Finite Group Databasebeta

← #25070509

Subgroups — #25070509

Order 11008.

1220 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1220 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×#14583480C211yesmaximal, characteristic
2#19145353C212yesmaximal, complemented
3#19140408C211yesmaximal, characteristic, complemented
4#19133634C214yesmaximal, complemented
5#19133566C214yesmaximal, complemented
6#19133164C212yesmaximal, complemented
7C2×C43×#324C211yesmaximal, characteristic, complemented
8C22×#10621898—21no—
9#14583480—24no—
10#14583480—22no—
11#14583480C2214yes—
12C2×#13063305—22no—
13C2×#13063304C2211yescharacteristic
14C2×#13063300—22no—
15C2×#13063299—21no—
16C2×#13063299C2212yes—
17#14583230—22no—
18#14583218C2214yescomplemented
19#14583128—24no—
20#14583052—22no—
21#14583040C2214yescomplemented
22#14582838C2212yescomplemented
23#14582832—22no—
24#14582819C2212yescomplemented
25#14582772—21no—
26#14582762—21no—
27#14582759C2211yescharacteristic
28#14582634C2218yescomplemented
29C22×C4⋊C4×C43C2211yescharacteristic
30C2×C22⋊C8×C43C2212yes—
31C43×#324C2214yescomplemented
32C2×#10621898—24no—
33C2×#10621898—24no—
34C2×#10621898—44no—
35C2×#10621898—44no—
36C22×Dic86—21no—
37C22×Dic86C2311yescharacteristic
38C22×D172—21no—
39C22×D172—21no—
40C22×D172D412yes—
41C22×C22⋊D43—22no—
42C22×C4×D43—22no—
43C2×#10621873—24no—
44#13063305—28no—
45#13063305—28no—
46#13063304—22no—
47#13063304—22no—
48#13063300—28no—
49#13063300—28no—
50#13063299—24no—
51#13063299—24no—
52#13063299—22no—
53#13063299—22no—
54#13063299D414yes—
55#13063299C2314yes—
56C2×C4⋊Dic43—21no—
57C2×C4⋊Dic43—21no—
58C2×C4⋊Dic43D412yes—
59C2×C4⋊Dic43C2311yescharacteristic
60C2×Dic43⋊C4—21no—
61C2×Dic43⋊C4—22no—
62C2×C22⋊Dic43—21no—
63C2×C22⋊Dic43—21no—
64C2×(C2×D43)⋊C4—22no—
65C2×(C2×D43)⋊C4—22no—
66#13063257—24no—
67#13063250—24no—
68#13063247—24no—
69C43⋊#95—22no—
70C43⋊#95—22no—
71C43⋊#95—22no—
72C43⋊#95C2312yes—
73C23×C4×C43D411yescharacteristic
74C23×C4×C43D411yescharacteristic
75C23×C4×C43C2311yescharacteristic
76C22×C8×C43—22no—
77C2×C4⋊C4×C43—22no—
78C2×C4⋊C4×C43—22no—
79C2×C4⋊C4×C43D412yes—
80C2×C4⋊C4×C43C2312yes—
81C22⋊C8×C43—22no—
82C22⋊C8×C43C2314yes—
83C24×C43C2×D411yescharacteristic
84C23×D43—22no—
85C22×C4×C43—21no—
86C22×C4×C43—21no—
87C22×C4×C43—21no—
88C22×C4×C43—21no—
89C22×C4×C43—22no—
90C22×C4×C43—81no—
91C22×C4×C43—81no—
92C22×C4×C43SD1614yescomplemented
93C22×C4×C43C2×D412yes—
94C22×C4×C43C2×D412yes—
95C22×C4×C43C2×D412yes—
96C22×C4×C43C2×D412yes—
97C22×C4×C43C2×D411yescharacteristic
98C22×C4×C43C2411yescharacteristic
99C2×C22⋊D43—24no—
100C2×C22⋊D43—24no—
101C2×C22⋊D43—44no—
102C2×C22⋊D43—44no—
103C22×Dic43—22no—
104C22×Dic43—21no—
105C22×Dic43—22no—
106C22×Dic43—21no—
107#10621898—48no—
108#10621898—416no—
109#10621898—48no—
110C2×D172—24no—
111C2×D172—24no—
112C2×D172—22no—
113C2×D172—44no—
114C2×D172—82no—
115C2×D172—82no—
116C2×D172C2×D414yes—
117C2×C4×D43—28no—
118C2×C4×D43—48no—
119C2×Dic86—24no—
120C2×Dic86—22no—
121C2×Dic86—44no—
122C2×Dic86—81no—
123C2×C8×C43—24no—
124C2×C8×C43—82no—
125C4⋊C4×C43—24no—
126C4⋊C4×C43—24no—
127C22⋊Dic43—41no—
128C22⋊Dic43—41no—
129C22⋊Dic43—42no—
130(C2×D43)⋊C4—28no—
131(C2×D43)⋊C4—24no—
132(C2×D43)⋊C4—24no—
133C4⋊Dic43—24no—
134C4⋊Dic43—22no—
135C4⋊Dic43—22no—
136C4⋊Dic43—81no—
137C4⋊Dic43C2×D414yes—
138Dic43⋊C4—42no—
139Dic43⋊C4—82no—
140#10621873—84no—
141C23×C43—41no—
142C23×C43—41no—
143C23×C43#12011yescharacteristic
144C23×C43#12011yescharacteristic
145C23×C43C2×SD1612yes—
146C23×C43AGL(1,ℤ/8)12yes—
147C23×C43C22×D411yescharacteristic
148C22×D43—24no—
149C22×D43—44no—
150C22×D43—44no—
151C22×D43—44no—
152C2×C4×C43—22no—
153C2×C4×C43—22no—
154C2×C4×C43—24no—
155C2×C4×C43—24no—
156C2×C4×C43—22no—
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—81no—
164C2×C4×C43#12012yes—
165C2×C4×C43C2×SD1614yes—
166C2×C4×C43C22×D412yes—
167C22⋊D43—44no—
168C22⋊D43—44no—
169C22⋊D43—48no—
170C2×Dic43—24no—
171C2×Dic43—44no—
172C2×Dic43—42no—
173C2×Dic43—42no—
174C2×Dic43—42no—
175C2×Dic43—42no—
176C2×Dic43—41no—
177C2×Dic43—41no—
178C2×Dic43—42no—
179D172—44no—
180D172—48no—
181D172—82no—
182D172—84no—
183C4×D43—416no—
184Dic86—48no—
185Dic86—44no—
186C8×C43—44no—
187#18293—431nomaximal, Sylow
188C22×C43—21no—
189C22×C43—21no—
190C22×C43—21no—
191C22×C43—21no—
192C22×C43—21no—
193C22×C43—21no—
194C22×C43—42no—
195C22×C43—42no—
196C22×C43#45012yes—
197C22×C43#45012yes—
198C22×C43C2×#12011yescharacteristic
199C22×C43C22×SD1611yescharacteristic
200C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
201C2×D43—44no—
202C2×D43—44no—
203C2×D43—44no—
204C2×D43—88no—
205C4×C43—24no—
206C4×C43—22no—
207C4×C43—22no—
208C4×C43—42no—
209Dic43—44no—
210Dic43—82no—
211Dic43—81no—
212Dic43—81no—
213C2×#537—431no—
214#3187—432no—
215#2627—431no—
216#1616—434no—
217#1604—434no—
218#1519—432no—
219C2×#324D4311yescharacteristic, complemented
220C2×C43—21no—
221C2×C43—22no—
222C2×C43—21no—
223C2×C43—22no—
224C2×C43C2×#45011yescharacteristic
225C2×C43#302011yescharacteristic
226C2×C43#302611yescharacteristic
227D43—84no—
228C22×#41—861no—
229#537—434no—
230#537—864no—
231#537—862no—
232C2×C22⋊Q8—431no—
233C2×C22⋊Q8—862no—
234C2×#121—432no—
235C2×#121—862no—
236C2×#121—861no—
237C22×C4⋊C4C2×D4311yescharacteristic
238#483—862no—
239#481—434no—
240#465—864no—
241#450—862no—
242#448—434no—
243C2×C22⋊C8C2×D4312yes—
244#400—432no—
245#398—862no—
246#395—432no—
247#385—861no—
248#381—861no—
249#380—431no—
250#327—438no—
251#324C2×D4314yescomplemented
252C43#1829311yescharacteristic, complemented, Sylow
253C2×#41—864no—
254C2×#41—864no—
255C2×#41—1724no—
256C2×#41—1724no—
257C22×Q8—431no—
258C22×Q8—861no—
259C22×D4—432no—
260C22×D4—861no—
261C22×D4—861no—
262C22×D4—862no—
263C23×C4—862no—
264C23×C4D17211yescharacteristic
265C23×C4D17211yescharacteristic
266C23×C4C22×D4311yescharacteristic
267C2×SD16—864no—
268C22×C8—22no—
269C22⋊Q8—868no—
270C22⋊Q8—868no—
271C22⋊Q8—862no—
272C22⋊Q8—862no—
273#121—434no—
274#121—434no—
275#121—868no—
276#121—868no—
277#121—864no—
278#121—862no—
279#121—864no—
280#121—862no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—432no—
284C2×C4⋊C4—431no—
285C2×C4⋊C4—861no—
286C2×C4⋊C4—862no—
287C2×C4⋊C4—861no—
288C2×C4⋊C4—861no—
289C2×C4⋊C4D17212yes—
290C2×C4⋊C4C22×D4312yes—
291C2×C22⋊C4—862no—
292C2×C22⋊C4—862no—
293C2×C22⋊C4—861no—
294C2×C22⋊C4—861no—
295#106—864no—
296#103—864no—
297#102—864no—
298C22⋊C8—22no—
299C22⋊C8C22×D4314yes—
300#95—432no—
301#95—862no—
302#95—862no—
303#95—862no—
304C24—862no—
305C24C2×D17211yescharacteristic
306#41—17216no—
307#41—1728no—
308#41—1728no—
309C2×Q8—864no—
310C2×Q8—862no—
311C2×Q8—3441no—
312C2×Q8—1724no—
313C2×D4—434no—
314C2×D4—864no—
315C2×D4—864no—
316C2×D4—864no—
317C2×D4—864no—
318C2×D4—862no—
319C2×D4—3442no—
320C2×D4—3442no—
321C2×D4—1724no—
322C2×D4—1724no—
323C2×D4—1724no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—868no—
332C22×C4—862no—
333C22×C4—862no—
334C22×C4—861no—
335C22×C4—861no—
336C22×C4—1728no—
337C22×C4#1062187314yescomplemented
338C22×C4C2×D17212yes—
339C22×C4C2×D17212yes—
340C22×C4C2×D17211yescharacteristic
341C22×C4D4×D4312yes—
342C22×C4D4×D4312yes—
343C22×C4C23×D4311yescharacteristic
344SD16—3444no—
345C2×C8—24no—
346C2×C8—82no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—434no—
350C4⋊C4—864no—
351C4⋊C4—862no—
352C4⋊C4—862no—
353C4⋊C4—3442no—
354C4⋊C4—3441no—
355C4⋊C4—1722no—
356C22⋊C4—868no—
357C22⋊C4—864no—
358C22⋊C4—864no—
359C22⋊C4—1721no—
360C22⋊C4—1721no—
361C22⋊C4—1722no—
362C23—41no—
363C23—41no—
364C23—864no—
365C23—1724no—
366C23—1724no—
367C23—1724no—
368C23#1306329511yescharacteristic
369C23#1306329511yescharacteristic
370C23C2×#1062187312yes—
371C23#1306335312yes—
372C23C22×D17211yescharacteristic
373Q8—1724no—
374Q8—1728no—
375D4—3442no—
376D4—3444no—
377D4—1724no—
378D4—1728no—
379D4—1728no—
380D4—1724no—
381D4—1724no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—864no—
395C2×C4—17216no—
396C2×C4—1722no—
397C2×C4—1722no—
398C2×C4—1722no—
399C2×C4—1722no—
400C2×C4—1721no—
401C2×C4—1721no—
402C2×C4—1722no—
403C2×C4—1724no—
404C2×C4#1306329512yes—
405C2×C4C2×#1062187314yes—
406C2×C4C2×D4×D4312yes—
407C8—44no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—3448no—
417C22—1724no—
418C22—1724no—
419C22—1724no—
420C22#1458305212yes—
421C22#1458305212yes—
422C22C2×#1306329511yescharacteristic
423C22C22×#1062187311yescharacteristic
424C22C2×#1306335311yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—3442no—
430C4—3441no—
431C4—3441no—
432C4—1724no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—3444no—
438C2C2×#1458305211yescharacteristic
439C2#1914396211yescharacteristic
440C2#1914399411yescharacteristic
subgroupG/H
centerC22C2×#13063295
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#324C2
socleC22×C43C2×#120
prime radicalC2×#14583480C2
hypercenterC2×#324D43
hypocenterC43#18293
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #19145353C2semidirect
3. #19140408C2semidirect
4. #19133634C2semidirect
5. #19133566C2semidirect
6. #19133164C2semidirect
7. C2×C43×#324C2semidirect
18. #14583218C22semidirect
21. #14583040C22semidirect
22. #14582838C22semidirect
24. #14582819C22semidirect
28. #14582634C22semidirect
31. C43×#324C22semidirect
92. C22×C4×C43SD16semidirect
219. C2×#324D43semidirect
251. #324C2×D43semidirect
252. C43#18293semidirect
337. C22×C4#10621873semidirect

Maximal subgroups

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

subgroupindex
C2×#145834802
#191453532
#191404082
#191336342
#191335662
#191331642
C2×C43×#3242
#1829343