Finite Group Databasebeta

← #25101065

Subgroups — #25101065

Order 11008.

1386 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic86C211yesmaximal, characteristic
2C2×#14582923C211yesmaximal, characteristic
3#19135535C214yesmaximal, complemented
4#19135531C214yesmaximal, complemented
5#19134288C214yesmaximal
6C2×C4×M16×C43C211yesmaximal, characteristic
7C23×Dic86C411yescharacteristic
8C22×#10621877C2212yes—
9C2×C4×Dic86—24no—
10C2×C4×Dic86—24no—
11C2×C4×Dic86C412yes—
12C2×C4×Dic86C412yes—
13C2×C4×Dic86C2212yes—
14C2×C4×Dic86C2212yes—
15C22×C4⋊Dic43C411yescharacteristic
16C22×Dic43⋊C4C412yes—
17C22×C4×Dic43C2212yes—
18#14582923—24no—
19C2×C4⋊(C43⋊C8)C2212yes—
20C2×C4⋊(C43⋊C8)C2212yes—
21#14582653C22116yes—
22C22×M16×C43C2212yes—
23C22×C42×C43C2211yescharacteristic
24C4×M16×C43—24no—
25C2×#97×C43C2212yes—
26C2×C4×C8×C43C2212yes—
27C22×Dic86—24no—
28C22×Dic86—28no—
29C22×Dic86C2×C412yes—
30C22×Dic86C2×C412yes—
31C23×Dic43C2×C412yes—
32C23×Dic43C2×C412yes—
33C2×#10621877—24no—
34C2×#10621877—28no—
35C22×C43⋊C8C2314yes—
36C4×Dic86—28no—
37C4×Dic86—28no—
38C4×Dic86—24no—
39C4×Dic86—24no—
40C4×Dic86—24no—
41C4×Dic86—24no—
42C2×C4⋊Dic43—24no—
43C2×C4⋊Dic43C2×C412yes—
44C2×C4⋊Dic43C2×C412yes—
45C2×Dic43⋊C4—28no—
46C2×Dic43⋊C4C2×C414yes—
47C2×Dic43⋊C4C2×C414yes—
48C2×C4×Dic43—24no—
49C2×C4×Dic43—24no—
50C2×C4×Dic43C2×C414yes—
51C2×C4×Dic43C2314yes—
52C4⋊(C43⋊C8)—24no—
53C4⋊(C43⋊C8)—24no—
54C23×C4×C43C2×C411yescharacteristic
55C23×C4×C43C2×C411yescharacteristic
56C23×C4×C43C2311yescharacteristic
57C2×M16×C43—24no—
58C2×M16×C43—24no—
59C2×M16×C43Q818yescomplemented
60C22×C8×C43C2314yes—
61C2×C42×C43C2×C411yescharacteristic
62C2×C42×C43C2×C411yescharacteristic
63C2×C42×C43D414yes—
64C2×C42×C43D414yes—
65C2×C42×C43C2311yescharacteristic
66C2×C42×C43C2311yescharacteristic
67#97×C43—24no—
68C4×C8×C43—24no—
69C24×C43C22×C411yescharacteristic
70C22×C4×C43—24no—
71C22×C4×C43—42no—
72C22×C4×C43C22⋊C414yes—
73C22×C4×C43C22⋊C414yes—
74C22×C4×C43C22×C411yescharacteristic
75C22×C4×C43C22×C411yescharacteristic
76C22×C4×C43C22×C411yescharacteristic
77C22×C4×C43C22×C411yescharacteristic
78C22×C4×C43C22×C411yescharacteristic
79C22×C4×C43C2×D412yes—
80C22×C4×C43C2×D412yes—
81C22×C4×C43C2×Q814yes—
82C22×C4×C43#4114yes—
83C22×C4×C43C2411yescharacteristic
84C22×Dic43—28no—
85C22×Dic43—24no—
86C22×Dic43—24no—
87C22×Dic43—44no—
88C22×Dic43C22×C414yes—
89C22×Dic43C22×C414yes—
90C2×Dic86—232no—
91C2×Dic86—24no—
92C2×Dic86—24no—
93C2×Dic86—24no—
94C2×Dic86—28no—
95C2×Dic86M1618yescomplemented
96M16×C43—216no—
97C2×C8×C43—24no—
98C2×C8×C43—24no—
99C2×C8×C43C2×Q818yescomplemented
100C42×C43C22⋊C414yes—
101C42×C43C22⋊C414yes—
102C42×C43C2×D412yes—
103C42×C43C2×D412yes—
104C42×C43C2×D412yes—
105C42×C43C2×D412yes—
106C4⋊Dic43—24no—
107C4⋊Dic43—24no—
108Dic43⋊C4—28no—
109Dic43⋊C4—28no—
110C4×Dic43—28no—
111C4×Dic43—24no—
112C4×Dic43—24no—
113#10621877—48no—
114C2×C43⋊C8—24no—
115C2×C43⋊C8—28no—
116C23×C43—22no—
117C23×C43—22no—
118C23×C43C2×C22⋊C412yes—
119C23×C43C4×Q814yes—
120C23×C43C23×C411yescharacteristic
121C2×C4×C43—22no—
122C2×C4×C43—22no—
123C2×C4×C43—22no—
124C2×C4×C43—28no—
125C2×C4×C43—44no—
126C2×C4×C43C2×C22⋊C412yes—
127C2×C4×C43C2×C22⋊C412yes—
128C2×C4×C43C2×C22⋊C412yes—
129C2×C4×C43C2×C22⋊C412yes—
130C2×C4×C43C2×C22⋊C412yes—
131C2×C4×C43C4×Q814yes—
132C2×C4×C43C22⋊Q818yes—
133C2×C4×C43#12414yes—
134C2×C4×C43C4⋊Q814yes—
135C2×C4×C43C2×M1614yes—
136C2×C4×C43C22×D411yescharacteristic
137C2×C4×C43C22×D411yescharacteristic
138C2×C4×C43C22×Q812yes—
139C2×C4×C43C2×#4112yes—
140C2×Dic43—216no—
141C2×Dic43—24no—
142C2×Dic43—24no—
143C2×Dic43—24no—
144C2×Dic43—28no—
145C2×Dic43—48no—
146C2×Dic43C2×M1618yes—
147Dic86—216no—
148Dic86—216no—
149C8×C43—28no—
150C43⋊C8—44no—
151#21303—431nomaximal, Sylow
152C22×C43—22no—
153C22×C43—22no—
154C22×C43—28no—
155C22×C43#39114yes—
156C22×C43C22×C22⋊C411yescharacteristic
157C22×C43C2×C4×Q812yes—
158C22×C43C22×M1612yes—
159C22×C43#56812yes—
160C4×C43—22no—
161C4×C43—22no—
162C4×C43—22no—
163C4×C43#39114yes—
164C4×C43#40714yes—
165C4×C43C2×C22⋊Q812yes—
166C4×C43C2×#12411yescharacteristic
167C4×C43C2×C4⋊Q811yescharacteristic
168Dic43—28no—
169Dic43—44no—
170C22×C4×Q8—431no—
171C2×#423—431no—
172C2×C4×M16D4311yescharacteristic
173#1985—434no—
174#1984—434no—
175#1728—434no—
176C2×C43—24no—
177C2×C43C2×#39111yescharacteristic
178C2×C43C2×#40711yescharacteristic
179C2×C43#287911yescharacteristic
180C2×C43Q8×M1614yes—
181C23×Q8—431no—
182C22×M16—432no—
183C22×M16C2×D4312yes—
184C2×C4×Q8—432no—
185C2×C4×Q8—432no—
186C2×C4×Q8—432no—
187C2×C4×Q8—432no—
188C2×C4×Q8—864no—
189C2×C4×Q8—864no—
190C22×C4⋊C4—432no—
191C22×C4⋊C4—431no—
192C22×C42—432no—
193C22×C42C2×D4311yescharacteristic
194#423—864no—
195C2×C4⋊C8—432no—
196C2×C4⋊C8—432no—
197C4×M16—24no—
198C2×#97C2×D4312yes—
199C2×C4×C8C2×D4312yes—
200#336—4316no—
201C43#2130311yescharacteristic, complemented, Sylow
202C22×Q8—432no—
203C22×Q8—432no—
204C22×Q8—868no—
205C22×Q8—864no—
206C23×C4—432no—
207C23×C4—432no—
208C23×C4C4×D4311yescharacteristic
209C23×C4C4×D4311yescharacteristic
210C23×C4C22×D4311yescharacteristic
211C2×M16—24no—
212C2×M16—24no—
213C2×M16—864no—
214C2×M16—868no—
215C2×M16Dic8618yescomplemented
216C22×C8—434no—
217C22×C8C22×D4314yes—
218C4×Q8—868no—
219C4×Q8—868no—
220C4×Q8—864no—
221C4×Q8—864no—
222C4×Q8—864no—
223C4×Q8—864no—
224C2×C4⋊C4—434no—
225C2×C4⋊C4—434no—
226C2×C4⋊C4—432no—
227C2×C4⋊C4—432no—
228C2×C4⋊C4—868no—
229C2×C4⋊C4—864no—
230C2×C42—434no—
231C2×C42—434no—
232C2×C42—864no—
233C2×C42—864no—
234C2×C42C4×D4311yescharacteristic
235C2×C42C4×D4311yescharacteristic
236C2×C42D17214yes—
237C2×C42C22⋊D4314yes—
238C2×C42C22×D4311yescharacteristic
239C2×C42C22×D4311yescharacteristic
240C4⋊C8—864no—
241C4⋊C8—864no—
242#97—24no—
243C4×C8—24no—
244C24C2×C4×D4311yescharacteristic
245C2×Q8—438no—
246C2×Q8—868no—
247C2×Q8—8632no—
248C2×Q8—864no—
249C2×Q8—864no—
250C2×Q8—864no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—434no—
254C22×C4—434no—
255C22×C4—868no—
256C22×C4—864no—
257C22×C4—864no—
258C22×C4—1724no—
259C22×C4(C2×D43)⋊C414yes—
260C22×C4(C2×D43)⋊C414yes—
261C22×C4C2×Dic8614yes—
262C22×C4C2×C4×D4311yescharacteristic
263C22×C4C2×C4×D4311yescharacteristic
264C22×C4C2×C4×D4311yescharacteristic
265C22×C4C2×C4×D4311yescharacteristic
266C22×C4C2×C4×D4311yescharacteristic
267C22×C4C2×D17212yes—
268C22×C4#1062189814yes—
269C22×C4C2×C22⋊D4312yes—
270C22×C4C23×D4311yescharacteristic
271M16—216no—
272M16—1728no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—864no—
276C2×C8—868no—
277C2×C8C2×Dic8618yescomplemented
278C4⋊C4—868no—
279C4⋊C4—868no—
280C4⋊C4—864no—
281C4⋊C4—864no—
282C42—868no—
283C42—864no—
284C42—864no—
285C42(C2×D43)⋊C414yes—
286C42(C2×D43)⋊C414yes—
287C42C2×D17212yes—
288C42C2×D17212yes—
289C42C2×C22⋊D4312yes—
290C42C2×C22⋊D4312yes—
291C23—22no—
292C23—22no—
293C23C2×(C2×D43)⋊C412yes—
294C23C4×Dic8614yes—
295C23C22×C4×D4311yescharacteristic
296Q8—8616no—
297Q8—8616no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—28no—
302C2×C4—44no—
303C2×C4—438no—
304C2×C4—8616no—
305C2×C4—864no—
306C2×C4—864no—
307C2×C4—864no—
308C2×C4—868no—
309C2×C4—1728no—
310C2×C4C2×(C2×D43)⋊C412yes—
311C2×C4C2×(C2×D43)⋊C412yes—
312C2×C4C2×(C2×D43)⋊C412yes—
313C2×C4C2×(C2×D43)⋊C412yes—
314C2×C4C2×(C2×D43)⋊C412yes—
315C2×C4C4×Dic8614yes—
316C2×C4#1306330418yes—
317C2×C4#1306331314yes—
318C2×C4#1306332514yes—
319C2×C4M16×D4314yes—
320C2×C4C22×C22⋊D4311yescharacteristic
321C2×C4C22×D17211yescharacteristic
322C2×C4C22×Dic8612yes—
323C2×C4C2×#1062189812yes—
324C8—28no—
325C8—1724no—
326C22—22no—
327C22—28no—
328C22—22no—
329C22#1458280514yes—
330C22C22×(C2×D43)⋊C411yescharacteristic
331C22C2×C4×Dic8612yes—
332C22C2×M16×D4312yes—
333C22#1458366512yes—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—868no—
338C4—1724no—
339C4#1458280514yes—
340C4#1458286214yes—
341C4C2×#1306330412yes—
342C4C2×#1306331311yescharacteristic
343C4C2×#1306332511yescharacteristic
344C2—24no—
345C2C2×#1458280511yescharacteristic
346C2C2×#1458286211yescharacteristic
347C2#1914303911yescharacteristic
348C2M16×Dic8614yes—
subgroupG/H
center, FrattiniC22×C4C23×D43
commutator, socleC23×C43C23×C4
FittingC2×C4×M16×C43C2
prime radicalC24×C43C22×C4
cosocleC22×C4×C43C24
hypercenterC2×C4×M16D43
hypocenterC43#21303
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C22×C4 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
3. #19135535C2semidirect
4. #19135531C2semidirect
59. C2×M16×C43Q8semidirect
95. C2×Dic86M16semidirect
99. C2×C8×C43C2×Q8semidirect
201. C43#21303semidirect
215. C2×M16Dic86semidirect
277. C2×C8C2×Dic86semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic862
C2×#145829232
#191355352
#191355312
#191342882
C2×C4×M16×C432
#2130343