Finite Group Databasebeta

← #24988201

Subgroups — #24988201

Order 11008.

788 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14583589C211yesmaximal, characteristic
2#19134692C212yesmaximal, complemented
3C4×#13063240C211yesmaximal, characteristic
4#19133116C212yesmaximal, complemented
5C43×#1465C211yesmaximal, characteristic
6#14583589—24no—
7#14583589—24no—
8C2×#13063318—21no—
9C2×#13063317C411yescharacteristic, complemented
10C2×#13063315—21no—
11C2×#13063306—21no—
12C2×#13063304C411yescharacteristic, complemented
13C2×#13063304C2211yescharacteristic
14C2×C4×Dic86C411yescharacteristic, complemented
15C2×#13063284C2211yescharacteristic
16C2×#13063281—21no—
17C22×C4⋊Dic43C411yescharacteristic, complemented
18C2×#13063240—21no—
19C4×C22⋊Dic43C2212yes—
20#14582666—21no—
21#14582660—24no—
22#14582637—24no—
23C2×C4×D4×C43C2211yescharacteristic
24C43×#407C2212yes—
25C43×#324—21no—
26C43×#323—21no—
27C22×Dic86C2×C411yescharacteristic
28C23×Dic43—21no—
29#13063318—28no—
30#13063317—22no—
31#13063317—22no—
32#13063315—28no—
33#13063306—28no—
34#13063304—22no—
35#13063304—22no—
36#13063304—22no—
37#13063304D414yes—
38C4×Dic86—22no—
39C4×Dic86—22no—
40#13063284—22no—
41#13063284D414yes—
42#13063281—28no—
43C2×C4⋊Dic43—21no—
44C2×C4⋊Dic43—22no—
45C2×C4⋊Dic43—22no—
46C2×C4⋊Dic43C2×C411yescharacteristic
47C2×C4⋊Dic43C2×C411yescharacteristic
48C2×C4⋊Dic43C2×C411yescharacteristic
49C2×Dic43⋊C4—21no—
50C2×Dic43⋊C4—21no—
51C2×Dic43⋊C4—21no—
52C2×Dic43⋊C4—21no—
53C2×Dic43⋊C4—21no—
54C2×C22⋊Dic43—21no—
55C2×C22⋊Dic43—21no—
56C2×C22⋊Dic43—21no—
57C2×C22⋊Dic43—22no—
58C2×C4×Dic43—21no—
59C2×C4×Dic43—21no—
60C2×C4×Dic43—22no—
61#13063240—24no—
62C43⋊#95—22no—
63C22×D4×C43D411yescharacteristic
64C23×C4×C43C2×C411yescharacteristic
65C23×C4×C43C2311yescharacteristic
66C2×M16×C43—22no—
67C4×D4×C43—22no—
68C4×D4×C43—22no—
69C2×C4⋊C4×C43D411yescharacteristic
70C2×C22⋊C4×C43D411yescharacteristic
71C2×C22⋊C4×C43D411yescharacteristic
72C2×C42×C43C2×C411yescharacteristic
73C22⋊C8×C43—22no—
74C22⋊C8×C43—22no—
75C24×C43C22⋊C411yescharacteristic
76C24×C43C2×D411yescharacteristic
77C2×D4×C43—22no—
78C2×D4×C43—22no—
79C2×D4×C43—41no—
80C22×C4×C43—21no—
81C22×C4×C43—21no—
82C22×C4×C43—42no—
83C22×C4×C43—42no—
84C22×C4×C43C22⋊C411yescharacteristic
85C22×C4×C43C22⋊C411yescharacteristic
86C22×C4×C43C22⋊C411yescharacteristic
87C22×C4×C43C22×C411yescharacteristic
88C22×C4×C43C2×D411yescharacteristic
89C22×C4×C43C2×D412yes—
90C22×C4×C43C2×D412yes—
91C22×Dic43—24no—
92C22×Dic43—21no—
93C22×Dic43—21no—
94C22×Dic43—21no—
95C22×Dic43—21no—
96C22×Dic43—42no—
97C22×Dic43—42no—
98C22×Dic43—42no—
99C2×Dic86—21no—
100C2×Dic86—42no—
101C2×Dic86C22⋊C412yes—
102M16×C43—44no—
103C2×C8×C43—42no—
104C4⋊C4×C43—22no—
105C22⋊C4×C43—22no—
106C22⋊C4×C43—22no—
107C42×C43—21no—
108C42×C43—21no—
109C22⋊Dic43—24no—
110C22⋊Dic43—24no—
111C22⋊Dic43—24no—
112C22⋊Dic43—24no—
113C22⋊Dic43—42no—
114C4⋊Dic43—24no—
115C4⋊Dic43—21no—
116C4⋊Dic43—21no—
117C4⋊Dic43—21no—
118C4⋊Dic43C22⋊C412yes—
119C4⋊Dic43C22⋊C412yes—
120C4⋊Dic43C22⋊C412yes—
121Dic43⋊C4—24no—
122Dic43⋊C4—24no—
123Dic43⋊C4—24no—
124Dic43⋊C4—24no—
125Dic43⋊C4—24no—
126C4×Dic43—24no—
127C4×Dic43—24no—
128C4×Dic43—44no—
129C23×C43—21no—
130C23×C43—21no—
131C23×C43—21no—
132C23×C43—21no—
133C23×C43—41no—
134C23×C43—41no—
135C23×C43—41no—
136C23×C43—41no—
137C23×C43#10412yes—
138C23×C43C2×C22⋊C411yescharacteristic
139C23×C43#12012yes—
140D4×C43—44no—
141C2×C4×C43—24no—
142C2×C4×C43—22no—
143C2×C4×C43—21no—
144C2×C4×C43—21no—
145C2×C4×C43—21no—
146C2×C4×C43—21no—
147C2×C4×C43—21no—
148C2×C4×C43—42no—
149C2×C4×C43—41no—
150C2×C4×C43—42no—
151C2×C4×C43—42no—
152C2×C4×C43—41no—
153C2×C4×C43—41no—
154C2×C4×C43C2×C22⋊C411yescharacteristic
155C2×C4×C43C2×C22⋊C411yescharacteristic
156C2×C4×C43#12012yes—
157C2×Dic43—22no—
158C2×Dic43—22no—
159C2×Dic43—22no—
160C2×Dic43—22no—
161C2×Dic43—42no—
162C2×Dic43—42no—
163C2×Dic43—48no—
164C2×Dic43—41no—
165C2×Dic43—41no—
166C2×Dic43—41no—
167C2×Dic43—41no—
168C2×Dic43—44no—
169C2×Dic43—82no—
170C2×Dic43—82no—
171Dic86—42no—
172Dic86—42no—
173C8×C43—82no—
174#8363—431nomaximal, Sylow
175C22×C43—21no—
176C22×C43—21no—
177C22×C43—22no—
178C22×C43—22no—
179C22×C43—22no—
180C22×C43—22no—
181C22×C43—42no—
182C22×C43—42no—
183C22×C43—41no—
184C22×C43—41no—
185C22×C43—81no—
186C22×C43—81no—
187C22×C43#37911yescharacteristic
188C22×C43C2×#10411yescharacteristic
189C22×C43#42111yescharacteristic
190C4×C43—22no—
191C4×C43—21no—
192C4×C43—41no—
193C4×C43—41no—
194C4×C43—41no—
195C4×C43#41012yes—
196Dic43—42no—
197Dic43—42no—
198Dic43—42no—
199Dic43—42no—
200Dic43—84no—
201C2×#552—431no—
202#1826—432no—
203C4×#99—431no—
204#1508—432no—
205#1465D4311yescharacteristic
206C2×C43—21no—
207C2×C43—21no—
208C2×C43—42no—
209C2×C43—42no—
210C2×C43#178211yescharacteristic
211C2×C43#178911yescharacteristic
212C2×C43#179611yescharacteristic
213#552—864no—
214#552—864no—
215C2×#126—861no—
216C2×#125—431no—
217C2×#125—861no—
218C2×#123—861no—
219C2×C22⋊Q8—431no—
220C2×C22⋊Q8—431no—
221C2×C4×Q8—431no—
222C2×C4×D4C2×D4311yescharacteristic
223C2×#117—431no—
224C2×#117—861no—
225C22×C4⋊C4—431no—
226C2×#99—861no—
227#407C2×D4312yes—
228C4×C22⋊C4—432no—
229#342—861no—
230#339—864no—
231#328—864no—
232#324—21no—
233#323—21no—
234C43#836311yescharacteristic, complemented, Sylow
235C22×Q8—431no—
236C22×D4D17211yescharacteristic
237C23×C4—861no—
238C23×C4C4×D4311yescharacteristic
239C23×C4C22×D4311yescharacteristic
240C2×M16—22no—
241#126—868no—
242#125—862no—
243#125—862no—
244#125—868no—
245#123—868no—
246C22⋊Q8—434no—
247C22⋊Q8—862no—
248C22⋊Q8—862no—
249C22⋊Q8—862no—
250C4×Q8—862no—
251C4×Q8—862no—
252C4×D4—22no—
253C4×D4—22no—
254#117—434no—
255#117—868no—
256#117—862no—
257C2×C4⋊C4—431no—
258C2×C4⋊C4—431no—
259C2×C4⋊C4—431no—
260C2×C4⋊C4—862no—
261C2×C4⋊C4—862no—
262C2×C4⋊C4—861no—
263C2×C4⋊C4—861no—
264C2×C4⋊C4—861no—
265C2×C4⋊C4—861no—
266C2×C4⋊C4—861no—
267C2×C4⋊C4—861no—
268C2×C4⋊C4C22⋊D4311yescharacteristic
269C2×C22⋊C4—861no—
270C2×C22⋊C4—861no—
271C2×C22⋊C4—861no—
272C2×C22⋊C4—862no—
273C2×C22⋊C4D17211yescharacteristic
274C2×C22⋊C4C22⋊D4311yescharacteristic
275C2×C42—861no—
276C2×C42—861no—
277C2×C42—862no—
278C2×C42C4×D4311yescharacteristic
279#99—864no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282#95—862no—
283C24(C2×D43)⋊C411yescharacteristic
284C24C2×D17211yescharacteristic
285C2×Q8—432no—
286C2×Q8—861no—
287C2×Q8—1722no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—41no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—864no—
296C22×C4—861no—
297C22×C4—861no—
298C22×C4—861no—
299C22×C4—861no—
300C22×C4—1722no—
301C22×C4—1722no—
302C22×C4—1722no—
303C22×C4(C2×D43)⋊C411yescharacteristic
304C22×C4(C2×D43)⋊C411yescharacteristic
305C22×C4(C2×D43)⋊C411yescharacteristic
306C22×C4C2×C4×D4311yescharacteristic
307C22×C4D4×D4312yes—
308C22×C4D4×D4312yes—
309C22×C4C2×C22⋊D4311yescharacteristic
310M16—44no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—432no—
314C4⋊C4—432no—
315C4⋊C4—432no—
316C4⋊C4—864no—
317C4⋊C4—864no—
318C4⋊C4—864no—
319C4⋊C4—864no—
320C4⋊C4—864no—
321C4⋊C4—861no—
322C4⋊C4—861no—
323C4⋊C4—861no—
324C4⋊C4—864no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—864no—
328C22⋊C4—864no—
329C22⋊C4—864no—
330C22⋊C4—864no—
331C22⋊C4—1722no—
332C42—21no—
333C42—21no—
334C42—864no—
335C42—864no—
336C42—1724no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23#1306325112yes—
346C23C2×(C2×D43)⋊C411yescharacteristic
347C23#1306329512yes—
348Q8—1722no—
349Q8—1722no—
350D4—44no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—862no—
365C2×C4—862no—
366C2×C4—862no—
367C2×C4—862no—
368C2×C4—3442no—
369C2×C4—3442no—
370C2×C4—1724no—
371C2×C4—1728no—
372C2×C4—1722no—
373C2×C4—1722no—
374C2×C4—1721no—
375C2×C4—1721no—
376C2×C4—1721no—
377C2×C4—1721no—
378C2×C4C22⋊C4×D4311yescharacteristic
379C2×C4C22⋊C4×D4311yescharacteristic
380C2×C4#1306329312yes—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22#1458275411yescharacteristic
395C22C2×#1306325111yescharacteristic
396C22#1458291511yescharacteristic
397C4—22no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—3444no—
403C4—1722no—
404C4—1722no—
405C4—1722no—
406C4—1722no—
407C4#1458287112yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—42no—
412C2#1913451111yescharacteristic
413C2#1913453811yescharacteristic
414C2#1913457111yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1465C2
socleC22×C43#379
prime radicalC22×D4×C43D4
cosocleC23×C4×C43C23
hypercenter#1465D43
hypocenterC43#8363
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #19134692C2semidirect
4. #19133116C2semidirect
9. C2×#13063317C4semidirect
12. C2×#13063304C4semidirect
14. C2×C4×Dic86C4semidirect
17. C22×C4⋊Dic43C4semidirect
234. C43#8363semidirect

Maximal subgroups

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

subgroupindex
C2×#145835892
#191346922
C4×#130632402
#191331162
C43×#14652
#836343