Finite Group Databasebeta

← #24988286

Subgroups — #24988286

Order 11008.

818 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583606C211yesmaximal, characteristic
2#19134683C212yesmaximal, complemented
3#19133166C212yesmaximal, complemented
4#19132913C211yesmaximal, characteristic
5C43×#1799C211yesmaximal, characteristic
6#14583606—24no—
7#14583606—24no—
8C2×#13063325C411yescharacteristic, complemented
9C2×#13063316—21no—
10C2×#13063304C411yescharacteristic, complemented
11C2×#13063304C2211yescharacteristic
12C2×#13063302—21no—
13C2×#13063302—21no—
14C2×C4×Dic86C411yescharacteristic, complemented
15C2×D4×Dic43—21no—
16C22×C4⋊Dic43C411yescharacteristic, complemented
17C22×C4⋊Dic43C2211yescharacteristic
18C2×C22⋊(C43⋊C8)C2212yes—
19#14582661—24no—
20#14582635—24no—
21#14582625—21no—
22#14582623—21no—
23C2×C4×D4×C43C2211yescharacteristic
24C2×#99×C43—21no—
25C43×#380C2212yes—
26C43×#342—21no—
27C22×Dic86—21no—
28C22×Dic86C2×C411yescharacteristic
29C23×Dic43—21no—
30C23×Dic43—21no—
31C22×C43⋊C8—22no—
32#13063325—22no—
33#13063325—22no—
34#13063316—28no—
35#13063304—22no—
36#13063304—22no—
37#13063304—22no—
38#13063304D414yes—
39#13063302—28no—
40#13063302—28no—
41C4×Dic86—22no—
42C4×Dic86—22no—
43D4×Dic43—28no—
44C2×C4⋊Dic43—21no—
45C2×C4⋊Dic43—21no—
46C2×C4⋊Dic43—22no—
47C2×C4⋊Dic43—21no—
48C2×C4⋊Dic43—22no—
49C2×C4⋊Dic43—22no—
50C2×C4⋊Dic43C2×C411yescharacteristic
51C2×C4⋊Dic43C2×C411yescharacteristic
52C2×C4⋊Dic43C2×C411yescharacteristic
53C2×C4⋊Dic43D414yes—
54C2×Dic43⋊C4—21no—
55C2×Dic43⋊C4—21no—
56C2×Dic43⋊C4—21no—
57C2×C22⋊Dic43—21no—
58C2×C22⋊Dic43—21no—
59C2×C4×Dic43—21no—
60C22⋊(C43⋊C8)—22no—
61C22⋊(C43⋊C8)—22no—
62C22×D4×C43D411yescharacteristic
63C23×C4×C43C2×C411yescharacteristic
64C23×C4×C43C2311yescharacteristic
65C4×D4×C43—22no—
66C4×D4×C43—22no—
67C2×C4⋊C4×C43—22no—
68C2×C4⋊C4×C43D411yescharacteristic
69C2×C22⋊C4×C43—22no—
70C2×C22⋊C4×C43D411yescharacteristic
71C2×C22⋊C4×C43D411yescharacteristic
72C2×C42×C43C2×C411yescharacteristic
73#99×C43—42no—
74#95×C43—22no—
75C24×C43C22⋊C411yescharacteristic
76C24×C43C2×D411yescharacteristic
77C2×D4×C43—22no—
78C2×D4×C43—22no—
79C2×D4×C43SD1612yes—
80C2×D4×C43Q1612yes—
81C22×C4×C43—21no—
82C22×C4×C43—21no—
83C22×C4×C43—42no—
84C22×C4×C43—42no—
85C22×C4×C43—42no—
86C22×C4×C43C22⋊C411yescharacteristic
87C22×C4×C43C22⋊C411yescharacteristic
88C22×C4×C43C22⋊C411yescharacteristic
89C22×C4×C43C22×C411yescharacteristic
90C22×C4×C43C2×D411yescharacteristic
91C22×C4×C43C2×D412yes—
92C22×C4×C43C2×D412yes—
93C22×Dic43—24no—
94C22×Dic43—21no—
95C22×Dic43—21no—
96C22×Dic43—21no—
97C22×Dic43—21no—
98C22×Dic43—24no—
99C22×Dic43—82no—
100C22×Dic43—82no—
101C2×Dic86—21no—
102C2×Dic86—24no—
103C2×Dic86—42no—
104C2×Dic86—82no—
105C2×Dic86C22⋊C412yes—
106C4⋊C4×C43—22no—
107C4⋊C4×C43—44no—
108C22⋊C4×C43—22no—
109C22⋊C4×C43—22no—
110C22⋊C4×C43—42no—
111C22⋊C4×C43—42no—
112C42×C43—21no—
113C42×C43—21no—
114C22⋊Dic43—24no—
115C22⋊Dic43—24no—
116C4⋊Dic43—24no—
117C4⋊Dic43—24no—
118C4⋊Dic43—21no—
119C4⋊Dic43—21no—
120C4⋊Dic43—21no—
121C4⋊Dic43—24no—
122C4⋊Dic43C22⋊C412yes—
123C4⋊Dic43C22⋊C412yes—
124C4⋊Dic43C22⋊C412yes—
125Dic43⋊C4—24no—
126Dic43⋊C4—24no—
127Dic43⋊C4—24no—
128C4×Dic43—24no—
129C2×C43⋊C8—44no—
130C2×C43⋊C8—42no—
131C23×C43—21no—
132C23×C43—21no—
133C23×C43—41no—
134C23×C43—41no—
135C23×C43—41no—
136C23×C43—41no—
137C23×C43#10312yes—
138C23×C43C2×C22⋊C411yescharacteristic
139C23×C43#12012yes—
140C23×C43C2×SD1611yescharacteristic
141C23×C43C2×Q1611yescharacteristic
142C23×C43#13712yes—
143D4×C43—24no—
144D4×C43—24no—
145C2×C4×C43—21no—
146C2×C4×C43—21no—
147C2×C4×C43—24no—
148C2×C4×C43—22no—
149C2×C4×C43—21no—
150C2×C4×C43—21no—
151C2×C4×C43—41no—
152C2×C4×C43—42no—
153C2×C4×C43—42no—
154C2×C4×C43—42no—
155C2×C4×C43—41no—
156C2×C4×C43—41no—
157C2×C4×C43—82no—
158C2×C4×C43—82no—
159C2×C4×C43—82no—
160C2×C4×C43#10312yes—
161C2×C4×C43C2×C22⋊C411yescharacteristic
162C2×C4×C43C2×C22⋊C411yescharacteristic
163C2×C4×C43#12012yes—
164C2×Dic43—22no—
165C2×Dic43—22no—
166C2×Dic43—22no—
167C2×Dic43—22no—
168C2×Dic43—42no—
169C2×Dic43—42no—
170C2×Dic43—42no—
171C2×Dic43—42no—
172C2×Dic43—84no—
173C2×Dic43—84no—
174Dic86—42no—
175Dic86—42no—
176Dic86—84no—
177C43⋊C8—82no—
178#8380—431nomaximal, Sylow
179C22×C43—21no—
180C22×C43—22no—
181C22×C43—21no—
182C22×C43—22no—
183C22×C43—22no—
184C22×C43—41no—
185C22×C43—41no—
186C22×C43—42no—
187C22×C43—42no—
188C22×C43—42no—
189C22×C43—41no—
190C22×C43—41no—
191C22×C43#37911yescharacteristic
192C22×C43C2×#10311yescharacteristic
193C22×C43#41911yescharacteristic
194C22×C43#44812yes—
195C22×C43#45112yes—
196C4×C43—22no—
197C4×C43—21no—
198C4×C43—22no—
199C4×C43—41no—
200C4×C43—41no—
201C4×C43—162no—
202C4×C43#41012yes—
203Dic43—42no—
204Dic43—42no—
205Dic43—42no—
206Dic43—42no—
207C2×#554—431no—
208#1824—432no—
209#1799D4311yescharacteristic
210#1520—432no—
211#1459—431no—
212C2×C43—22no—
213C2×C43—21no—
214C2×C43—21no—
215C2×C43—22no—
216C2×C43—42no—
217C2×C43#178211yescharacteristic
218C2×C43#178711yescharacteristic
219C2×C43#179511yescharacteristic
220#554—864no—
221#554—864no—
222C2×C4⋊Q8—431no—
223C2×#125—861no—
224C2×C22⋊Q8—431no—
225C2×C22⋊Q8—431no—
226C2×C22⋊Q8—861no—
227C2×C22⋊Q8—861no—
228C2×C4×Q8—431no—
229C2×C4×D4—861no—
230C2×C4×D4C2×D4311yescharacteristic
231C22×C4⋊C4—431no—
232C22×C4⋊C4—431no—
233C2×#99—21no—
234C2×C22⋊C8—432no—
235#380C2×D4312yes—
236#342—21no—
237#340—864no—
238#328—864no—
239#324—861no—
240#323—861no—
241C43#838011yescharacteristic, complemented, Sylow
242C22×Q8—431no—
243C22×Q8—861no—
244C22×D4C22⋊D4311yescharacteristic
245C23×C4—861no—
246C23×C4—861no—
247C23×C4C4×D4311yescharacteristic
248C23×C4C22×D4311yescharacteristic
249C22×C8—862no—
250C4⋊Q8—862no—
251C4⋊Q8—862no—
252#125—868no—
253C22⋊Q8—434no—
254C22⋊Q8—868no—
255C22⋊Q8—868no—
256C22⋊Q8—862no—
257C22⋊Q8—862no—
258C22⋊Q8—862no—
259C4×Q8—862no—
260C4×Q8—862no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—868no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—431no—
266C2×C4⋊C4—431no—
267C2×C4⋊C4—431no—
268C2×C4⋊C4—434no—
269C2×C4⋊C4—862no—
270C2×C4⋊C4—862no—
271C2×C4⋊C4—861no—
272C2×C4⋊C4—861no—
273C2×C4⋊C4—861no—
274C2×C4⋊C4—861no—
275C2×C4⋊C4—861no—
276C2×C4⋊C4—861no—
277C2×C4⋊C4—862no—
278C2×C4⋊C4D17211yescharacteristic
279C2×C22⋊C4—22no—
280C2×C22⋊C4—861no—
281C2×C22⋊C4—861no—
282C2×C22⋊C4D17211yescharacteristic
283C2×C22⋊C4C22⋊D4311yescharacteristic
284C2×C42—861no—
285C2×C42C4×D4311yescharacteristic
286#99—42no—
287C22⋊C8—862no—
288C22⋊C8—862no—
289#95—22no—
290C24(C2×D43)⋊C411yescharacteristic
291C24C2×C22⋊D4311yescharacteristic
292C2×Q8—432no—
293C2×Q8—864no—
294C2×Q8—861no—
295C2×Q8—3442no—
296C2×Q8—1722no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4#1062188312yes—
300C2×D4#1062188512yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—864no—
307C22×C4—864no—
308C22×C4—861no—
309C22×C4—861no—
310C22×C4—861no—
311C22×C4—861no—
312C22×C4—3442no—
313C22×C4—3442no—
314C22×C4(C2×D43)⋊C411yescharacteristic
315C22×C4(C2×D43)⋊C411yescharacteristic
316C22×C4(C2×D43)⋊C411yescharacteristic
317C22×C4C2×C4×D4311yescharacteristic
318C22×C4C2×D17211yescharacteristic
319C22×C4D4×D4312yes—
320C22×C4D4×D4312yes—
321C2×C8—1724no—
322C2×C8—1722no—
323C4⋊C4—22no—
324C4⋊C4—44no—
325C4⋊C4—432no—
326C4⋊C4—432no—
327C4⋊C4—432no—
328C4⋊C4—864no—
329C4⋊C4—864no—
330C4⋊C4—864no—
331C4⋊C4—864no—
332C4⋊C4—861no—
333C4⋊C4—861no—
334C4⋊C4—861no—
335C4⋊C4—864no—
336C4⋊C4—864no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—864no—
342C22⋊C4—864no—
343C42—21no—
344C42—21no—
345C42—864no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23#1306324812yes—
353C23C2×(C2×D43)⋊C411yescharacteristic
354C23#1306329312yes—
355C23C2×#1062188311yescharacteristic
356C23C2×#1062188511yescharacteristic
357C23#1306335512yes—
358Q8—3444no—
359Q8—1722no—
360Q8—1722no—
361D4—24no—
362D4—24no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—862no—
379C2×C4—862no—
380C2×C4—862no—
381C2×C4—862no—
382C2×C4—3444no—
383C2×C4—3444no—
384C2×C4—1722no—
385C2×C4—1722no—
386C2×C4—1722no—
387C2×C4—1722no—
388C2×C4#1306324812yes—
389C2×C4C22⋊C4×D4311yescharacteristic
390C2×C4C22⋊C4×D4311yescharacteristic
391C2×C4#1306329512yes—
392C8—3442no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22#1458275411yescharacteristic
406C22C2×#1306324811yescharacteristic
407C22#1458290711yescharacteristic
408C22#1458303712yes—
409C22#1458305412yes—
410C4—22no—
411C4—21no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—162no—
416C4—1722no—
417C4—1722no—
418C4—1722no—
419C4—1722no—
420C4#1458287112yes—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—22no—
425C2—42no—
426C2#1913451111yescharacteristic
427C2#1913452911yescharacteristic
428C2#1913456311yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1799C2
socleC22×C43#379
prime radicalC22×D4×C43D4
cosocleC23×C4×C43C23
hypercenter#1799D43
hypocenterC43#8380
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #19134683C2semidirect
3. #19133166C2semidirect
8. C2×#13063325C4semidirect
10. C2×#13063304C4semidirect
14. C2×C4×Dic86C4semidirect
16. C22×C4⋊Dic43C4semidirect
241. C43#8380semidirect

Maximal subgroups

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

subgroupindex
C2×#145836062
#191346832
#191331662
#191329132
C43×#17992
#838043