Finite Group Databasebeta

← #35564607

Subgroups — #35564607

Order 21248.

2414 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910412C216yesmaximal, complemented
2Q8×C4⋊Dic83C212yesmaximal, complemented
3#24910356C216yesmaximal, complemented
4#24910304C212yesmaximal
5#24910106C216yesmaximal, complemented
6#24909870C213yesmaximal, complemented
7#24909861C213yesmaximal, complemented
8#24909828C212yesmaximal, complemented
9C2×C4×#41×C83C211yesmaximal, characteristic
10C2×Q8×Dic83C2214yes—
11C2×#14581751C2216yes—
12#19127604C22112yes—
13#19127600C22112yes—
14#19127594C22124yes—
15#19127571C2216yes—
16#19127569C2216yes—
17#19127567C2211yescharacteristic
18#19127567C2213yes—
19#19127564C2216yes—
20#19127561C2216yes—
21#19127561C2216yes—
22#19127561C2216yes—
23C4⋊C4×Dic83C22112yes—
24C4×C4⋊Dic83C2216yes—
25C4×C4⋊Dic83C2216yes—
26C4×C4⋊Dic83C2216yes—
27C4×C22⋊Dic83C2216yes—
28#19127547C2216yes—
29C22×#41×C83C2211yescharacteristic
30C4×#41×C83C4116yescomplemented
31C2×C4×Q8×C83C2211yescharacteristic
32C2×C4×Q8×C83C2211yescharacteristic
33C2×C4×D4×C83C2213yes—
34C2×C4×D4×C83C2213yes—
35C2×#117×C83C2213yes—
36C22×C42×C83C2213yes—
37Q8×Dic83—216no—
38#14581751—412no—
39C2×C4⋊Dic83C2312yes—
40C2×C4⋊Dic83C23112yes—
41C2×C4⋊Dic83C2316yes—
42C2×C4⋊Dic83C23112yes—
43C2×C22⋊Dic83C23112yes—
44C2×C22⋊Dic83C23112yes—
45C2×C4×Dic83C23112yes—
46C2×C4×Dic83C23112yes—
47C83⋊#95C2316yes—
48C83⋊#95C2316yes—
49C83⋊#95C23112yes—
50C83⋊#95C23112yes—
51C83⋊#95C2312yes—
52C83⋊#95C2312yes—
53C2×#41×C83—28no—
54C2×#41×C83C2×C418yes—
55C22×Q8×C83C2311yescharacteristic
56C22×D4×C83C2313yes—
57C23×C4×C83C2313yes—
58C23×C4×C83C2313yes—
59C23×C4×C83C2313yes—
60C4×Q8×C83C2×C418yes—
61C4×Q8×C83C2×C418yes—
62C4×D4×C83C2×C4124yes—
63C4×D4×C83C2×C4124yes—
64#117×C83C2×C4124yes—
65C2×C4⋊C4×C83C2313yes—
66C2×C4⋊C4×C83C2313yes—
67C2×C22⋊C4×C83C2316yes—
68C2×C42×C83C2×C4124yes—
69C2×C42×C83C2313yes—
70C2×C42×C83C2313yes—
71C2×C42×C83C2313yes—
72C2×C42×C83C2311yescharacteristic
73C22×Dic83C2414yes—
74C22×Dic83C24112yes—
75C4⋊Dic83—224no—
76C4⋊Dic83—212no—
77C4⋊Dic83—24no—
78C4⋊Dic83—412no—
79C22⋊Dic83—412no—
80C22⋊Dic83—412no—
81C4×Dic83—224no—
82C4×Dic83—412no—
83C24×C83C2413yes—
84#41×C83—232no—
85C2×Q8×C83C22×C414yes—
86C2×Q8×C83#4118yes—
87C2×D4×C83—212no—
88C2×D4×C83C22×C4112yes—
89C22×C4×C83—212no—
90C22×C4×C83—26no—
91C22×C4×C83—26no—
92C22×C4×C83—46no—
93C22×C4×C83C22×C4112yes—
94C22×C4×C83C22×C4112yes—
95C22×C4×C83C22×C4112yes—
96C22×C4×C83C2411yescharacteristic
97C22×C4×C83C2411yescharacteristic
98C22×C4×C83C2413yes—
99C22×C4×C83C2413yes—
100C22×C4×C83C2413yes—
101C22×C4×C83C2411yescharacteristic
102C4⋊C4×C83C22×C4112yes—
103C4⋊C4×C83C22×C4112yes—
104C22⋊C4×C83C22×C4124yes—
105C42×C83C22×C414yes—
106C42×C83C22×C4112yes—
107C42×C83C22×C4112yes—
108C42×C83C22×C4112yes—
109C23×C83—26no—
110C23×C83—23no—
111C23×C83—23no—
112C23×C83—26no—
113C23×C83C23×C416yes—
114C23×C83C2511yescharacteristic
115Q8×C83#117116yescomplemented
116D4×C83—224no—
117C2×C4×C83—22no—
118C2×C4×C83—23no—
119C2×C4×C83—23no—
120C2×C4×C83—22no—
121C2×C4×C83—26no—
122C2×C4×C83—224no—
123C2×C4×C83—224no—
124C2×C4×C83—412no—
125C2×C4×C83C23×C412yes—
126C2×C4×C83C23×C412yes—
127C2×C4×C83C23×C416yes—
128C2×C4×C83C23×C416yes—
129C2×C4×C83C23×C416yes—
130C2×C4×C83C23×C412yes—
131C2×C4×C83C2×#41112yes—
132C2×C4×C83H(5,𝔽2)12yes—
133C2×C4×C832−1+412yes—
134C2×Dic83—24no—
135C2×Dic83—212no—
136C2×Dic83—212no—
137C2×Dic83—24no—
138C2×Dic83—24no—
139C2×Dic83—212no—
140C22×C83—26no—
141C22×C83—26no—
142C22×C83—26no—
143C22×C83—46no—
144C22×C83—46no—
145C22×C83C24×C411yescharacteristic
146C22×C83C22×#4112yes—
147C22×C83C2×H(5,𝔽2)11yescharacteristic
148C22×C83C2×2−1+411yescharacteristic
149C22×C83#58512yes—
150C4×C83—22no—
151C4×C83—26no—
152C4×C83—22no—
153C4×C83—26no—
154C4×C83C2×#117112yes—
155C4×C83#52014yes—
156Dic83—412no—
157Dic83—44no—
158#34193—831nomaximal, Sylow
159C2×C83—46no—
160C2×C83C22×#11711yescharacteristic
161C2×C83C2×#52011yescharacteristic
162C2×C83#342811yescharacteristic
163C2×C83#351712yes—
164C2×C83#351812yes—
165C2×C4×#41D8311yescharacteristic
166#2354—836no—
167Q8×C4⋊C4—832no—
168#2347—836no—
169#2341—832no—
170#2319—836no—
171#2285—833no—
172#2283—833no—
173#2276—832no—
174C83#3419311yescharacteristic, complemented, Sylow
175C22×#41C2×D8311yescharacteristic
176C4×#41Dic83116yescomplemented
177C2×C4×Q8—834no—
178C2×C4×Q8C2×D8311yescharacteristic
179C2×C4×Q8C2×D8311yescharacteristic
180C2×C4×D4C2×D8313yes—
181C2×C4×D4C2×D8313yes—
182C2×#117—836no—
183C2×#117C2×D8313yes—
184C22×C42C2×D8313yes—
185#390—8312no—
186#389—8312no—
187#388—8324no—
188#384—836no—
189#383—836no—
190#382—831no—
191#382—833no—
192#381—836no—
193#380—836no—
194#380—836no—
195#380—836no—
196C4×C4⋊C4—8312no—
197C4×C4⋊C4—836no—
198C4×C4⋊C4—836no—
199C4×C4⋊C4—836no—
200C4×C22⋊C4—836no—
201#376—836no—
202C2×#41—28no—
203C2×#41C2×Dic8318yes—
204C22×Q8C22×D8311yescharacteristic
205C22×D4C22×D8313yes—
206C23×C4C22×D8313yes—
207C23×C4C22×D8313yes—
208C23×C4C22×D8313yes—
209C4×Q8—16616no—
210C4×Q8C2×Dic8318yes—
211C4×Q8C2×Dic8318yes—
212C4×D4C2×Dic83124yes—
213C4×D4C2×Dic83124yes—
214#117—33212no—
215#117C2×Dic83124yes—
216C2×C4⋊C4—8312no—
217C2×C4⋊C4—836no—
218C2×C4⋊C4—8312no—
219C2×C4⋊C4—832no—
220C2×C4⋊C4C22×D8313yes—
221C2×C4⋊C4C22×D8313yes—
222C2×C22⋊C4—8312no—
223C2×C22⋊C4—8312no—
224C2×C22⋊C4C22×D8316yes—
225C2×C42—8312no—
226C2×C42—8312no—
227C2×C42C2×Dic83124yes—
228C2×C42C22×D8313yes—
229C2×C42C22×D8313yes—
230C2×C42C22×D8313yes—
231C2×C42C22×D8311yescharacteristic
232#95—832no—
233#95—836no—
234#95—8312no—
235#95—832no—
236#95—836no—
237#95—8312no—
238C24C23×D8313yes—
239#41—232no—
240C2×Q8C22×Dic8314yes—
241C2×Q8#1305059018yes—
242C2×D4—212no—
243C2×D4C22×Dic83112yes—
244C22×C4—212no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—46no—
248C22×C4—834no—
249C22×C4—8312no—
250C22×C4C22×Dic83112yes—
251C22×C4C22×Dic83112yes—
252C22×C4C22×Dic83112yes—
253C22×C4C23×D8311yescharacteristic
254C22×C4C23×D8311yescharacteristic
255C22×C4C23×D8313yes—
256C22×C4C23×D8313yes—
257C22×C4C23×D8313yes—
258C22×C4C23×D8311yescharacteristic
259C4⋊C4—33212no—
260C4⋊C4—16624no—
261C4⋊C4—16612no—
262C4⋊C4—1664no—
263C4⋊C4C22×Dic83112yes—
264C4⋊C4C22×Dic83112yes—
265C22⋊C4—33212no—
266C22⋊C4—33212no—
267C22⋊C4C22×Dic83124yes—
268C42—33212no—
269C42—16624no—
270C42C22×Dic8314yes—
271C42C22×Dic83112yes—
272C42C22×Dic83112yes—
273C42C22×Dic83112yes—
274C23—26no—
275C23—23no—
276C23—23no—
277C23—26no—
278C23C23×Dic8316yes—
279C23C24×D8311yescharacteristic
280Q8#14581751116yescomplemented
281D4—224no—
282C2×C4—22no—
283C2×C4—23no—
284C2×C4—23no—
285C2×C4—26no—
286C2×C4—22no—
287C2×C4—224no—
288C2×C4—224no—
289C2×C4—412no—
290C2×C4—1664no—
291C2×C4—16612no—
292C2×C4—16612no—
293C2×C4—1664no—
294C2×C4—1664no—
295C2×C4—16612no—
296C2×C4C23×Dic8312yes—
297C2×C4C23×Dic8312yes—
298C2×C4C23×Dic8316yes—
299C2×C4C23×Dic8316yes—
300C2×C4C23×Dic8316yes—
301C2×C4C23×Dic8312yes—
302C2×C4C2×#13050590112yes—
303C2×C4#1458183812yes—
304C2×C4#1458183912yes—
305C22—26no—
306C22—26no—
307C22—26no—
308C22—46no—
309C22—46no—
310C22C24×Dic8311yescharacteristic
311C22C22×#1305059012yes—
312C22C2×#1458183811yescharacteristic
313C22C2×#1458183911yescharacteristic
314C22#1912853712yes—
315C4—22no—
316C4—26no—
317C4—22no—
318C4—26no—
319C4—33212no—
320C4—3324no—
321C4C2×#14581751112yes—
322C4#1912818414yes—
323C2—46no—
324C2C22×#1458175111yescharacteristic
325C2C2×#1912818411yescharacteristic
326C2#2492066611yescharacteristic
327C2#2492146912yes—
328C2#2492148012yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC2×C4×#41×C83C2
socle, cosocleC23×C83C25
prime radicalC22×#41×C83C22
hypercenterC2×C4×#41D83
hypocenterC83#34193
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #24910412C2semidirect
2. Q8×C4⋊Dic83C2semidirect
3. #24910356C2semidirect
5. #24910106C2semidirect
6. #24909870C2semidirect
7. #24909861C2semidirect
8. #24909828C2semidirect
30. C4×#41×C83C4semidirect
115. Q8×C83#117semidirect
174. C83#34193semidirect
176. C4×#41Dic83semidirect
280. Q8#14581751semidirect

Maximal subgroups

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

subgroupindex
#249104122
Q8×C4⋊Dic832
#249103562
#249103042
#249101062
#249098702
#249098612
#249098282
C2×C4×#41×C832
#3419383