Finite Group Databasebeta

← #14673231

Subgroups — #14673231

Order 2816.

512 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 77 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×#10622180C212yesmaximal, complemented
2#13069020C212yesmaximal, complemented
3#13068830C211yesmaximal, characteristic, complemented
4#13066743C211yesmaximal, characteristic, complemented
5C11×#2733C211yesmaximal, characteristic
6C2×C11×#124C2212yes—
7#10623343C2211yescharacteristic
8C11×#394—21no—
9C11×#392—22no—
10C11×#391C2211yescharacteristic
11C11×#390—21no—
12C11×#388—22no—
13C2×#66900C2212yes—
14C2×#66900C2211yescharacteristic
15#10622828—22no—
16#10622577—21no—
17#10622180—24no—
18#10622161—21no—
19#10622161—22no—
20C22×Q8×C11C2311yescharacteristic
21C22×D4×C11—21no—
22C22×D4×C11—21no—
23C22×D4×C11D412yes—
24C11×#124—22no—
25C11×#124—22no—
26C11×#124—22no—
27C11×#124—22no—
28C2×C11×C4⋊C4—21no—
29C2×C11×C4⋊C4D411yescharacteristic
30C2×C11×C22⋊C4—22no—
31C2×C11×C22⋊C4—22no—
32C2×C11×C22⋊C4—22no—
33C2×C11×C22⋊C4—22no—
34C2×C42×C11D412yes—
35C2×C42×C11D411yescharacteristic
36Q8×Dic11—41no—
37Q8×Dic11—41no—
38#66979—22no—
39#66979—21no—
40C2×#4318—21no—
41C2×#4318—22no—
42C11×#95—21no—
43C11×#95—22no—
44#66900—21no—
45#66900—22no—
46#66900—22no—
47#66900D412yescomplemented
48#66896—42no—
49#66896—42no—
50#66872—41no—
51#66871—41no—
52#13426—111nomaximal, Sylow
53C24×C11—22no—
54C2×Q8×C11—41no—
55C2×Q8×C11—41no—
56C2×Q8×C11C2×D411yescharacteristic
57C2×Q8×C11#4111yescharacteristic
58C2×Q8×C11#4112yes—
59C2×D4×C11—22no—
60C2×D4×C11—22no—
61C2×D4×C11—42no—
62C2×D4×C11—42no—
63C2×D4×C11—42no—
64C2×D4×C11—81no—
65C2×D4×C11—81no—
66C2×D4×C11—82no—
67C2×D4×C11—81no—
68C22×C4×C11—21no—
69C22×C4×C11—21no—
70C22×C4×C11—22no—
71C22×C4×C11C2×D412yes—
72C22×C4×C11C2×D411yescharacteristic
73C11×C4⋊C4—41no—
74C11×C4⋊C4—81no—
75C11×C22⋊C4—44no—
76C11×C22⋊C4—42no—
77C11×C22⋊C4—42no—
78C11×C22⋊C4—42no—
79C11×C22⋊C4—42no—
80C11×C22⋊C4—82no—
81C42×C11—22no—
82C42×C11—22no—
83C42×C11—41no—
84C22⋊Dic11—41no—
85C4⋊Dic11—21no—
86C4⋊Dic11—21no—
87C4⋊Dic11—22no—
88C4⋊Dic11—22no—
89C4×Dic11—41no—
90C4×Dic11—42no—
91#4318—22no—
92#4318—41no—
93#4318—42no—
94#4318—42no—
95C2×C11⋊C8—41no—
96C2×C11⋊C8—42no—
97#2733D1111yescharacteristic
98C2×#355—112no—
99#1888—112no—
100#1857—111no—
101#1394—111no—
102C23×C11—42no—
103C23×C11—42no—
104C23×C11—42no—
105C23×C11—42no—
106C23×C11—42no—
107C23×C11—42no—
108C23×C11—42no—
109C23×C11#12011yescharacteristic
110Q8×C11—41no—
111Q8×C11—42no—
112Q8×C11—41no—
113D4×C11—44no—
114D4×C11—82no—
115D4×C11—84no—
116D4×C11—84no—
117C2×C4×C11—21no—
118C2×C4×C11—22no—
119C2×C4×C11—21no—
120C2×C4×C11—22no—
121C2×C4×C11—22no—
122C2×C4×C11—41no—
123C2×C4×C11—41no—
124C2×C4×C11—42no—
125C2×C4×C11—42no—
126C2×C4×C11—42no—
127C2×C4×C11—42no—
128C2×C4×C11—41no—
129C2×C4×C11#12111yescharacteristic
130C2×C4×C11#12112yes—
131C2×C4×C11#12312yes—
132C2×C4×C11#12411yescharacteristic
133C2×Dic11—41no—
134C2×Dic11—41no—
135C11⋊C8—42no—
136C11⋊C8—82no—
137C2×#124C2×D1112yes—
138#519—111no—
139#428—221no—
140#418—222no—
141C2×#101—111no—
142C2×#101—112no—
143#394—21no—
144#392—22no—
145#391C2×D1111yescharacteristic
146#390—21no—
147#388—22no—
148#355—224no—
149#339—221no—
150#339—222no—
151C22×C11—21no—
152C22×C11—21no—
153C22×C11—82no—
154C22×C11—82no—
155C22×C11—82no—
156C22×C11—82no—
157C22×C11—82no—
158C22×C11—82no—
159C22×C11—82no—
160C22×C11#39411yescharacteristic
161C22×C11B(4,𝔽2)11yescharacteristic
162C22×C11#45811yescharacteristic
163C4×C11—22no—
164C4×C11—21no—
165C4×C11—22no—
166C4×C11—21no—
167C4×C11—42no—
168C4×C11—82no—
169C4×C11—81no—
170Dic11—81no—
171Dic11—81no—
172C22×Q8C22×D1111yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C22⋊D1112yes—
176C2×M16—221no—
177C2×M16—222no—
178#124—22no—
179#124—22no—
180#124—22no—
181#124—22no—
182C4×Q8—441no—
183C4×Q8—441no—
184#117—222no—
185#117—221no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C22⋊D1111yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42C22⋊D1112yes—
193C2×C42C22⋊D1111yescharacteristic
194#107—441no—
195#106—441no—
196#102—442no—
197#102—442no—
198#101—112no—
199#101—221no—
200#101—222no—
201#101—222no—
202#95—21no—
203#95—22no—
204C2×C11—21no—
205C2×C11—21no—
206C2×C11—162no—
207C2×C11#204411yescharacteristic
208C2×C11#220211yescharacteristic
209C2×C11#220411yescharacteristic
210C24—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8D4×D1111yescharacteristic
214C2×Q8#434111yescharacteristic
215C2×Q8#434112yes—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—81no—
222C2×D4—81no—
223C2×D4—82no—
224C2×D4—81no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4C2×C22⋊D1112yes—
229C22×C4C2×C22⋊D1111yescharacteristic
230M16—222no—
231M16—441no—
232M16—442no—
233M16—442no—
234C2×C8—441no—
235C2×C8—442no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—222no—
239C4⋊C4—222no—
240C4⋊C4—221no—
241C4⋊C4—221no—
242C22⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C22⋊C4—441no—
249C42—22no—
250C42—22no—
251C42—41no—
252C42—441no—
253C42—442no—
254C11#1342611yescharacteristic, complemented, Sylow
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23#6700611yescharacteristic
263Q8—41no—
264Q8—41no—
265Q8—42no—
266D4—44no—
267D4—82no—
268D4—84no—
269D4—84no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—441no—
283C2×C4—441no—
284C2×C4#6698812yes—
285C2×C4#6698911yescharacteristic
286C2×C4#6699111yescharacteristic
287C2×C4#6699212yes—
288C8—442no—
289C8—882no—
290C22—21no—
291C22—21no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22#1062283511yescharacteristic
300C22#1062285611yescharacteristic
301C22#1062285711yescharacteristic
302C4—21no—
303C4—22no—
304C4—22no—
305C4—21no—
306C4—42no—
307C4—82no—
308C4—81no—
309C4—881no—
310C4—881no—
311C2—21no—
312C2—21no—
313C2—162no—
314C2#1306986411yescharacteristic
315C2#1307063311yescharacteristic
316C2#1307064211yescharacteristic
subgroupG/H
centerC22#10622857
commutator, cosocleC22×Q8×C11C23
FrattiniC22×Q8C22×D11
Fitting, prime radicalC11×#2733C2
socleC22×C11B(4,𝔽2)
hypercenter#2733D11
hypocenterC11#13426
LCSG ▷ C22×Q8×C11 ▷ C22×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. C2×#10622180C2semidirect
2. #13069020C2semidirect
3. #13068830C2semidirect
4. #13066743C2semidirect
47. #66900D4semidirect
254. C11#13426semidirect

Maximal subgroups

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

subgroupindex
C2×#106221802
#130690202
#130688302
#130667432
C11×#27332
#1342611