Finite Group Databasebeta

← #33084214

Subgroups — #33084214

Order 18688.

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×#17988648C211yesmaximal, characteristic
2C2×#17988543C211yesmaximal, characteristic
3#22609806C212yesmaximal, complemented
4C4⋊C4×#10596831C212yesmaximal, complemented
5#22609715C212yesmaximal
6C22⋊C4×#10596831C212yesmaximal
7#22609683C211yesmaximal, characteristic, complemented
8#22609680C211yesmaximal, characteristic, complemented
9#22609618C211yesmaximal, characteristic, complemented
10C42×#10596831C211yesmaximal, characteristic, complemented
11C73×#2298C211yesmaximal, characteristic
12C22×#11924012C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14#17988648C418yescomplemented
15C2×C4⋊#10596831C2212yes—
16C2×C22⋊#10596831C2212yes—
17#17988543C418yescomplemented
18C2×#14556502C2212yes—
19C2×#14556502C2211yescharacteristic
20C2×#14556502C2211yescharacteristic
21C2×#14556502C2211yescharacteristic
22C2×C4×#10596831C2212yes—
23C2×C4×#10596831C2212yes—
24C2×C4×#10596831C2212yes—
25C2×C4×#10596831C2211yescharacteristic
26C2×C4×#10596831C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28#17988342C2212yes—
29#17988342C2212yes—
30#17988342C2212yes—
31#17988342C2212yes—
32C2×#117×C73C2211yescharacteristic
33#388×C73C414yes—
34#387×C73C414yes—
35C4×C4⋊C4×C73C2212yes—
36C4×C22⋊C4×C73C2212yes—
37#376×C73C2211yescharacteristic
38C43×C73C2211yescharacteristic
39C2×#11924012—24no—
40C2×#11924012C2×C414yes—
41C22×#10596831—24no—
42C22×#10596831C2×C414yes—
43C22×#10596831C2312yes—
44C22×#10596831C2312yes—
45C22×#10596831C2311yescharacteristic
46C22×#10596831C2311yescharacteristic
47C22×#10596831C2311yescharacteristic
48C22×#10596831C2311yescharacteristic
49C4⋊#10596831C2×C418yes—
50C22⋊#10596831C2×C418yes—
51#14556502—24no—
52#14556502C2×C414yes—
53#14556502C2×C414yes—
54#14556502C2×C414yes—
55C4×#10596831—24no—
56C4×#10596831—24no—
57C4×#10596831—24no—
58C4×#10596831C2×C414yes—
59C4×#10596831C2×C414yes—
60C4×#10596831C2×C414yes—
61C23×C4×C73C2311yescharacteristic
62#117×C73C2×C414yes—
63#117×C73C2×C414yes—
64C2×C4⋊C4×C73C2×C414yes—
65C2×C4⋊C4×C73C2×C412yes—
66C2×C22⋊C4×C73C2×C414yes—
67C2×C22⋊C4×C73C2×C412yes—
68C2×C42×C73C2×C414yes—
69C2×C42×C73C2312yes—
70C2×C42×C73C2312yes—
71C2×C42×C73C2311yescharacteristic
72C2×C42×C73C2311yescharacteristic
73#95×C73C2×C414yes—
74#95×C73C2×C414yes—
75#95×C73C2×C414yes—
76#11924012—28no—
77C2×#10596831—22no—
78C2×#10596831—22no—
79C2×#10596831—22no—
80C2×#10596831—22no—
81C2×#10596831—22no—
82C2×#10596831—22no—
83C2×#10596831—28no—
84C2×#10596831C22×C412yes—
85C2×#10596831C22×C414yes—
86C2×#10596831C22×C412yes—
87C2×#10596831C22×C414yes—
88C2×#10596831C22×C412yes—
89C2×#10596831C22×C412yes—
90C2×#10596831#4114yes—
91C2×#10596831#4114yes—
92C24×C73C22×C411yescharacteristic
93C22×C4×C73—22no—
94C22×C4×C73—22no—
95C22×C4×C73C22×C412yes—
96C22×C4×C73C22×C412yes—
97C22×C4×C73C22×C414yes—
98C22×C4×C73C22×C411yescharacteristic
99C22×C4×C73C22×C412yes—
100C22×C4×C73C22×C412yes—
101C22×C4×C73C22×C414yes—
102C22×C4×C73C2411yescharacteristic
103C4⋊C4×C73—28no—
104C4⋊C4×C73C4218yes—
105C22⋊C4×C73—28no—
106C22⋊C4×C73C4218yes—
107C42×C73—28no—
108C42×C73—22no—
109C42×C73—22no—
110C42×C73—22no—
111C42×C73—22no—
112C42×C73C22×C412yes—
113C42×C73C22×C412yes—
114C42×C73C22×C412yes—
115C42×C73C22×C412yes—
116C23×C73—22no—
117C23×C73—24no—
118C23×C73C2×C4212yes—
119C23×C73C23×C411yescharacteristic
120C2×C4×C73—24no—
121C2×C4×C73—22no—
122C2×C4×C73—24no—
123C2×C4×C73—24no—
124C2×C4×C73—24no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—24no—
128C2×C4×C73—24no—
129C2×C4×C73—24no—
130C2×C4×C73—24no—
131C2×C4×C73C2×C4214yes—
132C2×C4×C73C2×C4214yes—
133C2×C4×C73C2×C4212yes—
134C2×C4×C73#13118yes—
135C2×C4×C73C23×C411yescharacteristic
136C2×C4×C73C23×C411yescharacteristic
137C2×C4×C73C2×#4112yes—
138C2×C4×C73C2×#4112yes—
139#10596831—22no—
140#10596831—22no—
141#10596831—24no—
142#10596831—24no—
143#10596831#11714yes—
144#10596831#11714yes—
145C22×C73—24no—
146C22×C73—22no—
147C22×C73—44no—
148C22×C73C22×C4211yescharacteristic
149C22×C73C4×#4112yes—
150C22×C73C2×#13114yes—
151C4×C73—24no—
152C4×C73—24no—
153C4×C73—28no—
154C4×C73—22no—
155C4×C73—44no—
156C4×C73C2×#11711yescharacteristic
157C4×C73C2×#11711yescharacteristic
158C4×C73C4×#4112yes—
159#20474—731nomaximal, Sylow
160C2×C73—42no—
161C2×C73C4×#11711yescharacteristic
162C2×C73C4×#13112yes—
163C2×C73#296614yes—
164C2×#433—731no—
165C2×#405—731no—
166#2298D7311yescharacteristic
167#1771—732no—
168C8×C4⋊C4—732no—
169#1753—732no—
170C8×C22⋊C4—732no—
171#1746—731no—
172#1745—731no—
173#1726—731no—
174C42×C8—731no—
175C73#2047411yescharacteristic, complemented, Sylow
176C22×M16—731no—
177C23×C8—731no—
178C2×#117C2×D7311yescharacteristic
179#433—738no—
180C2×C4⋊C8—732no—
181C2×C22⋊C8—732no—
182#405—738no—
183C2×#97—731no—
184C2×#97—731no—
185C2×#97—731no—
186C2×#97—732no—
187C2×C4×C8—731no—
188C2×C4×C8—731no—
189C2×C4×C8—731no—
190C2×C4×C8—732no—
191C2×C4×C8—732no—
192C2×C4×C8—732no—
193#388Dic7314yes—
194#387Dic7314yes—
195C4×C4⋊C4C2×D7312yes—
196C4×C22⋊C4C2×D7312yes—
197#376C2×D7311yescharacteristic
198C43C2×D7311yescharacteristic
199#336—732no—
200#336—732no—
201#336—732no—
202#336—732no—
203C23×C4C22×D7311yescharacteristic
204C2×M16—734no—
205C2×M16—1464no—
206C22×C8—731no—
207C22×C8—731no—
208C22×C8—734no—
209C22×C8—731no—
210C22×C8—731no—
211C22×C8—732no—
212C22×C8—732no—
213C22×C8—1464no—
214#117C4×D7314yes—
215#117C4×D7314yes—
216C2×C4⋊C4C2×Dic7314yes—
217C2×C4⋊C4C2×Dic7312yes—
218C2×C22⋊C4C2×Dic7314yes—
219C2×C22⋊C4C2×Dic7312yes—
220C2×C42C2×Dic7314yes—
221C2×C42C22×D7312yes—
222C2×C42C22×D7312yes—
223C2×C42C22×D7311yescharacteristic
224C2×C42C22×D7311yescharacteristic
225C4⋊C8—738no—
226C22⋊C8—738no—
227#97—734no—
228#97—734no—
229#97—734no—
230#97—1464no—
231C4×C8—734no—
232C4×C8—734no—
233C4×C8—734no—
234C4×C8—1464no—
235C4×C8—1464no—
236C4×C8—1464no—
237#95C2×Dic7314yes—
238#95C2×Dic7314yes—
239#95C2×Dic7314yes—
240C24C22×Dic7311yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4C2×C4×D7312yes—
244C22×C4C2×C4×D7312yes—
245C22×C4C22×Dic7314yes—
246C22×C4C22×Dic7311yescharacteristic
247C22×C4C22×Dic7312yes—
248C22×C4C22×Dic7312yes—
249C22×C4C22×Dic7314yes—
250C22×C4C23×D7311yescharacteristic
251M16—1468no—
252C2×C8—732no—
253C2×C8—732no—
254C2×C8—734no—
255C2×C8—734no—
256C2×C8—732no—
257C2×C8—734no—
258C2×C8—732no—
259C2×C8—734no—
260C2×C8—1468no—
261C2×C8—1462no—
262C2×C8—1462no—
263C2×C8—1462no—
264C2×C8—1462no—
265C2×C8—1462no—
266C2×C8—1462no—
267C4⋊C4—28no—
268C4⋊C4C4×Dic7318yes—
269C22⋊C4—28no—
270C22⋊C4C4×Dic7318yes—
271C42—28no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42C2×C4×D7312yes—
277C42C2×C4×D7312yes—
278C42C2×C4×D7312yes—
279C42C2×C4×D7312yes—
280C23—22no—
281C23—24no—
282C23C2×C4×Dic7312yes—
283C23C23×Dic7311yescharacteristic
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4C2×C4×Dic7314yes—
296C2×C4C2×C4×Dic7314yes—
297C2×C4C2×C4×Dic7312yes—
298C2×C4#1455659818yes—
299C2×C4C22×C4×D7311yescharacteristic
300C2×C4C22×C4×D7311yescharacteristic
301C2×C4#41×D7312yes—
302C2×C4#41×D7312yes—
303C8—734no—
304C8—734no—
305C8—1462no—
306C8—1462no—
307C8—1464no—
308C8—1464no—
309C22—22no—
310C22—24no—
311C22—44no—
312C22C22×C4×Dic7311yescharacteristic
313C22#41×Dic7312yes—
314C22C2×#1455659814yes—
315C4—28no—
316C4—22no—
317C4—24no—
318C4—24no—
319C4—44no—
320C4#117×D7311yescharacteristic
321C4#117×D7311yescharacteristic
322C4#1798905512yes—
323C2—42no—
324C2#117×Dic7311yescharacteristic
325C2C4×#1455659812yes—
326C2#2261890714yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C22×C42
FittingC73×#2298C2
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenter#2298D73
hypocenterC73#20474
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
3. #22609806C2semidirect
4. C4⋊C4×#10596831C2semidirect
7. #22609683C2semidirect
8. #22609680C2semidirect
9. #22609618C2semidirect
10. C42×#10596831C2semidirect
14. #17988648C4semidirect
17. #17988543C4semidirect
175. C73#20474semidirect

Maximal subgroups

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

subgroupindex
C2×#179886482
C2×#179885432
#226098062
C4⋊C4×#105968312
#226097152
C22⋊C4×#105968312
#226096832
#226096802
#226096182
C42×#105968312
C73×#22982
#2047473