Finite Group Databasebeta

← #24989102

Subgroups — #24989102

Order 11008.

754 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#14583520C211yesmaximal, characteristic
2#19134678C211yesmaximal, characteristic, complemented
3#19133127C212yesmaximal, complemented
4#19133123C212yesmaximal
5C43×#1820C211yesmaximal, characteristic, complemented
6#14583520—24no—
7#14583520—24no—
8C2×#13063325C411yescharacteristic, complemented
9C2×#13063320—21no—
10C2×#13063313C411yescharacteristic, complemented
11C2×#13063311—21no—
12C2×#13063307—21no—
13C2×#13063304C2211yescharacteristic
14C2×#13063303—21no—
15C2×#13063299C2211yescharacteristic
16C2×C4×Dic86C411yescharacteristic, complemented
17C2×C4×D172C411yescharacteristic, complemented
18#14582863C2212yes—
19#14582668—21no—
20#14582668—21no—
21#14582635—24no—
22#14582632—24no—
23C2×#117×C43C2211yescharacteristic
24C43×#381C2212yes—
25C43×#342—21no—
26C43×#342—21no—
27C22×Dic86—21no—
28C22×Dic86C2×C411yescharacteristic
29C22×D172C2×C411yescharacteristic
30C22×C22⋊D43—21no—
31C22×C4×D43—21no—
32C2×#10621877—22no—
33#13063325—22no—
34#13063325—22no—
35#13063320—28no—
36#13063313—22no—
37#13063313—22no—
38#13063311—28no—
39#13063307—28no—
40#13063304—22no—
41#13063304D414yes—
42#13063303—28no—
43#13063299—22no—
44#13063299D414yes—
45C4×Dic86—22no—
46C4×Dic86—22no—
47C4×D172—22no—
48C4×D172—22no—
49C2×C4⋊Dic43—21no—
50C2×C4⋊Dic43C2×C411yescharacteristic
51C2×C4⋊Dic43C2×C411yescharacteristic
52C2×Dic43⋊C4—21no—
53C2×Dic43⋊C4—21no—
54C2×C22⋊Dic43—21no—
55C2×(C2×D43)⋊C4—21no—
56C2×(C2×D43)⋊C4—21no—
57C2×(C2×D43)⋊C4—21no—
58C2×(C2×D43)⋊C4—21no—
59C2×C4×Dic43—21no—
60C22⋊(C43⋊C8)—22no—
61C22⋊(C43⋊C8)—22no—
62C23×C4×C43C2311yescharacteristic
63#117×C43—22no—
64#117×C43—22no—
65C2×C4⋊C4×C43D411yescharacteristic
66C2×C4⋊C4×C43D411yescharacteristic
67C2×C22⋊C4×C43—22no—
68C2×C22⋊C4×C43D411yescharacteristic
69C2×C22⋊C4×C43D411yescharacteristic
70C2×C42×C43C2×C411yescharacteristic
71C2×C42×C43C2×C411yescharacteristic
72#95×C43—22no—
73#95×C43—22no—
74C24×C43C2×D411yescharacteristic
75C23×D43—21no—
76C22×C4×C43—42no—
77C22×C4×C43—42no—
78C22×C4×C43C22⋊C411yescharacteristic
79C22×C4×C43C22⋊C411yescharacteristic
80C22×C4×C43C22⋊C411yescharacteristic
81C22×C4×C43C22⋊C411yescharacteristic
82C22×C4×C43C22×C411yescharacteristic
83C22×C4×C43C2×D411yescharacteristic
84C22×C4×C43C2×D412yes—
85C22×C4×C43C2×D412yes—
86C2×C22⋊D43—24no—
87C2×C22⋊D43—82no—
88C22×Dic43—21no—
89C22×Dic43—21no—
90C22×Dic43—21no—
91C2×D172—21no—
92C2×D172—42no—
93C2×D172C22⋊C412yes—
94C2×C4×D43—24no—
95C2×C4×D43—82no—
96C2×Dic86—21no—
97C2×Dic86—24no—
98C2×Dic86—42no—
99C2×Dic86—82no—
100C2×Dic86C22⋊C412yes—
101C4⋊C4×C43—22no—
102C4⋊C4×C43—22no—
103C22⋊C4×C43—22no—
104C22⋊C4×C43—22no—
105C22⋊C4×C43—44no—
106C42×C43—21no—
107C42×C43—21no—
108C42×C43—21no—
109C42×C43—21no—
110C22⋊Dic43—24no—
111(C2×D43)⋊C4—24no—
112(C2×D43)⋊C4—24no—
113(C2×D43)⋊C4—24no—
114(C2×D43)⋊C4—24no—
115C4⋊Dic43—24no—
116C4⋊Dic43—21no—
117C4⋊Dic43—21no—
118C4⋊Dic43C22⋊C412yes—
119C4⋊Dic43C22⋊C412yes—
120Dic43⋊C4—24no—
121Dic43⋊C4—24no—
122C4×Dic43—24no—
123#10621877—44no—
124C2×C43⋊C8—42no—
125C23×C43—22no—
126C23×C43—41no—
127C23×C43—41no—
128C23×C43C2×C22⋊C411yescharacteristic
129C23×C43#12012yes—
130C22×D43—22no—
131C22×D43—42no—
132C22×D43—42no—
133C22×D43—42no—
134C2×C4×C43—22no—
135C2×C4×C43—22no—
136C2×C4×C43—21no—
137C2×C4×C43—21no—
138C2×C4×C43—21no—
139C2×C4×C43—21no—
140C2×C4×C43—42no—
141C2×C4×C43—42no—
142C2×C4×C43—41no—
143C2×C4×C43—41no—
144C2×C4×C43—41no—
145C2×C4×C43—41no—
146C2×C4×C43—82no—
147C2×C4×C43—82no—
148C2×C4×C43—82no—
149C2×C4×C43C2×C22⋊C411yescharacteristic
150C2×C4×C43C2×C22⋊C411yescharacteristic
151C2×C4×C43#12012yes—
152C22⋊D43—84no—
153C2×Dic43—22no—
154C2×Dic43—22no—
155C2×Dic43—22no—
156C2×Dic43—42no—
157C2×Dic43—42no—
158C2×Dic43—42no—
159D172—42no—
160D172—42no—
161C4×D43—84no—
162Dic86—42no—
163Dic86—42no—
164Dic86—84no—
165C43⋊C8—82no—
166#8519—431nomaximal, Sylow
167C22×C43—21no—
168C22×C43—21no—
169C22×C43—22no—
170C22×C43—22no—
171C22×C43—22no—
172C22×C43—82no—
173C22×C43#37911yescharacteristic
174C22×C43#41012yes—
175C2×D43—42no—
176C2×D43—42no—
177C2×D43—42no—
178C2×D43—84no—
179C4×C43—22no—
180C4×C43—21no—
181C4×C43—21no—
182C4×C43—41no—
183C4×C43—41no—
184C4×C43—41no—
185C4×C43—41no—
186C4×C43—162no—
187Dic43—42no—
188Dic43—42no—
189Dic43—42no—
190C2×#542—431no—
191#1822—431no—
192#1820D4311yescharacteristic, complemented
193#1511—432no—
194#1510—432no—
195C2×C43—22no—
196C2×C43—42no—
197C2×C43#178211yescharacteristic
198C2×C43#179511yescharacteristic
199C2×C43#179611yescharacteristic
200D43—82no—
201#542—864no—
202#542—864no—
203C2×C4⋊Q8—431no—
204C2×#126—861no—
205C2×#124—431no—
206C2×#124—861no—
207C2×#123—861no—
208C2×C22⋊Q8—431no—
209C2×C22⋊Q8—861no—
210C2×#121—431no—
211C2×C4×Q8—431no—
212C2×C4×D4—431no—
213C2×#117C2×D4311yescharacteristic
214#407—432no—
215#381C2×D4312yes—
216#343—861no—
217#343—861no—
218#342—21no—
219#342—21no—
220#328—864no—
221#327—864no—
222C43#851911yescharacteristic, complemented, Sylow
223C22×Q8—431no—
224C22×Q8—861no—
225C22×D4—431no—
226C22×D4—861no—
227C23×C4—861no—
228C23×C4C22×D4311yescharacteristic
229C2×M16—862no—
230C4⋊Q8—862no—
231C4⋊Q8—862no—
232#126—868no—
233#124—862no—
234#124—862no—
235#124—868no—
236#123—868no—
237C22⋊Q8—434no—
238C22⋊Q8—868no—
239C22⋊Q8—862no—
240#121—434no—
241#121—862no—
242C4×Q8—862no—
243C4×Q8—862no—
244C4×D4—862no—
245C4×D4—862no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—431no—
249C2×C4⋊C4—431no—
250C2×C4⋊C4—861no—
251C2×C4⋊C4—861no—
252C2×C4⋊C4—861no—
253C2×C4⋊C4D17211yescharacteristic
254C2×C4⋊C4D17211yescharacteristic
255C2×C22⋊C4—22no—
256C2×C22⋊C4—861no—
257C2×C22⋊C4—861no—
258C2×C22⋊C4—861no—
259C2×C22⋊C4—861no—
260C2×C22⋊C4—861no—
261C2×C22⋊C4C22⋊D4311yescharacteristic
262C2×C22⋊C4C22⋊D4311yescharacteristic
263C2×C42—861no—
264C2×C42C4×D4311yescharacteristic
265C2×C42C4×D4311yescharacteristic
266C22⋊C8—862no—
267C22⋊C8—862no—
268#95—22no—
269#95—22no—
270C24—861no—
271C24C2×C22⋊D4311yescharacteristic
272C2×Q8—432no—
273C2×Q8—864no—
274C2×Q8—861no—
275C2×Q8—3442no—
276C2×Q8—1722no—
277C2×D4—432no—
278C2×D4—864no—
279C2×D4—861no—
280C2×D4—3442no—
281C2×D4—1722no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—864no—
285C22×C4—861no—
286C22×C4—861no—
287C22×C4—861no—
288C22×C4—3442no—
289C22×C4(C2×D43)⋊C411yescharacteristic
290C22×C4(C2×D43)⋊C411yescharacteristic
291C22×C4(C2×D43)⋊C411yescharacteristic
292C22×C4(C2×D43)⋊C411yescharacteristic
293C22×C4C2×C4×D4311yescharacteristic
294C22×C4C2×D17211yescharacteristic
295C22×C4D4×D4312yes—
296C22×C4D4×D4312yes—
297M16—1724no—
298C2×C8—1722no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—432no—
302C4⋊C4—432no—
303C4⋊C4—864no—
304C4⋊C4—864no—
305C4⋊C4—861no—
306C4⋊C4—861no—
307C4⋊C4—864no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—44no—
311C22⋊C4—864no—
312C22⋊C4—864no—
313C22⋊C4—864no—
314C22⋊C4—864no—
315C22⋊C4—864no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—864no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—862no—
325C23—1722no—
326C23—1722no—
327C23—1722no—
328C23C2×(C2×D43)⋊C411yescharacteristic
329C23#1306329312yes—
330Q8—3444no—
331Q8—1722no—
332Q8—1722no—
333D4—3444no—
334D4—1722no—
335D4—1722no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—862no—
352C2×C4—862no—
353C2×C4—862no—
354C2×C4—3444no—
355C2×C4—1722no—
356C2×C4—1722no—
357C2×C4—1722no—
358C2×C4C22⋊C4×D4311yescharacteristic
359C2×C4C22⋊C4×D4311yescharacteristic
360C2×C4#1306329512yes—
361C8—3442no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—82no—
368C22—3444no—
369C22—1722no—
370C22—1722no—
371C22—1722no—
372C22#1458275411yescharacteristic
373C22#1458287212yes—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—162no—
382C4—1722no—
383C4—1722no—
384C4—1722no—
385C2—22no—
386C2—42no—
387C2—3442no—
388C2#1913451211yescharacteristic
389C2#1913456311yescharacteristic
390C2#1913456811yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1820C2
socleC22×C43#379
prime radicalC2×#13063299C22
cosocleC23×C4×C43C23
hypercenter#1820D43
hypocenterC43#8519
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #19134678C2semidirect
3. #19133127C2semidirect
5. C43×#1820C2semidirect
8. C2×#13063325C4semidirect
10. C2×#13063313C4semidirect
16. C2×C4×Dic86C4semidirect
17. C2×C4×D172C4semidirect
192. #1820D43semidirect
222. C43#8519semidirect

Maximal subgroups

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

subgroupindex
C2×#145835202
#191346782
#191331272
#191331232
C43×#18202
#851943