Finite Group Databasebeta

← #25012574

Subgroups — #25012574

Order 11008.

512 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#19137269C211yesmaximal, characteristic
2C2×#14582690C211yesmaximal, characteristic
3C2×#14582688C211yesmaximal, characteristic
4#19134585C211yesmaximal, characteristic
5#19134581C211yesmaximal, characteristic
6#19132713C211yesmaximal, characteristic
7C43×#1390C211yesmaximal, characteristic
8C2×#13063306C2211yescharacteristic
9C2×D4×Dic43C2211yescharacteristic
10C22×Dic43⋊C4C411yescharacteristic, complemented
11#14582922—21no—
12C2×#13063242C2211yescharacteristic
13C2×#13063240C2211yescharacteristic
14#14582796—21no—
15#14582787—21no—
16#14582781—21no—
17#14582772C411yescharacteristic, complemented
18#14582771—21no—
19#14582760C411yescharacteristic, complemented
20#14582757—21no—
21C22⋊C4×Dic43C411yescharacteristic, complemented
22#14582690—22no—
23#14582688—22no—
24#14582667—21no—
25#14582665—21no—
26C2×#100×C43C2211yescharacteristic
27C2×#99×C43C2211yescharacteristic
28C43×#386C2211yescharacteristic
29C23×Dic43C2×C411yescharacteristic
30C23×Dic43C2×C411yescharacteristic
31C23×Dic43D411yescharacteristic
32C2×#10621877—21no—
33#13063306—21no—
34#13063306—21no—
35#13063306D412yes—
36#13063306Q812yes—
37D4×Dic43—41no—
38D4×Dic43—41no—
39C2×C4⋊Dic43D411yescharacteristic
40C2×Dic43⋊C4—21no—
41C2×Dic43⋊C4—21no—
42C2×Dic43⋊C4—21no—
43C2×Dic43⋊C4—21no—
44C2×Dic43⋊C4—41no—
45C2×Dic43⋊C4—41no—
46C2×Dic43⋊C4—41no—
47C2×C22⋊Dic43—21no—
48C2×C22⋊Dic43—21no—
49C2×C22⋊Dic43C2×C411yescharacteristic
50C2×C22⋊Dic43C2×C411yescharacteristic
51C2×C22⋊Dic43D411yescharacteristic
52C2×C4×Dic43—21no—
53C2×C4×Dic43D411yescharacteristic
54Dic43⋊C8—41no—
55Dic43⋊C8—41no—
56#13063242—21no—
57#13063242—21no—
58#13063240—21no—
59#13063240—21no—
60C43⋊#95—21no—
61C43⋊#95—21no—
62C43⋊#95—21no—
63C43⋊#95—21no—
64C43⋊#95—41no—
65C22×D4×C43C2311yescharacteristic
66C23×C4×C43C2×C411yescharacteristic
67C2×M16×C43—21no—
68C2×C22⋊C4×C43—21no—
69C2×C22⋊C4×C43—21no—
70C2×C22⋊C4×C43C2×C411yescharacteristic
71#100×C43—21no—
72#100×C43—21no—
73#99×C43—21no—
74#99×C43—21no—
75#95×C43—21no—
76C24×C43C22×C411yescharacteristic
77C24×C43C2×D411yescharacteristic
78C2×D4×C43—41no—
79C2×D4×C43—41no—
80C2×D4×C43C2×D411yescharacteristic
81C2×D4×C43C2×Q811yescharacteristic
82C2×D4×C43#4111yescharacteristic
83C2×D4×C43#4111yescharacteristic
84C22×C4×C43—21no—
85C22×C4×C43—21no—
86C22×C4×C43—41no—
87C22×C4×C43—41no—
88C22×C4×C43—81no—
89C22×C4×C43C2×D411yescharacteristic
90C22×Dic43—21no—
91C22×Dic43—21no—
92C22×Dic43—21no—
93C22×Dic43—21no—
94C22×Dic43—21no—
95C22×Dic43—21no—
96C22×Dic43—21no—
97C22×Dic43—41no—
98C22×Dic43—41no—
99C22×Dic43—41no—
100C22×Dic43—81no—
101C22×Dic43—81no—
102C22×Dic43C22⋊C411yescharacteristic
103C22×Dic43C22⋊C411yescharacteristic
104C22×Dic43C22⋊C411yescharacteristic
105C22×Dic43C22⋊C411yescharacteristic
106C22×Dic43C4⋊C412yescomplemented
107M16×C43—41no—
108M16×C43—41no—
109C2×C8×C43—41no—
110C22⋊C4×C43—41no—
111C22⋊C4×C43—41no—
112C22⋊C4×C43—41no—
113C22⋊C4×C43—81no—
114C22⋊Dic43—21no—
115C22⋊Dic43—22no—
116C22⋊Dic43—22no—
117C22⋊Dic43—41no—
118C22⋊Dic43—41no—
119C22⋊Dic43—41no—
120C22⋊Dic43—41no—
121C22⋊Dic43C4⋊C412yescomplemented
122C4⋊Dic43—41no—
123Dic43⋊C4—22no—
124Dic43⋊C4—22no—
125Dic43⋊C4—42no—
126Dic43⋊C4—41no—
127Dic43⋊C4—41no—
128Dic43⋊C4—81no—
129Dic43⋊C4—82no—
130C4×Dic43—21no—
131C4×Dic43—21no—
132C4×Dic43—81no—
133#10621877—41no—
134#10621877—41no—
135C2×C43⋊C8—41no—
136C23×C43—21no—
137C23×C43—21no—
138C23×C43—21no—
139C23×C43—21no—
140C23×C43—41no—
141C23×C43—41no—
142C23×C43—41no—
143C23×C43—41no—
144C23×C43C2×C22⋊C411yescharacteristic
145C23×C43C2×C4⋊C411yescharacteristic
146C23×C43#11711yescharacteristic
147C23×C43#12111yescharacteristic
148C23×C43C22⋊Q811yescharacteristic
149D4×C43—41no—
150D4×C43—41no—
151D4×C43—41no—
152D4×C43—41no—
153C2×C4×C43—21no—
154C2×C4×C43—21no—
155C2×C4×C43—41no—
156C2×C4×C43—41no—
157C2×C4×C43—41no—
158C2×C4×C43—41no—
159C2×C4×C43—41no—
160C2×C4×C43—41no—
161C2×C4×C43—82no—
162C2×C4×C43—81no—
163C2×C4×C43—81no—
164C2×C4×C43—81no—
165C2×C4×C43#12111yescharacteristic
166C2×C4×C43C22⋊Q811yescharacteristic
167C2×Dic43—22no—
168C2×Dic43—22no—
169C2×Dic43—21no—
170C2×Dic43—21no—
171C2×Dic43—21no—
172C2×Dic43—41no—
173C2×Dic43—42no—
174C2×Dic43—41no—
175C2×Dic43—41no—
176C2×Dic43—42no—
177C2×Dic43—41no—
178C2×Dic43—41no—
179C2×Dic43—41no—
180C2×Dic43—41no—
181C2×Dic43—42no—
182C2×Dic43—41no—
183C2×Dic43—81no—
184C2×Dic43—81no—
185C2×Dic43—81no—
186C2×Dic43—81no—
187C2×Dic43—81no—
188C2×Dic43—81no—
189C2×Dic43#9911yescharacteristic
190C2×Dic43#9911yescharacteristic
191C8×C43—81no—
192C43⋊C8—81no—
193#12796—431nomaximal, Sylow
194C22×C43—21no—
195C22×C43—21no—
196C22×C43—21no—
197C22×C43—21no—
198C22×C43—21no—
199C22×C43—21no—
200C22×C43—41no—
201C22×C43—41no—
202C22×C43—41no—
203C22×C43—41no—
204C22×C43—42no—
205C22×C43—41no—
206C22×C43—41no—
207C22×C43—41no—
208C22×C43—41no—
209C22×C43—81no—
210C22×C43#38011yescharacteristic
211C22×C43C2×#9911yescharacteristic
212C22×C43#41011yescharacteristic
213C4×C43—21no—
214C4×C43—21no—
215C4×C43—81no—
216C4×C43—81no—
217C4×C43—161no—
218Dic43—21no—
219Dic43—21no—
220Dic43—42no—
221Dic43—42no—
222Dic43—41no—
223Dic43—41no—
224Dic43—41no—
225Dic43—161no—
226#2322—431no—
227C2×#352—431no—
228C2×#352—431no—
229#1800—431no—
230#1799—431no—
231#1390—431no—
232#1390D4311yescharacteristic
233C2×C43—21no—
234C2×C43—21no—
235C2×C43—41no—
236C2×C43—41no—
237C2×C43—41no—
238C2×C43—41no—
239C2×C43#179911yescharacteristic
240C2×C43#212111yescharacteristic
241C2×C43#212311yescharacteristic
242C2×#123—431no—
243C2×C4×D4—431no—
244C22×C4⋊C4—431no—
245#423—861no—
246C2×#100—431no—
247C2×#100C2×D4311yescharacteristic
248C2×#99—431no—
249C2×#99C2×D4311yescharacteristic
250#389—861no—
251#388—861no—
252#387—861no—
253#386C2×D4311yescharacteristic
254#385—431no—
255#385—861no—
256#381—431no—
257#380—861no—
258C4×C22⋊C4—431no—
259#352—862no—
260#352—862no—
261#343—861no—
262#342—861no—
263C43#1279611yescharacteristic, complemented, Sylow
264C22×D4C22×D4311yescharacteristic
265C23×C4—431no—
266C23×C4—431no—
267C23×C4—431no—
268C23×C4C4×D4311yescharacteristic
269C2×M16—21no—
270C2×M16—861no—
271#123—432no—
272#123—432no—
273#123—861no—
274#123—861no—
275C4×D4—1721no—
276C4×D4—1721no—
277C2×C4⋊C4—431no—
278C2×C4⋊C4—861no—
279C2×C4⋊C4—861no—
280C2×C4⋊C4—861no—
281C2×C4⋊C4—861no—
282C2×C4⋊C4—1721no—
283C2×C4⋊C4—1721no—
284C2×C4⋊C4—1721no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—431no—
288C2×C22⋊C4—431no—
289C2×C22⋊C4—431no—
290C2×C22⋊C4—861no—
291C2×C22⋊C4—861no—
292C2×C22⋊C4C4×D4311yescharacteristic
293C2×C42—431no—
294C2×C42—861no—
295C4⋊C8—1721no—
296C4⋊C8—1721no—
297#100—21no—
298#100—21no—
299#100—861no—
300#100—861no—
301#99—21no—
302#99—21no—
303#99—861no—
304#99—861no—
305#95—21no—
306#95—861no—
307#95—861no—
308#95—861no—
309#95—861no—
310#95—1721no—
311C24C2×C4×D4311yescharacteristic
312C24D4×D4311yescharacteristic
313C2×D4—41no—
314C2×D4—41no—
315C2×D4#1062189811yescharacteristic
316C2×D4#1062189811yescharacteristic
317C2×D4D4×D4311yescharacteristic
318C2×D4Q8×D4311yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—81no—
324C22×C4—431no—
325C22×C4—431no—
326C22×C4—431no—
327C22×C4—431no—
328C22×C4—432no—
329C22×C4—861no—
330C22×C4—861no—
331C22×C4—861no—
332C22×C4—861no—
333C22×C4—861no—
334C22×C4—861no—
335C22×C4—861no—
336C22×C4—3441no—
337C22×C4—3441no—
338C22×C4—1721no—
339C22×C4—1721no—
340C22×C4—1721no—
341C22×C4D4×D4311yescharacteristic
342M16—41no—
343M16—41no—
344M16—1721no—
345M16—1721no—
346C2×C8—41no—
347C2×C8—1721no—
348C4⋊C4—862no—
349C4⋊C4—862no—
350C4⋊C4—3441no—
351C4⋊C4—3442no—
352C4⋊C4—1721no—
353C4⋊C4—1721no—
354C4⋊C4—1721no—
355C4⋊C4—1722no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—81no—
360C22⋊C4—432no—
361C22⋊C4—861no—
362C22⋊C4—862no—
363C22⋊C4—862no—
364C22⋊C4—1721no—
365C22⋊C4—1721no—
366C22⋊C4—1721no—
367C22⋊C4—1721no—
368C42—861no—
369C42—861no—
370C42—3441no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C22⋊C4×D4311yescharacteristic
380C23C4⋊C4×D4311yescharacteristic
381C23#1306328211yescharacteristic
382C23#1306329811yescharacteristic
383C23#1306330311yescharacteristic
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—431no—
401C2×C4—431no—
402C2×C4—861no—
403C2×C4—861no—
404C2×C4—861no—
405C2×C4—862no—
406C2×C4—862no—
407C2×C4—3441no—
408C2×C4—3441no—
409C2×C4—3441no—
410C2×C4—3441no—
411C2×C4—3441no—
412C2×C4—3441no—
413C2×C4—1721no—
414C2×C4—1722no—
415C2×C4—1721no—
416C2×C4—1722no—
417C2×C4—1721no—
418C2×C4—1721no—
419C2×C4—1721no—
420C2×C4—1721no—
421C2×C4—1721no—
422C2×C4—1722no—
423C2×C4—1721no—
424C2×C4#1306329811yescharacteristic
425C2×C4#1306330311yescharacteristic
426C8—81no—
427C8—3441no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—81no—
444C22#1458275811yescharacteristic
445C22#99×D4311yescharacteristic
446C22#1458287111yescharacteristic
447C4—21no—
448C4—21no—
449C4—81no—
450C4—81no—
451C4—161no—
452C4—861no—
453C4—861no—
454C4—1722no—
455C4—1722no—
456C4—1721no—
457C4—1721no—
458C4—1721no—
459C4—6881no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—41no—
464C2—41no—
465C2—41no—
466C2#1913458211yescharacteristic
467C2#1913623811yescharacteristic
468C2#1913624711yescharacteristic
subgroupG/H
centerC22#99×D43
commutatorC24×C43C22×C4
FrattiniC22×D4C22×D43
FittingC43×#1390C2
socleC22×C43C2×#99
prime radical, cosocleC22×D4×C43C23
hypercenter#1390D43
hypocenterC43#12796
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
10. C22×Dic43⋊C4C4semidirect
17. #14582772C4semidirect
19. #14582760C4semidirect
21. C22⋊C4×Dic43C4semidirect
106. C22×Dic43C4⋊C4semidirect
121. C22⋊Dic43C4⋊C4semidirect
263. C43#12796semidirect

Maximal subgroups

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

subgroupindex
#191372692
C2×#145826902
C2×#145826882
#191345852
#191345812
#191327132
C43×#13902
#1279643