Finite Group Databasebeta

← #25023761

Subgroups — #25023761

Order 11008.

4390 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10621900C211yesmaximal, characteristic
2C2×#14582758C212yesmaximal, complemented
3C2×#14582757C214yesmaximal, complemented
4#19132865C212yesmaximal, complemented
5#19132836C211yesmaximal, characteristic, complemented
6#19132834C214yesmaximal, complemented
7C43×#1438C211yesmaximal, characteristic, complemented
8C22×#10621900—216no—
9C22×#10621900—28no—
10C22×#10621900C418yescomplemented
11C23×Dic86C2211yescharacteristic
12C23×C22⋊D43C2212yes—
13C24×Dic43C2212yes—
14C23×C4×D43C2211yescharacteristic
15C22×Dic43⋊C4C2214yes—
16C22×C22⋊Dic43C2214yes—
17C22×(C2×D43)⋊C4C2214yes—
18#14582758—28no—
19#14582757—216no—
20C2×C43⋊#95C2218yes—
21C2×C43⋊#95C2212yes—
22C23×D4×C43C2211yescharacteristic
23C22×C22⋊C4×C43C2214yes—
24C2×#95×C43C2212yes—
25C24×D43C2311yescharacteristic
26C2×#10621900—2128no—
27C2×#10621900—264no—
28C2×#10621900—232no—
29C22×Dic86—24no—
30C22×Dic86—28no—
31C22×Dic86C2×C414yes—
32C22×C22⋊D43—28no—
33C22×C22⋊D43—28no—
34C22×C22⋊D43—28no—
35C22×C22⋊D43C2×C418yes—
36C23×Dic43—28no—
37C23×Dic43—28no—
38C23×Dic43C2×C418yes—
39C23×Dic43D4116yes—
40C23×Dic43C2314yes—
41C23×Dic43C2312yes—
42C23×Dic43C2311yescharacteristic
43C22×C4×D43—28no—
44C22×C4×D43C2×C414yes—
45C22×C4×D43D418yes—
46C2×Dic43⋊C4—216no—
47C2×Dic43⋊C4—28no—
48C2×C22⋊Dic43—28no—
49C2×C22⋊Dic43—48no—
50C2×(C2×D43)⋊C4—28no—
51C2×(C2×D43)⋊C4—48no—
52C43⋊#95—24no—
53C43⋊#95—216no—
54C25×C43C2312yes—
55C22×D4×C43—28no—
56C22×D4×C43—24no—
57C22×D4×C43C2×C414yes—
58C23×C4×C43C2314yes—
59C23×C4×C43C2311yescharacteristic
60C2×C22⋊C4×C43—28no—
61C2×C22⋊C4×C43—48no—
62#95×C43—24no—
63C24×C43—24no—
64C24×C43—24no—
65C24×C43—28no—
66C24×C43—28no—
67C24×C43C22×C414yes—
68C24×C43C2×D418yes—
69C24×C43C2411yescharacteristic
70C23×D43—24no—
71C23×D43—24no—
72C23×D43—24no—
73C23×D43C22×C412yes—
74C23×D43C2×D414yes—
75C2×D4×C43—232no—
76C2×D4×C43—28no—
77C2×D4×C43—216no—
78C22×C4×C43—24no—
79C22×C4×C43—24no—
80C22×C4×C43—28no—
81C22×C4×C43—24no—
82C22×C4×C43—28no—
83C22×C4×C43—24no—
84C22×C4×C43C22×C412yes—
85C22×C4×C43C2×D414yes—
86C2×C22⋊D43—264no—
87C2×C22⋊D43—216no—
88C2×C22⋊D43—216no—
89C2×C22⋊D43—216no—
90C22×Dic43—216no—
91C22×Dic43—216no—
92C22×Dic43—264no—
93C22×Dic43—24no—
94C22×Dic43—24no—
95C22×Dic43—24no—
96C22×Dic43—28no—
97C22×Dic43—28no—
98C22×Dic43—24no—
99C22×Dic43—24no—
100C22×Dic43C22⋊C4132yescomplemented
101C22×Dic43C22×C414yes—
102C22×Dic43C22×C412yes—
103C22×Dic43C2×D418yes—
104C22×Dic43C2×D418yes—
105C22×Dic43C2×D414yes—
106#10621900—2256no—
107C2×C4×D43—216no—
108C2×C4×D43—232no—
109C2×C4×D43C22⋊C4116yescomplemented
110C2×Dic86—232no—
111C2×Dic86—28no—
112C2×Dic86—216no—
113C22⋊C4×C43—416no—
114C22⋊Dic43—416no—
115(C2×D43)⋊C4—416no—
116Dic43⋊C4—416no—
117C23×C43—24no—
118C23×C43—24no—
119C23×C43—216no—
120C23×C43—216no—
121C23×C43—28no—
122C23×C43—28no—
123C23×C43—216no—
124C23×C43—44no—
125C23×C43—44no—
126C23×C43—44no—
127C23×C43—44no—
128C23×C43—48no—
129C23×C43—48no—
130C23×C43C2×C22⋊C418yes—
131C23×C43C23×C411yescharacteristic
132C23×C43C22×D412yes—
133C23×C43C22×D414yes—
134C23×C43H(5,𝔽2)14yes—
135C23×C432−1+414yes—
136C22×D43—24no—
137C22×D43—28no—
138C22×D43—28no—
139C22×D43—24no—
140C22×D43—24no—
141C22×D43—28no—
142C22×D43—44no—
143C22×D43—44no—
144C22×D43—48no—
145C22×D43C2×C22⋊C414yes—
146D4×C43—232no—
147C2×C4×C43—24no—
148C2×C4×C43—28no—
149C2×C4×C43—44no—
150C2×C4×C43—44no—
151C2×C4×C43—48no—
152C2×C4×C43—48no—
153C2×C4×C43—44no—
154C2×C4×C43C2×C22⋊C414yes—
155C22⋊D43—264no—
156C2×Dic43—24no—
157C2×Dic43—24no—
158C2×Dic43—216no—
159C2×Dic43—264no—
160C2×Dic43—44no—
161C2×Dic43—44no—
162C2×Dic43—48no—
163C2×Dic43—48no—
164C2×Dic43—44no—
165C2×Dic43C2×C22⋊C418yes—
166C2×Dic43C2×C22⋊C418yes—
167C2×Dic43C2×C22⋊C414yes—
168C2×Dic43#120116yes—
169C4×D43—232no—
170Dic86—232no—
171#14337—431nomaximal, Sylow
172C22×C43—216no—
173C22×C43—28no—
174C22×C43—48no—
175C22×C43—48no—
176C22×C43—416no—
177C22×C43—416no—
178C22×C43C22×C22⋊C412yes—
179C22×C43C22×C22⋊C411yescharacteristic
180C22×C43#52014yes—
181C22×C43C2×#12014yes—
182C22×C43#53512yes—
183C22×C43#53514yes—
184C22×C43#53612yes—
185C22×C43#53614yes—
186C22×C43#53718yes—
187C22×C43#53714yes—
188C2×D43—28no—
189C2×D43—24no—
190C2×D43—48no—
191C2×D43—48no—
192C2×D43—48no—
193C4×C43—24no—
194C4×C43—84no—
195Dic43—28no—
196Dic43—84no—
197Dic43#37918yes—
198C23×#41—431no—
199C2×#380—432no—
200C2×#380—434no—
201#1446—432no—
202#1438—431no—
203#1438—434no—
204#1438D4311yescharacteristic, complemented
205C2×C43—48no—
206C2×C43C2×#37911yescharacteristic
207C2×C43#231114yes—
208C2×C43#231112yes—
209C2×C43#261514yes—
210C2×C43#261614yes—
211D43—44no—
212C22×#41—438no—
213C22×#41—8616no—
214C22×#41—868no—
215C23×Q8—431no—
216C23×D4—432no—
217C23×D4C2×D4311yescharacteristic
218C24×C4—432no—
219C24×C4—431no—
220C22×C4⋊C4—434no—
221C22×C22⋊C4—434no—
222C22×C22⋊C4—434no—
223C22×C22⋊C4C2×D4314yes—
224#380—868no—
225#380—8616no—
226C2×#95—438no—
227C2×#95—432no—
228C2×#95C2×D4312yes—
229C43#1433711yescharacteristic, complemented, Sylow
230C25—431no—
231C25C22×D4312yes—
232C2×#41—8664no—
233C2×#41—8632no—
234C2×#41—86128no—
235C22×Q8—434no—
236C22×Q8—868no—
237C22×Q8—864no—
238C22×D4—28no—
239C22×D4—24no—
240C22×D4—438no—
241C22×D4—868no—
242C22×D4—868no—
243C22×D4—868no—
244C22×D4C4×D4314yes—
245C23×C4—431no—
246C23×C4—438no—
247C23×C4—4316no—
248C23×C4—432no—
249C23×C4—434no—
250C23×C4—438no—
251C23×C4—434no—
252C23×C4—868no—
253C23×C4—868no—
254C23×C4—868no—
255C23×C4C22×D4314yes—
256C23×C4C22×D4311yescharacteristic
257C2×C4⋊C4—868no—
258C2×C4⋊C4—8616no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—868no—
262C2×C22⋊C4—868no—
263C2×C22⋊C4—1728no—
264C2×C22⋊C4—1728no—
265#95—24no—
266#95—8616no—
267#95—864no—
268C24—24no—
269C24—24no—
270C24—28no—
271C24—28no—
272C24—432no—
273C24—434no—
274C24—864no—
275C24—864no—
276C24—864no—
277C24C2×C4×D4314yes—
278C24D4×D4318yes—
279C24C23×D4311yescharacteristic
280#41—86256no—
281C2×Q8—8616no—
282C2×Q8—8632no—
283C2×Q8—868no—
284C2×D4—232no—
285C2×D4—216no—
286C2×D4—28no—
287C2×D4—8616no—
288C2×D4—8616no—
289C2×D4—8664no—
290C2×D4—8616no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—432no—
298C22×C4—434no—
299C22×C4—438no—
300C22×C4—4332no—
301C22×C4—434no—
302C22×C4—438no—
303C22×C4—4316no—
304C22×C4—8616no—
305C22×C4—8632no—
306C22×C4—8616no—
307C22×C4—8616no—
308C22×C4—8664no—
309C22×C4—864no—
310C22×C4—864no—
311C22×C4—864no—
312C22×C4—868no—
313C22×C4—868no—
314C22×C4—864no—
315C22×C4—864no—
316C22×C4C2×C4×D4312yes—
317C22×C4D4×D4314yes—
318C4⋊C4—17216no—
319C22⋊C4—416no—
320C22⋊C4—17216no—
321C22⋊C4—17216no—
322C23—24no—
323C23—24no—
324C23—216no—
325C23—216no—
326C23—28no—
327C23—28no—
328C23—216no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—48no—
334C23—48no—
335C23—434no—
336C23—864no—
337C23—868no—
338C23—868no—
339C23—864no—
340C23—864no—
341C23—868no—
342C23—1724no—
343C23—1724no—
344C23—1728no—
345C23C22⋊C4×D4318yes—
346C23C22×C4×D4311yescharacteristic
347C23C2×D4×D4312yes—
348C23C2×D4×D4314yes—
349C23#1306337014yes—
350C23#1306337214yes—
351Q8—8632no—
352D4—232no—
353D4—8664no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—44no—
361C2×C4—438no—
362C2×C4—434no—
363C2×C4—4316no—
364C2×C4—438no—
365C2×C4—8632no—
366C2×C4—8664no—
367C2×C4—864no—
368C2×C4—864no—
369C2×C4—8616no—
370C2×C4—1724no—
371C2×C4—1724no—
372C2×C4—1728no—
373C2×C4—1728no—
374C2×C4—1724no—
375C2×C4C22⋊C4×D4314yes—
376C22—216no—
377C22—28no—
378C22—48no—
379C22—48no—
380C22—416no—
381C22—416no—
382C22—868no—
383C22—864no—
384C22—1728no—
385C22—1728no—
386C22—1728no—
387C22C2×C22⋊C4×D4312yes—
388C22C2×C22⋊C4×D4311yescharacteristic
389C22#1458337814yes—
390C22#120×D4314yes—
391C22#1458346212yes—
392C22#1458346414yes—
393C22#1458346912yes—
394C22#1458347114yes—
395C22#1458347618yes—
396C22#1458347814yes—
397C4—24no—
398C4—84no—
399C4—438no—
400C4—868no—
401C4—3444no—
402C2—48no—
403C2—1724no—
404C2#379×D4311yescharacteristic
405C2#1913718214yes—
406C2#1913718512yes—
407C2#1914031814yes—
408C2#1914032314yes—
subgroupG/H
center, FrattiniC24C23×D43
commutatorC23×C43C23×C4
FittingC43×#1438C2
socle, cosocleC24×C43C24
prime radicalC23×#10621900C2
hypercenter#1438D43
hypocenterC43#14337
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#14582758C2semidirect
3. C2×#14582757C2semidirect
4. #19132865C2semidirect
5. #19132836C2semidirect
6. #19132834C2semidirect
7. C43×#1438C2semidirect
10. C22×#10621900C4semidirect
100. C22×Dic43C22⋊C4semidirect
109. C2×C4×D43C22⋊C4semidirect
204. #1438D43semidirect
229. C43#14337semidirect

Maximal subgroups

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

subgroupindex
C23×#106219002
C2×#145827582
C2×#145827572
#191328652
#191328362
#191328342
C43×#14382
#1433743