Finite Group Databasebeta

← #33245459

Subgroups — #33245459

Order 18688.

2446 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612896C216yesmaximal, complemented
2#22612852C212yesmaximal, complemented
3#22612800C216yesmaximal
4Q8×C22⋊Dic73C212yesmaximal
5#22612409C216yesmaximal, complemented
6#22612329C213yesmaximal
7C4×Q8×Dic73C212yesmaximal, complemented
8C4×#14556550C213yesmaximal
9C22×C4×Q8×C73C211yesmaximal, characteristic
10C2×Q8×Dic73C2214yes—
11C2×#14556550C2216yes—
12#17988494C22112yes—
13#17988486C22112yes—
14#17988473C22124yes—
15#17988455C2216yes—
16#17988453C2216yes—
17#17988450C2216yes—
18#17988447C2216yes—
19C4⋊C4×Dic73C22112yes—
20C4×C4⋊Dic73C2216yes—
21C4×C4⋊Dic73C2216yes—
22C4×C4⋊Dic73C2216yes—
23C4×C22⋊Dic73C2216yes—
24C4×C22⋊Dic73C2211yescharacteristic
25C4×C22⋊Dic73C2213yes—
26#17988433C2216yes—
27#17988433C2216yes—
28C42×Dic73C2216yes—
29C23×Q8×C73C2211yescharacteristic
30C2×C4×Q8×C73C4116yescomplemented
31C2×C4×Q8×C73C2211yescharacteristic
32C2×C4×Q8×C73C2211yescharacteristic
33C2×C4×Q8×C73C2213yes—
34C2×C4×Q8×C73C2213yes—
35C22×C4⋊C4×C73C2213yes—
36C22×C42×C73C2213yes—
37Q8×Dic73—216no—
38#14556550—224no—
39C2×C4⋊Dic73C23112yes—
40C2×C4⋊Dic73C23112yes—
41C2×C22⋊Dic73C2316yes—
42C2×C22⋊Dic73C2312yes—
43C2×C4×Dic73C2312yes—
44C2×C4×Dic73C2316yes—
45C2×C4×Dic73C23112yes—
46C2×C4×Dic73C23112yes—
47#14556498C2312yes—
48#14556498C2316yes—
49#14556498C23112yes—
50#14556498C23112yes—
51#14556498C23112yes—
52#14556498C23112yes—
53C22×Q8×C73—28no—
54C22×Q8×C73C2×C418yes—
55C22×Q8×C73C2313yes—
56C22×Q8×C73C2311yescharacteristic
57C23×C4×C73C2311yescharacteristic
58C23×C4×C73C2313yes—
59C23×C4×C73C2313yes—
60C4×Q8×C73C2×C4124yes—
61C4×Q8×C73C2×C4124yes—
62C4×Q8×C73C2×C418yes—
63C4×Q8×C73C2×C418yes—
64C2×C4⋊C4×C73C2×C4124yes—
65C2×C4⋊C4×C73C2313yes—
66C2×C4⋊C4×C73C2316yes—
67C2×C4⋊C4×C73C2313yes—
68C2×C42×C73C2×C4124yes—
69C2×C42×C73C2313yes—
70C2×C42×C73C2313yes—
71C2×C42×C73C2313yes—
72C2×C42×C73C2313yes—
73C22×Dic73C24112yes—
74C22×Dic73C2414yes—
75C4⋊Dic73—224no—
76C4⋊Dic73—224no—
77C22⋊Dic73—212no—
78C22⋊Dic73#4118yes—
79C4×Dic73—224no—
80C4×Dic73—24no—
81C4×Dic73—212no—
82C4×Dic73—224no—
83C24×C73C2411yescharacteristic
84C2×Q8×C73—232no—
85C2×Q8×C73—212no—
86C2×Q8×C73C22×C4112yes—
87C2×Q8×C73C22×C414yes—
88C2×Q8×C73#4118yes—
89C22×C4×C73—212no—
90C22×C4×C73—212no—
91C22×C4×C73—24no—
92C22×C4×C73C22×C414yes—
93C22×C4×C73C22×C4112yes—
94C22×C4×C73C22×C4112yes—
95C22×C4×C73C2411yescharacteristic
96C22×C4×C73C2413yes—
97C22×C4×C73C2413yes—
98C22×C4×C73C2413yes—
99C22×C4×C73C2413yes—
100C22×C4×C73C2411yescharacteristic
101C4⋊C4×C73C22×C4124yes—
102C4⋊C4×C73C22×C4112yes—
103C4⋊C4×C73C22×C4112yes—
104C42×C73C22×C4112yes—
105C42×C73C22×C4112yes—
106C42×C73C22×C4112yes—
107C42×C73C22×C4112yes—
108C23×C73—22no—
109C23×C73—22no—
110C23×C73C23×C412yes—
111C23×C73C2×#4114yes—
112C23×C73C2511yescharacteristic
113Q8×C73—224no—
114Q8×C73#117116yescomplemented
115C2×C4×C73—224no—
116C2×C4×C73—26no—
117C2×C4×C73—26no—
118C2×C4×C73—26no—
119C2×C4×C73—22no—
120C2×C4×C73—224no—
121C2×C4×C73—28no—
122C2×C4×C73C23×C412yes—
123C2×C4×C73C23×C416yes—
124C2×C4×C73C23×C416yes—
125C2×C4×C73C23×C416yes—
126C2×C4×C73C23×C416yes—
127C2×C4×C73C23×C412yes—
128C2×C4×C73C2×#41112yes—
129C2×C4×C732−1+414yes—
130C2×Dic73—24no—
131C2×Dic73—24no—
132C2×Dic73—212no—
133C2×Dic73—212no—
134C2×Dic73—212no—
135C2×Dic73C2×#4118yes—
136C22×C73—22no—
137C22×C73—22no—
138C22×C73—28no—
139C22×C73C4×#4114yes—
140C22×C73C24×C411yescharacteristic
141C22×C73C22×#4112yes—
142C22×C73C22×#4112yes—
143C22×C73C2×2−1+412yes—
144C4×C73—26no—
145C4×C73—26no—
146C4×C73—26no—
147C4×C73—22no—
148C4×C73C2×#117112yes—
149C4×C73#51914yes—
150Dic73—28no—
151Dic73—412no—
152#34128—731nomaximal, Sylow
153C2×C73—24no—
154C2×C73C22×#11711yescharacteristic
155C2×C73C2×C4×#4111yescharacteristic
156C2×C73C2×#51911yescharacteristic
157C2×C73#348214yes—
158C22×C4×Q8D7311yescharacteristic
159#2356—736no—
160#2351—732no—
161#2344—736no—
162Q8×C22⋊C4—732no—
163#2295—736no—
164#2283—733no—
165C42×Q8—732no—
166C4×#117—733no—
167C73#3412811yescharacteristic, complemented, Sylow
168C23×Q8C2×D7311yescharacteristic
169C2×C4×Q8—734no—
170C2×C4×Q8Dic73116yescomplemented
171C2×C4×Q8C2×D7311yescharacteristic
172C2×C4×Q8C2×D7313yes—
173C2×C4×Q8C2×D7311yescharacteristic
174C2×C4×Q8C2×D7313yes—
175C2×#117—736no—
176C22×C4⋊C4C2×D7313yes—
177C22×C42C2×D7313yes—
178#391—7312no—
179#389—7312no—
180#387—7324no—
181#383—736no—
182#382—736no—
183#381—736no—
184#380—736no—
185C4×C4⋊C4—7312no—
186C4×C4⋊C4—736no—
187C4×C4⋊C4—736no—
188C4×C4⋊C4—736no—
189C4×C22⋊C4—736no—
190C4×C22⋊C4—731no—
191C4×C22⋊C4—733no—
192#376—736no—
193#376—736no—
194C43—736no—
195C22×Q8—28no—
196C22×Q8C2×Dic7318yes—
197C22×Q8C22×D7313yes—
198C22×Q8C22×D7311yescharacteristic
199C23×C4C22×D7313yes—
200C23×C4C22×D7313yes—
201C23×C4C22×D7311yescharacteristic
202C4×Q8—14616no—
203C4×Q8C2×Dic73124yes—
204C4×Q8C2×Dic73124yes—
205C4×Q8C2×Dic7318yes—
206C4×Q8C2×Dic7318yes—
207#117—14624no—
208C2×C4⋊C4—7312no—
209C2×C4⋊C4—7312no—
210C2×C4⋊C4C2×Dic73124yes—
211C2×C4⋊C4C22×D7313yes—
212C2×C4⋊C4C22×D7316yes—
213C2×C4⋊C4C22×D7313yes—
214C2×C22⋊C4—732no—
215C2×C22⋊C4—736no—
216C2×C42—7312no—
217C2×C42—7312no—
218C2×C42—732no—
219C2×C42—736no—
220C2×C42C2×Dic73124yes—
221C2×C42C22×D7313yes—
222C2×C42C22×D7313yes—
223C2×C42C22×D7313yes—
224C2×C42C22×D7313yes—
225#95—732no—
226#95—736no—
227#95—7312no—
228#95—7312no—
229#95—7312no—
230#95—7312no—
231C24C23×D7311yescharacteristic
232C2×Q8—232no—
233C2×Q8—212no—
234C2×Q8C22×Dic73112yes—
235C2×Q8C22×Dic7314yes—
236C2×Q8#1192402818yes—
237C22×C4—212no—
238C22×C4—212no—
239C22×C4—24no—
240C22×C4—734no—
241C22×C4—7312no—
242C22×C4C22×Dic7314yes—
243C22×C4C22×Dic73112yes—
244C22×C4C22×Dic73112yes—
245C22×C4C23×D7311yescharacteristic
246C22×C4C23×D7313yes—
247C22×C4C23×D7313yes—
248C22×C4C23×D7313yes—
249C22×C4C23×D7313yes—
250C22×C4C23×D7311yescharacteristic
251C4⋊C4—14624no—
252C4⋊C4—14624no—
253C4⋊C4C22×Dic73124yes—
254C4⋊C4C22×Dic73112yes—
255C4⋊C4C22×Dic73112yes—
256C22⋊C4—738no—
257C22⋊C4—14612no—
258C42—14624no—
259C42—14624no—
260C42—14612no—
261C42—1464no—
262C42C22×Dic73112yes—
263C42C22×Dic73112yes—
264C42C22×Dic73112yes—
265C42C22×Dic73112yes—
266C23—22no—
267C23—22no—
268C23C23×Dic7312yes—
269C23#41×D7314yes—
270C23C24×D7311yescharacteristic
271Q8—224no—
272Q8#14556550116yescomplemented
273C2×C4—26no—
274C2×C4—26no—
275C2×C4—26no—
276C2×C4—22no—
277C2×C4—224no—
278C2×C4—224no—
279C2×C4—28no—
280C2×C4—738no—
281C2×C4—14612no—
282C2×C4—14612no—
283C2×C4—14612no—
284C2×C4—1464no—
285C2×C4—1464no—
286C2×C4C23×Dic7312yes—
287C2×C4C23×Dic7316yes—
288C2×C4C23×Dic7316yes—
289C2×C4C23×Dic7316yes—
290C2×C4C23×Dic7316yes—
291C2×C4C23×Dic7312yes—
292C2×C4C2×#11924028112yes—
293C2×C4#1455663914yes—
294C22—28no—
295C22—22no—
296C22—22no—
297C22#41×Dic7314yes—
298C22C24×Dic7311yescharacteristic
299C22C2×#41×D7312yes—
300C22C22×#1192402812yes—
301C22C2×#1455663912yes—
302C4—26no—
303C4—26no—
304C4—26no—
305C4—22no—
306C4—29212no—
307C4—1468no—
308C4C2×#14556550112yes—
309C4#1798906514yes—
310C2—24no—
311C2C22×#1455655011yescharacteristic
312C2C2×#41×Dic7311yescharacteristic
313C2C2×#1798906511yescharacteristic
314C2#2262354814yes—
subgroupG/H
centerC22×C4C23×D73
commutatorC22×C73C24×C4
FrattiniC23C24×D73
FittingC22×C4×Q8×C73C2
socle, cosocleC23×C73C25
prime radicalC24×C73C24
hypercenterC22×C4×Q8D73
hypocenterC73#34128
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22612896C2semidirect
2. #22612852C2semidirect
5. #22612409C2semidirect
7. C4×Q8×Dic73C2semidirect
30. C2×C4×Q8×C73C4semidirect
114. Q8×C73#117semidirect
167. C73#34128semidirect
170. C2×C4×Q8Dic73semidirect
272. Q8#14556550semidirect

Maximal subgroups

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

subgroupindex
#226128962
#226128522
#226128002
Q8×C22⋊Dic732
#226124092
#226123292
C4×Q8×Dic732
C4×#145565502
C22×C4×Q8×C732
#3412873