Finite Group Databasebeta

← #24989233

Subgroups — #24989233

Order 11008.

1154 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1154 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×#13063311C211yesmaximal, characteristic
2#19134518C212yesmaximal, complemented
3#19132922C211yesmaximal, characteristic, complemented
4#19132921C212yesmaximal, complemented
5C43×#1462C211yesmaximal, characteristic, complemented
6C23×Dic86C2211yescharacteristic
7C23×C22⋊D43C411yescharacteristic
8C2×#13063311—24no—
9C2×#13063311—24no—
10C2×#13063311—22no—
11C2×#13063311—22no—
12C22×C22⋊Dic43C411yescharacteristic
13C22×(C2×D43)⋊C4C411yescharacteristic
14C22×(C2×D43)⋊C4C2211yescharacteristic
15C22×C4×Dic43C411yescharacteristic
16C2×#13063243—24no—
17#14582863C2212yes—
18#14582623—22no—
19#14582622—24no—
20C22×C22⋊C4×C43C2211yescharacteristic
21C43×#407C2212yes—
22C43×#323—22no—
23C24×D43C2×C411yescharacteristic
24C22×Dic86—21no—
25C22×Dic86—22no—
26C22×Dic86—21no—
27C22×Dic86—44no—
28C22×Dic86D414yes—
29C22×C22⋊D43—22no—
30C22×C22⋊D43—22no—
31C22×C22⋊D43—22no—
32C22×C22⋊D43—44no—
33C23×Dic43C2×C411yescharacteristic
34C23×Dic43C2×C411yescharacteristic
35C23×Dic43C2×C411yescharacteristic
36C2×#10621877—22no—
37#13063311—216no—
38#13063311—216no—
39C2×C22⋊Dic43—22no—
40C2×C22⋊Dic43—22no—
41C2×C22⋊Dic43—22no—
42C2×(C2×D43)⋊C4—22no—
43C2×(C2×D43)⋊C4—22no—
44C2×(C2×D43)⋊C4—21no—
45C2×(C2×D43)⋊C4—22no—
46C2×(C2×D43)⋊C4—21no—
47C2×(C2×D43)⋊C4—22no—
48C2×(C2×D43)⋊C4D414yes—
49C2×C4×Dic43—22no—
50C2×C4×Dic43—22no—
51C2×C4×Dic43—22no—
52#13063243—48no—
53C22⋊(C43⋊C8)—22no—
54C22⋊(C43⋊C8)—22no—
55C25×C43C2×C411yescharacteristic
56C23×C4×C43C2×C411yescharacteristic
57C23×C4×C43C2311yescharacteristic
58C2×M16×C43—22no—
59C2×C22⋊C4×C43—22no—
60C2×C22⋊C4×C43—22no—
61C2×C22⋊C4×C43D412yes—
62C2×C22⋊C4×C43D412yes—
63C22⋊C8×C43—22no—
64C22⋊C8×C43—22no—
65C24×C43—21no—
66C24×C43—21no—
67C24×C43—42no—
68C24×C43—42no—
69C24×C43—42no—
70C24×C43C22⋊C412yes—
71C24×C43C22×C411yescharacteristic
72C23×D43—21no—
73C23×D43—21no—
74C23×D43—42no—
75C23×D43—42no—
76C23×D43—42no—
77C23×D43C22⋊C412yes—
78C22×C4×C43—21no—
79C22×C4×C43—21no—
80C22×C4×C43—42no—
81C22×C4×C43—42no—
82C22×C4×C43C22⋊C412yes—
83C22×C4×C43C2×D411yescharacteristic
84C22×C4×C43C2×D412yes—
85C22×C4×C43C2×D412yes—
86C22×C4×C43C2×D411yescharacteristic
87C2×C22⋊D43—28no—
88C2×C22⋊D43—48no—
89C2×C22⋊D43—44no—
90C2×C22⋊D43—48no—
91C2×C22⋊D43—44no—
92C22×Dic43—24no—
93C22×Dic43—21no—
94C22×Dic43—21no—
95C22×Dic43—21no—
96C22×Dic43—21no—
97C22×Dic43—21no—
98C22×Dic43—21no—
99C22×Dic43—42no—
100C22×Dic43—42no—
101C22×Dic43C22⋊C412yes—
102C22×Dic43C22⋊C412yes—
103C22×Dic43C22⋊C412yes—
104C2×Dic86—24no—
105C2×Dic86—24no—
106C2×Dic86—28no—
107C2×Dic86—44no—
108C2×Dic86—44no—
109C2×Dic86—48no—
110C2×Dic86—44no—
111M16×C43—44no—
112C2×C8×C43—42no—
113C22⋊C4×C43—24no—
114C22⋊C4×C43—24no—
115C22⋊Dic43—28no—
116(C2×D43)⋊C4—28no—
117(C2×D43)⋊C4—24no—
118(C2×D43)⋊C4—24no—
119C4×Dic43—28no—
120#10621877—44no—
121C2×C43⋊C8—42no—
122C23×C43—22no—
123C23×C43—22no—
124C23×C43—22no—
125C23×C43—22no—
126C23×C43—24no—
127C23×C43—42no—
128C23×C43—42no—
129C23×C43—42no—
130C23×C43—42no—
131C23×C43—42no—
132C23×C43—42no—
133C23×C43—42no—
134C23×C43—42no—
135C23×C43—88no—
136C23×C43C2×C22⋊C411yescharacteristic
137C23×C43C2×C22⋊C411yescharacteristic
138C23×C43C2×C22⋊C411yescharacteristic
139C22×D43—22no—
140C22×D43—24no—
141C22×D43—42no—
142C22×D43—42no—
143C22×D43—42no—
144C22×D43—42no—
145C22×D43—42no—
146C22×D43—42no—
147C22×D43—42no—
148C22×D43—42no—
149C22×D43—88no—
150C2×C4×C43—24no—
151C2×C4×C43—22no—
152C2×C4×C43—41no—
153C2×C4×C43—42no—
154C2×C4×C43—41no—
155C2×C4×C43—42no—
156C2×C4×C43—42no—
157C2×C4×C43—42no—
158C2×C4×C43#12012yes—
159C2×C4×C43#12012yes—
160C22⋊D43—416no—
161C2×Dic43—24no—
162C2×Dic43—24no—
163C2×Dic43—22no—
164C2×Dic43—22no—
165C2×Dic43—22no—
166C2×Dic43—22no—
167C2×Dic43—24no—
168C2×Dic43—42no—
169C2×Dic43—42no—
170C2×Dic43—42no—
171C2×Dic43—42no—
172C2×Dic43—42no—
173C2×Dic43#9914yes—
174Dic86—48no—
175Dic86—48no—
176C8×C43—82no—
177C43⋊C8—82no—
178#8552—431nomaximal, Sylow
179C22×C43—28no—
180C22×C43—21no—
181C22×C43—21no—
182C22×C43—21no—
183C22×C43—21no—
184C22×C43—21no—
185C22×C43—21no—
186C22×C43—42no—
187C22×C43—42no—
188C22×C43—42no—
189C22×C43—84no—
190C22×C43—84no—
191C22×C43—84no—
192C22×C43#37911yescharacteristic
193C22×C43C2×#9912yes—
194C22×C43#41312yes—
195C22×C43#41312yes—
196C2×D43—42no—
197C2×D43—42no—
198C2×D43—42no—
199C2×D43—84no—
200C2×D43—84no—
201C2×D43—84no—
202C4×C43—22no—
203C4×C43—22no—
204C4×C43—42no—
205Dic43—24no—
206Dic43—42no—
207Dic43—42no—
208C22×#124—431no—
209#1784—432no—
210#1462—432no—
211#1462—431no—
212#1462D4311yescharacteristic, complemented
213C2×C43—22no—
214C2×C43—22no—
215C2×C43—22no—
216C2×C43—82no—
217C2×C43#178111yescharacteristic
218C2×C43#178511yescharacteristic
219C2×C43#178511yescharacteristic
220D43—82no—
221C23×Q8—431no—
222C23×D4—431no—
223C2×#124—864no—
224C2×#124—864no—
225C2×#124—862no—
226C2×#124—862no—
227C22×C22⋊C4—431no—
228C22×C22⋊C4—431no—
229C22×C22⋊C4—431no—
230C22×C22⋊C4C2×D4311yescharacteristic
231C22×C42—431no—
232C2×#101—864no—
233#407—432no—
234#407C2×D4312yes—
235#323—22no—
236#323—864no—
237#323—862no—
238C43#855211yescharacteristic, complemented, Sylow
239C25—431no—
240C25C4×D4311yescharacteristic
241C22×Q8—434no—
242C22×Q8—861no—
243C22×Q8—862no—
244C22×Q8—861no—
245C22×Q8—1724no—
246C22×D4—862no—
247C22×D4—862no—
248C22×D4—862no—
249C22×D4—1724no—
250C23×C4—431no—
251C23×C4—431no—
252C23×C4—431no—
253C23×C4C4×D4311yescharacteristic
254C23×C4C22×D4311yescharacteristic
255C2×M16—22no—
256C2×M16—862no—
257#124—8616no—
258#124—8616no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—434no—
262C2×C22⋊C4—862no—
263C2×C22⋊C4—862no—
264C2×C22⋊C4—862no—
265C2×C22⋊C4—862no—
266C2×C22⋊C4—862no—
267C2×C22⋊C4—862no—
268C2×C22⋊C4—861no—
269C2×C22⋊C4—862no—
270C2×C22⋊C4—861no—
271C2×C22⋊C4D17212yes—
272C2×C22⋊C4C22⋊D4312yes—
273C2×C42—862no—
274C2×C42—862no—
275C2×C42—862no—
276#101—1728no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—862no—
280C22⋊C8—862no—
281C24—21no—
282C24—21no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—432no—
287C24—861no—
288C24—861no—
289C24—1722no—
290C24—1722no—
291C24—1722no—
292C24(C2×D43)⋊C412yes—
293C24C2×C4×D4311yescharacteristic
294C2×Q8—868no—
295C2×Q8—864no—
296C2×Q8—864no—
297C2×Q8—1724no—
298C2×Q8—1724no—
299C2×Q8—1728no—
300C2×Q8—1724no—
301C2×D4—868no—
302C2×D4—1724no—
303C2×D4—1728no—
304C2×D4—1728no—
305C2×D4—1724no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—432no—
311C22×C4—432no—
312C22×C4—432no—
313C22×C4—864no—
314C22×C4—861no—
315C22×C4—861no—
316C22×C4—861no—
317C22×C4—861no—
318C22×C4—861no—
319C22×C4—861no—
320C22×C4—1722no—
321C22×C4—1722no—
322C22×C4(C2×D43)⋊C412yes—
323C22×C4C2×D17211yescharacteristic
324C22×C4D4×D4312yes—
325C22×C4D4×D4312yes—
326C22×C4C2×C22⋊D4311yescharacteristic
327M16—44no—
328M16—1724no—
329C2×C8—42no—
330C2×C8—1722no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—868no—
334C22⋊C4—868no—
335C22⋊C4—864no—
336C22⋊C4—864no—
337C42—868no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—88no—
352C23—864no—
353C23—862no—
354C23—3448no—
355C23—1722no—
356C23—1722no—
357C23—1722no—
358C23—1722no—
359C23—1722no—
360C23—1722no—
361C23—1722no—
362C23—1722no—
363C23C22⋊C4×D4311yescharacteristic
364C23C22⋊C4×D4311yescharacteristic
365C23C2×(C2×D43)⋊C411yescharacteristic
366Q8—1728no—
367Q8—1728no—
368D4—17216no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—434no—
378C2×C4—864no—
379C2×C4—864no—
380C2×C4—862no—
381C2×C4—862no—
382C2×C4—864no—
383C2×C4—862no—
384C2×C4—862no—
385C2×C4—1722no—
386C2×C4—1722no—
387C2×C4—1722no—
388C2×C4—1722no—
389C2×C4—1722no—
390C2×C4#1306329312yes—
391C2×C4#1306329512yes—
392C8—82no—
393C8—3442no—
394C22—28no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—84no—
405C22—84no—
406C22—84no—
407C22—3444no—
408C22—3444no—
409C22—3444no—
410C22—1722no—
411C22—1722no—
412C22—1722no—
413C22#1458275411yescharacteristic
414C22#99×D4312yes—
415C22#1458288012yes—
416C22#1458288112yes—
417C4—22no—
418C4—22no—
419C4—42no—
420C4—864no—
421C4—1722no—
422C4—1722no—
423C2—22no—
424C2—22no—
425C2—22no—
426C2—82no—
427C2—3442no—
428C2#1913450811yescharacteristic
429C2#1913452011yescharacteristic
430C2#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#8552
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. #19134518C2semidirect
3. #19132922C2semidirect
4. #19132921C2semidirect
5. C43×#1462C2semidirect
212. #1462D43semidirect
238. C43#8552semidirect

Maximal subgroups

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

subgroupindex
C22×#130633112
#191345182
#191329222
#191329212
C43×#14622
#855243