Finite Group Databasebeta

← #24988228

Subgroups — #24988228

Order 11008.

998 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14583558C211yesmaximal, characteristic
2#19133192C212yesmaximal, complemented
3#19133043C212yesmaximal, complemented
4#19132936C211yesmaximal, characteristic, complemented
5C4×#100×C43C211yesmaximal, characteristic, complemented
6C22×#10621898C411yescharacteristic
7C23×Dic86C411yescharacteristic
8#14583558—24no—
9#14583558—24no—
10C2×#13063313C2211yescharacteristic
11C2×#13063311—21no—
12C2×#13063304C411yescharacteristic
13C2×#13063303—21no—
14C2×#13063299C411yescharacteristic
15C2×#13063297—21no—
16C2×C4×Dic86C2211yescharacteristic
17C2×(C22⋊D43)⋊C4—21no—
18#14582956C2212yes—
19#14582638—24no—
20#14582631—24no—
21#14582623—21no—
22#14582623—21no—
23C2×C4×D4×C43C2211yescharacteristic
24C2×#100×C43—21no—
25C4×M16×C43C2212yes—
26C43×#343—21no—
27C2×#10621898—22no—
28C2×#10621898—22no—
29C2×#10621898—44no—
30C22×Dic86—21no—
31C22×Dic86—22no—
32C22×Dic86—22no—
33C22×Dic86—44no—
34C22×Dic86C2×C411yescharacteristic
35C22×Dic86C2×C411yescharacteristic
36C22×D172C2×C411yescharacteristic
37C22×C22⋊D43—21no—
38C22×C22⋊D43—21no—
39C23×Dic43—21no—
40C22×C4×D43—21no—
41#13063313—22no—
42#13063313D414yes—
43#13063311—28no—
44#13063304—22no—
45#13063304—22no—
46#13063303—28no—
47#13063299—22no—
48#13063299—22no—
49#13063297—28no—
50C4×Dic86—22no—
51C4×Dic86D414yes—
52(C22⋊D43)⋊C4—28no—
53C2×C4⋊Dic43C2×C411yescharacteristic
54C2×Dic43⋊C4—21no—
55C2×Dic43⋊C4—21no—
56C2×C22⋊Dic43—21no—
57C2×(C2×D43)⋊C4—21no—
58C2×(C2×D43)⋊C4—21no—
59C2×(C2×D43)⋊C4—21no—
60C2×C4×Dic43—21no—
61C4⋊(C43⋊C8)—22no—
62C22⋊(C43⋊C8)—22no—
63#13063236—22no—
64C22×D4×C43D411yescharacteristic
65C23×C4×C43C2×C411yescharacteristic
66C23×C4×C43C2×C411yescharacteristic
67C2×M16×C43—22no—
68C4×D4×C43—22no—
69C4×D4×C43—22no—
70C2×C4⋊C4×C43D411yescharacteristic
71C2×C22⋊C4×C43D411yescharacteristic
72C2×C22⋊C4×C43D411yescharacteristic
73C2×C42×C43C2311yescharacteristic
74#100×C43—24no—
75#97×C43—22no—
76C4×C8×C43—22no—
77C24×C43C22⋊C411yescharacteristic
78C24×C43C22⋊C411yescharacteristic
79C23×D43—21no—
80C2×D4×C43—22no—
81C2×D4×C43—22no—
82C2×D4×C43—41no—
83C22×C4×C43—21no—
84C22×C4×C43—21no—
85C22×C4×C43—21no—
86C22×C4×C43—21no—
87C22×C4×C43—42no—
88C22×C4×C43—42no—
89C22×C4×C43C22⋊C411yescharacteristic
90C22×C4×C43C22⋊C411yescharacteristic
91C22×C4×C43C22×C411yescharacteristic
92C22×C4×C43C2×D411yescharacteristic
93C22×C4×C43C2×D411yescharacteristic
94C2×C22⋊D43—24no—
95C2×C22⋊D43—24no—
96C2×C22⋊D43—42no—
97C2×C22⋊D43—44no—
98C2×C22⋊D43—42no—
99C22×Dic43—24no—
100C22×Dic43—21no—
101C22×Dic43—21no—
102C22×Dic43—21no—
103C22×Dic43—44no—
104#10621898—48no—
105#10621898—44no—
106#10621898—44no—
107C2×D172—21no—
108C2×D172—42no—
109C2×D172C22⋊C412yes—
110C2×C4×D43—24no—
111C2×C4×D43—44no—
112C2×Dic86—21no—
113C2×Dic86—24no—
114C2×Dic86—24no—
115C2×Dic86—21no—
116C2×Dic86—48no—
117C2×Dic86—44no—
118C2×Dic86—44no—
119C2×Dic86—44no—
120C2×Dic86—42no—
121C2×Dic86C22⋊C412yes—
122C2×Dic86C22⋊C412yes—
123M16×C43—24no—
124M16×C43—42no—
125C2×C8×C43—42no—
126C4⋊C4×C43—22no—
127C22⋊C4×C43—22no—
128C22⋊C4×C43—22no—
129C42×C43C2×D412yes—
130C42×C43C2×D412yes—
131C22⋊Dic43—24no—
132(C2×D43)⋊C4—24no—
133(C2×D43)⋊C4—24no—
134(C2×D43)⋊C4—24no—
135C4⋊Dic43—21no—
136C4⋊Dic43C22⋊C412yes—
137Dic43⋊C4—24no—
138Dic43⋊C4—24no—
139C4×Dic43—24no—
140C2×C43⋊C8—42no—
141C23×C43—21no—
142C23×C43—21no—
143C23×C43—21no—
144C23×C43—21no—
145C23×C43—21no—
146C23×C43—41no—
147C23×C43—41no—
148C23×C43—41no—
149C23×C43—41no—
150C23×C43#10412yes—
151C23×C43C2×C22⋊C411yescharacteristic
152C22×D43—22no—
153C22×D43—42no—
154C22×D43—42no—
155C22×D43—41no—
156C22×D43—41no—
157D4×C43—44no—
158C2×C4×C43—21no—
159C2×C4×C43—21no—
160C2×C4×C43—22no—
161C2×C4×C43—21no—
162C2×C4×C43—21no—
163C2×C4×C43—41no—
164C2×C4×C43—42no—
165C2×C4×C43—42no—
166C2×C4×C43—42no—
167C2×C4×C43—42no—
168C2×C4×C43—41no—
169C2×C4×C43—41no—
170C2×C4×C43C2×C22⋊C411yescharacteristic
171C2×C4×C43C2×C22⋊C411yescharacteristic
172C2×C4×C43#12012yes—
173C2×C4×C43#12012yes—
174C22⋊D43—48no—
175C22⋊D43—48no—
176C2×Dic43—24no—
177C2×Dic43—22no—
178C2×Dic43—22no—
179C2×Dic43—22no—
180C2×Dic43—48no—
181C2×Dic43—42no—
182C2×Dic43—41no—
183C2×Dic43—41no—
184D172—42no—
185D172—42no—
186C4×D43—48no—
187Dic86—24no—
188Dic86—22no—
189Dic86—48no—
190Dic86—42no—
191Dic86—42no—
192Dic86#10014yescomplemented
193C8×C43—44no—
194C43⋊C8—82no—
195#8369—431nomaximal, Sylow
196C22×C43—21no—
197C22×C43—21no—
198C22×C43—22no—
199C22×C43—42no—
200C22×C43—42no—
201C22×C43—41no—
202C22×C43—41no—
203C22×C43—41no—
204C22×C43—41no—
205C22×C43—41no—
206C22×C43—81no—
207C22×C43—81no—
208C22×C43#37911yescharacteristic
209C22×C43C2×#10411yescharacteristic
210C22×C43#42111yescharacteristic
211C2×D43—42no—
212C2×D43—42no—
213C2×D43—42no—
214C2×D43—82no—
215C2×D43—82no—
216C4×C43—22no—
217C4×C43—21no—
218C4×C43—22no—
219C4×C43—41no—
220C4×C43—41no—
221C4×C43C2×#10012yes—
222Dic43—24no—
223Dic43—42no—
224Dic43—42no—
225C2×#548—431no—
226C4×#100D4311yescharacteristic, complemented
227#1527—432no—
228#1491—432no—
229#1466—431no—
230C2×C43—21no—
231C2×C43—21no—
232C2×C43—42no—
233C2×C43—81no—
234C2×C43#178311yescharacteristic
235C2×C43#178911yescharacteristic
236C2×C43#179611yescharacteristic
237D43—82no—
238C22×#41—431no—
239C23×Q8—431no—
240#548—864no—
241#548—864no—
242C2×#124—431no—
243C2×#124—861no—
244C2×C22⋊Q8—431no—
245C2×C22⋊Q8—861no—
246C2×#121—431no—
247C2×#121—861no—
248C2×C4×Q8—431no—
249C2×C4×D4—861no—
250C2×C4×D4C2×D4311yescharacteristic
251#432—432no—
252C2×#100—21no—
253C4×M16C2×D4312yes—
254#343—21no—
255#329—864no—
256Q8⋊C8—864no—
257#323—861no—
258#323—861no—
259C43#836911yescharacteristic, complemented, Sylow
260C2×#41—862no—
261C2×#41—862no—
262C2×#41—1724no—
263C22×Q8—431no—
264C22×Q8—431no—
265C22×Q8—862no—
266C22×Q8—862no—
267C22×Q8—861no—
268C22×Q8—1724no—
269C22×D4—431no—
270C22×D4—861no—
271C22×D4—861no—
272C22×D4C22⋊D4311yescharacteristic
273C23×C4—861no—
274C23×C4—861no—
275C23×C4C4×D4311yescharacteristic
276C23×C4C4×D4311yescharacteristic
277C2×M16—22no—
278#124—434no—
279#124—868no—
280#124—862no—
281C22⋊Q8—868no—
282C22⋊Q8—862no—
283C22⋊Q8—862no—
284#121—862no—
285#121—862no—
286#121—868no—
287C4×Q8—434no—
288C4×Q8—862no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—868no—
292C2×C4⋊C4—431no—
293C2×C4⋊C4—861no—
294C2×C4⋊C4—861no—
295C2×C4⋊C4C22⋊D4311yescharacteristic
296C2×C22⋊C4—861no—
297C2×C22⋊C4—861no—
298C2×C22⋊C4—861no—
299C2×C22⋊C4—861no—
300C2×C22⋊C4D17211yescharacteristic
301C2×C22⋊C4D17211yescharacteristic
302C2×C42—861no—
303C2×C42C22×D4311yescharacteristic
304C4⋊C8—862no—
305#100—24no—
306C22⋊C8—862no—
307#97—22no—
308#97—862no—
309C4×C8—22no—
310C24—861no—
311C24(C2×D43)⋊C411yescharacteristic
312C24(C2×D43)⋊C411yescharacteristic
313#41—1724no—
314#41—1728no—
315#41—1724no—
316C2×Q8—432no—
317C2×Q8—432no—
318C2×Q8—864no—
319C2×Q8—864no—
320C2×Q8—861no—
321C2×Q8—861no—
322C2×Q8—1722no—
323C2×Q8—1724no—
324C2×Q8—1724no—
325C2×Q8—1728no—
326C2×Q8—1724no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—432no—
331C2×D4—864no—
332C2×D4—864no—
333C2×D4—861no—
334C2×D4—1722no—
335C2×D4—1724no—
336C2×D4—1722no—
337C2×D4—1722no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—864no—
345C22×C4—861no—
346C22×C4—864no—
347C22×C4—861no—
348C22×C4—861no—
349C22×C4—1724no—
350C22×C4—1724no—
351C22×C4(C2×D43)⋊C411yescharacteristic
352C22×C4(C2×D43)⋊C411yescharacteristic
353C22×C4C2×C4×D4311yescharacteristic
354C22×C4C2×D17211yescharacteristic
355C22×C4C2×C22⋊D4311yescharacteristic
356M16—24no—
357M16—42no—
358C2×C8—42no—
359C2×C8—1722no—
360C4⋊C4—22no—
361C4⋊C4—432no—
362C4⋊C4—864no—
363C4⋊C4—861no—
364C4⋊C4—864no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—864no—
368C22⋊C4—864no—
369C22⋊C4—864no—
370C22⋊C4—864no—
371C42—864no—
372C42D4×D4312yes—
373C42D4×D4312yes—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—862no—
384C23—1722no—
385C23—1722no—
386C23—1721no—
387C23—1721no—
388C23#1306325312yes—
389C23C2×(C2×D43)⋊C411yescharacteristic
390Q8—434no—
391Q8—864no—
392Q8—862no—
393Q8—1722no—
394Q8—1728no—
395Q8—1722no—
396D4—44no—
397D4—1722no—
398D4—1728no—
399D4—1728no—
400D4—1722no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—864no—
414C2×C4—862no—
415C2×C4—862no—
416C2×C4—862no—
417C2×C4—1728no—
418C2×C4—1728no—
419C2×C4—1722no—
420C2×C4—1721no—
421C2×C4—1721no—
422C2×C4C22⋊C4×D4311yescharacteristic
423C2×C4C22⋊C4×D4311yescharacteristic
424C2×C4#1306329312yes—
425C2×C4#1306329512yes—
426C8—44no—
427C8—3442no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—81no—
439C22—81no—
440C22—3442no—
441C22—3442no—
442C22—1722no—
443C22—1722no—
444C22—1722no—
445C22#1458275411yescharacteristic
446C22C2×#1306325311yescharacteristic
447C22#1458291711yescharacteristic
448C4—21no—
449C4—22no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—864no—
454C4—1722no—
455C4—1722no—
456C4#100×D4312yes—
457C2—21no—
458C2—21no—
459C2—42no—
460C2—81no—
461C2—3442no—
462C2#1913451411yescharacteristic
463C2#1913454011yescharacteristic
464C2#1913457111yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC4×#100×C43C2
socleC22×C43#379
prime radicalC2×#14583558C2
cosocleC2×C42×C43C23
hypercenterC4×#100D43
hypocenterC43#8369
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #19133192C2semidirect
3. #19133043C2semidirect
4. #19132936C2semidirect
5. C4×#100×C43C2semidirect
192. Dic86#100semidirect
226. C4×#100D43semidirect
259. C43#8369semidirect

Maximal subgroups

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

subgroupindex
C2×#145835582
#191331922
#191330432
#191329362
C4×#100×C432
#836943