Finite Group Databasebeta

← #27142093

Subgroups — #27142093

Order 12416.

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

normal subgroups only

This group has 374 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, complemented
2C2×#15690734C211yesmaximal, characteristic, complemented
3C2×#15690731C211yesmaximal, characteristic
4#20260638C211yesmaximal, characteristic
5#20260498C211yesmaximal, characteristic, complemented
6#20260489C211yesmaximal, characteristic, complemented
7C22×SD16×C97C211yesmaximal, characteristic
8C22×#11715895C2211yescharacteristic
9Q8×Dic97—22no—
10Q8×Dic97—22no—
11C2×C4⋊Dic97—21no—
12C2×C4⋊Dic97C2211yescharacteristic
13C2×C4⋊Dic97C2211yescharacteristic
14C2×C22⋊Dic97—21no—
15C2×C22⋊Dic97—21no—
16C2×C4×Dic97—21no—
17C2×C4×Dic97C2211yescharacteristic
18#15690734—22no—
19#15690731—22no—
20C22×Q8×C97C2211yescharacteristic
21C22×D4×C97C2211yescharacteristic
22C2×SD16×C97—22no—
23C2×SD16×C97—22no—
24C2×SD16×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—
38C22⋊Dic97—42no—
39C22⋊Dic97—42no—
40C4×Dic97—24no—
41C4×Dic97—21no—
42C4×Dic97—21no—
43C24×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97C2×C412yes—
48C2×Q8×C97D412yes—
49C2×Q8×C97D412yes—
50C2×D4×C97—21no—
51C2×D4×C97—21no—
52C2×D4×C97—21no—
53C2×D4×C97—21no—
54C2×D4×C97—21no—
55C2×D4×C97C2×C412yes—
56C22×C4×C97—21no—
57C22×C4×C97C2311yescharacteristic
58SD16×C97—28no—
59C2×C8×C97—21no—
60C2×C8×C97—21no—
61C2×C8×C97C2×C412yes—
62C23×C97—22no—
63C23×C97—41no—
64C23×C97—41no—
65C23×C97—41no—
66C23×C97—41no—
67C23×C97—41no—
68C23×C97—41no—
69C23×C97C2×D411yescharacteristic
70Q8×C97—22no—
71Q8×C97—22no—
72Q8×C97—22no—
73Q8×C97C22⋊C414yescomplemented
74D4×C97—22no—
75D4×C97—22no—
76D4×C97—41no—
77D4×C97—41no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97C22×C411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97#4111yescharacteristic
86C2×C4×C97#4111yescharacteristic
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—21no—
90C2×Dic97—22no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—41no—
94C2×Dic97—41no—
95C2×Dic97—41no—
96C2×Dic97—41no—
97C2×Dic97—41no—
98C2×Dic97SD1612yescomplemented
99C8×C97—22no—
100#11715895—41no—
101C22×C97—42no—
102C22×C97—42no—
103C22×C97—42no—
104C22×C97—81no—
105C22×C97—81no—
106C22×C97—81no—
107C22×C97—81no—
108C22×C97C4×D411yescharacteristic
109C22×C97#12011yescharacteristic
110C22×C97#12111yescharacteristic
111C22×C97C2×SD1611yescharacteristic
112C22×C97#13511yescharacteristic
113C22×C97AGL(1,ℤ/8)11yescharacteristic
114C22×C97#13711yescharacteristic
115C4×C97—24no—
116C4×C97C2×C22⋊C411yescharacteristic
117C4×C97C4×D411yescharacteristic
118C4×C97#12111yescharacteristic
119C4×C97#12311yescharacteristic
120Dic97—22no—
121Dic97—41no—
122Dic97—42no—
123Dic97—81no—
124C2×C97—82no—
125C2×C97#38611yescharacteristic
126C2×C97C4×SD1611yescharacteristic
127C2×C97#44011yescharacteristic
128C2×C97#44811yescharacteristic
129C2×C97#44911yescharacteristic
130C2×C97#46011yescharacteristic
131C2×C97#46411yescharacteristic
132#1880—971nomaximal, Sylow
133C97#188011yescharacteristic, complemented, Sylow
134C22×SD16D9711yescharacteristic
135C2×C4×Q8—971no—
136C2×#103—971no—
137C2×#102—971no—
138#390—971no—
139#340—971no—
140#336—971no—
141C22×Q8C2×D9711yescharacteristic
142C22×D4C2×D9711yescharacteristic
143C2×SD16—22no—
144C2×SD16—22no—
145C2×SD16Dic9714yescomplemented
146C22×C8—971no—
147C22×C8C2×D9711yescharacteristic
148C4×Q8—1942no—
149C4×Q8—1942no—
150C2×C4⋊C4—971no—
151C2×C4⋊C4—971no—
152C2×C4⋊C4—1941no—
153C2×C22⋊C4—1941no—
154C2×C22⋊C4—1941no—
155C2×C42—971no—
156C2×C42—1941no—
157#103—1942no—
158#102—1942no—
159C24—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8C2×Dic9712yes—
164C2×Q8C22⋊D9712yes—
165C2×Q8C22⋊D9712yes—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C2×Dic9712yes—
172C22×C4—21no—
173C22×C4—971no—
174C22×C4—971no—
175C22×C4—1941no—
176C22×C4—1941no—
177C22×C4C22×D9711yescharacteristic
178SD16—28no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—1941no—
182C2×C8—1941no—
183C2×C8—1941no—
184C2×C8C2×Dic9712yes—
185C4⋊C4—972no—
186C4⋊C4—1944no—
187C4⋊C4—1941no—
188C4⋊C4—1941no—
189C4⋊C4—1941no—
190C22⋊C4—3882no—
191C22⋊C4—3882no—
192C42—1944no—
193C42—1941no—
194C42—1941no—
195C23—22no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23D4×D9711yescharacteristic
203Q8—22no—
204Q8—22no—
205Q8—22no—
206Q8C22⋊Dic9714yescomplemented
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—972no—
215C2×C4—3881no—
216C2×C4—3881no—
217C2×C4—3881no—
218C2×C4—3881no—
219C2×C4—3881no—
220C2×C4—1942no—
221C2×C4—1941no—
222C2×C4—1941no—
223C2×C4—1941no—
224C2×C4—1941no—
225C2×C4—1941no—
226C2×C4C22×Dic9711yescharacteristic
227C2×C4D4×D9711yescharacteristic
228C2×C4C2×C22⋊D9711yescharacteristic
229C2×C4C2×C22⋊D9711yescharacteristic
230C2×C4#1309551011yescharacteristic
231C2×C4#1309551011yescharacteristic
232C8—22no—
233C8—3881no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22D4×Dic9711yescharacteristic
242C22#1569077811yescharacteristic
243C22#1569078211yescharacteristic
244C22SD16×D9711yescharacteristic
245C22#1569083111yescharacteristic
246C22#1569083711yescharacteristic
247C22#1569083911yescharacteristic
248C4—24no—
249C4—7761no—
250C4—3882no—
251C4—3881no—
252C4—1942no—
253C4C2×C22⋊Dic9711yescharacteristic
254C4D4×Dic9711yescharacteristic
255C4#1569078211yescharacteristic
256C4#1569079111yescharacteristic
257C2—82no—
258C2#2026061511yescharacteristic
259C2SD16×Dic9711yescharacteristic
260C2#2026083611yescharacteristic
261C2#2026087211yescharacteristic
262C2#2026087911yescharacteristic
263C2#2026093311yescharacteristic
264C2#2026095511yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
prime radicalC22×D4×C97C22
cosocleC22×C4×C97C23
hypercenterC22×SD16D97
hypocenterC97#1880
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic97C2semidirect
2. C2×#15690734C2semidirect
5. #20260498C2semidirect
6. #20260489C2semidirect
24. C2×SD16×C97C4semidirect
73. Q8×C97C22⋊C4semidirect
98. C2×Dic97SD16semidirect
133. C97#1880semidirect
145. C2×SD16Dic97semidirect
206. Q8C22⋊Dic97semidirect

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
C2×#156907312
#202606382
#202604982
#202604892
C22×SD16×C972
#188097