Finite Group Databasebeta

← #24989237

Subgroups — #24989237

Order 11008.

1166 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C22×#13063307C211yesmaximal, characteristic
2#19134673C212yesmaximal, complemented
3#19132922C211yesmaximal, characteristic, complemented
4#19132921C212yesmaximal, complemented
5C43×#1462C211yesmaximal, characteristic, complemented
6C23×C22⋊D43C411yescharacteristic
7C23×C4×D43C411yescharacteristic
8C2×#13063307—24no—
9C2×#13063307—24no—
10C2×#13063307—22no—
11C2×#13063307—22no—
12C22×C4⋊Dic43C2211yescharacteristic
13C22×Dic43⋊C4C411yescharacteristic
14C22×C22⋊Dic43C411yescharacteristic
15C22×(C2×D43)⋊C4C2211yescharacteristic
16#14582863C2212yes—
17#14582667—24no—
18#14582623—22no—
19#14582622—24no—
20C22×C22⋊C4×C43C2211yescharacteristic
21C43×#407C2212yes—
22C43×#323—22no—
23C24×D43C2×C411yescharacteristic
24C22×C22⋊D43—22no—
25C22×C22⋊D43—22no—
26C22×C22⋊D43—22no—
27C22×C22⋊D43—44no—
28C23×Dic43C2×C411yescharacteristic
29C23×Dic43C2×C411yescharacteristic
30C23×Dic43C2×C411yescharacteristic
31C22×C4×D43—22no—
32C22×C4×D43—22no—
33C22×C4×D43—22no—
34C22×C4×D43—44no—
35C2×#10621877—22no—
36#13063307—216no—
37#13063307—216no—
38C2×C4⋊Dic43—21no—
39C2×C4⋊Dic43—22no—
40C2×C4⋊Dic43—21no—
41C2×C4⋊Dic43D414yes—
42C2×Dic43⋊C4—22no—
43C2×Dic43⋊C4—22no—
44C2×Dic43⋊C4—22no—
45C2×C22⋊Dic43—22no—
46C2×C22⋊Dic43—22no—
47C2×C22⋊Dic43—22no—
48C2×(C2×D43)⋊C4—21no—
49C2×(C2×D43)⋊C4—22no—
50C2×(C2×D43)⋊C4—21no—
51C2×(C2×D43)⋊C4D414yes—
52C22⋊(C43⋊C8)—22no—
53C22⋊(C43⋊C8)—22no—
54C25×C43C2×C411yescharacteristic
55C23×C4×C43C2×C411yescharacteristic
56C23×C4×C43C2311yescharacteristic
57C2×M16×C43—22no—
58C2×C22⋊C4×C43—22no—
59C2×C22⋊C4×C43—22no—
60C2×C22⋊C4×C43D412yes—
61C2×C22⋊C4×C43D412yes—
62C22⋊C8×C43—22no—
63C22⋊C8×C43—22no—
64C24×C43—21no—
65C24×C43—21no—
66C24×C43—42no—
67C24×C43—42no—
68C24×C43—42no—
69C24×C43C22⋊C412yes—
70C24×C43C22×C411yescharacteristic
71C23×D43—21no—
72C23×D43—21no—
73C23×D43—24no—
74C23×D43—42no—
75C23×D43—42no—
76C23×D43C22⋊C412yes—
77C22×C4×C43—21no—
78C22×C4×C43—21no—
79C22×C4×C43—42no—
80C22×C4×C43—42no—
81C22×C4×C43C22⋊C412yes—
82C22×C4×C43C2×D411yescharacteristic
83C22×C4×C43C2×D412yes—
84C22×C4×C43C2×D412yes—
85C22×C4×C43C2×D411yescharacteristic
86C2×C22⋊D43—28no—
87C2×C22⋊D43—48no—
88C2×C22⋊D43—48no—
89C2×C22⋊D43—44no—
90C2×C22⋊D43—44no—
91C22×Dic43—21no—
92C22×Dic43—21no—
93C22×Dic43—21no—
94C22×Dic43—21no—
95C22×Dic43—21no—
96C22×Dic43—21no—
97C22×Dic43—42no—
98C22×Dic43—42no—
99C22×Dic43—42no—
100C22×Dic43C22⋊C412yes—
101C22×Dic43C22⋊C412yes—
102C22×Dic43C22⋊C412yes—
103C2×C4×D43—28no—
104C2×C4×D43—48no—
105C2×C4×D43—48no—
106C2×C4×D43—44no—
107C2×C4×D43—44no—
108M16×C43—44no—
109C2×C8×C43—42no—
110C22⋊C4×C43—24no—
111C22⋊C4×C43—24no—
112C22⋊Dic43—28no—
113(C2×D43)⋊C4—24no—
114(C2×D43)⋊C4—24no—
115C4⋊Dic43—24no—
116C4⋊Dic43—24no—
117Dic43⋊C4—28no—
118#10621877—44no—
119C2×C43⋊C8—42no—
120C23×C43—22no—
121C23×C43—22no—
122C23×C43—22no—
123C23×C43—22no—
124C23×C43—24no—
125C23×C43—42no—
126C23×C43—42no—
127C23×C43—42no—
128C23×C43—42no—
129C23×C43—42no—
130C23×C43—42no—
131C23×C43—42no—
132C23×C43—42no—
133C23×C43—88no—
134C23×C43C2×C22⋊C411yescharacteristic
135C23×C43C2×C22⋊C411yescharacteristic
136C23×C43C2×C22⋊C411yescharacteristic
137C22×D43—24no—
138C22×D43—24no—
139C22×D43—22no—
140C22×D43—22no—
141C22×D43—416no—
142C22×D43—42no—
143C22×D43—42no—
144C22×D43—42no—
145C22×D43—42no—
146C22×D43—42no—
147C22×D43—42no—
148C22×D43#10014yes—
149C2×C4×C43—24no—
150C2×C4×C43—22no—
151C2×C4×C43—41no—
152C2×C4×C43—42no—
153C2×C4×C43—41no—
154C2×C4×C43—42no—
155C2×C4×C43—42no—
156C2×C4×C43—42no—
157C2×C4×C43#12012yes—
158C2×C4×C43#12012yes—
159C22⋊D43—416no—
160C2×Dic43—22no—
161C2×Dic43—22no—
162C2×Dic43—22no—
163C2×Dic43—42no—
164C2×Dic43—42no—
165C2×Dic43—42no—
166C2×Dic43—42no—
167C2×Dic43—42no—
168C2×Dic43—42no—
169C2×Dic43—42no—
170C2×Dic43—42no—
171C2×Dic43—42no—
172C4×D43—416no—
173C8×C43—82no—
174C43⋊C8—82no—
175#8553—431nomaximal, Sylow
176C22×C43—28no—
177C22×C43—21no—
178C22×C43—21no—
179C22×C43—21no—
180C22×C43—21no—
181C22×C43—21no—
182C22×C43—21no—
183C22×C43—42no—
184C22×C43—42no—
185C22×C43—42no—
186C22×C43—84no—
187C22×C43—84no—
188C22×C43—84no—
189C22×C43#37911yescharacteristic
190C22×C43#41012yes—
191C22×C43C2×#10012yes—
192C22×C43#41312yes—
193C2×D43—24no—
194C2×D43—48no—
195C2×D43—44no—
196C2×D43—48no—
197C2×D43—42no—
198C2×D43—42no—
199C2×D43—44no—
200C4×C43—22no—
201C4×C43—22no—
202C4×C43—42no—
203Dic43—42no—
204Dic43—42no—
205Dic43—42no—
206C22×#123—431no—
207#1821—432no—
208#1462—432no—
209#1462—431no—
210#1462D4311yescharacteristic, complemented
211C2×C43—22no—
212C2×C43—22no—
213C2×C43—22no—
214C2×C43—82no—
215C2×C43#178211yescharacteristic
216C2×C43#178311yescharacteristic
217C2×C43#178511yescharacteristic
218D43—44no—
219C23×D4—431no—
220C24×C4—431no—
221C2×#123—864no—
222C2×#123—864no—
223C2×#123—862no—
224C2×#123—862no—
225C22×C4⋊C4—431no—
226C22×C4⋊C4—431no—
227C22×C22⋊C4—431no—
228C22×C22⋊C4—431no—
229C22×C22⋊C4C2×D4311yescharacteristic
230#407—432no—
231#407C2×D4312yes—
232#343—864no—
233#323—22no—
234#323—864no—
235#323—862no—
236C43#855311yescharacteristic, complemented, Sylow
237C25—431no—
238C25C4×D4311yescharacteristic
239C22×D4—862no—
240C22×D4—862no—
241C22×D4—862no—
242C22×D4—1724no—
243C23×C4—431no—
244C23×C4—431no—
245C23×C4—431no—
246C23×C4—862no—
247C23×C4—862no—
248C23×C4—862no—
249C23×C4—1724no—
250C23×C4C4×D4311yescharacteristic
251C23×C4C22×D4311yescharacteristic
252C2×M16—22no—
253C2×M16—862no—
254#123—8616no—
255#123—8616no—
256C2×C4⋊C4—434no—
257C2×C4⋊C4—862no—
258C2×C4⋊C4—862no—
259C2×C4⋊C4—862no—
260C2×C4⋊C4—861no—
261C2×C4⋊C4—862no—
262C2×C4⋊C4—861no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—434no—
266C2×C22⋊C4—862no—
267C2×C22⋊C4—862no—
268C2×C22⋊C4—862no—
269C2×C22⋊C4—861no—
270C2×C22⋊C4—862no—
271C2×C22⋊C4—861no—
272C2×C22⋊C4D17212yes—
273C2×C22⋊C4C22⋊D4312yes—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—862no—
277C22⋊C8—862no—
278C24—21no—
279C24—21no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—432no—
284C24—864no—
285C24—861no—
286C24—861no—
287C24—1722no—
288C24—1722no—
289C24(C2×D43)⋊C412yes—
290C24C2×C4×D4311yescharacteristic
291C2×D4—868no—
292C2×D4—1724no—
293C2×D4—1728no—
294C2×D4—1724no—
295C2×D4—1728no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—432no—
301C22×C4—432no—
302C22×C4—432no—
303C22×C4—868no—
304C22×C4—861no—
305C22×C4—861no—
306C22×C4—861no—
307C22×C4—861no—
308C22×C4—861no—
309C22×C4—861no—
310C22×C4—1724no—
311C22×C4—1728no—
312C22×C4—1724no—
313C22×C4—1728no—
314C22×C4—1722no—
315C22×C4—1722no—
316C22×C4—1722no—
317C22×C4(C2×D43)⋊C412yes—
318C22×C4C2×D17211yescharacteristic
319C22×C4D4×D4312yes—
320C22×C4D4×D4312yes—
321C22×C4C2×C22⋊D4311yescharacteristic
322M16—44no—
323M16—1724no—
324C2×C8—42no—
325C2×C8—1722no—
326C4⋊C4—868no—
327C4⋊C4—864no—
328C4⋊C4—864no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—868no—
332C22⋊C4—864no—
333C22⋊C4—864no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—88no—
348C23—434no—
349C23—864no—
350C23—864no—
351C23—862no—
352C23—862no—
353C23—1722no—
354C23—1722no—
355C23—17216no—
356C23—1722no—
357C23—1722no—
358C23—1722no—
359C23—1722no—
360C23C22⋊C4×D4311yescharacteristic
361C23C22⋊C4×D4311yescharacteristic
362C23C2×(C2×D43)⋊C411yescharacteristic
363D4—17216no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—862no—
373C2×C4—862no—
374C2×C4—862no—
375C2×C4—17216no—
376C2×C4—1722no—
377C2×C4—1722no—
378C2×C4—1722no—
379C2×C4—1722no—
380C2×C4—1722no—
381C2×C4—1722no—
382C2×C4—1722no—
383C2×C4—1722no—
384C2×C4—1722no—
385C2×C4#1306329312yes—
386C2×C4#1306329512yes—
387C8—82no—
388C8—3442no—
389C22—28no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—84no—
400C22—84no—
401C22—84no—
402C22—864no—
403C22—1724no—
404C22—1728no—
405C22—1724no—
406C22—1728no—
407C22—1722no—
408C22—1722no—
409C22#1458275411yescharacteristic
410C22#1458287112yes—
411C22#100×D4312yes—
412C22#1458288112yes—
413C4—22no—
414C4—22no—
415C4—42no—
416C4—1722no—
417C4—1722no—
418C4—1722no—
419C2—22no—
420C2—22no—
421C2—22no—
422C2—82no—
423C2—1724no—
424C2#1913451111yescharacteristic
425C2#1913451411yescharacteristic
426C2#1913452111yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC24×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1462C2
socleC22×C43#379
prime radicalC23×C22⋊D43C4
cosocleC23×C4×C43C23
hypercenter#1462D43
hypocenterC43#8553
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #19134673C2semidirect
3. #19132922C2semidirect
4. #19132921C2semidirect
5. C43×#1462C2semidirect
210. #1462D43semidirect
236. C43#8553semidirect

Maximal subgroups

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

subgroupindex
C22×#130633072
#191346732
#191329222
#191329212
C43×#14622
#855343