Finite Group Databasebeta

← #25253496

Subgroups — #25253496

Order 11008.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19146964C211yesmaximal, characteristic, complemented
2#19146338C211yesmaximal, characteristic
3#19136827C211yesmaximal, characteristic, complemented
4#19136804C211yesmaximal, characteristic, complemented
5#19136776C211yesmaximal, characteristic, complemented
6#19136760C211yesmaximal, characteristic, complemented
7#19136691C211yesmaximal, characteristic, complemented
8#19136686C211yesmaximal, characteristic, complemented
9#19136658C211yesmaximal, characteristic, complemented
10#19136537C211yesmaximal, characteristic, complemented
11#19136531C211yesmaximal, characteristic, complemented
12C4×Dic344C211yesmaximal, characteristic, complemented
13#19136303C211yesmaximal, characteristic, complemented
14C2×#14582704C211yesmaximal, characteristic, complemented
15D4×C16×C43C211yesmaximal, characteristic, complemented
16C2×#13063347C2211yescharacteristic
17C22×Dic172C2211yescharacteristic
18#14583589—21no—
19#14583558—21no—
20C2×Dic344C2211yescharacteristic
21C2×Dic344C2211yescharacteristic
22C2×#13063270C2211yescharacteristic
23#14583301C2211yescharacteristic
24#14583283C2211yescharacteristic
25#14583241—21no—
26#14583230—21no—
27#14583207—21no—
28#14583140C2211yescharacteristic
29#14583140C2211yescharacteristic
30#14583140C2211yescharacteristic
31#14583135C2211yescharacteristic
32#14583122—21no—
33#14583063—21no—
34#14583040—21no—
35C4×Dic172C2211yescharacteristic
36C4×Dic172C2211yescharacteristic
37#14582937C2211yescharacteristic
38C2×#13063259C2211yescharacteristic
39#14582727C2211yescharacteristic
40#14582725C2211yescharacteristic
41#14582725C2211yescharacteristic
42#14582704C2211yescharacteristic
43#14582704C2211yescharacteristic
44#14582704C2211yescharacteristic
45#14582704C2211yescharacteristic
46#14582704C2211yescharacteristic
47#14582704C2211yescharacteristic
48#14582704C2211yescharacteristic
49#14582701C2211yescharacteristic
50#14582701C2211yescharacteristic
51#14582701C2211yescharacteristic
52C22×C16×C43C2211yescharacteristic
53C22×C16×C43C2211yescharacteristic
54C8×D4×C43C2211yescharacteristic
55C43×C4⋊C16C2211yescharacteristic
56C43×C22⋊C16C2211yescharacteristic
57C43×C22⋊C16C2211yescharacteristic
58C4×C16×C43C2211yescharacteristic
59C2×#10621898—21no—
60C22×Dic86—21no—
61#13063347—22no—
62#13063347—41no—
63C2×Dic172—22no—
64C2×Dic172—21no—
65C2×Dic172—41no—
66C2×Dic172C2311yescharacteristic
67C2×Dic172C2311yescharacteristic
68C2×Dic172C2311yescharacteristic
69C2×#10621873—21no—
70C2×#10621873—21no—
71C2×D344C2311yescharacteristic
72#13063318—41no—
73#13063317—21no—
74#13063315—41no—
75#13063313—21no—
76#13063311—41no—
77#13063306—41no—
78#13063304—21no—
79#13063304—21no—
80#13063304—21no—
81#13063303—41no—
82#13063299—21no—
83#13063297—41no—
84C4×Dic86—21no—
85C4×Dic86—21no—
86(C22⋊D43)⋊C4—41no—
87#13063284—21no—
88#13063281—41no—
89C2×C4⋊Dic43—21no—
90Dic344—22no—
91Dic344—41no—
92#13063270—41no—
93#13063259—21no—
94#13063259C2311yescharacteristic
95#13063259C2311yescharacteristic
96#13063259C2311yescharacteristic
97#13063259C2311yescharacteristic
98#13063257—21no—
99#13063257—21no—
100#13063250—21no—
101#13063250—21no—
102#13063250—21no—
103#13063250—21no—
104#13063250—21no—
105#13063247—21no—
106C22×C8×C43C2311yescharacteristic
107C22×C8×C43C2311yescharacteristic
108C4×D4×C43D411yescharacteristic
109C2×C16×C43—41no—
110C2×C16×C43—41no—
111C2×C16×C43C2311yescharacteristic
112C2×C16×C43C2311yescharacteristic
113C2×C16×C43C2311yescharacteristic
114C2×C16×C43C2311yescharacteristic
115C4⋊C8×C43D411yescharacteristic
116C22⋊C8×C43D411yescharacteristic
117C22⋊C8×C43D411yescharacteristic
118C4×C8×C43C2311yescharacteristic
119C2×D4×C43D811yescharacteristic
120C22×C4×C43C2×D411yescharacteristic
121C22×C4×C43C2×D411yescharacteristic
122C2×C22⋊D43—41no—
123C2×C22⋊D43—41no—
124C22×Dic43—41no—
125C22×Dic43—41no—
126#10621898—42no—
127#10621898—81no—
128C2×D172—21no—
129C2×C4×D43—41no—
130C2×Dic86—21no—
131C2×Dic86—21no—
132C2×Dic86—21no—
133C2×Dic86—42no—
134C2×Dic86—41no—
135C2×Dic86—81no—
136C2×C8×C43—21no—
137C2×C8×C43—21no—
138C2×C8×C43—21no—
139C2×C8×C43—21no—
140C2×C8×C43C2×D411yescharacteristic
141C2×C8×C43C2×D411yescharacteristic
142C2×C8×C43C2×D411yescharacteristic
143C2×C8×C43C2411yescharacteristic
144C4⋊C4×C43D811yescharacteristic
145C22⋊C4×C43D811yescharacteristic
146C22⋊C4×C43D811yescharacteristic
147C42×C43C2×D411yescharacteristic
148C22⋊Dic43—41no—
149C22⋊Dic43—41no—
150C22⋊Dic43—41no—
151C22⋊Dic43—41no—
152(C2×D43)⋊C4—41no—
153(C2×D43)⋊C4—41no—
154(C2×D43)⋊C4—41no—
155C4⋊Dic43—21no—
156C4⋊Dic43—21no—
157C4⋊Dic43—21no—
158C4⋊Dic43—21no—
159C4⋊Dic43—41no—
160Dic43⋊C4—41no—
161Dic43⋊C4—41no—
162Dic43⋊C4—41no—
163Dic43⋊C4—41no—
164Dic43⋊C4—41no—
165Dic43⋊C4—41no—
166Dic43⋊C4—41no—
167C4×Dic43—41no—
168C4×Dic43—41no—
169C4×Dic43—41no—
170Dic172—21no—
171Dic172—21no—
172Dic172—21no—
173Dic172—21no—
174Dic172—21no—
175Dic172—42no—
176Dic172#4112yes—
177D344—21no—
178D344—21no—
179#10621873—42no—
180#10621873—81no—
181C16×C43—21no—
182C16×C43—21no—
183C16×C43—21no—
184C16×C43—21no—
185C23×C43C2×D811yescharacteristic
186C23×C43C2×D811yescharacteristic
187C22×D43—41no—
188D4×C43—22no—
189C2×C4×C43—21no—
190C2×C4×C43—21no—
191C2×C4×C43—21no—
192C2×C4×C43—21no—
193C2×C4×C43C2×D811yescharacteristic
194C2×C4×C43C2×D811yescharacteristic
195C2×C4×C43C2×D811yescharacteristic
196C2×C4×C43C2×D811yescharacteristic
197C2×C4×C43C22×D411yescharacteristic
198C22⋊D43—82no—
199C22⋊D43—82no—
200C2×Dic43—41no—
201C2×Dic43—41no—
202C2×Dic43—41no—
203C2×Dic43—41no—
204C2×Dic43—41no—
205C2×Dic43—41no—
206C2×Dic43—41no—
207C2×Dic43—82no—
208C2×Dic43—82no—
209D172—41no—
210D172—41no—
211C4×D43—82no—
212Dic86—22no—
213Dic86—41no—
214Dic86—41no—
215Dic86—41no—
216Dic86—41no—
217Dic86—41no—
218Dic86—82no—
219C8×C43—21no—
220C8×C43—21no—
221C8×C43—21no—
222C8×C43C2×#4111yescharacteristic
223C8×C43H(5,𝔽2)11yescharacteristic
224#33867—431nomaximal, Sylow
225C22×C43—21no—
226C22×C43—21no—
227C22×C43—21no—
228C22×C43—21no—
229C22×C43—21no—
230C22×C43#50812yes—
231C22×C43C22×D811yescharacteristic
232C2×D43—81no—
233C2×D43—81no—
234C2×D43—81no—
235C4×C43—21no—
236C4×C43—21no—
237C4×C43—21no—
238C4×C43—21no—
239C4×C43#54611yescharacteristic
240C4×C43#57711yescharacteristic
241Dic43—42no—
242Dic43—81no—
243Dic43—81no—
244Dic43—81no—
245Dic43—81no—
246Dic43—81no—
247Dic43—81no—
248#3336—431no—
249#3285—431no—
250#2242—431no—
251#2237—431no—
252#2232—431no—
253#2229—431no—
254#2217—431no—
255#2216—431no—
256#2211—431no—
257#2189—431no—
258#2188—431no—
259C4×Q32—431no—
260D4×C16D4311yescharacteristic, complemented
261#2138—431no—
262C2×#358—431no—
263C2×C43—22no—
264C2×C43—41no—
265C2×C43#329411yescharacteristic
266C2×C43C2×#50811yescharacteristic
267C2×C43#341711yescharacteristic
268D43—161no—
269C2×#135—431no—
270C22×Q16—431no—
271#552—861no—
272#548—861no—
273C2×Q32—431no—
274C2×Q32—431no—
275C2×SD32—431no—
276C22×C16C2×D4311yescharacteristic
277C22×C16C2×D4311yescharacteristic
278#499—431no—
279#495—431no—
280#485—861no—
281#483—861no—
282#479—861no—
283#467—431no—
284#467—431no—
285#467—431no—
286#466—431no—
287#464—861no—
288#452—861no—
289#448—861no—
290C4×Q16—431no—
291C4×Q16—431no—
292C8×D4C2×D4311yescharacteristic
293#427—431no—
294C2×#107—431no—
295#367—431no—
296#366—431no—
297#366—431no—
298C4⋊C16C2×D4311yescharacteristic
299#358—431no—
300#358—431no—
301#358—431no—
302#358—431no—
303#358—431no—
304#358—431no—
305#358—431no—
306#357—431no—
307#357—431no—
308#357—431no—
309C22⋊C16C2×D4311yescharacteristic
310C22⋊C16C2×D4311yescharacteristic
311C4×C16C2×D4311yescharacteristic
312C43#3386711yescharacteristic, complemented, Sylow
313C2×#41—861no—
314C22×Q8—861no—
315#135—862no—
316#135—1721no—
317C2×Q16—431no—
318C2×Q16—431no—
319C2×Q16—431no—
320C2×Q16—862no—
321C2×Q16—861no—
322C2×Q16—1721no—
323C2×SD16—861no—
324C2×SD16—861no—
325C2×D8—431no—
326C22×C8C22×D4311yescharacteristic
327C22×C8C22×D4311yescharacteristic
328#126—1721no—
329#125—861no—
330#125—1721no—
331#124—861no—
332#124—1721no—
333#123—1721no—
334C22⋊Q8—861no—
335C22⋊Q8—861no—
336C22⋊Q8—861no—
337C22⋊Q8—1721no—
338#121—861no—
339#121—1721no—
340C4×Q8—861no—
341C4×Q8—861no—
342C4×D4—1721no—
343C4×D4D17211yescharacteristic
344#117—861no—
345#117—1721no—
346C2×C4⋊C4—861no—
347Q32—862no—
348Q32—1721no—
349SD32—1721no—
350C2×C16—41no—
351C2×C16—41no—
352C2×C16C22×D4311yescharacteristic
353C2×C16C22×D4311yescharacteristic
354C2×C16C22×D4311yescharacteristic
355C2×C16C22×D4311yescharacteristic
356#107—431no—
357#107—431no—
358#107—431no—
359#107—431no—
360#107—861no—
361#106—861no—
362#106—861no—
363C4⋊C8D17211yescharacteristic
364#103—861no—
365#103—861no—
366#103—861no—
367#103—861no—
368#103—861no—
369#102—861no—
370C22⋊C8D17211yescharacteristic
371C22⋊C8D17211yescharacteristic
372C4×C8C22×D4311yescharacteristic
373#41—3441no—
374#41—1722no—
375C2×Q8—861no—
376C2×Q8—861no—
377C2×Q8—861no—
378C2×Q8—3441no—
379C2×Q8—1722no—
380C2×Q8—1721no—
381C2×D4—861no—
382C2×D4—1721no—
383C2×D4—1721no—
384C2×D4D34411yescharacteristic
385C22×C4—1721no—
386C22×C4—1721no—
387C22×C4—1721no—
388C22×C4C2×D17211yescharacteristic
389C22×C4C2×D17211yescharacteristic
390Q16—432no—
391Q16—861no—
392Q16—861no—
393Q16—861no—
394Q16—861no—
395Q16—861no—
396Q16—1722no—
397SD16—3441no—
398SD16—1722no—
399D8—861no—
400D8—861no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×D17211yescharacteristic
406C2×C8C2×D17211yescharacteristic
407C2×C8C2×D17211yescharacteristic
408C2×C8C23×D4311yescharacteristic
409C4⋊C4—861no—
410C4⋊C4—861no—
411C4⋊C4—861no—
412C4⋊C4—861no—
413C4⋊C4—1721no—
414C4⋊C4—1721no—
415C4⋊C4—1721no—
416C4⋊C4—1721no—
417C4⋊C4—1721no—
418C4⋊C4—1721no—
419C4⋊C4—1721no—
420C4⋊C4—1721no—
421C4⋊C4D34411yescharacteristic
422C22⋊C4—1721no—
423C22⋊C4—1721no—
424C22⋊C4—1721no—
425C22⋊C4—1721no—
426C22⋊C4—1721no—
427C22⋊C4—1721no—
428C22⋊C4—1721no—
429C22⋊C4D34411yescharacteristic
430C22⋊C4D34411yescharacteristic
431C42—1721no—
432C42—1721no—
433C42—1721no—
434C42C2×D17211yescharacteristic
435C16—21no—
436C16—21no—
437C16—21no—
438C16—21no—
439C23—1721no—
440C23C2×D34411yescharacteristic
441C23C2×D34411yescharacteristic
442Q8—862no—
443Q8—3442no—
444Q8—1721no—
445Q8—1721no—
446Q8—1721no—
447Q8—1721no—
448Q8—1721no—
449D4—22no—
450D4—3442no—
451D4—3442no—
452D4—1721no—
453D4—1721no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—3442no—
459C2×C4—3442no—
460C2×C4—3442no—
461C2×C4—1721no—
462C2×C4—1721no—
463C2×C4—1721no—
464C2×C4—1721no—
465C2×C4—1721no—
466C2×C4—1721no—
467C2×C4—1721no—
468C2×C4C2×D34411yescharacteristic
469C2×C4C2×D34411yescharacteristic
470C2×C4C2×D34411yescharacteristic
471C2×C4C2×D34411yescharacteristic
472C2×C4C22×D17211yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8#41×D4311yescharacteristic
477C8#1306337011yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—3441no—
484C22—3441no—
485C22—3441no—
486C22#1458333112yes—
487C22C22×D34411yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—3441no—
493C4—3441no—
494C4—3441no—
495C4—3441no—
496C4—3441no—
497C4—3441no—
498C4—1722no—
499C4#1458354411yescharacteristic
500C4#1458371111yescharacteristic
501C2—22no—
502C2—41no—
503C2—6881no—
504C2#1914640911yescharacteristic
505C2C2×#1458333111yescharacteristic
506C2#1914774811yescharacteristic
subgroupG/H
centerC22C22×D344
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingD4×C16×C43C2
socleC22×C43C22×D8
prime radical#19146338C2
hypercenterD4×C16D43
hypocenterC43#33867
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #19146964C2semidirect
3. #19136827C2semidirect
4. #19136804C2semidirect
5. #19136776C2semidirect
6. #19136760C2semidirect
7. #19136691C2semidirect
8. #19136686C2semidirect
9. #19136658C2semidirect
10. #19136537C2semidirect
11. #19136531C2semidirect
12. C4×Dic344C2semidirect
13. #19136303C2semidirect
14. C2×#14582704C2semidirect
15. D4×C16×C43C2semidirect
260. D4×C16D43semidirect
312. C43#33867semidirect

Maximal subgroups

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

subgroupindex
#191469642
#191463382
#191368272
#191368042
#191367762
#191367602
#191366912
#191366862
#191366582
#191365372
#191365312
C4×Dic3442
#191363032
C2×#145827042
D4×C16×C432
#3386743