Finite Group Databasebeta

← #25070512

Subgroups — #25070512

Order 11008.

1002 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 187 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×#14583530C211yesmaximal, characteristic, complemented
2#19145352C212yesmaximal
3#19133602C214yesmaximal, complemented
4#19133534C214yesmaximal, complemented
5#19133166C212yesmaximal, complemented
6C2×#14582625C211yesmaximal, characteristic, complemented
7C43×#2618C211yesmaximal, characteristic
8#14583530—24no—
9#14583530—22no—
10#14583530C2214yes—
11C2×#13063324—22no—
12C2×#13063316—22no—
13C2×#13063304—21no—
14C2×#13063304C2212yes—
15C2×#13063284—21no—
16#14583229C2214yes—
17#14583215—22no—
18#14583126—24no—
19#14583050C2214yes—
20#14583037—22no—
21C2×C22⋊(C43⋊C8)C2212yes—
22#14582635C2218yes—
23#14582625C2214yes—
24C2×#123×C43—21no—
25C2×#121×C43C2211yescharacteristic
26C22×C4⋊C4×C43C2211yescharacteristic
27C43×#398C2212yes—
28C43×#396—22no—
29C43×#394C2212yes—
30C43×#386—21no—
31C43×#380C2211yescharacteristic
32C22×Dic86—21no—
33C22×Dic86D412yes—
34C2×#10621883—24no—
35C22×C43⋊C8—22no—
36#13063324—28no—
37#13063324—28no—
38#13063316—28no—
39#13063316—28no—
40#13063304—24no—
41#13063304—24no—
42#13063304—22no—
43#13063304—22no—
44#13063304D414yes—
45#13063304C2314yes—
46#13063284—24no—
47#13063284—24no—
48C2×C4⋊Dic43—21no—
49C2×C4⋊Dic43—21no—
50C2×C4⋊Dic43—21no—
51C2×C4⋊Dic43D412yes—
52C2×Dic43⋊C4—22no—
53C2×Dic43⋊C4—22no—
54C2×C22⋊Dic43—22no—
55C2×C4×Dic43—22no—
56#13063256—24no—
57#13063248—24no—
58#13063246—24no—
59C22⋊(C43⋊C8)—22no—
60C22⋊(C43⋊C8)C2314yes—
61C22×D4×C43—21no—
62C22×D4×C43C2311yescharacteristic
63C23×C4×C43D411yescharacteristic
64C23×C4×C43D411yescharacteristic
65C23×C4×C43C2311yescharacteristic
66#123×C43—44no—
67#121×C43—22no—
68#121×C43—22no—
69C2×C4⋊C4×C43—22no—
70C2×C4⋊C4×C43—22no—
71C2×C4⋊C4×C43—22no—
72C2×C4⋊C4×C43—21no—
73C2×C4⋊C4×C43D412yes—
74C2×C4⋊C4×C43C2311yescharacteristic
75C2×C4⋊C4×C43C2312yes—
76C2×C22⋊C4×C43—21no—
77C2×C22⋊C4×C43—21no—
78C2×C22⋊C4×C43—22no—
79C2×C22⋊C4×C43—22no—
80#95×C43—22no—
81#95×C43C2312yes—
82C24×C43—21no—
83C24×C43C2×D411yescharacteristic
84C2×D4×C43—22no—
85C2×D4×C43—41no—
86C2×D4×C43—41no—
87C2×D4×C43—42no—
88C2×D4×C43—42no—
89C2×D4×C43—81no—
90C22×C4×C43—21no—
91C22×C4×C43—21no—
92C22×C4×C43—21no—
93C22×C4×C43—21no—
94C22×C4×C43—22no—
95C22×C4×C43—21no—
96C22×C4×C43—22no—
97C22×C4×C43—81no—
98C22×C4×C43—81no—
99C22×C4×C43SD1614yescomplemented
100C22×C4×C43C2×D412yes—
101C22×C4×C43C2×D412yes—
102C22×C4×C43C2×D412yes—
103C22×C4×C43C2×D412yes—
104C22×C4×C43C2×D411yescharacteristic
105C22×C4×C43C2411yescharacteristic
106C22×Dic43—22no—
107C22×Dic43—22no—
108C2×Dic86—24no—
109C2×Dic86—22no—
110C2×Dic86—82no—
111C2×Dic86—82no—
112C2×Dic86C2×D414yes—
113C4⋊C4×C43—24no—
114C4⋊C4×C43—24no—
115C4⋊C4×C43—22no—
116C4⋊C4×C43—44no—
117C4⋊C4×C43—81no—
118C22⋊C4×C43—42no—
119C22⋊C4×C43—44no—
120C22⋊C4×C43—42no—
121C22⋊C4×C43—82no—
122C22⋊Dic43—24no—
123C22⋊Dic43—24no—
124#10621883—84no—
125C4⋊Dic43—24no—
126C4⋊Dic43—24no—
127C4⋊Dic43—24no—
128C4⋊Dic43—22no—
129C4⋊Dic43C2×D414yes—
130Dic43⋊C4—28no—
131Dic43⋊C4—24no—
132Dic43⋊C4—24no—
133C4×Dic43—28no—
134C2×C43⋊C8—24no—
135C2×C43⋊C8—82no—
136C23×C43—41no—
137C23×C43—41no—
138C23×C43—42no—
139C23×C43—42no—
140C23×C43—41no—
141C23×C43—41no—
142C23×C43—41no—
143C23×C43#12011yescharacteristic
144C23×C43#12011yescharacteristic
145C23×C43C2×SD1612yes—
146C23×C43#13712yes—
147C23×C43C22×D411yescharacteristic
148D4×C43—44no—
149D4×C43—82no—
150D4×C43—82no—
151C2×C4×C43—22no—
152C2×C4×C43—22no—
153C2×C4×C43—24no—
154C2×C4×C43—24no—
155C2×C4×C43—22no—
156C2×C4×C43—42no—
157C2×C4×C43—41no—
158C2×C4×C43—42no—
159C2×C4×C43—41no—
160C2×C4×C43—42no—
161C2×C4×C43—42no—
162C2×C4×C43—42no—
163C2×C4×C43—42no—
164C2×C4×C43—41no—
165C2×C4×C43—42no—
166C2×C4×C43—41no—
167C2×C4×C43—41no—
168C2×C4×C43—81no—
169C2×C4×C43#12012yes—
170C2×C4×C43C2×SD1614yes—
171C2×C4×C43C22×D412yes—
172C2×Dic43—24no—
173C2×Dic43—24no—
174C2×Dic43—44no—
175C2×Dic43—44no—
176Dic86—44no—
177Dic86—82no—
178Dic86—84no—
179C43⋊C8—44no—
180#18294—431nomaximal, Sylow
181C22×C43—21no—
182C22×C43—21no—
183C22×C43—21no—
184C22×C43—21no—
185C22×C43—21no—
186C22×C43—21no—
187C22×C43—42no—
188C22×C43—42no—
189C22×C43—82no—
190C22×C43—82no—
191C22×C43—81no—
192C22×C43—81no—
193C22×C43—81no—
194C22×C43#44812yes—
195C22×C43#44812yes—
196C22×C43C2×#12011yescharacteristic
197C22×C43C22×SD1611yescharacteristic
198C22×C43C2×#13711yescharacteristic
199C4×C43—24no—
200C4×C43—22no—
201C4×C43—22no—
202C4×C43—42no—
203C4×C43—82no—
204C4×C43—81no—
205Dic43—44no—
206Dic43—44no—
207C2×#544—431no—
208#3187—432no—
209#2618D4311yescharacteristic
210#1611—434no—
211#1599—434no—
212#1520—432no—
213C2×#324—431no—
214C2×C43—21no—
215C2×C43—21no—
216C2×C43—22no—
217C2×C43—22no—
218C2×C43—161no—
219C2×C43C2×#44811yescharacteristic
220C2×C43#302011yescharacteristic
221C2×C43#302611yescharacteristic
222#544—434no—
223#544—864no—
224#544—862no—
225C2×C4⋊Q8—862no—
226C2×#125—862no—
227C2×#123—21no—
228C2×C22⋊Q8—432no—
229C2×C22⋊Q8—861no—
230C2×#121C2×D4311yescharacteristic
231C2×#117—861no—
232C22×C4⋊C4C2×D4311yescharacteristic
233#483—434no—
234#481—862no—
235#465—864no—
236#450—434no—
237#448—862no—
238C2×C22⋊C8—432no—
239#398C2×D4312yes—
240#396—22no—
241#394C2×D4312yes—
242#386—21no—
243#380C2×D4311yescharacteristic
244#328—438no—
245#324—434no—
246C43#1829411yescharacteristic, complemented, Sylow
247C22×Q8—432no—
248C22×Q8—861no—
249C22×D4—21no—
250C22×D4C22×D4311yescharacteristic
251C23×C4C22⋊D4311yescharacteristic
252C23×C4C22⋊D4311yescharacteristic
253C23×C4C22×D4311yescharacteristic
254C2×SD16—864no—
255C22×C8—862no—
256C4⋊Q8—868no—
257C4⋊Q8—868no—
258#125—868no—
259#125—868no—
260#123—44no—
261C22⋊Q8—434no—
262C22⋊Q8—434no—
263C22⋊Q8—864no—
264C22⋊Q8—862no—
265C22⋊Q8—862no—
266C22⋊Q8—864no—
267#121—22no—
268#121—22no—
269#117—864no—
270#117—864no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—432no—
276C2×C4⋊C4—861no—
277C2×C4⋊C4—861no—
278C2×C4⋊C4—861no—
279C2×C4⋊C4—862no—
280C2×C4⋊C4—862no—
281C2×C4⋊C4C22⋊D4312yes—
282C2×C4⋊C4C22×D4311yescharacteristic
283C2×C4⋊C4C22×D4312yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—862no—
289C2×C42—862no—
290#106—864no—
291#103—864no—
292#102—864no—
293C22⋊C8—434no—
294C22⋊C8—862no—
295#95—22no—
296#95C22×D4312yes—
297C24—21no—
298C24C2×C22⋊D4311yescharacteristic
299C2×Q8—434no—
300C2×Q8—864no—
301C2×Q8—862no—
302C2×Q8—3442no—
303C2×Q8—3442no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—81no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—862no—
320C22×C4—862no—
321C22×C4#1062188314yescomplemented
322C22×C4D4×D4312yes—
323C22×C4D4×D4312yes—
324C22×C4C2×C22⋊D4312yes—
325C22×C4C2×C22⋊D4312yes—
326C22×C4C2×C22⋊D4311yescharacteristic
327C22×C4C23×D4311yescharacteristic
328SD16—3444no—
329C2×C8—864no—
330C2×C8—3442no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—81no—
336C4⋊C4—434no—
337C4⋊C4—864no—
338C4⋊C4—864no—
339C4⋊C4—864no—
340C4⋊C4—868no—
341C4⋊C4—862no—
342C4⋊C4—864no—
343C4⋊C4—864no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—864no—
349C22⋊C4—864no—
350C42—868no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#1306329311yescharacteristic
359C23#1306329311yescharacteristic
360C23C2×#1062188312yes—
361C23#1306335512yes—
362C23C22×C22⋊D4311yescharacteristic
363Q8—3442no—
364Q8—3444no—
365Q8—1724no—
366D4—44no—
367D4—82no—
368D4—82no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—864no—
388C2×C4—864no—
389C2×C4—1724no—
390C2×C4—1724no—
391C2×C4#1306329312yes—
392C2×C4C2×#1062188314yes—
393C2×C4C2×D4×D4312yes—
394C8—1724no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#1458303712yes—
409C22#1458303712yes—
410C22C2×#1306329311yescharacteristic
411C22C22×#1062188311yescharacteristic
412C22C2×#1306335511yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—1724no—
420C4—1724no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×#1458303711yescharacteristic
427C2#1914395811yescharacteristic
428C2#1914399111yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2618C2
socleC22×C43C2×#120
prime radicalC2×#121×C43C22
hypercenter#2618D43
hypocenterC43#18294
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#14583530C2semidirect
3. #19133602C2semidirect
4. #19133534C2semidirect
5. #19133166C2semidirect
6. C2×#14582625C2semidirect
99. C22×C4×C43SD16semidirect
246. C43#18294semidirect
321. C22×C4#10621883semidirect

Maximal subgroups

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

subgroupindex
C2×#145835302
#191453522
#191336022
#191335342
#191331662
C2×#145826252
C43×#26182
#1829443