Finite Group Databasebeta

← #24997115

Subgroups — #24997115

Order 11008.

650 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 109 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#19145352C211yesmaximal, characteristic
2C2×#14583229C211yesmaximal, characteristic, complemented
3#19135580C211yesmaximal, characteristic, complemented
4#19134681C211yesmaximal, characteristic, complemented
5#19133163C211yesmaximal, characteristic
6#19132443C211yesmaximal, characteristic, complemented
7C43×#2501C211yesmaximal, characteristic
8C2×#13063304C2211yescharacteristic
9#14583229—22no—
10#14583229—22no—
11#14583229—21no—
12#14583215—21no—
13#14583126—22no—
14#14583050—21no—
15#14583037—21no—
16C2×#13063256—21no—
17C2×#13063248—21no—
18C2×C22⋊(C43⋊C8)C2211yescharacteristic
19C2×C22⋊(C43⋊C8)C2211yescharacteristic
20#14582759C2211yescharacteristic
21#14582665—22no—
22#14582661—22no—
23#14582633—22no—
24C2×#123×C43—22no—
25C2×#121×C43C2211yescharacteristic
26C22×C4⋊C4×C43C2211yescharacteristic
27C43×#399—22no—
28C43×#390—21no—
29C43×#384—21no—
30C4×C22⋊C4×C43C2211yescharacteristic
31C22×Dic86D411yescharacteristic
32C2×#10621883—22no—
33C22×C43⋊C8—21no—
34C22×C43⋊C8D412yes—
35#13063304—21no—
36#13063304—21no—
37#13063304—21no—
38#13063304—21no—
39C2×C4⋊Dic43—21no—
40C2×C4⋊Dic43D411yescharacteristic
41C2×Dic43⋊C4—21no—
42C2×C22⋊Dic43—21no—
43C2×C22⋊Dic43—21no—
44#13063256—22no—
45#13063256—24no—
46#13063248—22no—
47#13063248—24no—
48#13063246—22no—
49C22⋊(C43⋊C8)—21no—
50C22⋊(C43⋊C8)—21no—
51C22⋊(C43⋊C8)—21no—
52C22⋊(C43⋊C8)D412yes—
53C43⋊#95—21no—
54C22×D4×C43—21no—
55C22×D4×C43D411yescharacteristic
56C23×C4×C43—21no—
57C23×C4×C43C2311yescharacteristic
58#123×C43—44no—
59#123×C43—44no—
60#121×C43—21no—
61#121×C43—21no—
62#121×C43—21no—
63#121×C43—21no—
64C2×C4⋊C4×C43—21no—
65C2×C4⋊C4×C43—21no—
66C2×C4⋊C4×C43—21no—
67C2×C4⋊C4×C43—21no—
68C2×C4⋊C4×C43—24no—
69C2×C4⋊C4×C43—21no—
70C2×C4⋊C4×C43D412yes—
71C2×C4⋊C4×C43D411yescharacteristic
72C2×C22⋊C4×C43—24no—
73C2×C22⋊C4×C43—21no—
74C2×C22⋊C4×C43D412yes—
75C2×C42×C43—21no—
76#95×C43—21no—
77C24×C43—21no—
78C24×C43C2×D411yescharacteristic
79C2×D4×C43—21no—
80C2×D4×C43—21no—
81C2×D4×C43—42no—
82C2×D4×C43—41no—
83C2×D4×C43—41no—
84C2×D4×C43—42no—
85C2×D4×C43—42no—
86C22×C4×C43—22no—
87C22×C4×C43—22no—
88C22×C4×C43—21no—
89C22×C4×C43—42no—
90C22×C4×C43—41no—
91C22×C4×C43—41no—
92C22×C4×C43—42no—
93C22×C4×C43—42no—
94C22×C4×C43C2×D411yescharacteristic
95C22×C4×C43C2×D411yescharacteristic
96C22×C4×C43C2×D411yescharacteristic
97C22×C4×C43C2×D411yescharacteristic
98C22×C4×C43C2×D411yescharacteristic
99C22×C4×C43#4111yescharacteristic
100C22×Dic43—21no—
101C22×Dic43—21no—
102C22×Dic43—41no—
103C2×Dic86—21no—
104C2×Dic86—21no—
105C2×Dic86—42no—
106C4⋊C4×C43—22no—
107C4⋊C4×C43—21no—
108C4⋊C4×C43—22no—
109C4⋊C4×C43—21no—
110C4⋊C4×C43—48no—
111C4⋊C4×C43—41no—
112C4⋊C4×C43—41no—
113C22⋊C4×C43—22no—
114C22⋊C4×C43—48no—
115C22⋊C4×C43—41no—
116C22⋊C4×C43—41no—
117C22⋊C4×C43SD1614yescomplemented
118C42×C43—42no—
119C22⋊Dic43—41no—
120C22⋊Dic43—41no—
121C22⋊Dic43—42no—
122#10621883—82no—
123C4⋊Dic43—21no—
124C4⋊Dic43—21no—
125C4⋊Dic43—42no—
126Dic43⋊C4—41no—
127Dic43⋊C4—41no—
128C2×C43⋊C8—22no—
129C2×C43⋊C8—22no—
130C2×C43⋊C8—42no—
131C2×C43⋊C8—42no—
132C23×C43—41no—
133C23×C43—41no—
134C23×C43—42no—
135C23×C43—42no—
136C23×C43—41no—
137C23×C43—42no—
138C23×C43#12011yescharacteristic
139C23×C43#12111yescharacteristic
140C23×C43#12711yescharacteristic
141C23×C43C2×SD1612yes—
142C23×C43AGL(1,ℤ/8)12yes—
143D4×C43—42no—
144D4×C43—81no—
145D4×C43—82no—
146D4×C43—81no—
147D4×C43—81no—
148C2×C4×C43—22no—
149C2×C4×C43—22no—
150C2×C4×C43—22no—
151C2×C4×C43—24no—
152C2×C4×C43—41no—
153C2×C4×C43—41no—
154C2×C4×C43—41no—
155C2×C4×C43—42no—
156C2×C4×C43—42no—
157C2×C4×C43—41no—
158C2×C4×C43—42no—
159C2×C4×C43—44no—
160C2×C4×C43—42no—
161C2×C4×C43—82no—
162C2×C4×C43—81no—
163C2×C4×C43—81no—
164C2×C4×C43#12011yescharacteristic
165C2×C4×C43#12011yescharacteristic
166C2×C4×C43#12111yescharacteristic
167C2×C4×C43#12111yescharacteristic
168C2×Dic43—41no—
169C2×Dic43—41no—
170C2×Dic43—81no—
171C2×Dic43—81no—
172C2×Dic43—81no—
173C2×Dic43—81no—
174C2×Dic43—81no—
175Dic86—42no—
176Dic86—42no—
177C43⋊C8—42no—
178C43⋊C8—42no—
179#9960—431nomaximal, Sylow
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—84no—
189C22×C43—81no—
190C22×C43—81no—
191C22×C43—81no—
192C22×C43#39211yescharacteristic
193C22×C43#45012yes—
194C22×C43#46012yes—
195C22×C43#49211yescharacteristic
196C22×C43#49611yescharacteristic
197C4×C43—21no—
198C4×C43—21no—
199C4×C43—21no—
200C4×C43—44no—
201C4×C43—82no—
202C4×C43—81no—
203C4×C43#45812yes—
204Dic43—81no—
205Dic43—81no—
206Dic43—161no—
207#3187—431no—
208C2×#483—431no—
209#2501D4311yescharacteristic
210#1996—431no—
211#1823—431no—
212#1519—431no—
213#1282—431no—
214C2×C43—21no—
215C2×C43—22no—
216C2×C43—22no—
217C2×C43—21no—
218C2×C43—161no—
219C2×C43#200911yescharacteristic
220C2×C43#201411yescharacteristic
221C2×C43#202211yescharacteristic
222C2×#123—22no—
223C2×C22⋊Q8—431no—
224C2×#121C2×D4311yescharacteristic
225C22×C4⋊C4C2×D4311yescharacteristic
226#483—862no—
227#483—862no—
228#483—861no—
229#481—861no—
230#465—862no—
231#450—861no—
232#448—861no—
233C2×#106—861no—
234C2×#103—861no—
235C2×C22⋊C8—431no—
236C2×C22⋊C8—431no—
237#399—22no—
238#390—21no—
239#384—21no—
240#380—431no—
241C4×C22⋊C4C2×D4311yescharacteristic
242#342—862no—
243#340—862no—
244#327—862no—
245C43#996011yescharacteristic, complemented, Sylow
246C22×Q8—431no—
247C22×D4—21no—
248C22×D4C22⋊D4311yescharacteristic
249C23×C4—21no—
250C23×C4C22×D4311yescharacteristic
251C2×SD16—862no—
252C22×C8—432no—
253C22×C8—861no—
254#123—44no—
255#123—44no—
256C22⋊Q8—861no—
257C22⋊Q8—861no—
258C22⋊Q8—861no—
259C22⋊Q8—861no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—431no—
271C2×C4⋊C4—861no—
272C2×C4⋊C4—861no—
273C2×C4⋊C4C22⋊D4312yes—
274C2×C4⋊C4C22⋊D4311yescharacteristic
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—861no—
278C2×C22⋊C4—861no—
279C2×C22⋊C4C22⋊D4312yes—
280C2×C42—21no—
281#106—864no—
282#106—862no—
283#103—864no—
284#103—862no—
285#102—862no—
286C22⋊C8—432no—
287C22⋊C8—861no—
288C22⋊C8—861no—
289C22⋊C8—861no—
290#95—21no—
291#95—861no—
292C24—21no—
293C24C2×C22⋊D4311yescharacteristic
294C2×Q8—861no—
295C2×Q8—861no—
296C2×Q8—1722no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—861no—
313C22×C4—861no—
314C22×C4—1721no—
315C22×C4D4×D4311yescharacteristic
316C22×C4D4×D4311yescharacteristic
317C22×C4D4×D4311yescharacteristic
318C22×C4#1062190011yescharacteristic
319C22×C4C2×C22⋊D4311yescharacteristic
320C22×C4C2×C22⋊D4311yescharacteristic
321SD16—3442no—
322C2×C8—862no—
323C2×C8—862no—
324C2×C8—1722no—
325C2×C8—1722no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—48no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—861no—
334C4⋊C4—861no—
335C4⋊C4—1722no—
336C4⋊C4—1721no—
337C4⋊C4—1721no—
338C22⋊C4—22no—
339C22⋊C4—48no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—1722no—
343C22⋊C4—1721no—
344C22⋊C4—1721no—
345C22⋊C4#1062188314yescomplemented
346C42—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23#1306329411yescharacteristic
354C23#1306329611yescharacteristic
355C23#1306332111yescharacteristic
356C23C2×#1062188312yes—
357C23#1306335112yes—
358Q8—1722no—
359Q8—1722no—
360D4—42no—
361D4—81no—
362D4—82no—
363D4—81no—
364D4—81no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—3441no—
382C2×C4—3441no—
383C2×C4—3441no—
384C2×C4—3441no—
385C2×C4—3441no—
386C2×C4—1721no—
387C2×C4—1721no—
388C2×C4#1306329311yescharacteristic
389C2×C4#1306329311yescharacteristic
390C2×C4#1306329711yescharacteristic
391C2×C4#1306329711yescharacteristic
392C8—1722no—
393C8—1722no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1458280711yescharacteristic
407C22#1458305012yes—
408C22#1458309612yes—
409C22#1458327111yescharacteristic
410C22#1458328611yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—44no—
415C4—82no—
416C4—81no—
417C4—3441no—
418C4—3441no—
419C4—6881no—
420C4#1458308812yes—
421C2—22no—
422C2—21no—
423C2—22no—
424C2—21no—
425C2—161no—
426C2#1913565511yescharacteristic
427C2#1913568311yescharacteristic
428C2#1913572311yescharacteristic
subgroupG/H
centerC22#14582807
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC43×#2501C2
socleC22×C43#392
prime radicalC2×#121×C43C22
hypercenter#2501D43
hypocenterC43#9960
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. C2×#14583229C2semidirect
3. #19135580C2semidirect
4. #19134681C2semidirect
6. #19132443C2semidirect
117. C22⋊C4×C43SD16semidirect
245. C43#9960semidirect
345. C22⋊C4#10621883semidirect

Maximal subgroups

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

subgroupindex
#191453522
C2×#145832292
#191355802
#191346812
#191331632
#191324432
C43×#25012
#996043