Finite Group Databasebeta

← C22×#5124 #11658879

Subgroups — C22×#5124 #11658879

Order 768.

2718 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic
2C23×#208C211yesmaximal, characteristic
3C23×#208C211yesmaximal, characteristic
4C2×#5124C2124yesmaximal, direct product factor
5C22×#822C214yesmaximal, complemented
6C22×#430—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×M16—21no—
9C22×C3×M16—26no—
10C22×C3×M16C2212yes—
11C22×C3×M16C22112yes—
12C23×C3×C8—21no—
13C22×#208—21no—
14C22×#208—26no—
15C22×#208C412yes—
16C22×#208C412yes—
17C22×#208C4112yes—
18C22×#208C2212yes—
19C22×#208C22112yes—
20C22×#208C22112yes—
21C23×C3⋊C8—21no—
22C23×C3⋊C8C2212yes—
23#5124C22164yesdirect product factor
24C2×#822C22148yescomplemented
25C23×M16—31no—
26C23×M16—31no—
27C23×M16D311yescharacteristic
28C2×#430—324no—
29C22×#108—34no—
30C25×C3Q811yescharacteristic
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C416yes—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D416yes—
37C23×C3×C4Q811yescharacteristic
38C23×C3×C4Q816yes—
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2316yes—
41C2×C3×M16—212no—
42C2×C3×M16—26no—
43C2×C3×M16—224no—
44C2×C3×M16C23112yes—
45C2×C3×M16C23116yes—
46C22×C3×C8—212no—
47C22×C3×C8—22no—
48C2×#208—212no—
49C2×#208—26no—
50C2×#208—224no—
51C2×#208C2×C4124yes—
52C2×#208C2×C4124yes—
53C2×#208C2×C4148yes—
54C2×#208C23112yes—
55C2×#208C23116yes—
56C2×#208C23116yes—
57C22×C3⋊C8—212no—
58C22×C3⋊C8—22no—
59C22×C3⋊C8C2×C4112yes—
60C22×C3⋊C8C2×C414yes—
61C22×C3⋊C8C23112yes—
62#822C23164yescomplemented
63C24×C4C2×D311yescharacteristic
64C22×M16—26no—
65C22×M16—21no—
66C22×M16—32no—
67C22×M16—32no—
68C22×M16—312no—
69C22×M16—312no—
70C22×M16—32no—
71C22×M16—312no—
72C22×M16—66no—
73C22×M16—61no—
74C22×M16C2×D312yes—
75C22×M16C2×D3112yes—
76C23×C8—21no—
77C23×C8—32no—
78C23×C8—61no—
79#430—364no—
80C2×#108—348no—
81C24×C3—44no—
82C24×C3C4⋊C411yescharacteristic
83C24×C3C4⋊C411yescharacteristic
84C24×C3C4⋊C416yes—
85C24×C3C2×Q811yescharacteristic
86C24×C3C2×Q816yes—
87C22×C3×C4C4⋊C416yes—
88C22×C3×C4C4⋊C416yes—
89C22×C3×C4C4⋊C416yes—
90C22×C3×C4C4⋊C416yes—
91C22×C3×C4C4⋊C416yes—
92C22×C3×C4C4⋊C4124yes—
93C22×C3×C4C4⋊C411yescharacteristic
94C22×C3×C4C4⋊C411yescharacteristic
95C22×C3×C4C22×C413yes—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C22×C416yes—
98C22×C3×C4C22×C416yes—
99C22×C3×C4C22×C4112yes—
100C22×C3×C4C2×D416yes—
101C22×C3×C4C2×D416yes—
102C22×C3×C4C2×D413yes—
103C22×C3×C4C2×D4112yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×Q813yes—
106C22×C3×C4C2×Q816yes—
107C22×C3×C4C2×Q8112yes—
108C22×C3×C4C2413yes—
109C22×C3×C4C2414yes—
110C3×M16—216no—
111C3×M16—212no—
112C3×M16C2418yes—
113C2×C3×C8—212no—
114C2×C3×C8—216no—
115#208—216no—
116#208—212no—
117#208C22×C4132yes—
118#208C22×C4132yes—
119#208C2418yes—
120C2×C3⋊C8—212no—
121C2×C3⋊C8—216no—
122C2×C3⋊C8C22×C4124yes—
123C2×C3⋊C8C22×C4124yes—
124C2×C3⋊C8C2418yes—
125C25Dic611yescharacteristic
126C23×C4Dic611yescharacteristic
127C23×C4Dic616yes—
128C23×C4C4×D311yescharacteristic
129C23×C4C4×D311yescharacteristic
130C23×C4C4×D316yes—
131C23×C4#6716yes—
132C23×C4#6711yescharacteristic
133C23×C4#6711yescharacteristic
134C23×C4C22×D316yes—
135C23×C4C22×D311yescharacteristic
136C2×M16—212no—
137C2×M16—26no—
138C2×M16—224no—
139C2×M16—324no—
140C2×M16—324no—
141C2×M16—348no—
142C2×M16—316no—
143C2×M16—312no—
144C2×M16—316no—
145C2×M16—612no—
146C2×M16—66no—
147C2×M16—624no—
148C2×M16C22×D3112yes—
149C2×M16C22×D3116yes—
150C22×C8—212no—
151C22×C8—22no—
152C22×C8—34no—
153C22×C8—312no—
154C22×C8—312no—
155C22×C8—612no—
156C22×C8—62no—
157#108—364no—
158C23×C3—24no—
159C23×C3—24no—
160C23×C3—24no—
161C23×C3—412no—
162C23×C3—412no—
163C23×C3C2×C4⋊C413yes—
164C23×C3C2×C4⋊C416yes—
165C23×C3C2×C4⋊C416yes—
166C23×C3C2×C4⋊C4112yes—
167C23×C3C2×C4⋊C411yescharacteristic
168C23×C3C22×Q813yes—
169C23×C3C22×Q814yes—
170C2×C3×C4C2×C4⋊C416yes—
171C2×C3×C4C2×C4⋊C4112yes—
172C2×C3×C4C2×C4⋊C4112yes—
173C2×C3×C4C2×C4⋊C4112yes—
174C2×C3×C4C2×C4⋊C4112yes—
175C2×C3×C4C2×C4⋊C4112yes—
176C2×C3×C4C2×C4⋊C413yes—
177C2×C3×C4C2×C4⋊C416yes—
178C2×C3×C4C2×C4⋊C416yes—
179C2×C3×C4C2×C4⋊C413yes—
180C2×C3×C4C23×C413yes—
181C2×C3×C4C23×C413yes—
182C2×C3×C4C23×C414yes—
183C2×C3×C4C23×C414yes—
184C2×C3×C4C22×D413yes—
185C2×C3×C4C22×D414yes—
186C2×C3×C4C22×D414yes—
187C2×C3×C4C22×D413yes—
188C2×C3×C4C22×Q813yes—
189C2×C3×C4C22×Q814yes—
190C2×C3×C4C2511yescharacteristic
191C3×C8—28no—
192C3⋊C8—28no—
193C3⋊C8C23×C4116yes—
194C24—44no—
195C24Dic3⋊C411yescharacteristic
196C24Dic3⋊C411yescharacteristic
197C24Dic3⋊C416yes—
198C24C2×Dic611yescharacteristic
199C24C2×Dic616yes—
200C22×C4Dic3⋊C416yes—
201C22×C4Dic3⋊C416yes—
202C22×C4Dic3⋊C416yes—
203C22×C4Dic3⋊C4124yes—
204C22×C4Dic3⋊C411yescharacteristic
205C22×C4Dic3⋊C411yescharacteristic
206C22×C4Dic3⋊C416yes—
207C22×C4Dic3⋊C416yes—
208C22×C4C2×Dic616yes—
209C22×C4C2×Dic6112yes—
210C22×C4C2×Dic613yes—
211C22×C4C2×C4×D311yescharacteristic
212C22×C4C2×C4×D313yes—
213C22×C4C2×C4×D316yes—
214C22×C4C2×C4×D316yes—
215C22×C4C2×C4×D3112yes—
216C22×C4C2×#6716yes—
217C22×C4C2×#6716yes—
218C22×C4C2×#6713yes—
219C22×C4C2×#67112yes—
220C22×C4C2×#6711yescharacteristic
221C22×C4C23×D313yes—
222C22×C4C23×D314yes—
223M16—216no—
224M16—212no—
225M16—332no—
226M16—332no—
227M16—38no—
228M16—616no—
229M16—612no—
230M16C23×D318yes—
231C2×C8—212no—
232C2×C8—216no—
233C2×C8—324no—
234C2×C8—324no—
235C2×C8—38no—
236C2×C8—612no—
237C2×C8—616no—
238C22×C3—212no—
239C22×C3—212no—
240C22×C3—26no—
241C22×C3—26no—
242C22×C3—412no—
243C22×C3—44no—
244C22×C3#43014yescomplemented
245C22×C3C22×C4⋊C413yes—
246C22×C3C22×C4⋊C414yes—
247C22×C3C22×C4⋊C414yes—
248C22×C3C22×C4⋊C413yes—
249C22×C3C23×Q811yescharacteristic
250C3×C4C22×C4⋊C413yes—
251C3×C4C22×C4⋊C413yes—
252C3×C4C22×C4⋊C414yes—
253C3×C4C22×C4⋊C414yes—
254C3×C4C24×C411yescharacteristic
255C3×C4C23×D411yescharacteristic
256C23—24no—
257C23—24no—
258C23—24no—
259C23—412no—
260C23—412no—
261C23C2×Dic3⋊C413yes—
262C23C2×Dic3⋊C416yes—
263C23C2×Dic3⋊C416yes—
264C23C2×Dic3⋊C4112yes—
265C23C2×Dic3⋊C411yescharacteristic
266C23C22×Dic613yes—
267C23C22×Dic614yes—
268C2×C4C2×Dic3⋊C416yes—
269C2×C4C2×Dic3⋊C4112yes—
270C2×C4C2×Dic3⋊C4112yes—
271C2×C4C2×Dic3⋊C4112yes—
272C2×C4C2×Dic3⋊C4112yes—
273C2×C4C2×Dic3⋊C4112yes—
274C2×C4C2×Dic3⋊C413yes—
275C2×C4C2×Dic3⋊C416yes—
276C2×C4C2×Dic3⋊C416yes—
277C2×C4C2×Dic3⋊C413yes—
278C2×C4C22×Dic613yes—
279C2×C4C22×Dic614yes—
280C2×C4C22×C4×D313yes—
281C2×C4C22×C4×D313yes—
282C2×C4C22×C4×D314yes—
283C2×C4C22×C4×D314yes—
284C2×C4C22×#6713yes—
285C2×C4C22×#6714yes—
286C2×C4C22×#6714yes—
287C2×C4C22×#6713yes—
288C2×C4C24×D311yescharacteristic
289C8—28no—
290C8—316no—
291C8—68no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—23no—
295C2×C3—21no—
296C2×C3C2×#43016yescomplemented
297C2×C3C23×C4⋊C411yescharacteristic
298C22—212no—
299C22—212no—
300C22—26no—
301C22—26no—
302C22—412no—
303C22—44no—
304C22#512414yesdirect product factor
305C22C22×Dic3⋊C413yes—
306C22C22×Dic3⋊C414yes—
307C22C22×Dic3⋊C414yes—
308C22C22×Dic3⋊C413yes—
309C22C23×Dic611yescharacteristic
310C4C22×Dic3⋊C413yes—
311C4C22×Dic3⋊C414yes—
312C4C22×Dic3⋊C414yes—
313C4C22×Dic3⋊C413yes—
314C4C23×C4×D311yescharacteristic
315C4C23×#6711yescharacteristic
316C3C22×#43011yescharacteristic, complemented, Sylow
317C2—21no—
318C2—24no—
319C2—24no—
320C2—23no—
321C2C2×#512416yesdirect product factor
322C2C23×Dic3⋊C411yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC23×C3×M16C2
socleC23×C3C2×C4⋊C4
prime radicalC25×C3Q8
cosocleC2×C3×C4C25
hypercenterC23×M16D3
hypocenterC3C22×#430
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
4. C2×#5124C2direct (N × H)
5. C22×#822C2semidirect
23. #5124C22direct (N × H)
24. C2×#822C22semidirect
62. #822C23semidirect
244. C22×C3#430semidirect
296. C2×C3C2×#430semidirect
304. C22#5124direct (N × H)
316. C3C22×#430semidirect
321. C2C2×#5124direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×M162
C23×#2082
C2×#51242
C22×#8222
C22×#4303