Finite Group Databasebeta

← #27143045

Subgroups — #27143045

Order 12416.

360 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×#13095496C211yesmaximal, characteristic
2C2×#15690739C211yesmaximal, characteristic, complemented
3C2×#15690732C211yesmaximal, characteristic, complemented
4#20260638C211yesmaximal, characteristic, complemented
5#20260636C211yesmaximal, characteristic, complemented
6#20260497C211yesmaximal, characteristic, complemented
7C2×#102×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095496—22no—
10C2×#13095496—22no—
11C2×#13095496—22no—
12C22×#11715895C2211yescharacteristic
13C2×C4⋊Dic97C2211yescharacteristic
14C2×C22⋊Dic97—21no—
15C2×C22⋊Dic97—21no—
16C2×#13095488—21no—
17C2×#13095488—21no—
18C2×C4×Dic97C2211yescharacteristic
19#15690739—22no—
20#15690732—22no—
21C22×D4×C97C2211yescharacteristic
22C22×C8×C97C2211yescharacteristic
23C2×C4⋊C4×C97C2211yescharacteristic
24#102×C97—22no—
25C23×D97—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388D412yes—
32C22×Dic97—21no—
33C22×Dic97D411yescharacteristic
34C22×Dic97D411yescharacteristic
35#13095496—28no—
36C2×#11715895—21no—
37C2×#11715895—21no—
38C2×#11715895D412yes—
39C4⋊Dic97—21no—
40C4⋊Dic97—21no—
41C22⋊Dic97—42no—
42C22⋊Dic97—42no—
43#13095488—42no—
44#13095488—42no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C24×C97—21no—
48C2×D4×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97—21no—
51C2×D4×C97—21no—
52C2×D4×C97—21no—
53C2×D4×C97D412yes—
54C22×C4×C97—21no—
55C22×C4×C97C2311yescharacteristic
56C2×C8×C97—21no—
57C2×C8×C97—21no—
58C2×C8×C97—21no—
59C4⋊C4×C97—21no—
60C4⋊C4×C97—21no—
61C23×C97—22no—
62C23×C97—41no—
63C23×C97—41no—
64C23×C97—41no—
65C23×C97—41no—
66C23×C97—41no—
67C23×C97—41no—
68C23×C97C2×D411yescharacteristic
69C22×D97—22no—
70C22×D97—41no—
71C22×D97—41no—
72C22×D97—41no—
73C22×D97—41no—
74C22×D97—41no—
75C22×D97—41no—
76D4×C97—22no—
77D4×C97—22no—
78D4×C97—41no—
79D4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97—41no—
82C2×C4×C97—41no—
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D411yescharacteristic
86C2×C4×C97#4111yescharacteristic
87C2×C4×C97#4111yescharacteristic
88C2×C4×C97#4111yescharacteristic
89C2×Dic97—21no—
90C2×Dic97—21no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—21no—
94C2×Dic97—41no—
95C2×Dic97—41no—
96C2×Dic97—41no—
97C2×Dic97D812yescomplemented
98D388—22no—
99D388—22no—
100D388—41no—
101D388—41no—
102C8×C97—41no—
103#11715895—22no—
104C22×C97—42no—
105C22×C97—42no—
106C22×C97—42no—
107C22×C97—81no—
108C22×C97—81no—
109C22×C97—81no—
110C22×C97—81no—
111C22×C97#12111yescharacteristic
112C22×C97#12111yescharacteristic
113C22×C97#12111yescharacteristic
114C22×C97C2×D811yescharacteristic
115C22×C97#13511yescharacteristic
116C22×C97AGL(1,ℤ/8)11yescharacteristic
117C22×C97AGL(1,ℤ/8)11yescharacteristic
118C2×D97—42no—
119C2×D97—42no—
120C2×D97—42no—
121C2×D97—81no—
122C2×D97—81no—
123C2×D97—81no—
124C2×D97—81no—
125C4×C97—81no—
126C4×C97#12011yescharacteristic
127C4×C97#12311yescharacteristic
128C4×C97#12311yescharacteristic
129C4×C97#12411yescharacteristic
130Dic97—22no—
131Dic97—41no—
132Dic97—81no—
133C2×C97—82no—
134C2×C97#39411yescharacteristic
135C2×C97#45911yescharacteristic
136C2×C97#45911yescharacteristic
137C2×C97#46311yescharacteristic
138C2×C97#46311yescharacteristic
139C2×C97#46611yescharacteristic
140C2×C97#46911yescharacteristic
141D97—82no—
142#2054—971nomaximal, Sylow
143C97#205411yescharacteristic, complemented, Sylow
144C22×D8—971no—
145C2×C4⋊C8—971no—
146C2×#102—971no—
147C2×#102D9711yescharacteristic, complemented
148#390—971no—
149#390—971no—
150#340—971no—
151C22×D4—971no—
152C22×D4C2×D9711yescharacteristic
153C2×D8—1942no—
154C2×D8—1942no—
155C2×D8—1942no—
156C22×C8—971no—
157C22×C8C2×D9711yescharacteristic
158C2×C4⋊C4—971no—
159C2×C4⋊C4C2×D9711yescharacteristic
160C2×C22⋊C4—1941no—
161C2×C22⋊C4—1941no—
162C2×C22⋊C4—1941no—
163C2×C22⋊C4—1941no—
164C2×C42—971no—
165C4⋊C8—1942no—
166#102—22no—
167#102—1942no—
168C24—21no—
169C24—1941no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—972no—
176C2×D4—1941no—
177C2×D4—1941no—
178C2×D4—1941no—
179C2×D4—1941no—
180C2×D4—1941no—
181C2×D4C22⋊D9712yes—
182C22×C4—21no—
183C22×C4—971no—
184C22×C4—971no—
185C22×C4—1941no—
186C22×C4C22×D9711yescharacteristic
187D8—1948no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—972no—
192C2×C8—1941no—
193C2×C8—1941no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—1941no—
197C4⋊C4—1941no—
198C22⋊C4—3882no—
199C22⋊C4—3882no—
200C22⋊C4—3882no—
201C22⋊C4—3882no—
202C42—1941no—
203C42—1941no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—3881no—
212C23—3881no—
213C23—3881no—
214C23—3881no—
215C23—3881no—
216C23—3881no—
217C23—1942no—
218C23D4×D9711yescharacteristic
219D4—22no—
220D4—22no—
221D4—41no—
222D4—41no—
223D4—3881no—
224D4—3881no—
225D4—1942no—
226D4—1942no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—972no—
231C2×C4—3881no—
232C2×C4—3881no—
233C2×C4—3881no—
234C2×C4—1941no—
235C2×C4—1941no—
236C2×C4—1941no—
237C2×C4—1941no—
238C2×C4—1941no—
239C2×C4D4×D9711yescharacteristic
240C2×C4D4×D9711yescharacteristic
241C2×C4C2×C22⋊D9711yescharacteristic
242C2×C4#1309551011yescharacteristic
243C2×C4#1309551111yescharacteristic
244C2×C4#1309551111yescharacteristic
245C8—41no—
246C8—1942no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—7761no—
255C22—7761no—
256C22—7761no—
257C22—7761no—
258C22—3882no—
259C22—3882no—
260C22—3882no—
261C22#1569078211yescharacteristic
262C22#1569078311yescharacteristic
263C22#1569078311yescharacteristic
264C22D8×D9711yescharacteristic
265C22#1569083111yescharacteristic
266C22#1569083411yescharacteristic
267C22#1569083711yescharacteristic
268C4—81no—
269C4—7761no—
270C4—3881no—
271C4—1942no—
272C4#1569077811yescharacteristic
273C4#1569079211yescharacteristic
274C4#1569079311yescharacteristic
275C4#1569079611yescharacteristic
276C2—82no—
277C2—7762no—
278C2#2026065011yescharacteristic
279C2#2026092811yescharacteristic
280C2#2026092911yescharacteristic
281C2#2026094911yescharacteristic
282C2#2026095211yescharacteristic
283C2#2026097011yescharacteristic
284C2#2026098711yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#102×C97C2
socleC23×C97C2×D4
prime radicalC22×#13095496C2
hypercenterC2×#102D97
hypocenterC97#2054
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#15690739C2semidirect
3. C2×#15690732C2semidirect
4. #20260638C2semidirect
5. #20260636C2semidirect
6. #20260497C2semidirect
7. C2×#102×C97C2semidirect
97. C2×Dic97D8semidirect
143. C97#2054semidirect
147. C2×#102D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954962
C2×#156907392
C2×#156907322
#202606382
#202606362
#202604972
C2×#102×C972
#205497