Finite Group Databasebeta

← #24997114

Subgroups — #24997114

Order 11008.

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

normal subgroups only

This group has 650 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
1#19145348C211yesmaximal, characteristic
2C2×#14583227C211yesmaximal, characteristic, complemented
3#19139066C211yesmaximal, characteristic
4#19134679C211yesmaximal, characteristic, complemented
5#19133162C211yesmaximal, characteristic
6#19132443C211yesmaximal, characteristic, complemented
7C43×#1996C211yesmaximal, characteristic, complemented
8C2×#13063309—22no—
9C2×#13063299C2211yescharacteristic
10C22×C4⋊Dic43C2211yescharacteristic
11#14583227—22no—
12#14583227—21no—
13#14583227—22no—
14#14583217—21no—
15#14583130—22no—
16#14583049—21no—
17#14583038—21no—
18C2×#13063256—21no—
19C2×#13063249—21no—
20C2×C22⋊(C43⋊C8)C2211yescharacteristic
21C2×C22⋊(C43⋊C8)C2211yescharacteristic
22#14582836—22no—
23#14582801—21no—
24#14582769—21no—
25#14582661—22no—
26#14582632—22no—
27C2×C22⋊Q8×C43C2211yescharacteristic
28C43×#380C2211yescharacteristic
29C4×C22⋊C4×C43C2211yescharacteristic
30C43×#342—22no—
31C22×D172D411yescharacteristic
32C22×C22⋊D43—21no—
33C23×Dic43—21no—
34C2×#10621884—22no—
35C22×C43⋊C8—21no—
36C22×C43⋊C8D412yes—
37#13063309—44no—
38#13063309—44no—
39#13063299—21no—
40#13063299—21no—
41#13063299—21no—
42#13063299—21no—
43C2×C4⋊Dic43—21no—
44C2×C4⋊Dic43—21no—
45C2×C4⋊Dic43—21no—
46C2×C4⋊Dic43—21no—
47C2×C4⋊Dic43—24no—
48C2×C4⋊Dic43—21no—
49C2×C4⋊Dic43D412yes—
50C2×C4⋊Dic43D411yescharacteristic
51C2×(C2×D43)⋊C4—24no—
52C2×(C2×D43)⋊C4—21no—
53#13063256—22no—
54#13063256—24no—
55#13063249—22no—
56#13063249—24no—
57#13063245—22no—
58C22⋊(C43⋊C8)—21no—
59C22⋊(C43⋊C8)—21no—
60C22⋊(C43⋊C8)—21no—
61C22⋊(C43⋊C8)D412yes—
62C22×Q8×C43D411yescharacteristic
63C23×C4×C43C2311yescharacteristic
64C22⋊Q8×C43—21no—
65C22⋊Q8×C43—21no—
66C22⋊Q8×C43—21no—
67C22⋊Q8×C43—21no—
68C2×C4⋊C4×C43—21no—
69C2×C4⋊C4×C43—21no—
70C2×C4⋊C4×C43D411yescharacteristic
71C2×C22⋊C4×C43—21no—
72C2×C22⋊C4×C43—21no—
73C2×C22⋊C4×C43D412yes—
74C2×C42×C43—21no—
75#95×C43—21no—
76#95×C43—21no—
77C24×C43C2×D411yescharacteristic
78C23×D43—21no—
79C2×Q8×C43—21no—
80C2×Q8×C43—21no—
81C2×Q8×C43—42no—
82C22×C4×C43—22no—
83C22×C4×C43—21no—
84C22×C4×C43—21no—
85C22×C4×C43—42no—
86C22×C4×C43—41no—
87C22×C4×C43C2×D411yescharacteristic
88C22×C4×C43C2×D411yescharacteristic
89C22×C4×C43C2×D411yescharacteristic
90C22×C4×C43C2×D411yescharacteristic
91C22×C4×C43C2×D411yescharacteristic
92C22×C4×C43#4111yescharacteristic
93C2×C22⋊D43—41no—
94C2×C22⋊D43—41no—
95C2×C22⋊D43—42no—
96C2×C22⋊D43—42no—
97C22×Dic43—22no—
98C22×Dic43—21no—
99C22×Dic43—41no—
100C22×Dic43—41no—
101C22×Dic43—42no—
102C22×Dic43—42no—
103C2×D172—21no—
104C2×D172—21no—
105C2×D172—42no—
106C4⋊C4×C43—21no—
107C4⋊C4×C43—21no—
108C4⋊C4×C43—42no—
109C4⋊C4×C43—41no—
110C4⋊C4×C43—41no—
111C22⋊C4×C43—22no—
112C22⋊C4×C43—41no—
113C22⋊C4×C43—41no—
114C22⋊C4×C43—42no—
115C22⋊C4×C43SD1614yescomplemented
116C42×C43—42no—
117#10621884—82no—
118(C2×D43)⋊C4—48no—
119(C2×D43)⋊C4—41no—
120(C2×D43)⋊C4—41no—
121C4⋊Dic43—22no—
122C4⋊Dic43—21no—
123C4⋊Dic43—22no—
124C4⋊Dic43—21no—
125C4⋊Dic43—48no—
126C4⋊Dic43—41no—
127C4⋊Dic43—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#12011yescharacteristic
135C23×C43#12111yescharacteristic
136C23×C43#12711yescharacteristic
137C23×C43C2×SD1612yes—
138C23×C43AGL(1,ℤ/8)12yes—
139C22×D43—42no—
140C22×D43—42no—
141C22×D43—41no—
142C22×D43—42no—
143Q8×C43—42no—
144Q8×C43—42no—
145C2×C4×C43—22no—
146C2×C4×C43—22no—
147C2×C4×C43—22no—
148C2×C4×C43—24no—
149C2×C4×C43—41no—
150C2×C4×C43—41no—
151C2×C4×C43—41no—
152C2×C4×C43—42no—
153C2×C4×C43—42no—
154C2×C4×C43—41no—
155C2×C4×C43—41no—
156C2×C4×C43—81no—
157C2×C4×C43—81no—
158C2×C4×C43—81no—
159C2×C4×C43—81no—
160C2×C4×C43—81no—
161C2×C4×C43#12011yescharacteristic
162C2×C4×C43#12011yescharacteristic
163C2×C4×C43#12111yescharacteristic
164C2×C4×C43#12111yescharacteristic
165C22⋊D43—82no—
166C22⋊D43—81no—
167C22⋊D43—81no—
168C2×Dic43—42no—
169C2×Dic43—41no—
170C2×Dic43—42no—
171C2×Dic43—44no—
172C2×Dic43—82no—
173C2×Dic43—81no—
174C2×Dic43—81no—
175D172—42no—
176D172—81no—
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#39211yescharacteristic
189C22×C43#45012yes—
190C22×C43#46012yes—
191C22×C43#49211yescharacteristic
192C22×C43#49611yescharacteristic
193C2×D43—84no—
194C2×D43—81no—
195C2×D43—81no—
196C2×D43—81no—
197C4×C43—21no—
198C4×C43—21no—
199C4×C43—21no—
200C4×C43—44no—
201C4×C43—81no—
202C4×C43—81no—
203C4×C43—161no—
204C4×C43#45812yes—
205Dic43—82no—
206Dic43—81no—
207#3187—431no—
208C2×#483—431no—
209#2501—431no—
210#1996D4311yescharacteristic, complemented
211#1823—431no—
212#1519—431no—
213#1282—431no—
214C2×C43—22no—
215C2×C43—21no—
216C2×C43—22no—
217C2×C43—21no—
218C2×C43#200911yescharacteristic
219C2×C43#201411yescharacteristic
220C2×C43#202211yescharacteristic
221D43—161no—
222C2×#123—862no—
223C2×C22⋊Q8C2×D4311yescharacteristic
224C2×#121—431no—
225C22×C4⋊C4—431no—
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—862no—
238#390—861no—
239#384—861no—
240#380C2×D4311yescharacteristic
241C4×C22⋊C4C2×D4311yescharacteristic
242#342—22no—
243#340—862no—
244#327—862no—
245C43#996011yescharacteristic, complemented, Sylow
246C22×Q8D17211yescharacteristic
247C22×D4—431no—
248C22×D4—861no—
249C23×C4—861no—
250C23×C4C22×D4311yescharacteristic
251C2×SD16—862no—
252C22×C8—432no—
253C22×C8—861no—
254#123—1724no—
255#123—1724no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260#121—861no—
261#121—861no—
262#121—861no—
263#121—861no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—431no—
268C2×C4⋊C4—861no—
269C2×C4⋊C4—861no—
270C2×C4⋊C4—861no—
271C2×C4⋊C4—861no—
272C2×C4⋊C4—864no—
273C2×C4⋊C4—861no—
274C2×C4⋊C4D17211yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—864no—
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—21no—
292C24—861no—
293C24C2×C22⋊D4311yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—861no—
298C2×D4—861no—
299C2×D4—1722no—
300C2×D4—1721no—
301C2×D4—1721no—
302C2×D4—1722no—
303C2×D4—1722no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—862no—
310C22×C4—861no—
311C22×C4—1721no—
312C22×C4—1721no—
313C22×C4—1722no—
314C22×C4—1722no—
315C22×C4C2×D17211yescharacteristic
316C22×C4#1062189811yescharacteristic
317C22×C4D4×D4311yescharacteristic
318C22×C4D4×D4311yescharacteristic
319C22×C4D4×D4311yescharacteristic
320C22×C4D4×D4311yescharacteristic
321SD16—3442no—
322C2×C8—862no—
323C2×C8—862no—
324C2×C8—1722no—
325C2×C8—1722no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—862no—
332C4⋊C4—861no—
333C4⋊C4—862no—
334C4⋊C4—861no—
335C4⋊C4—1728no—
336C4⋊C4—1721no—
337C4⋊C4—1721no—
338C22⋊C4—22no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—1728no—
343C22⋊C4—1721no—
344C22⋊C4—1721no—
345C22⋊C4#1062188414yescomplemented
346C42—42no—
347C23—41no—
348C23—41no—
349C23—1722no—
350C23—1722no—
351C23—1722no—
352C23—1721no—
353C23#1306329311yescharacteristic
354C23#1306329911yescharacteristic
355C23#1306332111yescharacteristic
356C23C2×#1062188412yes—
357C23#1306335012yes—
358Q8—42no—
359Q8—42no—
360D4—3441no—
361D4—3442no—
362D4—3441no—
363D4—3441no—
364D4—1722no—
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—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—3442no—
382C2×C4—3441no—
383C2×C4—3441no—
384C2×C4—1724no—
385C2×C4—1722no—
386C2×C4—1721no—
387C2×C4—1722no—
388C2×C4#1306329511yescharacteristic
389C2×C4#1306329511yescharacteristic
390C2×C4#1306329811yescharacteristic
391C2×C4#1306329811yescharacteristic
392C8—1722no—
393C8—1722no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—3444no—
403C22—3441no—
404C22—3441no—
405C22—3441no—
406C22#1458280811yescharacteristic
407C22#1458304912yes—
408C22#1458309912yes—
409C22#1458327311yescharacteristic
410C22#1458328911yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—44no—
415C4—81no—
416C4—81no—
417C4—161no—
418C4—3442no—
419C4—3441no—
420C4#1458309012yes—
421C2—22no—
422C2—21no—
423C2—22no—
424C2—21no—
425C2—6881no—
426C2#1913565411yescharacteristic
427C2#1913568211yescharacteristic
428C2#1913572511yescharacteristic
subgroupG/H
centerC22#14582808
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC43×#1996C2
socleC22×C43#392
prime radicalC2×#13063299C22
hypercenter#1996D43
hypocenterC43#9960
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. C2×#14583227C2semidirect
4. #19134679C2semidirect
6. #19132443C2semidirect
7. C43×#1996C2semidirect
115. C22⋊C4×C43SD16semidirect
210. #1996D43semidirect
245. C43#9960semidirect
345. C22⋊C4#10621884semidirect

Maximal subgroups

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

subgroupindex
#191453482
C2×#145832272
#191390662
#191346792
#191331622
#191324432
C43×#19962
#996043