Finite Group Databasebeta

← #31995685

Subgroups — #31995685

Order 18176.

946 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985955C211yesmaximal, characteristic
2#22590838C211yesmaximal, characteristic
3#22588217C212yesmaximal, complemented
4#22588205C212yesmaximal, complemented
5#22588126C212yesmaximal
6#22588114C212yesmaximal
7#22588091C211yesmaximal, characteristic, complemented
8#22588088C211yesmaximal, characteristic, complemented
9#22588028C211yesmaximal, characteristic, complemented
10C4×C8×Dic71C211yesmaximal, characteristic, complemented
11C2×#433×C71C211yesmaximal, characteristic
12C22×#11765849C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14#17985955C418yescomplemented
15C2×#14556039C2211yescharacteristic
16C2×#14556039C2211yescharacteristic
17C2×#14556038C2212yes—
18C2×C4×C71⋊C8C2211yescharacteristic
19C2×C4×C71⋊C8C2211yescharacteristic
20C2×C8×Dic71C2212yes—
21C2×C8×Dic71C2212yes—
22C2×C8×Dic71C2212yes—
23#17985892C414yes—
24#17985885C414yes—
25C4×C4⋊Dic71C2212yes—
26C4×C22⋊Dic71C2212yes—
27#17985845C2211yescharacteristic
28C42×Dic71C2211yescharacteristic
29#17985753C2212yes—
30#17985753C2212yes—
31#17985753C2212yes—
32#17985753C2212yes—
33C2×#117×C71C2211yescharacteristic
34#433×C71C418yescomplemented
35C2×C4⋊C8×C71C2212yes—
36C2×C22⋊C8×C71C2212yes—
37C2×#97×C71C2211yescharacteristic
38C2×C4×C8×C71C2211yescharacteristic
39C2×#11765849—24no—
40C2×#11765849C2×C414yes—
41C22×C71⋊C8—24no—
42C22×C71⋊C8C2×C414yes—
43C22×C71⋊C8C2311yescharacteristic
44C22×C71⋊C8C2311yescharacteristic
45C22×C71⋊C8C2311yescharacteristic
46C22×C71⋊C8C2311yescharacteristic
47C2×C4⋊Dic71C2×C414yes—
48C2×C22⋊Dic71C2×C414yes—
49C2×C4×Dic71C2×C414yes—
50C2×C4×Dic71C2312yes—
51C2×C4×Dic71C2312yes—
52#14556039C2×C414yes—
53#14556039C2×C414yes—
54#14556038—24no—
55C4×C71⋊C8C2×C414yes—
56C4×C71⋊C8C2×C414yes—
57C8×Dic71—24no—
58C8×Dic71—24no—
59C8×Dic71—24no—
60C71⋊#95C2×C414yes—
61C71⋊#95C2×C414yes—
62C71⋊#95C2×C414yes—
63C23×C4×C71C2311yescharacteristic
64C22×C8×C71C2312yes—
65C22×C8×C71C2312yes—
66#117×C71C2×C414yes—
67#117×C71C2×C414yes—
68C2×C4⋊C4×C71C2×C412yes—
69C2×C22⋊C4×C71C2×C412yes—
70C2×C42×C71C2311yescharacteristic
71C2×C42×C71C2311yescharacteristic
72C4⋊C8×C71C2×C418yes—
73C22⋊C8×C71C2×C418yes—
74#97×C71C2×C414yes—
75C4×C8×C71C2×C414yes—
76C22×Dic71C22×C414yes—
77C22×Dic71C22×C414yes—
78#11765849—28no—
79C2×C71⋊C8—22no—
80C2×C71⋊C8—22no—
81C2×C71⋊C8—28no—
82C2×C71⋊C8C22×C412yes—
83C2×C71⋊C8C22×C412yes—
84C2×C71⋊C8C22×C412yes—
85C2×C71⋊C8C22×C412yes—
86C2×C71⋊C8#4114yes—
87C2×C71⋊C8#4114yes—
88C4⋊Dic71—28no—
89C22⋊Dic71—28no—
90C4×Dic71—22no—
91C4×Dic71—22no—
92C4×Dic71—22no—
93C4×Dic71—22no—
94C4×Dic71—28no—
95C24×C71C22×C411yescharacteristic
96C22×C4×C71—22no—
97C22×C4×C71—22no—
98C22×C4×C71C22×C412yes—
99C22×C4×C71C22×C412yes—
100C22×C4×C71C22×C411yescharacteristic
101C22×C4×C71C22×C412yes—
102C22×C4×C71C22×C412yes—
103C22×C4×C71C2411yescharacteristic
104C2×C8×C71—22no—
105C2×C8×C71—22no—
106C2×C8×C71—22no—
107C2×C8×C71—22no—
108C2×C8×C71C22×C414yes—
109C2×C8×C71C22×C414yes—
110C4⋊C4×C71C4218yes—
111C22⋊C4×C71C4218yes—
112C42×C71C22×C412yes—
113C42×C71C22×C412yes—
114C42×C71C22×C412yes—
115C42×C71C22×C412yes—
116C23×C71—22no—
117C23×C71—24no—
118C23×C71C2×C4212yes—
119C23×C71C23×C411yescharacteristic
120C2×C4×C71—22no—
121C2×C4×C71—24no—
122C2×C4×C71—24no—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—24no—
126C2×C4×C71C2×C4214yes—
127C2×C4×C71C2×C4214yes—
128C2×C4×C71C2×C4212yes—
129C2×C4×C71C23×C411yescharacteristic
130C2×C4×C71C23×C411yescharacteristic
131C2×C4×C71C2×#4112yes—
132C2×C4×C71C2×#4112yes—
133C2×Dic71—24no—
134C2×Dic71—24no—
135C2×Dic71—24no—
136C2×Dic71—24no—
137C2×Dic71—24no—
138C2×Dic71#13118yes—
139C8×C71—24no—
140C8×C71—24no—
141C71⋊C8—22no—
142C71⋊C8—22no—
143C71⋊C8#11714yes—
144C71⋊C8#11714yes—
145C22×C71—24no—
146C22×C71—22no—
147C22×C71—44no—
148C22×C71C22×C4211yescharacteristic
149C22×C71C4×#4112yes—
150C22×C71C2×#13114yes—
151C4×C71—24no—
152C4×C71—24no—
153C4×C71—22no—
154C4×C71C2×#11711yescharacteristic
155C4×C71C2×#11711yescharacteristic
156C4×C71C4×#4112yes—
157Dic71—28no—
158Dic71—44no—
159#20474—711nomaximal, Sylow
160C2×C71—42no—
161C2×C71C4×#11711yescharacteristic
162C2×C71C4×#13112yes—
163C2×C71#296614yes—
164C2×#433D7111yescharacteristic
165C2×#405—711no—
166#2298—711no—
167#1771—712no—
168C8×C4⋊C4—712no—
169#1753—712no—
170C8×C22⋊C4—712no—
171#1746—711no—
172#1745—711no—
173#1726—711no—
174C42×C8—711no—
175C71#2047411yescharacteristic, complemented, Sylow
176C22×M16—711no—
177C23×C8—711no—
178C2×#117C2×D7111yescharacteristic
179#433Dic7118yescomplemented
180C2×C4⋊C8C2×D7112yes—
181C2×C22⋊C8C2×D7112yes—
182#405—718no—
183C2×#97—711no—
184C2×#97—711no—
185C2×#97—712no—
186C2×#97C2×D7111yescharacteristic
187C2×C4×C8—711no—
188C2×C4×C8—711no—
189C2×C4×C8—712no—
190C2×C4×C8—712no—
191C2×C4×C8—712no—
192C2×C4×C8C2×D7111yescharacteristic
193#388—714no—
194#387—714no—
195C4×C4⋊C4—712no—
196C4×C22⋊C4—712no—
197#376—711no—
198C43—711no—
199#336—712no—
200#336—712no—
201#336—712no—
202#336—712no—
203C23×C4C22×D7111yescharacteristic
204C2×M16—714no—
205C2×M16—1424no—
206C22×C8—711no—
207C22×C8—711no—
208C22×C8—714no—
209C22×C8—711no—
210C22×C8—711no—
211C22×C8—1424no—
212C22×C8C22×D7112yes—
213C22×C8C22×D7112yes—
214#117C4×D7114yes—
215#117C2×Dic7114yes—
216C2×C4⋊C4—714no—
217C2×C4⋊C4C4×D7112yes—
218C2×C22⋊C4—714no—
219C2×C22⋊C4C4×D7112yes—
220C2×C42—714no—
221C2×C42—712no—
222C2×C42—712no—
223C2×C42C22×D7111yescharacteristic
224C2×C42C22×D7111yescharacteristic
225C4⋊C8C2×Dic7118yes—
226C22⋊C8C2×Dic7118yes—
227#97—714no—
228#97—714no—
229#97—1424no—
230#97C2×Dic7114yes—
231C4×C8—714no—
232C4×C8—714no—
233C4×C8—1424no—
234C4×C8—1424no—
235C4×C8—1424no—
236C4×C8C2×Dic7114yes—
237#95—714no—
238#95—714no—
239#95—714no—
240C24C2×C4×D7111yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—714no—
244C22×C4—714no—
245C22×C4C2×C4×D7112yes—
246C22×C4C2×C4×D7111yescharacteristic
247C22×C4C2×C4×D7112yes—
248C22×C4C2×C4×D7112yes—
249C22×C4C22×Dic7112yes—
250C22×C4C23×D7111yescharacteristic
251M16—1428no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—712no—
257C2×C8—712no—
258C2×C8—714no—
259C2×C8—714no—
260C2×C8—712no—
261C2×C8—712no—
262C2×C8—1428no—
263C2×C8—1422no—
264C2×C8—1422no—
265C2×C8C22×Dic7114yes—
266C2×C8C22×Dic7114yes—
267C4⋊C4—1428no—
268C4⋊C4C4×Dic7118yes—
269C22⋊C4—1428no—
270C22⋊C4C4×Dic7118yes—
271C42—1428no—
272C42—1422no—
273C42—1422no—
274C42—1422no—
275C42—1422no—
276C42C2×C4×D7112yes—
277C42C2×C4×D7112yes—
278C42C22×Dic7112yes—
279C42C22×Dic7112yes—
280C23—22no—
281C23—24no—
282C23C2×C4×Dic7112yes—
283C23C22×C4×D7111yescharacteristic
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—718no—
291C2×C4—1424no—
292C2×C4—1424no—
293C2×C4—1424no—
294C2×C4—1424no—
295C2×C4—1424no—
296C2×C4C2×C4×Dic7114yes—
297C2×C4C2×C4×Dic7114yes—
298C2×C4C2×C4×Dic7112yes—
299C2×C4C22×C4×D7111yescharacteristic
300C2×C4C23×Dic7111yescharacteristic
301C2×C4#41×D7112yes—
302C2×C4#41×D7112yes—
303C8—24no—
304C8—24no—
305C8—714no—
306C8—714no—
307C8—1422no—
308C8—1422no—
309C22—24no—
310C22—22no—
311C22—44no—
312C22C22×C4×Dic7111yescharacteristic
313C22#1798646712yes—
314C22#131×D7114yes—
315C4—24no—
316C4—24no—
317C4—22no—
318C4—2844no—
319C4—1428no—
320C4#117×D7111yescharacteristic
321C4#117×D7111yescharacteristic
322C4#41×Dic7112yes—
323C2—42no—
324C2#117×Dic7111yescharacteristic
325C2#131×Dic7112yes—
326C2#2259731314yes—
subgroupG/H
center, FrattiniC22×C4C23×D71
commutatorC22×C71C22×C42
FittingC2×#433×C71C2
socleC23×C71C23×C4
prime radicalC24×C71C22×C4
cosocleC22×C4×C71C24
hypercenterC2×#433D71
hypocenterC71#20474
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #22588217C2semidirect
4. #22588205C2semidirect
7. #22588091C2semidirect
8. #22588088C2semidirect
9. #22588028C2semidirect
10. C4×C8×Dic71C2semidirect
14. #17985955C4semidirect
34. #433×C71C4semidirect
175. C71#20474semidirect
179. #433Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859552
#225908382
#225882172
#225882052
#225881262
#225881142
#225880912
#225880882
#225880282
C4×C8×Dic712
C2×#433×C712
#2047471