Finite Group Databasebeta

← #34031875

Subgroups — #34031875

Order 18688.

1974 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626834C218yesmaximal, complemented
2#22626827C216yesmaximal, complemented
3#22626824C212yesmaximal, complemented
4Q8×#11924037C218yesmaximal, complemented
5#22626607C216yesmaximal
6C22×Q8×Dic73C211yesmaximal, characteristic
7#17989638C22148yescomplemented
8Q8×#10596833C22116yescomplemented
9#17989608C22112yes—
10#17989607C22112yes—
11#17989604C22112yes—
12#17989603C22112yes—
13#17989590C22112yes—
14#17989557C2214yes—
15C4×#11924037C22112yes—
16C2×Q8×Dic73C414yes—
17C2×Q8×Dic73C414yes—
18C2×Q8×Dic73C413yes—
19C2×Q8×Dic73C411yescharacteristic
20C2×Q8×Dic73C2211yescharacteristic
21C2×Q8×Dic73C2213yes—
22C2×Q8×Dic73C2214yes—
23C2×Q8×Dic73C2214yes—
24C22×C4⋊Dic73C413yes—
25C22×C4×Dic73C2213yes—
26C23×Q8×C73C411yescharacteristic
27C2×#11924037C2318yes—
28#14556658C23124yes—
29C4⋊#10596833C23124yes—
30C22⋊#10596833C2318yes—
31C22⋊#10596833C2318yes—
32#14556646C23124yes—
33C4×#10596833C23124yes—
34C23×Dic73C2×C413yes—
35C23×Dic73C2311yescharacteristic
36Q8×Dic73C2×C4112yes—
37Q8×Dic73C2×C4112yes—
38Q8×Dic73C2×C4112yes—
39Q8×Dic73C2×C414yes—
40Q8×Dic73C2×C414yes—
41Q8×Dic73C2×C414yes—
42Q8×Dic73C2314yes—
43Q8×Dic73C23112yes—
44C2×C4⋊Dic73C2×C4112yes—
45C2×C4⋊Dic73C2×C416yes—
46C2×C4⋊Dic73C2×C4112yes—
47C2×C4⋊Dic73C2×C413yes—
48C2×C4⋊Dic73C2×C413yes—
49C2×C4×Dic73C2×C416yes—
50C2×C4×Dic73C2×C413yes—
51C2×C4×Dic73C2×C416yes—
52C2×C4×Dic73C2×C413yes—
53C2×C4×Dic73C2316yes—
54C2×C4×Dic73C2313yes—
55C2×C4×Dic73C2316yes—
56C2×C4×Dic73C2313yes—
57C22×Q8×C73—28no—
58C22×Q8×C73C2×C414yes—
59C22×Q8×C73C2×C414yes—
60C22×Q8×C73C2×C413yes—
61C22×Q8×C73C2×C411yescharacteristic
62C23×C4×C73C2×C413yes—
63#11924037—216no—
64C2×#10596833C24116yes—
65C22×Dic73—24no—
66C22×Dic73—46no—
67C22×Dic73C22×C411yescharacteristic
68C22×Dic73C22×C411yescharacteristic
69C22×Dic73C22×C411yescharacteristic
70C22×Dic73C22×C416yes—
71C22×Dic73C22×C416yes—
72C22×Dic73C22×C413yes—
73C22×Dic73C22×C413yes—
74C22×Dic73C2411yescharacteristic
75C22×Dic73C2411yescharacteristic
76C22×Dic73C2411yescharacteristic
77C4⋊Dic73C22×C4124yes—
78C4⋊Dic73C22×C4112yes—
79C4⋊Dic73C22×C4112yes—
80C4×Dic73C22×C416yes—
81C4×Dic73C22×C416yes—
82C4×Dic73C22×C416yes—
83C4×Dic73C22×C416yes—
84C4×Dic73C22×C416yes—
85C4×Dic73C22×C416yes—
86C4×Dic73C2416yes—
87C4×Dic73C2416yes—
88C24×C73C22×C411yescharacteristic
89C2×Q8×C73—216no—
90C2×Q8×C73—216no—
91C2×Q8×C73—212no—
92C2×Q8×C73M1618yescomplemented
93C2×Q8×C73C22×C4112yes—
94C2×Q8×C73C22×C414yes—
95C22×C4×C73—212no—
96C22×C4×C73C22×C413yes—
97C22×C4×C73C22×C416yes—
98C22×C4×C73C22×C416yes—
99C22×C4×C73C22×C413yes—
100C23×C73—22no—
101C23×C73—22no—
102C23×C73—22no—
103C23×C73C23×C411yescharacteristic
104C23×C73C23×C411yescharacteristic
105C23×C73C23×C411yescharacteristic
106Q8×C73—224no—
107Q8×C73C2×M16116yes—
108C2×C4×C73—26no—
109C2×C4×C73—212no—
110C2×C4×C73—212no—
111C2×C4×C73C2×M16112yes—
112C2×C4×C73C23×C416yes—
113C2×C4×C73C23×C416yes—
114C2×Dic73—22no—
115C2×Dic73—212no—
116C2×Dic73—22no—
117C2×Dic73—22no—
118C2×Dic73—22no—
119C2×Dic73—22no—
120C2×Dic73—46no—
121C2×Dic73—43no—
122C2×Dic73—43no—
123C2×Dic73C23×C411yescharacteristic
124C2×Dic73C23×C411yescharacteristic
125C2×Dic73C23×C416yes—
126C2×Dic73C23×C416yes—
127C2×Dic73C23×C411yescharacteristic
128C2×Dic732−1+414yes—
129C2×Dic73C2511yescharacteristic
130#10596833—216no—
131C22×C73—28no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73#51912yes—
139C22×C73C22×M1612yes—
140C22×C73#56812yes—
141C22×C73C24×C411yescharacteristic
142C4×C73—26no—
143C4×C73C22×M16112yes—
144Dic73—21no—
145Dic73—21no—
146Dic73—26no—
147Dic73—26no—
148Dic73#51912yes—
149Dic73C2×2−1+412yes—
150#60248—731nomaximal, Sylow
151C2×C73—22no—
152C2×C73—22no—
153C2×C73—22no—
154C2×C73C2×#51911yescharacteristic
155C2×C73C23×M1611yescharacteristic
156C2×C73C2×#56811yescharacteristic
157C22×C4×Q8—731no—
158#2984—738no—
159#2983—736no—
160#2982—732no—
161Q8×M16—738no—
162#2920—736no—
163C73#6024811yescharacteristic, complemented, Sylow
164C23×Q8#6440811yescharacteristic
165C2×C4×Q8—734no—
166C2×C4×Q8—731no—
167C2×C4×Q8—733no—
168C2×C4×Q8—734no—
169C2×C4×Q8—734no—
170C2×C4×Q8—733no—
171C2×C4×Q8—734no—
172C2×C4×Q8—731no—
173C22×C4⋊C4—733no—
174C22×C42—733no—
175#446—7348no—
176C8×Q8—7316no—
177#432—7312no—
178#432—7312no—
179#431—7312no—
180#431—7312no—
181#423—7312no—
182#407—734no—
183C4×M16—7312no—
184C22×Q8—28no—
185C22×Q8C2×#6440814yes—
186C22×Q8C2×#6440814yes—
187C22×Q8C2×#6440813yes—
188C22×Q8C2×#6440811yescharacteristic
189C23×C4—733no—
190C23×C4—731no—
191C23×C4C2×#6440813yes—
192C2×M16—738no—
193C4×Q8—7312no—
194C4×Q8—7312no—
195C4×Q8—7312no—
196C4×Q8—734no—
197C4×Q8—734no—
198C4×Q8—734no—
199C4×Q8—734no—
200C4×Q8—7312no—
201C2×C4⋊C4—7312no—
202C2×C4⋊C4—736no—
203C2×C4⋊C4—733no—
204C2×C4⋊C4—733no—
205C2×C4⋊C4—7312no—
206C2×C42—733no—
207C2×C42—736no—
208C2×C42—733no—
209C2×C42—736no—
210C2×C42—736no—
211C2×C42—733no—
212C2×C42—736no—
213C2×C42—733no—
214C4⋊C8—7324no—
215C4⋊C8—7324no—
216C22⋊C8—738no—
217C22⋊C8—738no—
218#97—7324no—
219C4×C8—7324no—
220C24C22×#6440811yescharacteristic
221C2×Q8—216no—
222C2×Q8—216no—
223C2×Q8—212no—
224C2×Q8#1192403718yescomplemented
225C2×Q8C22×#64408112yes—
226C2×Q8C22×#6440814yes—
227C22×C4—212no—
228C22×C4—2926no—
229C22×C4—736no—
230C22×C4—733no—
231C22×C4—733no—
232C22×C4—731no—
233C22×C4—731no—
234C22×C4—731no—
235C22×C4—731no—
236C22×C4—736no—
237C22×C4—731no—
238C22×C4—731no—
239C22×C4—1464no—
240C22×C4C22×#6440813yes—
241C22×C4C22×#6440816yes—
242C22×C4C22×#6440816yes—
243C22×C4C22×#6440813yes—
244M16—14616no—
245C2×C8—7316no—
246C4⋊C4—7324no—
247C4⋊C4—7312no—
248C4⋊C4—7312no—
249C42—736no—
250C42—736no—
251C42—736no—
252C42—736no—
253C42—736no—
254C42—736no—
255C42—736no—
256C42—736no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23C23×#6440811yescharacteristic
261C23C23×#6440811yescharacteristic
262C23C23×#6440811yescharacteristic
263Q8—224no—
264Q8C2×#11924037116yes—
265C2×C4—26no—
266C2×C4—212no—
267C2×C4—212no—
268C2×C4—2926no—
269C2×C4—2923no—
270C2×C4—2923no—
271C2×C4—734no—
272C2×C4—731no—
273C2×C4—731no—
274C2×C4—731no—
275C2×C4—731no—
276C2×C4—736no—
277C2×C4—736no—
278C2×C4—1462no—
279C2×C4—1462no—
280C2×C4—1462no—
281C2×C4—1462no—
282C2×C4—1462no—
283C2×C4—14612no—
284C2×C4C2×#11924037112yes—
285C2×C4C23×#6440816yes—
286C2×C4C23×#6440816yes—
287C8—14616no—
288C22—28no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22#1798965512yes—
296C22C22×#1192403712yes—
297C22#1798966712yes—
298C22C24×#6440811yescharacteristic
299C4—26no—
300C4—732no—
301C4—732no—
302C4—1461no—
303C4—1461no—
304C4—1466no—
305C4—1466no—
306C4C22×#11924037112yes—
307C2—22no—
308C2—22no—
309C2—22no—
310C2C2×#1798965511yescharacteristic
311C2C23×#1192403711yescharacteristic
312C2C2×#1798966711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
FittingC23×Q8×C73C4
prime radicalC24×C73C22×C4
cosocleC2×Dic73C25
hypercenterC23×Q8#64408
hypocenterC73#60248
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. #22626834C2semidirect
2. #22626827C2semidirect
3. #22626824C2semidirect
4. Q8×#11924037C2semidirect
7. #17989638C22semidirect
8. Q8×#10596833C22semidirect
92. C2×Q8×C73M16semidirect
163. C73#60248semidirect
224. C2×Q8#11924037semidirect

Maximal subgroups

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

subgroupindex
#226268342
#226268272
#226268242
Q8×#119240372
#226266072
C22×Q8×Dic732
#6024873