Finite Group Databasebeta

← C2×Q8×#13050574 #36268792

Subgroups — C2×Q8×#13050574 #36268792

Order 21248.

2154 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#13050574C2116yesmaximal, direct product factor
2C2×#19127832C216yesmaximal, complemented
3C2×Q8×C83⋊C8C212yesmaximal, complemented
4C2×#19127725C213yesmaximal
5C2×C4×#13050574C213yesmaximal
6C22×C4×Q8×C83C211yesmaximal, characteristic
7C22×#13050574C2213yes—
8C22×#13050574C2211yescharacteristic
9#19127832C22148yescomplemented
10Q8×C83⋊C8C22116yescomplemented
11#19127725C22124yes—
12C2×C4⋊(C83⋊C8)C2216yes—
13C2×C4⋊(C83⋊C8)C2216yes—
14C4×#13050574C22124yes—
15C2×#14581703C2216yes—
16C2×C4×C83⋊C8C2216yes—
17C23×Q8×C83C411yescharacteristic
18C2×C4×Q8×C83C413yes—
19C2×C4×Q8×C83C418yes—
20C2×C4×Q8×C83C411yescharacteristic
21C2×C4×Q8×C83C2211yescharacteristic
22C2×C4×Q8×C83C2213yes—
23C2×C4×Q8×C83C2218yes—
24C22×C4⋊C4×C83C413yes—
25C22×C42×C83C2213yes—
26C2×#13050574—212no—
27C2×#13050574Q818yesdirect product factor
28C2×#13050574C23112yes—
29C2×#13050574C2314yes—
30C22×C83⋊C8C2316yes—
31C22×C83⋊C8C2312yes—
32C4⋊(C83⋊C8)C23124yes—
33C4⋊(C83⋊C8)C23124yes—
34#14581703C23124yes—
35C4×C83⋊C8C23124yes—
36C22×Q8×C83—28no—
37C22×Q8×C83C2×C418yes—
38C22×Q8×C83C2×C413yes—
39C22×Q8×C83C2×C411yescharacteristic
40C23×C4×C83C2×C413yes—
41C23×C4×C83C2×C413yes—
42C23×C4×C83C2311yescharacteristic
43C4×Q8×C83C2×C4112yes—
44C4×Q8×C83C2×C4124yes—
45C4×Q8×C83C2×C414yes—
46C4×Q8×C83C2×C418yes—
47C4×Q8×C83C2314yes—
48C4×Q8×C83C23112yes—
49C2×C4⋊C4×C83C2×C416yes—
50C2×C4⋊C4×C83C2×C413yes—
51C2×C4⋊C4×C83C2×C413yes—
52C2×C4⋊C4×C83C2×C4124yes—
53C2×C42×C83C2×C413yes—
54C2×C42×C83C2×C413yes—
55C2×C42×C83C2×C4112yes—
56C2×C42×C83C2313yes—
57C2×C42×C83C2313yes—
58C2×C42×C83C23112yes—
59#13050574—224no—
60#13050574C2×Q8116yesdirect product factor
61C2×C83⋊C8—212no—
62C2×C83⋊C8C2×Q818yescomplemented
63C2×C83⋊C8C24112yes—
64C2×C83⋊C8C2414yes—
65C24×C83C22×C411yescharacteristic
66C2×Q8×C83—232no—
67C2×Q8×C83—212no—
68C2×Q8×C83M1618yescomplemented
69C2×Q8×C83C22×C4112yes—
70C2×Q8×C83C22×C414yes—
71C22×C4×C83—212no—
72C22×C4×C83—46no—
73C22×C4×C83C22×C413yes—
74C22×C4×C83C22×C413yes—
75C22×C4×C83C22×C413yes—
76C22×C4×C83C22×C4112yes—
77C22×C4×C83C22×C4112yes—
78C22×C4×C83C22×C412yes—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C22×C413yes—
81C22×C4×C83C2×Q814yes—
82C22×C4×C83#4114yes—
83C22×C4×C83C2412yes—
84C22×C4×C83C2411yescharacteristic
85C4⋊C4×C83C22×C4124yes—
86C4⋊C4×C83C22×C4112yes—
87C4⋊C4×C83C22×C4112yes—
88C42×C83C22×C416yes—
89C42×C83C22×C4112yes—
90C42×C83C22×C416yes—
91C42×C83C22×C4112yes—
92C42×C83C2416yes—
93C42×C83C2416yes—
94C23×C83—22no—
95C23×C83—22no—
96C23×C83C4×Q814yes—
97C23×C83C23×C412yes—
98C23×C83C23×C411yescharacteristic
99Q8×C83—224no—
100Q8×C83C2×M16116yescomplemented
101C2×C4×C83—26no—
102C2×C4×C83—26no—
103C2×C4×C83—26no—
104C2×C4×C83—224no—
105C2×C4×C83—412no—
106C2×C4×C83C4×Q818yes—
107C2×C4×C83C4×Q814yes—
108C2×C4×C83C2×M16112yes—
109C2×C4×C83C23×C416yes—
110C2×C4×C83C23×C416yes—
111C2×C4×C83C23×C416yes—
112C2×C4×C83C23×C412yes—
113C2×C4×C83C23×C411yescharacteristic
114C2×C4×C83C23×C416yes—
115C2×C4×C83C22×Q812yes—
116C2×C4×C83C22×Q814yes—
117C2×C4×C83C2×#4112yes—
118C2×C4×C83C2×#4114yes—
119C2×C4×C83C2511yescharacteristic
120C83⋊C8—212no—
121C83⋊C8C22×Q818yescomplemented
122C22×C83—28no—
123C22×C83—22no—
124C22×C83—22no—
125C22×C83C2×C4×Q814yes—
126C22×C83C2×C4×Q812yes—
127C22×C83C22×M1612yes—
128C22×C83#56812yes—
129C22×C83C24×C411yescharacteristic
130C4×C83—26no—
131C4×C83—26no—
132C4×C83—26no—
133C4×C83C2×C4×Q812yes—
134C4×C83C2×C4×Q814yes—
135C4×C83C22×M16112yes—
136C4×C83C23×Q811yescharacteristic
137C4×C83C22×#4111yescharacteristic
138C2×Q8×M16—831nomaximal, Sylow
139C2×C83—24no—
140C2×C83Q8×M1614yescomplemented
141C2×C83C22×C4×Q811yescharacteristic
142C2×C83C23×M1611yescharacteristic
143C2×C83C2×#56811yescharacteristic
144C22×C4×Q8D8311yescharacteristic
145Q8×M16—8316no—
146C2×#446—836no—
147C2×C8×Q8—832no—
148C2×#423—833no—
149C2×C4×M16—833no—
150C83C2×Q8×M1611yescharacteristic, complemented, Sylow
151C23×Q8Dic8311yescharacteristic
152C22×M16—831no—
153C22×M16—833no—
154C2×C4×Q8Dic8313yes—
155C2×C4×Q8Dic8318yes—
156C2×C4×Q8Dic8311yescharacteristic
157C2×C4×Q8C2×D8311yescharacteristic
158C2×C4×Q8C2×D8313yes—
159C2×C4×Q8C2×D8318yes—
160C22×C4⋊C4Dic8313yes—
161C22×C42C2×D8313yes—
162#446—8348no—
163C8×Q8—8316no—
164#423—8324no—
165C2×C4⋊C8—836no—
166C2×C4⋊C8—836no—
167C4×M16—8324no—
168C2×#97—836no—
169C2×C4×C8—836no—
170C22×Q8—28no—
171C22×Q8C2×Dic8318yes—
172C22×Q8C2×Dic8313yes—
173C22×Q8C2×Dic8311yescharacteristic
174C23×C4C2×Dic8313yes—
175C23×C4C2×Dic8313yes—
176C23×C4C22×D8311yescharacteristic
177C2×M16—838no—
178C2×M16—834no—
179C2×M16—8312no—
180C2×M16—16612no—
181C22×C8—832no—
182C22×C8—836no—
183C4×Q8C2×Dic83112yes—
184C4×Q8C2×Dic83124yes—
185C4×Q8C2×Dic8314yes—
186C4×Q8C2×Dic8318yes—
187C4×Q8C22×D8314yes—
188C4×Q8C22×D83112yes—
189C2×C4⋊C4C2×Dic8316yes—
190C2×C4⋊C4C2×Dic8313yes—
191C2×C4⋊C4C2×Dic8313yes—
192C2×C4⋊C4C2×Dic83124yes—
193C2×C42C2×Dic8313yes—
194C2×C42C2×Dic8313yes—
195C2×C42C2×Dic83112yes—
196C2×C42C22×D83112yes—
197C2×C42C22×D8313yes—
198C2×C42C22×D8313yes—
199C4⋊C8—8324no—
200C4⋊C8—8324no—
201#97—8324no—
202C4×C8—8324no—
203C24C22×Dic8311yescharacteristic
204C2×Q8—232no—
205C2×Q8—212no—
206C2×Q8#1305057418yesdirect product factor
207C2×Q8C22×Dic83112yes—
208C2×Q8C22×Dic8314yes—
209C22×C4—212no—
210C22×C4—46no—
211C22×C4C22×Dic8313yes—
212C22×C4C22×Dic8313yes—
213C22×C4C22×Dic8313yes—
214C22×C4C22×Dic83112yes—
215C22×C4C22×Dic83112yes—
216C22×C4C22×Dic8313yes—
217C22×C4C22×Dic8311yescharacteristic
218C22×C4C22×Dic8312yes—
219C22×C4Q8×D8314yes—
220C22×C4#1305059114yes—
221C22×C4C23×D8312yes—
222C22×C4C23×D8311yescharacteristic
223M16—8316no—
224M16—16624no—
225C2×C8—838no—
226C2×C8—834no—
227C2×C8—8312no—
228C2×C8—16612no—
229C4⋊C4C22×Dic83124yes—
230C4⋊C4C22×Dic83112yes—
231C4⋊C4C22×Dic83112yes—
232C42C22×Dic83112yes—
233C42C22×Dic8316yes—
234C42C22×Dic83112yes—
235C42C22×Dic8316yes—
236C42C23×D8316yes—
237C42C23×D8316yes—
238C23—22no—
239C23—22no—
240C23Q8×Dic8314yes—
241C23C23×Dic8312yes—
242C23C23×Dic8311yescharacteristic
243Q8—224no—
244Q8C2×#13050574116yesdirect product factor
245C2×C4—26no—
246C2×C4—26no—
247C2×C4—26no—
248C2×C4—224no—
249C2×C4—412no—
250C2×C4Q8×Dic8318yes—
251C2×C4Q8×Dic8314yes—
252C2×C4C2×#13050574112yes—
253C2×C4C23×Dic8316yes—
254C2×C4C23×Dic8316yes—
255C2×C4C23×Dic8316yes—
256C2×C4C23×Dic8312yes—
257C2×C4C23×Dic8311yescharacteristic
258C2×C4C23×Dic8316yes—
259C2×C4C2×Q8×D8312yes—
260C2×C4C2×Q8×D8314yes—
261C2×C4C2×#1305059112yes—
262C2×C4C2×#1305059114yes—
263C2×C4C24×D8311yescharacteristic
264C8—838no—
265C8—16612no—
266C22—22no—
267C22—22no—
268C22—28no—
269C22C2×Q8×Dic8314yes—
270C22C2×Q8×Dic8312yes—
271C22C22×#1305057412yes—
272C22#1912846812yes—
273C22C24×Dic8311yescharacteristic
274C4—26no—
275C4—26no—
276C4—26no—
277C4C2×Q8×Dic8314yes—
278C4C2×Q8×Dic8312yes—
279C4C22×#13050574112yes—
280C4C22×Q8×D8311yescharacteristic
281C4C22×#1305059111yescharacteristic
282C2—24no—
283C2Q8×#1305057414yesdirect product factor
284C2C22×Q8×Dic8311yescharacteristic
285C2C23×#1305057411yescharacteristic
286C2C2×#1912846811yescharacteristic
subgroupG/H
centerC22×C4C23×D83
commutatorC22×C83C24×C4
FrattiniC2×C4C24×D83
FittingC22×C4×Q8×C83C2
socleC23×C83C23×C4
prime radicalC24×C83C22×C4
cosocleC2×C4×C83C25
hypercenterC22×C4×Q8D83
hypocenterC83C2×Q8×M16
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#13050574C2direct (N × H)
2. C2×#19127832C2semidirect
3. C2×Q8×C83⋊C8C2semidirect
9. #19127832C22semidirect
10. Q8×C83⋊C8C22semidirect
27. C2×#13050574Q8direct (N × H)
60. #13050574C2×Q8direct (N × H)
62. C2×C83⋊C8C2×Q8semidirect
68. C2×Q8×C83M16semidirect
100. Q8×C83C2×M16semidirect
121. C83⋊C8C22×Q8semidirect
140. C2×C83Q8×M16semidirect
150. C83C2×Q8×M16semidirect
206. C2×Q8#13050574direct (N × H)
244. Q8C2×#13050574direct (N × H)
283. C2Q8×#13050574direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#130505742
C2×#191278322
C2×Q8×C83⋊C82
C2×#191277252
C2×C4×#130505742
C22×C4×Q8×C832
C2×Q8×M1683