Finite Group Databasebeta

← #24992909

Subgroups — #24992909

Order 11008.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14583240C211yesmaximal, characteristic
2#19143253C211yesmaximal, characteristic
3#19133114C211yesmaximal, characteristic
4#19132960C211yesmaximal, characteristic
5#19132471C211yesmaximal, characteristic
6#19132443C211yesmaximal, characteristic
7C43×#1826C211yesmaximal, characteristic
8C2×#13063304C2211yescharacteristic
9#14583240—21no—
10#14583240—21no—
11#14583240—21no—
12#14583240—21no—
13#14582960—22no—
14#14582956—21no—
15#14582952—21no—
16C2×#13063258C411yescharacteristic, complemented
17C2×#13063256C411yescharacteristic, complemented
18#14582926—21no—
19C2×#13063248C411yescharacteristic, complemented
20C2×#13063248C411yescharacteristic, complemented
21#14582863C2211yescharacteristic
22C2×C22⋊(C43⋊C8)C2211yescharacteristic
23C2×C22⋊(C43⋊C8)C2211yescharacteristic
24#14582855—21no—
25#14582635—22no—
26#14582624—21no—
27#14582624—21no—
28C2×#117×C43C2211yescharacteristic
29C43×#407C2211yescharacteristic
30C4×C22⋊C4×C43C2211yescharacteristic
31C43×#339—21no—
32C43×#339—21no—
33C22×Dic86C2×C411yescharacteristic
34C2×#10621877—21no—
35C22×C43⋊C8—21no—
36C22×C43⋊C8C2×C411yescharacteristic
37C22×C43⋊C8C2×C411yescharacteristic
38#13063304—21no—
39#13063304—21no—
40#13063304—21no—
41#13063304—21no—
42C2×C4⋊Dic43C2×C411yescharacteristic
43C2×Dic43⋊C4—21no—
44C2×C22⋊Dic43—21no—
45#13063258—21no—
46#13063258—21no—
47#13063256—21no—
48#13063256—21no—
49C4⋊(C43⋊C8)—22no—
50C4⋊(C43⋊C8)—22no—
51#13063248—21no—
52#13063248—21no—
53#13063248—21no—
54#13063248—21no—
55C22⋊(C43⋊C8)—21no—
56C22⋊(C43⋊C8)—21no—
57C22⋊(C43⋊C8)—21no—
58C22⋊(C43⋊C8)—21no—
59C22⋊(C43⋊C8)D411yescharacteristic
60C22⋊(C43⋊C8)D411yescharacteristic
61C22⋊(C43⋊C8)D412yes—
62#13063236—22no—
63C4×C43⋊C8—22no—
64C23×C4×C43C2311yescharacteristic
65C2×M16×C43—21no—
66#117×C43—21no—
67#117×C43—21no—
68#117×C43D411yescharacteristic
69#117×C43D411yescharacteristic
70#117×C43D411yescharacteristic
71#117×C43D411yescharacteristic
72C2×C4⋊C4×C43C2×C411yescharacteristic
73C2×C4⋊C4×C43C2×C411yescharacteristic
74C2×C22⋊C4×C43—21no—
75C2×C22⋊C4×C43—21no—
76C2×C42×C43—21no—
77C2×C42×C43—21no—
78C22⋊C8×C43—21no—
79C22⋊C8×C43—21no—
80#95×C43—21no—
81C24×C43#4111yescharacteristic
82C22×C4×C43—21no—
83C22×C4×C43—21no—
84C22×C4×C43—41no—
85C22×C4×C43—41no—
86C22×C4×C43—41no—
87C22×C4×C43C22×C411yescharacteristic
88C22×C4×C43C2×D411yescharacteristic
89C22×C4×C43C2×D411yescharacteristic
90C22×C4×C43C2×D411yescharacteristic
91C22×C4×C43C2×D411yescharacteristic
92C22×C4×C43#4111yescharacteristic
93C22×Dic43—21no—
94C22×Dic43—21no—
95C2×Dic86—21no—
96C2×Dic86—21no—
97C2×Dic86—41no—
98C2×Dic86—41no—
99M16×C43—41no—
100M16×C43—41no—
101C2×C8×C43—41no—
102C4⋊C4×C43—21no—
103C4⋊C4×C43—21no—
104C4⋊C4×C43C22⋊C411yescharacteristic
105C4⋊C4×C43C22⋊C411yescharacteristic
106C4⋊C4×C43C22⋊C411yescharacteristic
107C4⋊C4×C43C22⋊C411yescharacteristic
108C22⋊C4×C43—22no—
109C22⋊C4×C43—21no—
110C22⋊C4×C43—21no—
111C22⋊C4×C43—21no—
112C22⋊C4×C43—21no—
113C22⋊C4×C43—41no—
114C42×C43—21no—
115C42×C43—21no—
116C42×C43—21no—
117C42×C43—21no—
118C42×C43—41no—
119C42×C43—41no—
120C22⋊Dic43—41no—
121C22⋊Dic43—41no—
122C4⋊Dic43—21no—
123C4⋊Dic43—21no—
124Dic43⋊C4—41no—
125Dic43⋊C4—41no—
126#10621877—41no—
127#10621877—41no—
128C2×C43⋊C8—21no—
129C2×C43⋊C8—21no—
130C2×C43⋊C8—21no—
131C2×C43⋊C8—22no—
132C2×C43⋊C8—21no—
133C2×C43⋊C8—41no—
134C2×C43⋊C8—41no—
135C2×C43⋊C8—41no—
136C2×C43⋊C8—41no—
137C23×C43—21no—
138C23×C43—21no—
139C23×C43—41no—
140C23×C43—41no—
141C23×C43C4×D411yescharacteristic
142C23×C43#12111yescharacteristic
143C23×C43#12311yescharacteristic
144C2×C4×C43—21no—
145C2×C4×C43—21no—
146C2×C4×C43—21no—
147C2×C4×C43—21no—
148C2×C4×C43—21no—
149C2×C4×C43—21no—
150C2×C4×C43—21no—
151C2×C4×C43—21no—
152C2×C4×C43—21no—
153C2×C4×C43—21no—
154C2×C4×C43—41no—
155C2×C4×C43—41no—
156C2×C4×C43—41no—
157C2×C4×C43—41no—
158C2×C4×C43—41no—
159C2×C4×C43—42no—
160C2×C4×C43—81no—
161C2×C4×C43—81no—
162C2×C4×C43—81no—
163C2×C4×C43C2×C22⋊C411yescharacteristic
164C2×C4×C43C4×D411yescharacteristic
165C2×C4×C43#12011yescharacteristic
166C2×C4×C43#12111yescharacteristic
167C2×Dic43—41no—
168C2×Dic43—41no—
169C2×Dic43—41no—
170C2×Dic43—41no—
171C2×Dic43—81no—
172Dic86—41no—
173Dic86—41no—
174Dic86—81no—
175C8×C43—81no—
176C43⋊C8—41no—
177C43⋊C8—41no—
178C43⋊C8—42no—
179C43⋊C8—41no—
180C43⋊C8—41no—
181#9240—431nomaximal, Sylow
182C22×C43—21no—
183C22×C43—21no—
184C22×C43—21no—
185C22×C43—21no—
186C22×C43—21no—
187C22×C43—21no—
188C22×C43—21no—
189C22×C43—21no—
190C22×C43—81no—
191C22×C43—81no—
192C22×C43#38611yescharacteristic
193C22×C43#44311yescharacteristic
194C22×C43#44411yescharacteristic
195C4×C43—21no—
196C4×C43—21no—
197C4×C43—21no—
198C4×C43—41no—
199C4×C43—41no—
200C4×C43—41no—
201C4×C43—41no—
202C4×C43—82no—
203C4×C43#41011yescharacteristic
204C4×C43#41011yescharacteristic
205Dic43—81no—
206Dic43—81no—
207C2×#485—431no—
208#2924—431no—
209#1826D4311yescharacteristic
210#1508—431no—
211#1472—431no—
212#1293—431no—
213#1282—431no—
214C2×C43—21no—
215C2×C43—21no—
216C2×C43—41no—
217C2×C43—41no—
218C2×C43#187611yescharacteristic
219C2×C43#189911yescharacteristic
220C2×C43#190211yescharacteristic
221C2×C22⋊Q8—431no—
222C2×#117C2×D4311yescharacteristic
223#485—861no—
224#485—861no—
225#485—861no—
226#485—861no—
227#433—862no—
228#432—861no—
229#431—861no—
230C2×#107—431no—
231C2×#106—431no—
232#424—861no—
233C2×#103—431no—
234C2×#103—431no—
235#407—431no—
236#407C2×D4311yescharacteristic
237C2×C22⋊C8—431no—
238C2×C22⋊C8—431no—
239#405—861no—
240C4×C22⋊C4C2×D4311yescharacteristic
241#339—21no—
242#339—21no—
243#328—862no—
244#324—861no—
245#324—861no—
246C43#924011yescharacteristic, complemented, Sylow
247C22×Q8—431no—
248C23×C4C22×D4311yescharacteristic
249C2×M16—21no—
250C2×M16—861no—
251C22×C8—431no—
252C22×C8—431no—
253C22×C8—861no—
254C22⋊Q8—861no—
255C22⋊Q8—861no—
256C22⋊Q8—861no—
257C22⋊Q8—861no—
258#117—21no—
259#117—21no—
260#117D17211yescharacteristic
261#117D17211yescharacteristic
262#117C22⋊D4311yescharacteristic
263#117C22⋊D4311yescharacteristic
264C2×C4⋊C4—431no—
265C2×C4⋊C4—861no—
266C2×C4⋊C4C4×D4311yescharacteristic
267C2×C4⋊C4C4×D4311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—861no—
271C2×C42—21no—
272C2×C42—21no—
273#107—861no—
274#107—861no—
275#106—861no—
276#106—861no—
277C4⋊C8—862no—
278C4⋊C8—862no—
279#103—861no—
280#103—861no—
281#103—861no—
282#103—861no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—431no—
286C22⋊C8—431no—
287C22⋊C8—432no—
288C22⋊C8—861no—
289C22⋊C8—861no—
290C22⋊C8—861no—
291C22⋊C8—861no—
292#97—862no—
293C4×C8—862no—
294#95—21no—
295C24#1062190011yescharacteristic
296C2×Q8—861no—
297C2×Q8—861no—
298C2×Q8—1721no—
299C2×Q8—1721no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—861no—
306C22×C4—861no—
307C22×C4C2×C4×D4311yescharacteristic
308C22×C4C2×D17211yescharacteristic
309C22×C4D4×D4311yescharacteristic
310C22×C4D4×D4311yescharacteristic
311C22×C4#1062190011yescharacteristic
312C22×C4C2×C22⋊D4311yescharacteristic
313M16—41no—
314M16—41no—
315M16—1721no—
316M16—1721no—
317C2×C8—41no—
318C2×C8—861no—
319C2×C8—861no—
320C2×C8—861no—
321C2×C8—862no—
322C2×C8—861no—
323C2×C8—1721no—
324C2×C8—1721no—
325C2×C8—1721no—
326C2×C8—1721no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—861no—
330C4⋊C4—861no—
331C4⋊C4—1721no—
332C4⋊C4—1721no—
333C4⋊C4(C2×D43)⋊C411yescharacteristic
334C4⋊C4(C2×D43)⋊C411yescharacteristic
335C4⋊C4(C2×D43)⋊C411yescharacteristic
336C4⋊C4(C2×D43)⋊C411yescharacteristic
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—22no—
342C22⋊C4—41no—
343C22⋊C4—1721no—
344C22⋊C4—1721no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—41no—
350C42—41no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23(C22⋊D43)⋊C411yescharacteristic
356C23#1306329711yescharacteristic
357C23#1306330911yescharacteristic
358Q8—3441no—
359Q8—1721no—
360Q8—1721no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—3441no—
381C2×C4—1721no—
382C2×C4—1721no—
383C2×C4—1721no—
384C2×C4—1721no—
385C2×C4C2×(C2×D43)⋊C411yescharacteristic
386C2×C4(C22⋊D43)⋊C411yescharacteristic
387C2×C4#1306329511yescharacteristic
388C2×C4#1306329711yescharacteristic
389C8—81no—
390C8—1721no—
391C8—1721no—
392C8—1722no—
393C8—1721no—
394C8—1721no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22#1458277611yescharacteristic
406C22#1458301311yescharacteristic
407C22#1458301811yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—82no—
416C4—3441no—
417C4—3441no—
418C4#1458287211yescharacteristic
419C4#1458287211yescharacteristic
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#1913492011yescharacteristic
425C2#1913505111yescharacteristic
426C2#1913506611yescharacteristic
subgroupG/H
centerC22#14582776
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1826C2
socleC22×C43#386
prime radicalC24×C43#41
cosocleC23×C4×C43C23
hypercenter#1826D43
hypocenterC43#9240
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
16. C2×#13063258C4semidirect
17. C2×#13063256C4semidirect
19. C2×#13063248C4semidirect
20. C2×#13063248C4semidirect
246. C43#9240semidirect

Maximal subgroups

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

subgroupindex
C2×#145832402
#191432532
#191331142
#191329602
#191324712
#191324432
C43×#18262
#924043