Finite Group Databasebeta

← #24997168

Subgroups — #24997168

Order 11008.

658 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14583234C211yesmaximal, characteristic, complemented
2#19140528C211yesmaximal, characteristic
3C2×#14582637C211yesmaximal, characteristic, complemented
4C2×#14582633C211yesmaximal, characteristic
5#19132863C211yesmaximal, characteristic, complemented
6#19132437C211yesmaximal, characteristic, complemented
7C2×C43×#481C211yesmaximal, characteristic
8C2×#13063304C2211yescharacteristic
9#14583234—22no—
10#14583234—22no—
11C2×#13063258—21no—
12C2×#13063248—21no—
13C2×C22⋊(C43⋊C8)C2211yescharacteristic
14#14582837—21no—
15#14582824—21no—
16#14582813—21no—
17#14582811—21no—
18#14582779—22no—
19C2×C43⋊#95—22no—
20C2×C43⋊#95C2211yescharacteristic
21C2×C43⋊#95C2211yescharacteristic
22#14582637—22no—
23#14582633—22no—
24C2×#121×C43C2211yescharacteristic
25C22×C4⋊C4×C43C2211yescharacteristic
26C43×#481—22no—
27C43×#481—22no—
28C2×#106×C43—21no—
29C2×#102×C43—21no—
30C2×C22⋊C8×C43C2211yescharacteristic
31C22×Dic86D411yescharacteristic
32C23×Dic43—21no—
33C23×Dic43D412yes—
34C22×C43⋊C8—21no—
35#13063304—21no—
36#13063304—21no—
37#13063304—21no—
38#13063304—21no—
39C2×C4⋊Dic43D411yescharacteristic
40C2×Dic43⋊C4—21no—
41C2×C22⋊Dic43—21no—
42C2×C22⋊Dic43—22no—
43C2×C22⋊Dic43—22no—
44#13063258—24no—
45#13063248—24no—
46C22⋊(C43⋊C8)—21no—
47C22⋊(C43⋊C8)D412yes—
48C43⋊#95—22no—
49C43⋊#95—22no—
50C43⋊#95—21no—
51C43⋊#95—21no—
52C43⋊#95—21no—
53C43⋊#95—21no—
54C43⋊#95—44no—
55C22×D4×C43—21no—
56C22×D4×C43D411yescharacteristic
57C23×C4×C43—21no—
58C23×C4×C43C2311yescharacteristic
59C22×C8×C43—21no—
60#121×C43—21no—
61#121×C43—21no—
62#121×C43—21no—
63#121×C43—21no—
64C2×C4⋊C4×C43—21no—
65C2×C4⋊C4×C43—21no—
66C2×C4⋊C4×C43—21no—
67C2×C4⋊C4×C43—21no—
68C2×C4⋊C4×C43—21no—
69C2×C4⋊C4×C43D412yes—
70C2×C4⋊C4×C43D411yescharacteristic
71C2×C22⋊C4×C43—21no—
72#106×C43—24no—
73#102×C43—24no—
74C22⋊C8×C43—21no—
75C22⋊C8×C43D412yes—
76C24×C43—21no—
77C24×C43C2×D411yescharacteristic
78C2×D4×C43—21no—
79C2×D4×C43—21no—
80C2×D4×C43—41no—
81C2×D4×C43—41no—
82C2×D4×C43—42no—
83C2×D4×C43—42no—
84C2×D4×C43—42no—
85C22×C4×C43—21no—
86C22×C4×C43—22no—
87C22×C4×C43—42no—
88C22×C4×C43—42no—
89C22×C4×C43—42no—
90C22×C4×C43—41no—
91C22×C4×C43—41no—
92C22×C4×C43C2×D411yescharacteristic
93C22×C4×C43C2×D411yescharacteristic
94C22×C4×C43C2×D411yescharacteristic
95C22×C4×C43C2×D411yescharacteristic
96C22×C4×C43C2×D411yescharacteristic
97C22×C4×C43#4111yescharacteristic
98C22×Dic43—22no—
99C22×Dic43—22no—
100C22×Dic43—21no—
101C22×Dic43—21no—
102C22×Dic43—22no—
103C22×Dic43—22no—
104C22×Dic43—42no—
105C22×Dic43—42no—
106C22×Dic43—41no—
107C22×Dic43—42no—
108C22×Dic43—41no—
109C22×Dic43—42no—
110C2×Dic86—21no—
111C2×Dic86—21no—
112C2×Dic86—42no—
113C2×C8×C43—22no—
114C2×C8×C43—42no—
115C4⋊C4×C43—22no—
116C4⋊C4×C43—21no—
117C4⋊C4×C43—22no—
118C4⋊C4×C43—21no—
119C4⋊C4×C43—41no—
120C4⋊C4×C43—41no—
121C22⋊C4×C43—41no—
122C22⋊C4×C43—41no—
123C22⋊Dic43—41no—
124C22⋊Dic43—41no—
125C22⋊Dic43—82no—
126C22⋊Dic43—82no—
127C4⋊Dic43—21no—
128C4⋊Dic43—21no—
129Dic43⋊C4—41no—
130Dic43⋊C4—41no—
131C2×C43⋊C8—22no—
132C2×C43⋊C8—42no—
133C23×C43—21no—
134C23×C43—21no—
135C23×C43—22no—
136C23×C43—41no—
137C23×C43—42no—
138C23×C43—41no—
139C23×C43—41no—
140C23×C43—42no—
141C23×C43#12011yescharacteristic
142C23×C43#12111yescharacteristic
143C23×C43#12711yescharacteristic
144C23×C43#13512yes—
145C23×C43#13712yes—
146D4×C43—42no—
147D4×C43—81no—
148D4×C43—82no—
149D4×C43—81no—
150D4×C43—81no—
151C2×C4×C43—22no—
152C2×C4×C43—22no—
153C2×C4×C43—41no—
154C2×C4×C43—41no—
155C2×C4×C43—41no—
156C2×C4×C43—41no—
157C2×C4×C43—44no—
158C2×C4×C43—41no—
159C2×C4×C43—42no—
160C2×C4×C43—42no—
161C2×C4×C43—82no—
162C2×C4×C43—81no—
163C2×C4×C43—81no—
164C2×C4×C43#12011yescharacteristic
165C2×C4×C43#12011yescharacteristic
166C2×C4×C43#12111yescharacteristic
167C2×C4×C43#12111yescharacteristic
168C2×Dic43—44no—
169C2×Dic43—42no—
170C2×Dic43—44no—
171C2×Dic43—41no—
172C2×Dic43—41no—
173C2×Dic43—42no—
174C2×Dic43—41no—
175C2×Dic43—42no—
176C2×Dic43—41no—
177C2×Dic43—42no—
178C2×Dic43—42no—
179C2×Dic43—82no—
180C2×Dic43—81no—
181C2×Dic43—81no—
182C2×Dic43—82no—
183Dic86—42no—
184Dic86—81no—
185C8×C43—42no—
186C43⋊C8—42no—
187#9969—431nomaximal, Sylow
188C22×C43—22no—
189C22×C43—22no—
190C22×C43—22no—
191C22×C43—22no—
192C22×C43—21no—
193C22×C43—21no—
194C22×C43—21no—
195C22×C43—21no—
196C22×C43—82no—
197C22×C43—82no—
198C22×C43—81no—
199C22×C43—81no—
200C22×C43—81no—
201C22×C43#39211yescharacteristic
202C22×C43#45212yes—
203C22×C43#45311yescharacteristic
204C22×C43#45411yescharacteristic
205C22×C43#45812yes—
206C22×C43#46412yes—
207C22×C43#49511yescharacteristic
208C22×C43#49711yescharacteristic
209C4×C43—21no—
210C4×C43—21no—
211C4×C43—21no—
212C4×C43—21no—
213C4×C43—82no—
214C4×C43—81no—
215Dic43—82no—
216Dic43—82no—
217Dic43—81no—
218Dic43—81no—
219C2×#484—431no—
220C2×#481D4311yescharacteristic
221#2639—431no—
222C2×#328—431no—
223C2×#327—431no—
224#1445—431no—
225#1280—431no—
226C2×C43—21no—
227C2×C43—22no—
228C2×C43—21no—
229C2×C43—161no—
230C2×C43#161512yes—
231C2×C43#161812yes—
232C2×C43#200611yescharacteristic
233C2×C43#201511yescharacteristic
234C2×C43#202211yescharacteristic
235C2×C22⋊Q8—431no—
236C2×#121C2×D4311yescharacteristic
237C22×C4⋊C4C2×D4311yescharacteristic
238#484—862no—
239#484—862no—
240#481—22no—
241#481—22no—
242C2×#107—861no—
243C2×#106—21no—
244C2×#103—861no—
245C2×#102—21no—
246C2×C22⋊C8—431no—
247C2×C22⋊C8C2×D4311yescharacteristic
248#400—861no—
249#397—861no—
250#394—861no—
251#393—861no—
252#386—862no—
253C2×#95—431no—
254C2×#95—431no—
255C2×#95—862no—
256#328—862no—
257#327—862no—
258C43#996911yescharacteristic, complemented, Sylow
259C22×Q8—431no—
260C22×D4—21no—
261C22×D4C22⋊D4311yescharacteristic
262C23×C4—21no—
263C23×C4—432no—
264C23×C4—861no—
265C23×C4C22×D4311yescharacteristic
266C22×C8—21no—
267C22×C8—861no—
268C22⋊Q8—861no—
269C22⋊Q8—861no—
270C22⋊Q8—861no—
271C22⋊Q8—861no—
272#121—21no—
273#121—21no—
274#121—21no—
275#121—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—431no—
282C2×C4⋊C4—861no—
283C2×C4⋊C4C22⋊D4312yes—
284C2×C4⋊C4C22⋊D4311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—862no—
287C2×C22⋊C4—862no—
288C2×C22⋊C4—861no—
289#107—864no—
290#106—24no—
291#103—864no—
292#102—24no—
293C22⋊C8—21no—
294C22⋊C8—432no—
295C22⋊C8—861no—
296C22⋊C8C22⋊D4312yes—
297#95—862no—
298#95—862no—
299#95—861no—
300#95—861no—
301#95—861no—
302#95—861no—
303#95—1724no—
304C24—21no—
305C24D4×D4311yescharacteristic
306C2×Q8—861no—
307C2×Q8—861no—
308C2×Q8—1722no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—862no—
324C22×C4—862no—
325C22×C4—862no—
326C22×C4—861no—
327C22×C4—861no—
328C22×C4—862no—
329C22×C4—1721no—
330C22×C4—1722no—
331C22×C4—1721no—
332C22×C4—1722no—
333C22×C4—1722no—
334C22×C4—1722no—
335C22×C4D4×D4311yescharacteristic
336C22×C4D4×D4311yescharacteristic
337C22×C4#1062190011yescharacteristic
338C22×C4C2×C22⋊D4311yescharacteristic
339C22×C4C2×C22⋊D4311yescharacteristic
340C22×C4C2×C22⋊D4311yescharacteristic
341C2×C8—22no—
342C2×C8—42no—
343C2×C8—862no—
344C2×C8—1722no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—861no—
352C4⋊C4—861no—
353C4⋊C4—1721no—
354C4⋊C4—1721no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—3442no—
358C22⋊C4—3442no—
359C22⋊C4—1721no—
360C22⋊C4—1721no—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23#1306329311yescharacteristic
370C23#1306329711yescharacteristic
371C23#1306332111yescharacteristic
372C23#1306334612yes—
373C23#1306335412yes—
374Q8—3441no—
375Q8—1722no—
376D4—42no—
377D4—81no—
378D4—82no—
379D4—81no—
380D4—81no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—3442no—
395C2×C4—3441no—
396C2×C4—3441no—
397C2×C4—3442no—
398C2×C4—1722no—
399C2×C4—1724no—
400C2×C4—1724no—
401C2×C4—1722no—
402C2×C4—1721no—
403C2×C4—1721no—
404C2×C4—1721no—
405C2×C4—1722no—
406C2×C4—1721no—
407C2×C4—1722no—
408C2×C4—1722no—
409C2×C4#1306329311yescharacteristic
410C2×C4#1306329411yescharacteristic
411C2×C4#1306329611yescharacteristic
412C2×C4#1306329711yescharacteristic
413C8—42no—
414C8—1722no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—81no—
424C22—82no—
425C22—81no—
426C22—82no—
427C22—81no—
428C22#1458280711yescharacteristic
429C22#1458305812yes—
430C22#1458306511yescharacteristic
431C22#1458307011yescharacteristic
432C22#1458308812yes—
433C22#1458311912yes—
434C22#1458328011yescharacteristic
435C22#1458329011yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—81no—
442C4—3442no—
443C4—3441no—
444C4—3442no—
445C4—3441no—
446C2—21no—
447C2—22no—
448C2—21no—
449C2—161no—
450C2#1913362312yes—
451C2#1913364012yes—
452C2#1913563511yescharacteristic
453C2#1913568711yescharacteristic
454C2#1913572311yescharacteristic
subgroupG/H
centerC23#13063293
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC2×C43×#481C2
socleC23×C43#120
prime radicalC2×#121×C43C22
hypercenterC2×#481D43
hypocenterC43#9969
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C2×#14583234C2semidirect
3. C2×#14582637C2semidirect
5. #19132863C2semidirect
6. #19132437C2semidirect
258. C43#9969semidirect

Maximal subgroups

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

subgroupindex
C2×#145832342
#191405282
C2×#145826372
C2×#145826332
#191328632
#191324372
C2×C43×#4812
#996943