Finite Group Databasebeta

← #27142079

Subgroups — #27142079

Order 12416.

338 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q8×Dic97C211yesmaximal, characteristic
2C2×#15690734C211yesmaximal, characteristic
3C2×#15690734C211yesmaximal, characteristic
4#20260642C211yesmaximal, characteristic
5#20260498C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic
7C22×Q16×C97C211yesmaximal, characteristic
8C22×#11715895C2211yescharacteristic
9Q8×Dic97—22no—
10Q8×Dic97—22no—
11C2×C4⋊Dic97—21no—
12C2×C4⋊Dic97C2211yescharacteristic
13C2×C4⋊Dic97C2211yescharacteristic
14C2×C4×Dic97—21no—
15C2×C4×Dic97C2211yescharacteristic
16#15690734—22no—
17#15690734—22no—
18#15690717—21no—
19#15690717—21no—
20C22×Q8×C97C2211yescharacteristic
21C22×Q8×C97C2211yescharacteristic
22C2×Q16×C97—22no—
23C2×Q16×C97—22no—
24C2×Q16×C97C414yescomplemented
25C22×C8×C97C2211yescharacteristic
26C22×Dic97—21no—
27C22×Dic97—21no—
28C22×Dic97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C2×#11715895—21no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—24no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97D412yes—
38C4×Dic97—24no—
39C4×Dic97—21no—
40C4×Dic97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97C2×C412yes—
50C2×Q8×C97C2×C412yes—
51C2×Q8×C97D412yes—
52C2×Q8×C97D412yes—
53C22×C4×C97—21no—
54C22×C4×C97—21no—
55C22×C4×C97C2311yescharacteristic
56Q16×C97—28no—
57C2×C8×C97—21no—
58C2×C8×C97—21no—
59C2×C8×C97C2×C412yes—
60C23×C97C2×D411yescharacteristic
61Q8×C97—22no—
62Q8×C97—22no—
63Q8×C97—22no—
64Q8×C97—22no—
65Q8×C97—22no—
66Q8×C97—41no—
67Q8×C97—41no—
68Q8×C97C22⋊C414yes—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97C22×C411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97#4111yescharacteristic
80C2×C4×C97#4111yescharacteristic
81C2×Dic97—21no—
82C2×Dic97—21no—
83C2×Dic97—21no—
84C2×Dic97—22no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—41no—
88C2×Dic97—41no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97—41no—
92C2×Dic97Q1612yescomplemented
93C8×C97—22no—
94#11715895—41no—
95C22×C97C4×D411yescharacteristic
96C22×C97#12011yescharacteristic
97C22×C97#12111yescharacteristic
98C22×C97C2×Q1611yescharacteristic
99C22×C97#13511yescharacteristic
100C22×C97#13711yescharacteristic
101C22×C97#13711yescharacteristic
102C4×C97—24no—
103C4×C97—42no—
104C4×C97C2×C22⋊C411yescharacteristic
105C4×C97C4×D411yescharacteristic
106C4×C97#12111yescharacteristic
107C4×C97#12311yescharacteristic
108Dic97—22no—
109Dic97—41no—
110Dic97—42no—
111Dic97—81no—
112C2×C97#38611yescharacteristic
113C2×C97C4×Q1611yescharacteristic
114C2×C97#44111yescharacteristic
115C2×C97#45111yescharacteristic
116C2×C97#45211yescharacteristic
117C2×C97#46211yescharacteristic
118C2×C97#46411yescharacteristic
119#1878—971nomaximal, Sylow
120C97#187811yescharacteristic, complemented, Sylow
121C22×Q16D9711yescharacteristic
122C2×C4×Q8—971no—
123C2×#103—971no—
124C2×#103—971no—
125#391—971no—
126#340—971no—
127#336—971no—
128C22×Q8C2×D9711yescharacteristic
129C22×Q8C2×D9711yescharacteristic
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16Dic9714yescomplemented
133C22×C8—971no—
134C22×C8C2×D9711yescharacteristic
135C4×Q8—1942no—
136C4×Q8—1942no—
137C2×C4⋊C4—971no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—1941no—
140C2×C42—971no—
141C2×C42—1941no—
142#103—1942no—
143#103—1942no—
144#95—1941no—
145#95—1941no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×Dic9712yes—
155C2×Q8C2×Dic9712yes—
156C2×Q8C22⋊D9712yes—
157C2×Q8C22⋊D9712yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—1941no—
163C22×C4—1941no—
164C22×C4C22×D9711yescharacteristic
165Q16—28no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—1941no—
169C2×C8—1941no—
170C2×C8—1941no—
171C2×C8C2×Dic9712yes—
172C4⋊C4—972no—
173C4⋊C4—1944no—
174C4⋊C4—1941no—
175C4⋊C4—1941no—
176C4⋊C4—1941no—
177C42—1944no—
178C42—1941no—
179C42—1941no—
180C23D4×D9711yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—22no—
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8C22⋊Dic9714yes—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—972no—
196C2×C4—3881no—
197C2×C4—3881no—
198C2×C4—3881no—
199C2×C4—3881no—
200C2×C4—3881no—
201C2×C4—1942no—
202C2×C4—1941no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4—1941no—
206C2×C4—1941no—
207C2×C4C22×Dic9711yescharacteristic
208C2×C4D4×D9711yescharacteristic
209C2×C4C2×C22⋊D9711yescharacteristic
210C2×C4C2×C22⋊D9711yescharacteristic
211C2×C4#1309551011yescharacteristic
212C2×C4#1309551011yescharacteristic
213C8—22no—
214C8—3881no—
215C22D4×Dic9711yescharacteristic
216C22#1569077811yescharacteristic
217C22#1569078211yescharacteristic
218C22Q16×D9711yescharacteristic
219C22#1569083311yescharacteristic
220C22#1569084211yescharacteristic
221C22#1569084211yescharacteristic
222C4—24no—
223C4—42no—
224C4—7761no—
225C4—3882no—
226C4—3881no—
227C4—1942no—
228C4C2×C22⋊Dic9711yescharacteristic
229C4D4×Dic9711yescharacteristic
230C4#1569078211yescharacteristic
231C4#1569079111yescharacteristic
232C2#2026061511yescharacteristic
233C2Q16×Dic9711yescharacteristic
234C2#2026084211yescharacteristic
235C2#2026088911yescharacteristic
236C2#2026089611yescharacteristic
237C2#2026094311yescharacteristic
238C2#2026095911yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×Q16×C97C2
socle, prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenterC22×Q16D97
hypocenterC97#1878
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
24. C2×Q16×C97C4semidirect
92. C2×Dic97Q16semidirect
120. C97#1878semidirect
132. C2×Q16Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic972
C2×#156907342
#202606422
#202604982
#202604892
C22×Q16×C972
#187897