Finite Group Databasebeta

← C2×C31×#2906 #21516857

Subgroups — C2×C31×#2906 #21516857

Order 7936.

2810 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#131C212yesmaximal
2C2×C4×#41×C31C211yesmaximal, characteristic
3C31×#2906C2116yesmaximal, direct product factor
4C2×C31×#424C216yesmaximal, complemented
5C2×C31×#423C213yesmaximal, complemented
6C22×C31×C4⋊C8C213yesmaximal, complemented
7C22×#41×C31C2211yescharacteristic
8C2×C31×#131—216no—
9C2×C31×#131C22116yes—
10C22×M16×C31C2216yes—
11C23×C8×C31C2216yes—
12C4×#41×C31C418yes—
13C4×#41×C31C2218yes—
14C2×C4×Q8×C31C411yescharacteristic
15C2×C4×Q8×C31C411yescharacteristic
16C2×C4×D4×C31C413yes—
17C2×C4×D4×C31C413yes—
18C2×C31×#117C2213yes—
19C22×C42×C31C2213yes—
20C31×#424C22148yescomplemented
21C31×#423C22124yescomplemented
22C2×C31×C4⋊C8C2211yescharacteristic
23C2×C31×C4⋊C8C2213yes—
24C2×C31×C4⋊C8C2216yes—
25C2×C31×C4⋊C8C2216yes—
26C2×C31×C4⋊C8C22124yescomplemented
27C2×#41×C31C2×C414yes—
28C2×#41×C31D418yes—
29C2×#41×C31Q818yes—
30C2×#41×C31C2314yes—
31C22×Q8×C31C2×C411yescharacteristic
32C22×D4×C31C2×C413yes—
33C23×C4×C31C2×C413yes—
34C23×C4×C31C2×C413yes—
35C23×C4×C31C2313yes—
36C31×#131—264no—
37C2×M16×C31—224no—
38C2×M16×C31C23124yes—
39C22×C8×C31—224no—
40C22×C8×C31C23124yes—
41C22×C8×C31C2312yes—
42C22×C8×C31C2316yes—
43C4×Q8×C31C2×C418yes—
44C4×Q8×C31C2×C418yes—
45C4×D4×C31C2×C4124yes—
46C4×D4×C31C2×C4124yes—
47C31×#117C2×C4112yes—
48C31×#117C23112yes—
49C2×C4⋊C4×C31C2×C413yes—
50C2×C4⋊C4×C31C2×C413yes—
51C2×C22⋊C4×C31C2×C416yes—
52C2×C42×C31C2×C413yes—
53C2×C42×C31C2×C413yes—
54C2×C42×C31C2×C4112yes—
55C2×C42×C31C23112yes—
56C2×C42×C31C2311yescharacteristic
57C2×C42×C31C2313yes—
58C31×C4⋊C8C2314yes—
59C31×C4⋊C8C23112yes—
60C31×C4⋊C8C23124yes—
61C31×C4⋊C8C23124yes—
62C24×C31C22×C413yes—
63#41×C31C4⋊C4132yes—
64#41×C31C2×D4116yes—
65#41×C31C2×Q8116yes—
66C2×Q8×C31C4⋊C418yes—
67C2×Q8×C31C22×C414yes—
68C2×D4×C31C4⋊C4124yes—
69C2×D4×C31C22×C4112yes—
70C22×C4×C31—212no—
71C22×C4×C31—46no—
72C22×C4×C31C22×C411yescharacteristic
73C22×C4×C31C22×C413yes—
74C22×C4×C31C22×C413yes—
75C22×C4×C31C22×C4112yes—
76C22×C4×C31C22×C4112yes—
77C22×C4×C31C22×C413yes—
78C22×C4×C31C22×C416yes—
79C22×C4×C31C22×C411yescharacteristic
80C22×C4×C31C2×D4112yes—
81C22×C4×C31C2×Q8112yes—
82C22×C4×C31C2416yes—
83C22×C4×C31C2411yescharacteristic
84M16×C31—248no—
85C2×C8×C31—24no—
86C2×C8×C31—212no—
87C2×C8×C31—248no—
88C2×C8×C31C2414yes—
89C2×C8×C31C24112yes—
90C4⋊C4×C31C22×C4112yes—
91C4⋊C4×C31C22×C4112yes—
92C22⋊C4×C31C22×C4124yes—
93C42×C31C22×C412yes—
94C42×C31C22×C416yes—
95C42×C31C22×C4112yes—
96C42×C31C22×C4112yes—
97C42×C31C2412yes—
98C42×C31C2416yes—
99C2×#2906C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31—26no—
101C23×C31—26no—
102C23×C31C2×C4⋊C4112yes—
103C23×C31C23×C416yes—
104C23×C31C23×C411yescharacteristic
105Q8×C31C2×C4⋊C4116yes—
106D4×C31C2×C4⋊C4148yes—
107C2×C4×C31—26no—
108C2×C4×C31—22no—
109C2×C4×C31—26no—
110C2×C4×C31—224no—
111C2×C4×C31—412no—
112C2×C4×C31C2×C4⋊C4112yes—
113C2×C4×C31C2×C4⋊C4124yes—
114C2×C4×C31#13114yescomplemented
115C2×C4×C31C23×C412yes—
116C2×C4×C31C23×C416yes—
117C2×C4×C31C23×C416yes—
118C2×C4×C31C23×C411yescharacteristic
119C2×C4×C31C23×C416yes—
120C2×C4×C31C23×C412yes—
121C2×C4×C31C22×D412yes—
122C2×C4×C31C22×D4112yes—
123C2×C4×C31C22×Q812yes—
124C2×C4×C31C22×Q8112yes—
125C2×C4×C31C2511yescharacteristic
126C8×C31—24no—
127C8×C31—212no—
128C22×#131C2×C3112yes—
129C2×C4×#41C2×C3111yescharacteristic
130#2906C2×C31116yesdirect product factor
131C2×#424C2×C3116yescomplemented
132C2×#423C2×C3113yescomplemented
133C22×C4⋊C8C2×C3113yescomplemented
134C22×C31—224no—
135C22×C31—26no—
136C22×C31—26no—
137C22×C31C22×C4⋊C4112yes—
138C22×C31C22×C4⋊C412yes—
139C22×C31C2×#13112yes—
140C22×C31#56812yes—
141C22×C31C24×C411yescharacteristic
142C4×C31—26no—
143C4×C31—22no—
144C4×C31—26no—
145C4×C31C22×C4⋊C412yes—
146C4×C31C22×C4⋊C4112yes—
147C4×C31C2×#13114yescomplemented
148C4×C31C23×D411yescharacteristic
149C4×C31C23×Q811yescharacteristic
150C22×#41C22×C3111yescharacteristic
151C2×#131—216no—
152C2×#131C22×C31116yes—
153C22×M16C22×C3116yes—
154C23×C8C22×C3116yes—
155C4×#41C4×C3118yes—
156C4×#41C22×C3118yes—
157C2×C4×Q8C4×C3111yescharacteristic
158C2×C4×Q8C4×C3111yescharacteristic
159C2×C4×D4C4×C3113yes—
160C2×C4×D4C4×C3113yes—
161C2×#117C22×C3113yes—
162C22×C42C22×C3113yes—
163#424C22×C31148yescomplemented
164#423C22×C31124yescomplemented
165C2×C4⋊C8C22×C3111yescharacteristic
166C2×C4⋊C8C22×C3113yes—
167C2×C4⋊C8C22×C3116yes—
168C2×C4⋊C8C22×C3116yes—
169C2×C4⋊C8C22×C31124yescomplemented
170C2×C31—212no—
171C2×C31#290614yesdirect product factor
172C2×C31C23×C4⋊C411yescharacteristic
173C2×C31C22×#13111yescharacteristic
174C2×C31C2×#56811yescharacteristic
175C2×#41C2×C4×C3114yes—
176C2×#41D4×C3118yes—
177C2×#41Q8×C3118yes—
178C2×#41C23×C3114yes—
179C22×Q8C2×C4×C3111yescharacteristic
180C22×D4C2×C4×C3113yes—
181C23×C4C2×C4×C3113yes—
182C23×C4C2×C4×C3113yes—
183C23×C4C23×C3113yes—
184#131—264no—
185C2×M16—224no—
186C2×M16C23×C31124yes—
187C22×C8—224no—
188C22×C8C23×C3112yes—
189C22×C8C23×C3116yes—
190C22×C8C23×C31124yes—
191C4×Q8C2×C4×C3118yes—
192C4×Q8C2×C4×C3118yes—
193C4×D4C2×C4×C31124yes—
194C4×D4C2×C4×C31124yes—
195#117C2×C4×C31112yes—
196#117C23×C31112yes—
197C2×C4⋊C4C2×C4×C3113yes—
198C2×C4⋊C4C2×C4×C3113yes—
199C2×C22⋊C4C2×C4×C3116yes—
200C2×C42C2×C4×C3113yes—
201C2×C42C2×C4×C3113yes—
202C2×C42C2×C4×C31112yes—
203C2×C42C23×C3113yes—
204C2×C42C23×C31112yes—
205C2×C42C23×C3111yescharacteristic
206C4⋊C8C23×C3114yes—
207C4⋊C8C23×C31112yes—
208C4⋊C8C23×C31124yes—
209C4⋊C8C23×C31124yes—
210C31C2×#290611yescharacteristic, direct product factor, Sylow
211C24C22×C4×C3113yes—
212#41C4⋊C4×C31132yes—
213#41C2×D4×C31116yes—
214#41C2×Q8×C31116yes—
215C2×Q8C4⋊C4×C3118yes—
216C2×Q8C22×C4×C3114yes—
217C2×D4C4⋊C4×C31124yes—
218C2×D4C22×C4×C31112yes—
219C22×C4—212no—
220C22×C4—46no—
221C22×C4C22×C4×C3111yescharacteristic
222C22×C4C22×C4×C3113yes—
223C22×C4C22×C4×C3113yes—
224C22×C4C22×C4×C31112yes—
225C22×C4C22×C4×C31112yes—
226C22×C4C22×C4×C3113yes—
227C22×C4C22×C4×C3116yes—
228C22×C4C22×C4×C3111yescharacteristic
229C22×C4C2×D4×C31112yes—
230C22×C4C2×Q8×C31112yes—
231C22×C4C24×C3116yes—
232C22×C4C24×C3111yescharacteristic
233M16—248no—
234C2×C8—24no—
235C2×C8—212no—
236C2×C8—248no—
237C2×C8C24×C3114yes—
238C2×C8C24×C31112yes—
239C4⋊C4C22×C4×C31112yes—
240C4⋊C4C22×C4×C31112yes—
241C22⋊C4C22×C4×C31124yes—
242C42C22×C4×C3112yes—
243C42C22×C4×C3116yes—
244C42C22×C4×C31112yes—
245C42C22×C4×C31112yes—
246C42C24×C3112yes—
247C42C24×C3116yes—
248C23—26no—
249C23—26no—
250C23C2×C4⋊C4×C31112yes—
251C23C23×C4×C3116yes—
252C23C23×C4×C3111yescharacteristic
253Q8C2×C4⋊C4×C31116yes—
254D4C2×C4⋊C4×C31148yes—
255C2×C4—26no—
256C2×C4—22no—
257C2×C4—26no—
258C2×C4—224no—
259C2×C4—412no—
260C2×C4C2×C4⋊C4×C31112yes—
261C2×C4C2×C4⋊C4×C31124yes—
262C2×C4C31×#13114yescomplemented
263C2×C4C23×C4×C3112yes—
264C2×C4C23×C4×C3116yes—
265C2×C4C23×C4×C3116yes—
266C2×C4C23×C4×C3111yescharacteristic
267C2×C4C23×C4×C3116yes—
268C2×C4C23×C4×C3112yes—
269C2×C4C22×D4×C3112yes—
270C2×C4C22×D4×C31112yes—
271C2×C4C22×Q8×C3112yes—
272C2×C4C22×Q8×C31112yes—
273C2×C4C25×C3111yescharacteristic
274C8—24no—
275C8—212no—
276C22—224no—
277C22—26no—
278C22—26no—
279C22C22×C4⋊C4×C31112yes—
280C22C22×C4⋊C4×C3112yes—
281C22C2×C31×#13112yes—
282C22C31×#56812yes—
283C22C24×C4×C3111yescharacteristic
284C4—26no—
285C4—22no—
286C4—26no—
287C4C22×C4⋊C4×C3112yes—
288C4C22×C4⋊C4×C31112yes—
289C4C2×C31×#13114yescomplemented
290C4C23×D4×C3111yescharacteristic
291C4C23×Q8×C3111yescharacteristic
292C2—212no—
293C2C31×#290614yesdirect product factor
294C2C23×C4⋊C4×C3111yescharacteristic
295C2C22×C31×#13111yescharacteristic
296C2C2×C31×#56811yescharacteristic
subgroupG/H
centerC22×C4×C31C24
commutatorC22C24×C4×C31
Frattini, cosocleC2×C4C25×C31
socleC23×C31C23×C4
prime radicalC22×#41×C31C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2906C2direct (N × H)
4. C2×C31×#424C2semidirect
5. C2×C31×#423C2semidirect
6. C22×C31×C4⋊C8C2semidirect
20. C31×#424C22semidirect
21. C31×#423C22semidirect
26. C2×C31×C4⋊C8C22semidirect
99. C2×#2906C31direct (N × H)
114. C2×C4×C31#131semidirect
130. #2906C2×C31direct (N × H)
131. C2×#424C2×C31semidirect
132. C2×#423C2×C31semidirect
133. C22×C4⋊C8C2×C31semidirect
147. C4×C31C2×#131semidirect
163. #424C22×C31semidirect
164. #423C22×C31semidirect
169. C2×C4⋊C8C22×C31semidirect
171. C2×C31#2906direct (N × H)
210. C31C2×#2906direct (N × H)
262. C2×C4C31×#131semidirect
289. C4C2×C31×#131semidirect
293. C2C31×#2906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1312
C2×C4×#41×C312
C31×#29062
C2×C31×#4242
C2×C31×#4232
C22×C31×C4⋊C82
C2×#290631