8C2×C22⋊D8314yes—
264C2×Q8C2×Dic16614yes—
265C2×D4C2×C4×D83112yes—
266C2×D4C2×C22⋊D83112yes—
267C2×D4C2×Dic166112yes—
268C22×C4—212no—
269C22×C4—412no—
270C22×C4—3326no—
271C22×C4—3326no—
272C22×C4—33212no—
273C22×C4—8324no—
274C22×C4—832no—
275C22×C4—836no—
276C22×C4—836no—
277C22×C4—832no—
278C22×C4—16612no—
279C22×C4C2×C4×D83112yes—
280C22×C4C2×C22⋊D83112yes—
281C22×C4C2×Dic166112yes—
282C22×C4C23×D8316yes—
283C22×C4C23×D8312yes—
284C22×C4C23×D8313yes—
285C22×C4C23×D8311yescharacteristic
286C4⋊C4—26no—
287C4⋊C4—212no—
288C4⋊C4—22no—
289C4⋊C4—33212no—
290C4⋊C4—33212no—
291C4⋊C4—8324no—
292C4⋊C4—1666no—
293C4⋊C4—16612no—
294C4⋊C4—1662no—
295C22⋊C4—212no—
296C22⋊C4—8324no—
297C22⋊C4—16612no—
298C42—212no—
299C42—834no—
300C42—8312no—
301C42—8324no—
302C42—16612no—
303C23—26no—
304C23—26no—
305C23C22×C4×D8316yes—
306C23C22×C22⋊D8316yes—
307C23C22×Dic16616yes—
308C23C24×D8311yescharacteristic
309Q8C2×Dic83⋊C4116yes—
310D4C2×Dic83⋊C4148yes—
311C2×C4—22no—
312C2×C4—26no—
313C2×C4—212no—
314C2×C4—24no—
315C2×C4—812no—
316C2×C4—33224no—
317C2×C4—834no—
318C2×C4—8312no—
319C2×C4—66412no—
320C2×C4—1662no—
321C2×C4—1666no—
322C2×C4—1664no—
323C2×C4—16612no—
324C2×C4—1666no—
325C2×C4—1662no—
326C2×C4—1666no—
327C2×C4—1662no—
328C2×C4C2×Dic83⋊C4148yes—
329C2×C4C22×C4×D8312yes—
330C2×C4C22×C4×D8316yes—
331C2×C4C22×C22⋊D8316yes—
332C2×C4C22×C22⋊D8312yes—
333C2×C4C22×Dic16612yes—
334C2×C4C22×Dic16616yes—
335C22—212no—
336C22—26no—
337C22—26no—
338C22—212no—
339C22C22×Dic83⋊C4112yes—
340C22C23×C4×D8311yescharacteristic
341C22C23×C22⋊D8311yescharacteristic
342C22C23×Dic16611yescharacteristic
343C22#1912853612yes—
344C22#1912853812yes—
345C4—46no—
346C4—42no—
347C4—3322no—
348C4—3326no—
349C4—1664no—
350C4—16612no—
351C4C22×Dic83⋊C414yes—
352C4C22×Dic83⋊C4112yes—
353C2—212no—
354C2C23×Dic83⋊C411yescharacteristic
355C2#2492066312yes—
356C2#2492125111yescharacteristic
357C2#2492125711yescharacteristic
358C2#2492128512yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC2×#520×C83C2
socle, cosocleC23×C83C25
prime radicalC22×#41×C83C22
hypercenterC2×#520D83
hypocenterC83#34184
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. #24910381C2semidirect
4. #24910368C2semidirect
5. #24910087C2semidirect
6. #24910083C2semidirect
7. #24909866C2semidirect
8. C2×#19127566C2semidirect
11. #41×Dic83C4semidirect
42. Q8×Dic83C2×C4semidirect
44. D4×Dic83C2×C4semidirect
46. #14581751C2×C4semidirect
56. C2×C4×Dic83C2×C4semidirect
102. #41×C83C4⋊C4semidirect
189. C83#34184semidirect
261. #41Dic83⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#191281842
C2×#41×Dic832
#249103812
#249103682
#249100872
#249100832
#249098662
C2×#191275662
C2×#520×C832
#3418483