Finite Group Databasebeta

← #33084235

Subgroups — #33084235

Order 18688.

990 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988644C211yesmaximal, characteristic
2C22×C4×#10596831C211yesmaximal, characteristic
3#22609800C212yesmaximal, complemented
4C4⋊C4×#10596831C212yesmaximal, complemented
5#22609709C212yesmaximal
6C22⋊C4×#10596831C212yesmaximal
7#22609683C212yesmaximal, complemented
8#22609618C212yesmaximal, complemented
9C73×#2283C211yesmaximal, characteristic
10C23×#10596831C2211yescharacteristic
11C23×#10596831C2211yescharacteristic
12#17988644C418yescomplemented
13C2×C4⋊#10596831C2212yes—
14C2×C22⋊#10596831C2212yes—
15C2×#14556502C2214yes—
16C2×#14556502C2212yes—
17C2×C4×#10596831C418yescomplemented
18C2×C4×#10596831C2214yes—
19C2×C4×#10596831C2211yescharacteristic
20C2×C4×#10596831C2211yescharacteristic
21C2×C4×#10596831C2211yescharacteristic
22C2×C4×#10596831C2211yescharacteristic
23#17988342C2214yes—
24#17988342C2214yes—
25C22×C42×C73C2211yescharacteristic
26#383×C73C412yes—
27#382×C73C412yes—
28#381×C73C412yes—
29#380×C73C412yes—
30C4×C4⋊C4×C73C2212yes—
31C4×C22⋊C4×C73C2212yes—
32#376×C73C2212yes—
33C22×#10596831—24no—
34C22×#10596831—24no—
35C22×#10596831C2×C414yes—
36C22×#10596831C2×C414yes—
37C22×#10596831C2314yes—
38C22×#10596831C2311yescharacteristic
39C22×#10596831C2311yescharacteristic
40C22×#10596831C2311yescharacteristic
41C22×#10596831C2311yescharacteristic
42C4⋊#10596831C2×C418yes—
43C22⋊#10596831C2×C418yes—
44#14556502—28no—
45#14556502C2×C418yes—
46C4×#10596831—28no—
47C4×#10596831C2×C414yes—
48C4×#10596831C2×C414yes—
49C4×#10596831C2×C414yes—
50C4×#10596831C2×C414yes—
51C23×C4×C73C2×C411yescharacteristic
52C23×C4×C73C2×C411yescharacteristic
53C23×C4×C73C2311yescharacteristic
54C2×C4⋊C4×C73C2×C414yes—
55C2×C22⋊C4×C73C2×C414yes—
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73C2×C411yescharacteristic
58C2×C42×C73C2×C414yes—
59C2×C42×C73C2×C414yes—
60C2×C42×C73C2314yes—
61C2×C42×C73C2311yescharacteristic
62C2×C42×C73C2311yescharacteristic
63#95×C73C2×C414yes—
64#95×C73C2×C414yes—
65#95×C73C2×C414yes—
66#95×C73C2×C414yes—
67C2×#10596831—24no—
68C2×#10596831—24no—
69C2×#10596831—22no—
70C2×#10596831—22no—
71C2×#10596831—28no—
72C2×#10596831—28no—
73C2×#10596831C22×C412yes—
74C2×#10596831C22×C412yes—
75C2×#10596831C22×C418yes—
76C2×#10596831C22×C412yes—
77C2×#10596831C22×C412yes—
78C2×#10596831#4114yes—
79C2×#10596831#4114yes—
80C24×C73C22×C411yescharacteristic
81C22×C4×C73—24no—
82C22×C4×C73—24no—
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73C4214yes—
86C22×C4×C73C4214yes—
87C22×C4×C73C22×C412yes—
88C22×C4×C73C22×C412yes—
89C22×C4×C73C22×C418yes—
90C22×C4×C73C22×C411yescharacteristic
91C22×C4×C73C22×C411yescharacteristic
92C22×C4×C73C22×C411yescharacteristic
93C22×C4×C73C22×C411yescharacteristic
94C22×C4×C73C22×C411yescharacteristic
95C22×C4×C73C2411yescharacteristic
96C4⋊C4×C73—28no—
97C22⋊C4×C73—28no—
98C42×C73—24no—
99C42×C73—24no—
100C42×C73C4214yes—
101C42×C73C4214yes—
102C42×C73C22×C412yes—
103C42×C73C22×C412yes—
104C42×C73C22×C412yes—
105C42×C73C22×C412yes—
106C23×C73—22no—
107C23×C73—22no—
108C23×C73C2×C4212yes—
109C23×C73#13114yes—
110C23×C73C23×C411yescharacteristic
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73—28no—
116C2×C4×C73—22no—
117C2×C4×C73—22no—
118C2×C4×C73—24no—
119C2×C4×C73—28no—
120C2×C4×C73—28no—
121C2×C4×C73—28no—
122C2×C4×C73—28no—
123C2×C4×C73C2×C4212yes—
124C2×C4×C73C2×C4212yes—
125C2×C4×C73C2×C4212yes—
126C2×C4×C73C2×C4212yes—
127C2×C4×C73C2×C4212yes—
128C2×C4×C73#13114yes—
129C2×C4×C73C23×C411yescharacteristic
130C2×C4×C73C23×C411yescharacteristic
131C2×C4×C73C2×#4112yes—
132C2×C4×C73C2×#4112yes—
133#10596831—22no—
134#10596831—22no—
135#10596831—28no—
136#10596831#11714yes—
137#10596831#11714yes—
138C22×C73—22no—
139C22×C73—28no—
140C22×C73—22no—
141C22×C73#40514yes—
142C22×C73C22×C4211yescharacteristic
143C22×C73C4×#4112yes—
144C22×C73C2×#13112yes—
145C22×C73C2×#13112yes—
146C4×C73—22no—
147C4×C73—22no—
148C4×C73—22no—
149C4×C73—22no—
150C4×C73—48no—
151C4×C73#40514yes—
152C4×C73C2×#11711yescharacteristic
153C4×C73C2×#11711yescharacteristic
154C4×C73C4×#4112yes—
155#20476—731nomaximal, Sylow
156C2×C73—24no—
157C2×C73C4×#11711yescharacteristic
158C2×C73C2×#40511yescharacteristic
159C2×C73C2×#40511yescharacteristic
160C2×C73#296614yes—
161C2×#432—731no—
162C22×C4×C8—731no—
163#2283D7311yescharacteristic
164#1770—732no—
165C8×C4⋊C4—732no—
166#1752—732no—
167C8×C22⋊C4—732no—
168#1746—732no—
169#1726—732no—
170C73#2047611yescharacteristic, complemented, Sylow
171C23×C8—731no—
172C23×C8—731no—
173C22×C42C2×D7311yescharacteristic
174#432—738no—
175C2×C4⋊C8—732no—
176C2×C22⋊C8—732no—
177C2×#97—732no—
178C2×#97—734no—
179C2×C4×C8—738no—
180C2×C4×C8—731no—
181C2×C4×C8—731no—
182C2×C4×C8—731no—
183C2×C4×C8—731no—
184C2×C4×C8—734no—
185#383Dic7312yes—
186#382Dic7312yes—
187#381Dic7312yes—
188#380Dic7312yes—
189C4×C4⋊C4C2×D7312yes—
190C4×C22⋊C4C2×D7312yes—
191#376C2×D7312yes—
192#336—734no—
193#336—734no—
194C23×C4C2×Dic7311yescharacteristic
195C23×C4C2×Dic7311yescharacteristic
196C23×C4C22×D7311yescharacteristic
197C22×C8—731no—
198C22×C8—731no—
199C22×C8—734no—
200C22×C8—734no—
201C22×C8—731no—
202C22×C8—731no—
203C22×C8—734no—
204C22×C8—1464no—
205C22×C8—1464no—
206C2×C4⋊C4C2×Dic7314yes—
207C2×C22⋊C4C2×Dic7314yes—
208C2×C42C4×D7314yes—
209C2×C42C4×D7314yes—
210C2×C42C2×Dic7311yescharacteristic
211C2×C42C2×Dic7311yescharacteristic
212C2×C42C22×D7314yes—
213C2×C42C22×D7311yescharacteristic
214C2×C42C22×D7311yescharacteristic
215C4⋊C8—738no—
216C22⋊C8—738no—
217#97—738no—
218#97—1468no—
219C4×C8—734no—
220C4×C8—734no—
221C4×C8—734no—
222C4×C8—734no—
223C4×C8—1468no—
224#95C2×Dic7314yes—
225#95C2×Dic7314yes—
226#95C2×Dic7314yes—
227#95C2×Dic7314yes—
228C24C22×Dic7311yescharacteristic
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4C4×Dic7314yes—
234C22×C4C4×Dic7314yes—
235C22×C4C2×C4×D7312yes—
236C22×C4C2×C4×D7312yes—
237C22×C4C22×Dic7318yes—
238C22×C4C22×Dic7311yescharacteristic
239C22×C4C22×Dic7311yescharacteristic
240C22×C4C22×Dic7311yescharacteristic
241C22×C4C22×Dic7311yescharacteristic
242C22×C4C22×Dic7311yescharacteristic
243C22×C4C23×D7311yescharacteristic
244C2×C8—732no—
245C2×C8—732no—
246C2×C8—734no—
247C2×C8—734no—
248C2×C8—732no—
249C2×C8—732no—
250C2×C8—738no—
251C2×C8—1468no—
252C2×C8—1468no—
253C2×C8—1464no—
254C2×C8—1464no—
255C2×C8—1462no—
256C2×C8—1462no—
257C4⋊C4—28no—
258C22⋊C4—28no—
259C42—24no—
260C42—24no—
261C42C4×Dic7314yes—
262C42C4×Dic7314yes—
263C42C2×C4×D7312yes—
264C42C2×C4×D7312yes—
265C42C2×C4×D7312yes—
266C42C2×C4×D7312yes—
267C23—22no—
268C23—22no—
269C23C2×C4×Dic7312yes—
270C23#1455659814yes—
271C23C23×Dic7311yescharacteristic
272C2×C4—28no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—28no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4C2×C4×Dic7312yes—
285C2×C4C2×C4×Dic7312yes—
286C2×C4C2×C4×Dic7312yes—
287C2×C4C2×C4×Dic7312yes—
288C2×C4C2×C4×Dic7312yes—
289C2×C4#1455659814yes—
290C2×C4C22×C4×D7311yescharacteristic
291C2×C4C22×C4×D7311yescharacteristic
292C2×C4#41×D7312yes—
293C2×C4#41×D7312yes—
294C8—734no—
295C8—734no—
296C8—1462no—
297C8—1462no—
298C8—1468no—
299C22—28no—
300C22—22no—
301C22—22no—
302C22#1798854314yes—
303C22C22×C4×Dic7311yescharacteristic
304C22#41×Dic7312yes—
305C22C2×#1455659812yes—
306C22C2×#1455659812yes—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—22no—
311C4—48no—
312C4#1798854314yes—
313C4#117×D7311yescharacteristic
314C4#117×D7311yescharacteristic
315C4#1798905512yes—
316C2—24no—
317C2#117×Dic7311yescharacteristic
318C2C2×#1798854311yescharacteristic
319C2C2×#1798854311yescharacteristic
320C2#2261890714yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C22×C42
FittingC73×#2283C2
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenter#2283D73
hypocenterC73#20476
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #2283

Complements

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

NH ≅ G/Ntype
3. #22609800C2semidirect
4. C4⋊C4×#10596831C2semidirect
7. #22609683C2semidirect
8. #22609618C2semidirect
12. #17988644C4semidirect
17. C2×C4×#10596831C4semidirect
170. C73#20476semidirect

Maximal subgroups

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

subgroupindex
C2×#179886442
C22×C4×#105968312
#226098002
C4⋊C4×#105968312
#226097092
C22⋊C4×#105968312
#226096832
#226096182
C73×#22832
#2047673