Finite Group Databasebeta

← #25094571

Subgroups — #25094571

Order 11008.

1110 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582922C211yesmaximal, characteristic
2C2×#14582854C211yesmaximal, characteristic
3C22×#13063235C211yesmaximal, characteristic
4#19136998C211yesmaximal, characteristic
5#19134471C214yesmaximal, complemented
6#19134469C214yesmaximal, complemented
7#19134293C211yesmaximal, characteristic
8#19134292C211yesmaximal, characteristic
9C2×C43×#405C211yesmaximal, characteristic
10C22×#10621877C2211yescharacteristic
11C23×C43⋊C8C2211yescharacteristic
12C2×#13063284C2211yescharacteristic
13C22×C4×Dic43C2211yescharacteristic
14#14582922C418yescomplemented
15C2×Dic43⋊C8C2214yes—
16#14582854—24no—
17C2×#13063235C418yescomplemented
18C2×#13063235C2212yes—
19C2×#13063235C2212yes—
20C2×#13063235C2212yes—
21C2×C8×Dic43C2212yes—
22#14582768C412yes—
23#14582764C412yes—
24#14582757C414yes—
25C4×Dic43⋊C4C2214yes—
26#14582654C2214yes—
27#14582653C2214yes—
28C22×M16×C43C2211yescharacteristic
29C23×C8×C43C2211yescharacteristic
30C2×#117×C43C2211yescharacteristic
31C43×#405—24no—
32C2×#97×C43C2212yes—
33C2×C4×C8×C43C2212yes—
34C23×Dic43C2×C412yes—
35C2×#10621877—22no—
36C2×#10621877—22no—
37C2×#10621877C2×C414yes—
38C22×C43⋊C8—22no—
39C22×C43⋊C8—22no—
40C22×C43⋊C8C2×C414yes—
41C22×C43⋊C8C2312yes—
42C22×C43⋊C8C2312yes—
43#13063284—22no—
44#13063284—22no—
45C2×C4⋊Dic43C2×C412yes—
46C2×Dic43⋊C4C2×C418yes—
47C2×C22⋊Dic43C2×C412yes—
48C2×C4×Dic43C2×C411yescharacteristic
49C2×C4×Dic43C2×C411yescharacteristic
50C2×C4×Dic43C2×C414yes—
51C2×C4×Dic43C2×C414yes—
52C2×C4×Dic43C2312yes—
53C2×C4×Dic43C2312yes—
54Dic43⋊C8C2×C4116yes—
55#13063235—24no—
56#13063235C2×C418yes—
57#13063235C2×C418yes—
58C8×Dic43—24no—
59C43⋊#95C2×C418yes—
60C23×C4×C43C2311yescharacteristic
61C2×M16×C43—22no—
62C2×M16×C43—22no—
63C2×M16×C43C2×C414yes—
64C22×C8×C43C2×C414yes—
65C22×C8×C43D414yes—
66C22×C8×C43Q814yes—
67C22×C8×C43C2312yes—
68C22×C8×C43C2311yescharacteristic
69C22×C8×C43C2311yescharacteristic
70#117×C43—22no—
71#117×C43—22no—
72C2×C4⋊C4×C43C2×C412yes—
73C2×C22⋊C4×C43C2×C412yes—
74C2×C42×C43C2312yes—
75#97×C43—24no—
76C4×C8×C43—24no—
77C24×C43C22×C411yescharacteristic
78C22×C4×C43C22×C412yes—
79C22×C4×C43C22×C412yes—
80C22×C4×C43C22×C414yes—
81C22×C4×C43C22×C411yescharacteristic
82C22×C4×C43C2×D412yes—
83C22×C4×C43C2×Q812yes—
84C22×C4×C43#4112yes—
85C22×C4×C43#4112yes—
86C22×C4×C43C2411yescharacteristic
87C22×Dic43—44no—
88C22×Dic43C4218yes—
89C22×Dic43C22×C414yes—
90C22×Dic43C22×C412yes—
91C22×Dic43C22×C412yes—
92M16×C43—28no—
93C2×C8×C43—22no—
94C2×C8×C43—22no—
95C2×C8×C43C4⋊C4116yescomplemented
96C2×C8×C43C22×C414yes—
97C2×C8×C43C22×C412yes—
98C2×C8×C43C22×C412yes—
99C2×C8×C43C2×D412yes—
100C2×C8×C43C2×D412yes—
101C2×C8×C43C2×Q812yes—
102C2×C8×C43C2×Q812yes—
103C4⋊C4×C43—24no—
104C22⋊C4×C43—24no—
105C42×C43—22no—
106C42×C43—22no—
107C22⋊Dic43—24no—
108C4⋊Dic43—24no—
109Dic43⋊C4—48no—
110C4×Dic43—22no—
111C4×Dic43—22no—
112C4×Dic43C4214yes—
113C4×Dic43C4214yes—
114C4×Dic43C22×C414yes—
115C4×Dic43C22×C414yes—
116#10621877—28no—
117C2×C43⋊C8—28no—
118C2×C43⋊C8—22no—
119C2×C43⋊C8—22no—
120C2×C43⋊C8—22no—
121C2×C43⋊C8—22no—
122C2×C43⋊C8C22×C414yes—
123C2×C43⋊C8C22×C414yes—
124C23×C43—24no—
125C23×C43C2×C4212yes—
126C23×C43C4×D412yes—
127C23×C43C4×Q812yes—
128C23×C43C23×C411yescharacteristic
129C2×C4×C43—24no—
130C2×C4×C43—28no—
131C2×C4×C43C2×C4212yes—
132C2×C4×C43C2×C4⋊C414yes—
133C2×C4×C43#11714yes—
134C2×C4×C43C4×D414yes—
135C2×C4×C43C4×D412yes—
136C2×C4×C43C4×Q814yes—
137C2×C4×C43C4×Q812yes—
138C2×C4×C43C23×C411yescharacteristic
139C2×C4×C43C23×C411yescharacteristic
140C2×C4×C43C22×D411yescharacteristic
141C2×C4×C43C22×Q811yescharacteristic
142C2×C4×C43C2×#4111yescharacteristic
143C2×C4×C43C2×#4111yescharacteristic
144C2×Dic43—24no—
145C2×Dic43—24no—
146C2×Dic43—24no—
147C2×Dic43—28no—
148C2×Dic43—48no—
149C2×Dic43C2×C4214yes—
150C2×Dic43C2×C4214yes—
151C8×C43—24no—
152C8×C43C2×C4⋊C414yes—
153C8×C43C2×C4⋊C414yes—
154C43⋊C8—24no—
155C43⋊C8—24no—
156#20438—431nomaximal, Sylow
157C22×C43—24no—
158C22×C43—24no—
159C22×C43—24no—
160C22×C43C4×C4⋊C414yes—
161C22×C43C22×C4211yescharacteristic
162C22×C43C2×C4×D411yescharacteristic
163C22×C43C2×C4×Q811yescharacteristic
164C22×C43#56814yes—
165C4×C43—44no—
166C4×C43C4×C4⋊C414yes—
167C4×C43C22×C4⋊C411yescharacteristic
168C4×C43C2×#11711yescharacteristic
169C4×C43C2×C4×D411yescharacteristic
170C4×C43C2×C4×Q811yescharacteristic
171Dic43—24no—
172Dic43—24no—
173Dic43—44no—
174C2×#423—431no—
175C2×#405—431no—
176C2×#405D4311yescharacteristic
177C22×#97—431no—
178#2285—431no—
179#1771—434no—
180#1771—434no—
181#1729—431no—
182#1729—431no—
183C2×C43—24no—
184C2×C43C2×C4×C4⋊C411yescharacteristic
185C2×C43#287312yes—
186C2×C43#293212yes—
187C2×C43#296112yes—
188C22×M16—431no—
189C22×M16C2×D4311yescharacteristic
190C23×C8—431no—
191C23×C8C2×D4311yescharacteristic
192C2×#117—431no—
193C2×#117C2×D4311yescharacteristic
194C22×C42—431no—
195#423—438no—
196C2×C4⋊C8—434no—
197#405—24no—
198#405—864no—
199C2×#97—432no—
200C2×#97—432no—
201C2×#97—438no—
202C2×#97—432no—
203C2×#97C2×D4312yes—
204C2×C4×C8—432no—
205C2×C4×C8C2×D4312yes—
206#384—432no—
207#382—432no—
208#380—434no—
209C4×C4⋊C4—434no—
210#336—434no—
211#336—434no—
212C43#2043811yescharacteristic, complemented, Sylow
213C23×C4—432no—
214C23×C4C22×D4311yescharacteristic
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—434no—
218C2×M16—862no—
219C2×M16—862no—
220C2×M16C4×D4314yes—
221C22×C8—434no—
222C22×C8—432no—
223C22×C8—432no—
224C22×C8—862no—
225C22×C8—862no—
226C22×C8Dic8614yes—
227C22×C8C4×D4314yes—
228C22×C8C22⋊D4314yes—
229C22×C8C22×D4312yes—
230C22×C8C22×D4311yescharacteristic
231C22×C8C22×D4311yescharacteristic
232#117—22no—
233#117—22no—
234#117—862no—
235#117—862no—
236C2×C4⋊C4—432no—
237C2×C4⋊C4—438no—
238C2×C4⋊C4C4×D4312yes—
239C2×C22⋊C4—432no—
240C2×C22⋊C4C4×D4312yes—
241C2×C42—431no—
242C2×C42—431no—
243C2×C42—434no—
244C2×C42—434no—
245C2×C42—432no—
246C2×C42—432no—
247C2×C42C22×D4312yes—
248C4⋊C8—4316no—
249#97—24no—
250#97—438no—
251#97—438no—
252#97—864no—
253C4×C8—24no—
254C4×C8—864no—
255#95—438no—
256C24C2×C4×D4311yescharacteristic
257C22×C4—432no—
258C22×C4—432no—
259C22×C4—438no—
260C22×C4—434no—
261C22×C4—1724no—
262C22×C4C2×Dic8612yes—
263C22×C4C2×C4×D4314yes—
264C22×C4C2×C4×D4311yescharacteristic
265C22×C4C2×C4×D4312yes—
266C22×C4C2×C4×D4312yes—
267C22×C4#1062189812yes—
268C22×C4#1062189812yes—
269C22×C4C2×C22⋊D4312yes—
270C22×C4C23×D4311yescharacteristic
271M16—28no—
272M16—868no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—434no—
276C2×C8—434no—
277C2×C8—868no—
278C2×C8—862no—
279C2×C8—862no—
280C2×C8—862no—
281C2×C8—862no—
282C2×C8Dic43⋊C4116yescomplemented
283C2×C8C2×Dic8612yes—
284C2×C8C2×Dic8612yes—
285C2×C8C2×C4×D4314yes—
286C2×C8C2×C4×D4312yes—
287C2×C8C2×C4×D4312yes—
288C2×C8C2×C22⋊D4312yes—
289C2×C8C2×C22⋊D4312yes—
290C4⋊C4—24no—
291C4⋊C4—864no—
292C4⋊C4—1728no—
293C22⋊C4—24no—
294C22⋊C4—864no—
295C42—22no—
296C42—22no—
297C42—434no—
298C42—434no—
299C42—434no—
300C42—434no—
301C42—862no—
302C42—862no—
303C23—24no—
304C23C42×D4312yes—
305C23C4×C22⋊D4312yes—
306C23C4×Dic8612yes—
307C23C22×C4×D4311yescharacteristic
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—434no—
311C2×C4—434no—
312C2×C4—864no—
313C2×C4—864no—
314C2×C4—864no—
315C2×C4—868no—
316C2×C4—1728no—
317C2×C4C42×D4312yes—
318C2×C4C2×Dic43⋊C414yes—
319C2×C4#1306328214yes—
320C2×C4C4×C22⋊D4314yes—
321C2×C4C4×C22⋊D4312yes—
322C2×C4C4×Dic8614yes—
323C2×C4C4×Dic8612yes—
324C2×C4C22×C4×D4311yescharacteristic
325C2×C4C22×C4×D4311yescharacteristic
326C2×C4C22×C22⋊D4311yescharacteristic
327C2×C4C22×Dic8611yescharacteristic
328C2×C4C2×#1062189811yescharacteristic
329C2×C4C2×#1062189811yescharacteristic
330C8—24no—
331C8—864no—
332C8—864no—
333C8C2×Dic43⋊C414yes—
334C8C2×Dic43⋊C414yes—
335C22—24no—
336C22—24no—
337C22—24no—
338C22C4×Dic43⋊C414yes—
339C22C2×C42×D4311yescharacteristic
340C22C2×C4×C22⋊D4311yescharacteristic
341C22C2×C4×Dic8611yescharacteristic
342C22#1458366414yes—
343C4—44no—
344C4—864no—
345C4—864no—
346C4—1724no—
347C4C4×Dic43⋊C414yes—
348C4C22×Dic43⋊C411yescharacteristic
349C4C2×#1306328211yescharacteristic
350C4C2×C4×C22⋊D4311yescharacteristic
351C4C2×C4×Dic8611yescharacteristic
352C2—24no—
353C2C2×C4×Dic43⋊C411yescharacteristic
354C2#1914301112yes—
355C2#1914331112yes—
356C2#1914353312yes—
subgroupG/H
center, FrattiniC22×C4C23×D43
commutatorC22×C43C22×C42
FittingC2×C43×#405C2
socleC23×C43C23×C4
prime radicalC24×C43C22×C4
cosocleC22×C4×C43C24
hypercenterC2×#405D43
hypocenterC43#20438
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
5. #19134471C2semidirect
6. #19134469C2semidirect
14. #14582922C4semidirect
17. C2×#13063235C4semidirect
95. C2×C8×C43C4⋊C4semidirect
212. C43#20438semidirect
282. C2×C8Dic43⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#145829222
C2×#145828542
C22×#130632352
#191369982
#191344712
#191344692
#191342932
#191342922
C2×C43×#4052
#2043843