Finite Group Databasebeta

← C2×#20260917 #27151077

Subgroups — C2×#20260917 #27151077

Order 12416.

708 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690858C211yesmaximal, characteristic
2C2×#15690841C212yesmaximal, complemented
3#20260917C218yesmaximal, direct product factor
4C2×#15690738C212yesmaximal, complemented
5C2×#15690729C211yesmaximal, characteristic, complemented
6C2×C4⋊Q8×C97C211yesmaximal, characteristic, complemented
7#15690858—24no—
8#15690858—22no—
9#15690858C2214yes—
10C2×#13095512—22no—
11C2×#13095511—22no—
12C2×#13095511C2212yes—
13C2×Q8×D97—22no—
14#15690841—24no—
15#15690841C2218yescomplemented
16C2×#13095501—22no—
17C2×#13095498—22no—
18C2×#13095495C2212yes—
19#15690738C2218yescomplemented
20#15690729C2214yescomplemented
21C22×Q8×C97C2211yescharacteristic
22C22×Q8×C97C2211yescharacteristic
23C4⋊Q8×C97—22no—
24C4⋊Q8×C97C2214yescomplemented
25C2×C4⋊C4×C97—22no—
26C2×C42×C97C2211yescharacteristic
27#13095512—28no—
28#13095512—28no—
29#13095511—28no—
30#13095511—28no—
31#13095511—22no—
32#13095511—22no—
33#13095511D414yes—
34#13095511C2314yes—
35C2×Dic194—22no—
36C2×Dic194D412yes—
37Q8×D97—28no—
38Q8×D97—28no—
39C2×D388—22no—
40C2×D388D412yes—
41C2×C22⋊D97—22no—
42C2×C4×D97—24no—
43C2×C4×D97—22no—
44#13095501—28no—
45#13095498—28no—
46#13095495—22no—
47#13095495C2314yes—
48C2×#11715895—22no—
49C2×Q8×C97—22no—
50C2×Q8×C97—22no—
51C2×Q8×C97—21no—
52C2×Q8×C97—22no—
53C2×Q8×C97—22no—
54C2×Q8×C97D412yes—
55C2×Q8×C97C2312yes—
56C2×Q8×C97C2312yes—
57C22×C4×C97—21no—
58C22×C4×C97—21no—
59C22×C4×C97D412yes—
60C22×C4×C97C2311yescharacteristic
61C4⋊C4×C97—28no—
62C42×C97—21no—
63C42×C97C2312yes—
64C23×C97C2×D411yescharacteristic
65C22×D97—22no—
66Q8×C97—24no—
67Q8×C97—24no—
68Q8×C97—24no—
69Q8×C97—42no—
70C2×C4×C97—24no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97C2×D412yes—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D414yes—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2411yescharacteristic
82C22⋊D97—24no—
83C22⋊D97—24no—
84C2×Dic97—22no—
85D388—28no—
86D388—22no—
87D388C2×D414yes—
88C4×D97—216no—
89C4×D97—24no—
90C4×D97—24no—
91Dic194—28no—
92Dic194—22no—
93Dic194C2×D414yes—
94#11715895—24no—
95C22×C97—22no—
96C22×C97#12011yescharacteristic
97C22×C97#12011yescharacteristic
98C22×C97C22×D411yescharacteristic
99C2×D97—24no—
100C2×D97—44no—
101C4×C97—24no—
102C4×C97—24no—
103C4×C97—42no—
104C4×C97#12012yes—
105C4×C97C22×D412yes—
106Dic97—24no—
107C2×C97—21no—
108C2×C97—21no—
109C2×C97#45612yescomplemented
110C2×C97C2×#12011yescharacteristic
111D97—44no—
112C2×#456—971nomaximal, Sylow
113C97C2×#45611yescharacteristic, complemented, Sylow
114C2×2−1+4—971no—
115C2×#137—972no—
116C2×C4⋊Q8D9711yescharacteristic, complemented
117#456—978no—
118C2×#104—972no—
119C2×#101—971no—
1202−1+4—974no—
1212−1+4—1944no—
1222−1+4—1942no—
123C2×#41—972no—
124C2×#41—1942no—
125C2×#41—1942no—
126C22×Q8—1942no—
127C22×Q8C2×D9711yescharacteristic
128C22×Q8C2×D9711yescharacteristic
129#137—978no—
130#137—1944no—
131C2×Q16—1942no—
132C2×SD16—1942no—
133C2×M16—972no—
134C4⋊Q8—22no—
135C4⋊Q8C2×D9714yescomplemented
136C2×C4⋊C4—22no—
137C2×C42C2×D9711yescharacteristic
138#104—978no—
139#101—974no—
140#41—974no—
141#41—974no—
142#41—1948no—
143#41—1948no—
144#41—1948no—
145#41—1942no—
146#41—1942no—
147#41—1948no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—972no—
154C2×Q8—1942no—
155C2×Q8—1948no—
156C2×Q8—1948no—
157C2×Q8C22⋊D9712yes—
158C2×Q8C22×D9712yes—
159C2×Q8C22×D9712yes—
160C2×D4—972no—
161C2×D4—1942no—
162C2×D4—1942no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—1944no—
166C22×C4—1942no—
167C22×C4C22⋊D9712yes—
168C22×C4C22×D9711yescharacteristic
169Q16—1948no—
170SD16—1948no—
171M16—974no—
172M16—1942no—
173C2×C8—1942no—
174C4⋊C4—28no—
175C42—21no—
176C42C22×D9712yes—
177C23—1942no—
178C23C2×C22⋊D9711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—24no—
182Q8—42no—
183Q8—974no—
184Q8—1948no—
185Q8—1942no—
186D4—974no—
187D4—1948no—
188D4—1944no—
189D4—1944no—
190D4—1942no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—19416no—
199C2×C4—1942no—
200C2×C4—1944no—
201C2×C4—1944no—
202C2×C4D4×D9712yes—
203C2×C4D4×D9712yes—
204C2×C4C2×C22⋊D9714yes—
205C2×C4C2×C22⋊D9711yescharacteristic
206C2×C4C23×D9711yescharacteristic
207C8—1944no—
208C22—22no—
209C22—3884no—
210C22—1944no—
211C22#1569077811yescharacteristic
212C22#1569077811yescharacteristic
213C22C22×C22⋊D9711yescharacteristic
214C4—24no—
215C4—24no—
216C4—42no—
217C4—1944no—
218C4#1569077812yes—
219C4C2×D4×D9712yes—
220C2—21no—
221C2—21no—
222C2—3884no—
223C2#2026091712yesdirect product factor
224C2C2×#1569077811yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C4⋊Q8×C97C2
socleC22×C97#120
prime radicalC2×#15690858C2
hypercenterC2×C4⋊Q8D97
hypocenterC97C2×#456
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15690841C2semidirect
3. #20260917C2direct (N × H)
4. C2×#15690738C2semidirect
5. C2×#15690729C2semidirect
6. C2×C4⋊Q8×C97C2semidirect
15. #15690841C22semidirect
19. #15690738C22semidirect
20. #15690729C22semidirect
24. C4⋊Q8×C97C22semidirect
109. C2×C97#456semidirect
113. C97C2×#456semidirect
116. C2×C4⋊Q8D97semidirect
135. C4⋊Q8C2×D97semidirect
223. C2#20260917direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908582
C2×#156908412
#202609172
C2×#156907382
C2×#156907292
C2×C4⋊Q8×C972
C2×#45697