Finite Group Databasebeta

← C22×#11718785 #16726611

Subgroups — C22×#11718785 #16726611

Order 3328.

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×#6099C211yesmaximal, characteristic
2C23×#6099C211yesmaximal, characteristic
3C2×#11718785C2124yesmaximal, direct product factor
4C22×#88062C214yesmaximal, complemented
5C23×C13×M16C211yesmaximal, characteristic
6C24×C4×C13C2211yescharacteristic
7C22×C13×M16—21no—
8C22×C13×M16—26no—
9C22×C13×M16C2212yes—
10C22×C13×M16C22112yes—
11C23×C8×C13—21no—
12C22×#6099—21no—
13C22×#6099—26no—
14C22×#6099C412yes—
15C22×#6099C412yes—
16C22×#6099C4112yes—
17C22×#6099C2212yes—
18C22×#6099C22112yes—
19C22×#6099C22112yes—
20C23×#1055—21no—
21C23×#1055C2212yes—
22#11718785C22164yesdirect product factor
23C2×#88062C22148yescomplemented
24C25×C13Q811yescharacteristic
25C23×C4×C13C2×C411yescharacteristic
26C23×C4×C13C2×C411yescharacteristic
27C23×C4×C13C2×C416yes—
28C23×C4×C13D411yescharacteristic
29C23×C4×C13D411yescharacteristic
30C23×C4×C13D416yes—
31C23×C4×C13Q811yescharacteristic
32C23×C4×C13Q816yes—
33C23×C4×C13C2311yescharacteristic
34C23×C4×C13C2316yes—
35C2×C13×M16—212no—
36C2×C13×M16—26no—
37C2×C13×M16—224no—
38C2×C13×M16C23112yes—
39C2×C13×M16C23116yes—
40C22×C8×C13—212no—
41C22×C8×C13—22no—
42C2×#6099—212no—
43C2×#6099—26no—
44C2×#6099—224no—
45C2×#6099C2×C4124yes—
46C2×#6099C2×C4124yes—
47C2×#6099C2×C4148yes—
48C2×#6099C23112yes—
49C2×#6099C23116yes—
50C2×#6099C23116yes—
51C22×#1055—212no—
52C22×#1055—22no—
53C22×#1055C2×C4112yes—
54C22×#1055C2×C414yes—
55C22×#1055C23112yes—
56#88062C23164yescomplemented
57C22×#430—131nomaximal, Sylow
58C24×C13—44no—
59C24×C13C4⋊C411yescharacteristic
60C24×C13C4⋊C411yescharacteristic
61C24×C13C4⋊C416yes—
62C24×C13C2×Q811yescharacteristic
63C24×C13C2×Q816yes—
64C22×C4×C13C4⋊C416yes—
65C22×C4×C13C4⋊C416yes—
66C22×C4×C13C4⋊C416yes—
67C22×C4×C13C4⋊C416yes—
68C22×C4×C13C4⋊C416yes—
69C22×C4×C13C4⋊C4124yes—
70C22×C4×C13C4⋊C411yescharacteristic
71C22×C4×C13C4⋊C411yescharacteristic
72C22×C4×C13C22×C413yes—
73C22×C4×C13C22×C411yescharacteristic
74C22×C4×C13C22×C416yes—
75C22×C4×C13C22×C416yes—
76C22×C4×C13C22×C4112yes—
77C22×C4×C13C2×D416yes—
78C22×C4×C13C2×D416yes—
79C22×C4×C13C2×D413yes—
80C22×C4×C13C2×D4112yes—
81C22×C4×C13C2×D411yescharacteristic
82C22×C4×C13C2×Q813yes—
83C22×C4×C13C2×Q816yes—
84C22×C4×C13C2×Q8112yes—
85C22×C4×C13C2413yes—
86C22×C4×C13C2414yes—
87C13×M16—216no—
88C13×M16—212no—
89C13×M16C2418yes—
90C2×C8×C13—212no—
91C2×C8×C13—216no—
92#6099—216no—
93#6099—212no—
94#6099C22×C4132yes—
95#6099C22×C4132yes—
96#6099C2418yes—
97C2×#1055—212no—
98C2×#1055—216no—
99C2×#1055C22×C4124yes—
100C2×#1055C22×C4124yes—
101C2×#1055C2418yes—
102C23×M16—131no—
103C23×M16—131no—
104C23×M16D1311yescharacteristic
105C2×#430—1324no—
106C22×#108—134no—
107C23×C13—24no—
108C23×C13—24no—
109C23×C13—24no—
110C23×C13—412no—
111C23×C13—412no—
112C23×C13C2×C4⋊C413yes—
113C23×C13C2×C4⋊C416yes—
114C23×C13C2×C4⋊C416yes—
115C23×C13C2×C4⋊C4112yes—
116C23×C13C2×C4⋊C411yescharacteristic
117C23×C13C22×Q813yes—
118C23×C13C22×Q814yes—
119C2×C4×C13C2×C4⋊C416yes—
120C2×C4×C13C2×C4⋊C4112yes—
121C2×C4×C13C2×C4⋊C4112yes—
122C2×C4×C13C2×C4⋊C4112yes—
123C2×C4×C13C2×C4⋊C4112yes—
124C2×C4×C13C2×C4⋊C4112yes—
125C2×C4×C13C2×C4⋊C413yes—
126C2×C4×C13C2×C4⋊C416yes—
127C2×C4×C13C2×C4⋊C416yes—
128C2×C4×C13C2×C4⋊C413yes—
129C2×C4×C13C23×C413yes—
130C2×C4×C13C23×C413yes—
131C2×C4×C13C23×C414yes—
132C2×C4×C13C23×C414yes—
133C2×C4×C13C22×D413yes—
134C2×C4×C13C22×D414yes—
135C2×C4×C13C22×D414yes—
136C2×C4×C13C22×D413yes—
137C2×C4×C13C22×Q813yes—
138C2×C4×C13C22×Q814yes—
139C2×C4×C13C2511yescharacteristic
140C8×C13—28no—
141#1055—28no—
142#1055C23×C4116yes—
143C24×C4C2×D1311yescharacteristic
144C22×M16—26no—
145C22×M16—21no—
146C22×M16—132no—
147C22×M16—132no—
148C22×M16—1312no—
149C22×M16—1312no—
150C22×M16—132no—
151C22×M16—1312no—
152C22×M16—266no—
153C22×M16—261no—
154C22×M16C2×D1312yes—
155C22×M16C2×D13112yes—
156C23×C8—21no—
157C23×C8—132no—
158C23×C8—261no—
159#430—1364no—
160C2×#108—1348no—
161C22×C13—212no—
162C22×C13—212no—
163C22×C13—26no—
164C22×C13—26no—
165C22×C13—412no—
166C22×C13—44no—
167C22×C13#43014yescomplemented
168C22×C13C22×C4⋊C413yes—
169C22×C13C22×C4⋊C414yes—
170C22×C13C22×C4⋊C414yes—
171C22×C13C22×C4⋊C413yes—
172C22×C13C23×Q811yescharacteristic
173C4×C13C22×C4⋊C413yes—
174C4×C13C22×C4⋊C413yes—
175C4×C13C22×C4⋊C414yes—
176C4×C13C22×C4⋊C414yes—
177C4×C13C24×C411yescharacteristic
178C4×C13C23×D411yescharacteristic
179C25Dic2611yescharacteristic
180C23×C4Dic2611yescharacteristic
181C23×C4Dic2616yes—
182C23×C4C4×D1311yescharacteristic
183C23×C4C4×D1311yescharacteristic
184C23×C4C4×D1316yes—
185C23×C4C22⋊D1316yes—
186C23×C4C22⋊D1311yescharacteristic
187C23×C4C22⋊D1311yescharacteristic
188C23×C4C22×D1316yes—
189C23×C4C22×D1311yescharacteristic
190C2×M16—212no—
191C2×M16—26no—
192C2×M16—224no—
193C2×M16—1324no—
194C2×M16—1324no—
195C2×M16—1348no—
196C2×M16—1316no—
197C2×M16—1312no—
198C2×M16—1316no—
199C2×M16—2612no—
200C2×M16—266no—
201C2×M16—2624no—
202C2×M16C22×D13112yes—
203C2×M16C22×D13116yes—
204C22×C8—212no—
205C22×C8—22no—
206C22×C8—134no—
207C22×C8—1312no—
208C22×C8—1312no—
209C22×C8—2612no—
210C22×C8—262no—
211#108—1364no—
212C2×C13—24no—
213C2×C13—24no—
214C2×C13—23no—
215C2×C13—21no—
216C2×C13C2×#43016yescomplemented
217C2×C13C23×C4⋊C411yescharacteristic
218C24—44no—
219C24#610111yescharacteristic
220C24#610111yescharacteristic
221C24#610116yes—
222C24C2×Dic2611yescharacteristic
223C24C2×Dic2616yes—
224C22×C4#610116yes—
225C22×C4#610116yes—
226C22×C4#610116yes—
227C22×C4#6101124yes—
228C22×C4#610111yescharacteristic
229C22×C4#610111yescharacteristic
230C22×C4#610116yes—
231C22×C4#610116yes—
232C22×C4C2×Dic2616yes—
233C22×C4C2×Dic26112yes—
234C22×C4C2×Dic2613yes—
235C22×C4C2×C4×D1311yescharacteristic
236C22×C4C2×C4×D1313yes—
237C22×C4C2×C4×D1316yes—
238C22×C4C2×C4×D1316yes—
239C22×C4C2×C4×D13112yes—
240C22×C4C2×C22⋊D1316yes—
241C22×C4C2×C22⋊D1316yes—
242C22×C4C2×C22⋊D1313yes—
243C22×C4C2×C22⋊D13112yes—
244C22×C4C2×C22⋊D1311yescharacteristic
245C22×C4C23×D1313yes—
246C22×C4C23×D1314yes—
247M16—216no—
248M16—212no—
249M16—1332no—
250M16—1332no—
251M16—138no—
252M16—2616no—
253M16—2612no—
254M16C23×D1318yes—
255C2×C8—212no—
256C2×C8—216no—
257C2×C8—1324no—
258C2×C8—1324no—
259C2×C8—138no—
260C2×C8—2612no—
261C2×C8—2616no—
262C13C22×#43011yescharacteristic, complemented, Sylow
263C23—24no—
264C23—24no—
265C23—24no—
266C23—412no—
267C23—412no—
268C23C2×#610113yes—
269C23C2×#610116yes—
270C23C2×#610116yes—
271C23C2×#6101112yes—
272C23C2×#610111yescharacteristic
273C23C22×Dic2613yes—
274C23C22×Dic2614yes—
275C2×C4C2×#610116yes—
276C2×C4C2×#6101112yes—
277C2×C4C2×#6101112yes—
278C2×C4C2×#6101112yes—
279C2×C4C2×#6101112yes—
280C2×C4C2×#6101112yes—
281C2×C4C2×#610113yes—
282C2×C4C2×#610116yes—
283C2×C4C2×#610116yes—
284C2×C4C2×#610113yes—
285C2×C4C22×Dic2613yes—
286C2×C4C22×Dic2614yes—
287C2×C4C22×C4×D1313yes—
288C2×C4C22×C4×D1313yes—
289C2×C4C22×C4×D1314yes—
290C2×C4C22×C4×D1314yes—
291C2×C4C22×C22⋊D1313yes—
292C2×C4C22×C22⋊D1314yes—
293C2×C4C22×C22⋊D1314yes—
294C2×C4C22×C22⋊D1313yes—
295C2×C4C24×D1311yescharacteristic
296C8—28no—
297C8—1316no—
298C8—268no—
299C22—212no—
300C22—212no—
301C22—26no—
302C22—26no—
303C22—412no—
304C22—44no—
305C22#1171878514yesdirect product factor
306C22C22×#610113yes—
307C22C22×#610114yes—
308C22C22×#610114yes—
309C22C22×#610113yes—
310C22C23×Dic2611yescharacteristic
311C4C22×#610113yes—
312C4C22×#610114yes—
313C4C22×#610114yes—
314C4C22×#610113yes—
315C4C23×C4×D1311yescharacteristic
316C4C23×C22⋊D1311yescharacteristic
317C2—24no—
318C2—24no—
319C2—23no—
320C2—21no—
321C2C2×#1171878516yesdirect product factor
322C2C23×#610111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D13
commutatorC4×C13C24×C4
FrattiniC2×C4C24×D13
FittingC23×C13×M16C2
socleC23×C13C2×C4⋊C4
prime radicalC25×C13Q8
cosocleC2×C4×C13C25
hypercenterC23×M16D13
hypocenterC13C22×#430
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#11718785C2direct (N × H)
4. C22×#88062C2semidirect
22. #11718785C22direct (N × H)
23. C2×#88062C22semidirect
56. #88062C23semidirect
167. C22×C13#430semidirect
216. C2×C13C2×#430semidirect
262. C13C22×#430semidirect
305. C22#11718785direct (N × H)
321. C2C2×#11718785direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#60992
C2×#117187852
C22×#880622
C23×C13×M162
C22×#43013