Finite Group Databasebeta

← #24992870

Subgroups — #24992870

Order 11008.

604 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#19144627C211yesmaximal, characteristic
2C4×#13063290C211yesmaximal, characteristic, complemented
3#19135422C211yesmaximal, characteristic
4#19133089C211yesmaximal, characteristic
5#19133023C211yesmaximal, characteristic, complemented
6#19132427C211yesmaximal, characteristic, complemented
7C43×#1303C211yesmaximal, characteristic, complemented
8#14583449C2211yescharacteristic
9C2×C4×Dic86C2211yescharacteristic
10C2×#13063290—21no—
11#14583218—21no—
12#14583185C412yescomplemented
13#14583100C412yescomplemented
14#14583040—21no—
15C4×#10621873—22no—
16#14582952C2211yescharacteristic
17C2×#13063251—21no—
18C2×#13063251—21no—
19C4⋊C4×Dic43—21no—
20C4×Dic43⋊C4—21no—
21#14582750—22no—
22C42×Dic43—21no—
23C42×Dic43C2211yescharacteristic
24#14582632—22no—
25#14582629—22no—
26C43×#431C2211yescharacteristic
27C43×#423C2211yescharacteristic
28C4×C4⋊C4×C43C2211yescharacteristic
29C2×#10621898—21no—
30C22×Dic86D411yescharacteristic
31C2×#10621873—22no—
32#13063325C2×C411yescharacteristic
33#13063322C2×C411yescharacteristic
34#13063313—21no—
35#13063299—21no—
36#13063299—21no—
37C4×Dic86—21no—
38C4×Dic86—21no—
39C4×Dic86C2×C412yes—
40C4×Dic86D412yes—
41#13063290—28no—
42C4×D172—21no—
43C4×D172—21no—
44C2×C4⋊Dic43D411yescharacteristic
45C2×Dic43⋊C4—22no—
46C2×Dic43⋊C4—21no—
47C2×C4×Dic43—22no—
48C2×C4×Dic43—21no—
49C2×C4×Dic43—21no—
50C2×C4×Dic43—22no—
51C2×C4×Dic43—21no—
52C2×C4×Dic43D411yescharacteristic
53C2×C4×Dic43D411yescharacteristic
54#13063257—22no—
55C4⋊(C43⋊C8)—21no—
56#13063251—42no—
57#13063251—42no—
58#13063250—22no—
59#13063247—22no—
60C22⋊(C43⋊C8)—21no—
61C4×C43⋊C8—21no—
62C43⋊#95—22no—
63C2×M16×C43—21no—
64C2×C4⋊C4×C43—21no—
65C2×C42×C43—21no—
66C2×C42×C43C2311yescharacteristic
67C4⋊C8×C43—21no—
68C4⋊C8×C43—21no—
69C4⋊C8×C43C2×C412yes—
70C22⋊C8×C43—21no—
71C4×C8×C43—21no—
72#95×C43D412yes—
73C22×C4×C43—22no—
74C22×C4×C43C2×D411yescharacteristic
75C22×C4×C43C2×D411yescharacteristic
76C22×C4×C43C2×D411yescharacteristic
77C2×C22⋊D43—41no—
78C22×Dic43—21no—
79C22×Dic43—21no—
80C22×Dic43—21no—
81C22×Dic43—21no—
82#10621898—41no—
83#10621898—41no—
84#10621898—41no—
85#10621898—41no—
86C2×D172—21no—
87C2×D172—21no—
88C2×D172—21no—
89C2×C4×D43—41no—
90C2×Dic86—21no—
91C2×Dic86—24no—
92C2×Dic86—21no—
93C2×Dic86C22⋊C412yes—
94M16×C43—41no—
95M16×C43—41no—
96C2×C8×C43—22no—
97C2×C8×C43—41no—
98C4⋊C4×C43—24no—
99C42×C43—41no—
100C42×C43—41no—
101C42×C43C22×C411yescharacteristic
102C42×C43C2×D411yescharacteristic
103C42×C43#4111yescharacteristic
104C42×C43#4111yescharacteristic
105(C2×D43)⋊C4—41no—
106(C2×D43)⋊C4—41no—
107C4⋊Dic43—21no—
108C4⋊Dic43—21no—
109C4⋊Dic43—21no—
110C4⋊Dic43C22⋊C412yes—
111Dic43⋊C4—28no—
112Dic43⋊C4—22no—
113Dic43⋊C4—22no—
114C4×Dic43—28no—
115C4×Dic43—24no—
116C4×Dic43—22no—
117C4×Dic43—22no—
118C4×Dic43—21no—
119C4×Dic43—21no—
120C4×Dic43—21no—
121C4×Dic43—42no—
122C4×Dic43—44no—
123C4×Dic43SD1612yescomplemented
124C2×C43⋊C8—41no—
125#10621873—44no—
126C23×C43#12011yescharacteristic
127C22×D43—41no—
128C2×C4×C43—22no—
129C2×C4×C43—24no—
130C2×C4×C43—41no—
131C2×C4×C43—42no—
132C2×C4×C43—42no—
133C2×C4×C43C2×C22⋊C411yescharacteristic
134C2×C4×C43C4×D411yescharacteristic
135C2×C4×C43C4×D411yescharacteristic
136C2×C4×C43#12111yescharacteristic
137C2×C4×C43#12111yescharacteristic
138C2×C4×C43#12311yescharacteristic
139C2×C4×C43C2×SD1611yescharacteristic
140C2×C4×C43#13511yescharacteristic
141C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C43#13711yescharacteristic
143C22⋊D43—81no—
144C22⋊D43—81no—
145C2×Dic43—22no—
146C2×Dic43—24no—
147C2×Dic43—22no—
148C2×Dic43—22no—
149C2×Dic43—22no—
150C2×Dic43—22no—
151C2×Dic43—22no—
152C2×Dic43—41no—
153C2×Dic43—42no—
154C2×Dic43—41no—
155C2×Dic43—41no—
156D172—42no—
157D172—41no—
158D172—41no—
159D172—81no—
160C4×D43—81no—
161C4×D43—81no—
162Dic86—24no—
163Dic86—22no—
164Dic86—41no—
165Dic86—41no—
166Dic86#10414yescomplemented
167C8×C43—42no—
168C8×C43—81no—
169C43⋊C8—81no—
170#9234—431nomaximal, Sylow
171C22×C43—21no—
172C22×C43—21no—
173C22×C43#38611yescharacteristic
174C22×C43#44811yescharacteristic
175C22×C43#44911yescharacteristic
176C2×D43—81no—
177C2×D43—81no—
178C2×D43—81no—
179C4×C43—21no—
180C4×C43—21no—
181C4×C43—21no—
182C4×C43—44no—
183C4×C43C2×#10412yes—
184C4×C43C4×SD1611yescharacteristic
185C4×C43#44011yescharacteristic
186C4×C43#46011yescharacteristic
187C4×C43#46411yescharacteristic
188Dic43—24no—
189Dic43—24no—
190Dic43—42no—
191Dic43—42no—
192Dic43—81no—
193#3103—431no—
194C42×Q8—431no—
195#1962—431no—
196#1502—431no—
197#1487—431no—
198#1303D4311yescharacteristic, complemented
199#1276—431no—
200C2×C43—21no—
201C2×C43—21no—
202C2×C43#188011yescharacteristic
203C2×C43#189611yescharacteristic
204C2×C43#190611yescharacteristic
205D43—161no—
206#532—431no—
207C2×C4×Q8—431no—
208C2×C4×Q8—861no—
209#481—861no—
210#475—432no—
211#460—432no—
212#448—861no—
213C4×SD16—862no—
214#431—431no—
215#431C2×D4311yescharacteristic
216#423C2×D4311yescharacteristic
217C2×#104—861no—
218C2×#104—861no—
219C4×C4⋊C4—861no—
220C4×C4⋊C4—861no—
221C4×C4⋊C4—862no—
222C4×C4⋊C4C2×D4311yescharacteristic
223C43—431no—
224C43—861no—
225#327—862no—
226Q8⋊C8—862no—
227C43#923411yescharacteristic, complemented, Sylow
228C2×#41—861no—
229C22×Q8—431no—
230C2×SD16—862no—
231C2×M16—21no—
232C4⋊Q8—431no—
233#127—431no—
234#124—861no—
235#121—861no—
236#121—861no—
237C4×Q8—432no—
238C4×Q8—432no—
239C4×Q8—868no—
240C4×Q8—861no—
241C4×Q8—861no—
242C4×D4—861no—
243C4×D4—861no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—431no—
246C2×C4⋊C4—862no—
247C2×C4⋊C4—861no—
248C2×C42—21no—
249C2×C42—431no—
250C2×C42—431no—
251C2×C42—862no—
252C2×C42—861no—
253C2×C42—861no—
254C2×C42—862no—
255C2×C42—861no—
256C2×C42C22×D4311yescharacteristic
257#106—862no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—861no—
261C4⋊C8C4×D4312yes—
262#104—1722no—
263#104—1722no—
264#103—862no—
265#102—862no—
266C22⋊C8—21no—
267C22⋊C8—861no—
268C4×C8—21no—
269C4×C8—861no—
270#95—862no—
271#95D17212yes—
272#41—1721no—
273#41—1721no—
274#41—1721no—
275#41—1721no—
276C2×Q8—432no—
277C2×Q8—864no—
278C2×Q8—861no—
279C2×Q8—861no—
280C2×D4—861no—
281C2×D4—861no—
282C2×D4—861no—
283C2×D4—1721no—
284C22×C4—22no—
285C22×C4—861no—
286C22×C4—861no—
287C22×C4—861no—
288C22×C4—861no—
289C22×C4—1721no—
290C22×C4C2×D17211yescharacteristic
291C22×C4D4×D4311yescharacteristic
292C22×C4D4×D4311yescharacteristic
293SD16—1724no—
294M16—41no—
295M16—41no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—1721no—
299C4⋊C4—24no—
300C4⋊C4—432no—
301C4⋊C4—868no—
302C4⋊C4—862no—
303C4⋊C4—862no—
304C4⋊C4—861no—
305C4⋊C4—861no—
306C4⋊C4—861no—
307C22⋊C4—1721no—
308C22⋊C4—1721no—
309C42—41no—
310C42—41no—
311C42—432no—
312C42—864no—
313C42—868no—
314C42—862no—
315C42—862no—
316C42—861no—
317C42—861no—
318C42—861no—
319C42—1722no—
320C42—1724no—
321C42C2×C4×D4311yescharacteristic
322C42#1062189811yescharacteristic
323C42D4×D4311yescharacteristic
324C42#1062190211yescharacteristic
325C23—1721no—
326C23#1306329511yescharacteristic
327Q8—434no—
328Q8—864no—
329Q8—862no—
330Q8—1721no—
331Q8—1721no—
332D4—3441no—
333D4—3441no—
334D4—3441no—
335D4—1722no—
336D4—1721no—
337D4—1721no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—862no—
344C2×C4—864no—
345C2×C4—862no—
346C2×C4—862no—
347C2×C4—862no—
348C2×C4—862no—
349C2×C4—862no—
350C2×C4—3441no—
351C2×C4—3441no—
352C2×C4—1721no—
353C2×C4—1721no—
354C2×C4—1722no—
355C2×C4—1721no—
356C2×C4C22⋊C4×D4311yescharacteristic
357C2×C4#1306328711yescharacteristic
358C2×C4C4×D17211yescharacteristic
359C2×C4#1306329811yescharacteristic
360C2×C4#1306330011yescharacteristic
361C2×C4#1306331011yescharacteristic
362C2×C4SD16×D4311yescharacteristic
363C2×C4#1306334611yescharacteristic
364C2×C4#1306334911yescharacteristic
365C2×C4#1306335711yescharacteristic
366C8—42no—
367C8—81no—
368C8—3441no—
369C22—21no—
370C22—21no—
371C22—3441no—
372C22—3441no—
373C22—3441no—
374C22#1458277711yescharacteristic
375C22#1458303911yescharacteristic
376C22#1458304511yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—44no—
381C4—864no—
382C4—864no—
383C4—3441no—
384C4—1722no—
385C4—1722no—
386C4#104×D4312yes—
387C4#1458298611yescharacteristic
388C4#1458299811yescharacteristic
389C4#1458309811yescharacteristic
390C4#1458312411yescharacteristic
391C2—21no—
392C2—21no—
393C2—6881no—
394C2#1913495111yescharacteristic
395C2#1913503711yescharacteristic
396C2#1913509711yescharacteristic
subgroupG/H
centerC2×C4#13063298
commutatorC42×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1303C2
socleC22×C43#386
prime radical#14583449C22
cosocleC2×C42×C43C23
hypercenter#1303D43
hypocenterC43#9234
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
2. C4×#13063290C2semidirect
5. #19133023C2semidirect
6. #19132427C2semidirect
7. C43×#1303C2semidirect
12. #14583185C4semidirect
13. #14583100C4semidirect
123. C4×Dic43SD16semidirect
166. Dic86#104semidirect
198. #1303D43semidirect
227. C43#9234semidirect

Maximal subgroups

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

subgroupindex
#191446272
C4×#130632902
#191354222
#191330892
#191330232
#191324272
C43×#13032
#923443