Finite Group Databasebeta

← #36365330

Subgroups — #36365330

Order 22144.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012303C211yesmaximal, characteristic, complemented
2#26012298C211yesmaximal, characteristic, complemented
3#26012296C211yesmaximal, characteristic, complemented
4#26012293C211yesmaximal, characteristic, complemented
5#26011880C211yesmaximal, characteristic, complemented
6#26011762C211yesmaximal, characteristic, complemented
7#26011761C211yesmaximal, characteristic, complemented
8#26011755C211yesmaximal, characteristic, complemented
9#26011754C211yesmaximal, characteristic, complemented
10#26011670C211yesmaximal, characteristic, complemented
11#26011665C211yesmaximal, characteristic, complemented
12#26011660C211yesmaximal, characteristic, complemented
13#26011660C211yesmaximal, characteristic, complemented
14#26011614C211yesmaximal, characteristic, complemented
15#444×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic, complemented
17#19149725C2211yescharacteristic
18#19149725C2211yescharacteristic, complemented
19#19149725C2211yescharacteristic, complemented
20#19149712—21no—
21#19149712C2211yescharacteristic, complemented
22#19149711—21no—
23#19149709C2211yescharacteristic, complemented
24#19149707—21no—
25#19149707—21no—
26#19149707—21no—
27#19149707C2211yescharacteristic
28#19149707C2211yescharacteristic, complemented
29#19149707C2211yescharacteristic, complemented
30#19149706—21no—
31#19149706—21no—
32#19149704C2211yescharacteristic
33#19149704C2211yescharacteristic, complemented
34#19149704C2211yescharacteristic, complemented
35#19149703—21no—
36#19149702—21no—
37#19149701—21no—
38#19149701—21no—
39#19149701C2211yescharacteristic
40#19149700—21no—
41C2×#14583815—21no—
42C2×#14583815C2211yescharacteristic, complemented
43C2×#14583815C2211yescharacteristic, complemented
44SD16×D173—21no—
45C2×D1384—21no—
46C2×D1384C2211yescharacteristic, complemented
47C2×D1384C2211yescharacteristic, complemented
48D8×D173—21no—
49#19149685C2211yescharacteristic, complemented
50#19149685C2211yescharacteristic, complemented
51#19149676C2211yescharacteristic, complemented
52#19149667C2211yescharacteristic, complemented
53#19149615C2211yescharacteristic, complemented
54#19149607C2211yescharacteristic, complemented
55#19149607C2211yescharacteristic, complemented
56#19149597C2211yescharacteristic, complemented
57#19149595C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59#19149595C2211yescharacteristic, complemented
60#135×C173C2211yescharacteristic
61#131×C173C2211yescharacteristic, complemented
62#131×C173C2211yescharacteristic, complemented
63#108×C173C2211yescharacteristic, complemented
64#104×C173C2211yescharacteristic, complemented
65#104×C173C2211yescharacteristic, complemented
66#97×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583827—21no—
69#14583827—21no—
70#14583826—21no—
71#14583826—21no—
72#14583826—21no—
73#14583826—21no—
74#14583826C2311yescharacteristic
75#14583826C2311yescharacteristic
76#14583825—21no—
77#14583825—41no—
78C2×Dic346—21no—
79C2×Dic346C2311yescharacteristic
80Q8×D173—21no—
81C2×D692—21no—
82C2×D692—21no—
83C2×D692—21no—
84C2×D692—41no—
85C2×D692C2311yescharacteristic
86C2×D692C2311yescharacteristic
87C2×D692C2311yescharacteristic
88D4×D173—21no—
89D4×D173—21no—
90D4×D173—21no—
91D4×D173—41no—
92D4×D173—41no—
93D4×D173—41no—
94Dic692—21no—
95Dic692D411yescharacteristic, complemented
96#14583816—21no—
97#14583815—21no—
98#14583815—21no—
99#14583815—21no—
100#14583815—21no—
101#14583815—21no—
102#14583815—21no—
103#14583815—21no—
104#14583815—21no—
105#14583815D411yescharacteristic, complemented
106#14583815D411yescharacteristic, complemented
107#14583814—21no—
108#14583813—21no—
109#14583813—21no—
110#14583813—21no—
111D1384—21no—
112D1384—21no—
113D1384—21no—
114D1384—21no—
115D1384—21no—
116D1384—21no—
117D1384—21no—
118D1384D411yescharacteristic, complemented
119#14583811—21no—
120#14583811—21no—
121#14583811—21no—
122#14583810C2311yescharacteristic
123#14583810C2311yescharacteristic
124#14583809—21no—
125#14583809—21no—
126C8×D173—21no—
127C8×D173—21no—
128#14583803—41no—
129#41×C173C2311yescharacteristic
130#41×C173C2311yescharacteristic
131Q16×C173D411yescharacteristic, complemented
132SD16×C173D411yescharacteristic, complemented
133SD16×C173D411yescharacteristic, complemented
134D8×C173D411yescharacteristic, complemented
135M16×C173—21no—
136M16×C173—21no—
137M16×C173C2311yescharacteristic
138M16×C173C2311yescharacteristic
139C2×C8×C173—21no—
140C2×C8×C173—21no—
141C2×C8×C173C2311yescharacteristic
142C2×C8×C173C2311yescharacteristic
143C42×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—21no—
146C22×D173—21no—
147C22×D173—41no—
148C22×D173—41no—
149C22×D173—41no—
150Dic346—21no—
151Dic346—21no—
152Dic346—41no—
153Dic346C2×D411yescharacteristic
154Dic346C2×D411yescharacteristic
155D692—21no—
156D692—21no—
157D692—21no—
158D692—21no—
159D692—21no—
160D692—21no—
161D692—41no—
162D692—41no—
163D692—41no—
164D692—82no—
165D692C2×D411yescharacteristic
166D692C2×D411yescharacteristic
167C22⋊D173—21no—
168C22⋊D173—21no—
169C22⋊D173—41no—
170C22⋊D173—41no—
171C22⋊D173—41no—
172C22⋊D173—41no—
173C2×Dic173—21no—
174C2×Dic173—41no—
175C4×D173—21no—
176C4×D173—21no—
177C4×D173—41no—
178C4×D173—41no—
179C4×D173—41no—
180C4×D173—41no—
181#13063436—21no—
182#13063436—21no—
183Q8×C173C2×D411yescharacteristic
184Q8×C173C2×D411yescharacteristic
185D4×C173—21no—
186D4×C173—21no—
187D4×C173C2×D411yescharacteristic
188D4×C173C2×D411yescharacteristic
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173C2411yescharacteristic
193C8×C173—21no—
194C8×C173—21no—
195C8×C173C2×D411yescharacteristic
196C8×C173C2×D411yescharacteristic
197C8×C173C2×D411yescharacteristic
198C8×C173C2×D411yescharacteristic
199C22×C173—21no—
200C22×C173—21no—
201C22×C173H(5,𝔽2)11yescharacteristic
202C2×D173—21no—
203C2×D173—21no—
204C2×D173—41no—
205C2×D173—41no—
206C2×D173—41no—
207C2×D173—81no—
208C2×D173—81no—
209C2×D173—81no—
210C2×D173—161no—
211C2×D173—161no—
212C2×D173—161no—
213C4×C173—21no—
214C4×C173—21no—
215C4×C173—41no—
216C4×C173C22×D411yescharacteristic
217C4×C173C22×D411yescharacteristic
218Dic173—21no—
219Dic173—21no—
220Dic173—41no—
221C2×C173—21no—
222C2×C173—41no—
223C2×C173—41no—
224C2×C173D4211yescharacteristic
225D173—41no—
226D173—41no—
227D173—81no—
228D173—81no—
229D173—81no—
230C173#328811yescharacteristic, complemented, Sylow
231#3288—1731nomaximal, Sylow
232#577—1731no—
233#577—1731no—
234#576—1731no—
235#576—1731no—
236#496—1731no—
237#472—1731no—
238#472—1731no—
239#471—1731no—
240#471—1731no—
241#455—1731no—
242#454—1731no—
243#453—1731no—
244#453—1731no—
245#444—1731no—
246#444D17311yescharacteristic, complemented
2472−1+4—1731no—
248H(5,𝔽2)—1731no—
249H(5,𝔽2)—1731no—
250H(5,𝔽2)—1731no—
251#137—3461no—
252#137—3461no—
253#137—1731no—
254#137—1731no—
255AGL(1,ℤ/8)—3461no—
256AGL(1,ℤ/8)—3461no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—3461no—
259AGL(1,ℤ/8)—3461no—
260AGL(1,ℤ/8)—3461no—
261AGL(1,ℤ/8)—1731no—
262AGL(1,ℤ/8)—1731no—
263AGL(1,ℤ/8)—1731no—
264AGL(1,ℤ/8)—1731no—
265AGL(1,ℤ/8)—1731no—
266AGL(1,ℤ/8)—1731no—
267#135—3461no—
268#135—3461no—
269#135—3461no—
270#135—3461no—
271#135—1731no—
272#135C2×D17311yescharacteristic
273C2×SD16—3461no—
274C2×SD16—3461no—
275C2×SD16—1731no—
276C2×SD16—1731no—
277C2×D8—3461no—
278C2×D8—3461no—
279C2×D8—1731no—
280C2×D8—1731no—
281#131—1731no—
282#131—1731no—
283#131C2×D17311yescharacteristic, complemented
284#131C2×D17311yescharacteristic, complemented
285#127—1731no—
286#124—1731no—
287#108—1731no—
288#108C2×D17311yescharacteristic, complemented
289#104—1731no—
290#104—1731no—
291#104C2×D17311yescharacteristic, complemented
292#104C2×D17311yescharacteristic, complemented
293#101—1731no—
294#100—1731no—
295#100—1731no—
296#100—1731no—
297#97C2×D17311yescharacteristic, complemented
298#41—3461no—
299#41—3461no—
300#41—3461no—
301#41—3461no—
302#41—3461no—
303#41—3461no—
304#41—3461no—
305#41—3461no—
306#41—1731no—
307#41—1731no—
308#41—6921no—
309#41C22×D17311yescharacteristic
310#41C22×D17311yescharacteristic
311C2×Q8—3461no—
312C2×Q8—3461no—
313C2×Q8—1731no—
314C2×D4—3461no—
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—3461no—
318C2×D4—3461no—
319C2×D4—3461no—
320C2×D4—1731no—
321C2×D4—1731no—
322C2×D4—1731no—
323C2×D4—6921no—
324C2×D4—6921no—
325C2×D4—6921no—
326C2×D4—6921no—
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16D69211yescharacteristic, complemented
331SD16—3461no—
332SD16—3461no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—3461no—
343SD16—1731no—
344SD16—1731no—
345SD16D69211yescharacteristic, complemented
346SD16D69211yescharacteristic, complemented
347D8—3461no—
348D8—3461no—
349D8—3461no—
350D8—3461no—
351D8—3461no—
352D8—3461no—
353D8—3461no—
354D8—3461no—
355D8—3461no—
356D8—3461no—
357D8—1731no—
358D8D69211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—3461no—
362M16—3461no—
363M16—1731no—
364M16—1731no—
365M16C22×D17311yescharacteristic
366M16C22×D17311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—3461no—
370C2×C8—3461no—
371C2×C8C22×D17311yescharacteristic
372C2×C8C22×D17311yescharacteristic
373C22⋊C4—6921no—
374C42C22×D17311yescharacteristic
375C23—3461no—
376C23—3461no—
377C23—3461no—
378C23—6921no—
379C23—6921no—
380C23—6921no—
381Q8—3461no—
382Q8—3461no—
383Q8—1731no—
384Q8—1731no—
385Q8—6921no—
386Q8C2×D69211yescharacteristic
387Q8C2×D69211yescharacteristic
388D4—21no—
389D4—21no—
390D4—3461no—
391D4—3461no—
392D4—3461no—
393D4—3461no—
394D4—3461no—
395D4—3461no—
396D4—3461no—
397D4—3461no—
398D4—13842no—
399D4—1731no—
400D4—1731no—
401D4—6921no—
402D4—6921no—
403D4—6921no—
404D4—6921no—
405D4—6921no—
406D4—6921no—
407D4—6921no—
408D4C2×D69211yescharacteristic
409D4C2×D69211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—3461no—
414C2×C4—3461no—
415C2×C4—3461no—
416C2×C4—6921no—
417C2×C4—6921no—
418C2×C4—6921no—
419C2×C4—6921no—
420C2×C4—6921no—
421C2×C4C23×D17311yescharacteristic
422C8—21no—
423C8—21no—
424C8—3461no—
425C8—3461no—
426C8D4×D17311yescharacteristic
427C8D4×D17311yescharacteristic
428C8C2×D69211yescharacteristic
429C8C2×D69211yescharacteristic
430C22—21no—
431C22—21no—
432C22—3461no—
433C22—3461no—
434C22—13841no—
435C22—13841no—
436C22—13841no—
437C22—6921no—
438C22—6921no—
439C22—6921no—
440C22—27681no—
441C22—27681no—
442C22—27681no—
443C22#1914972511yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3461no—
448C4—3461no—
449C4—6921no—
450C4C2×D4×D17311yescharacteristic
451C4C22×D69211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—13841no—
456C2—13841no—
457C2—13841no—
458C2—6921no—
459C2—6921no—
460C2D4×D69211yescharacteristic
subgroupG/H
centerC2D4×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#444×C173C2
socleC2×C173D42
hypercenter#444D173
hypocenterC173#3288
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #26012303C2semidirect
2. #26012298C2semidirect
3. #26012296C2semidirect
4. #26012293C2semidirect
5. #26011880C2semidirect
6. #26011762C2semidirect
7. #26011761C2semidirect
8. #26011755C2semidirect
9. #26011754C2semidirect
10. #26011670C2semidirect
11. #26011665C2semidirect
12. #26011660C2semidirect
13. #26011660C2semidirect
14. #26011614C2semidirect
15. #444×C173C2semidirect
16. #19149726C22semidirect
18. #19149725C22semidirect
19. #19149725C22semidirect
21. #19149712C22semidirect
23. #19149709C22semidirect
28. #19149707C22semidirect
29. #19149707C22semidirect
33. #19149704C22semidirect
34. #19149704C22semidirect
42. C2×#14583815C22semidirect
43. C2×#14583815C22semidirect
46. C2×D1384C22semidirect
47. C2×D1384C22semidirect
49. #19149685C22semidirect
50. #19149685C22semidirect
51. #19149676C22semidirect
52. #19149667C22semidirect
53. #19149615C22semidirect
54. #19149607C22semidirect
55. #19149607C22semidirect
56. #19149597C22semidirect
57. #19149595C22semidirect
58. #19149595C22semidirect
59. #19149595C22semidirect
61. #131×C173C22semidirect
62. #131×C173C22semidirect
63. #108×C173C22semidirect
64. #104×C173C22semidirect
65. #104×C173C22semidirect
66. #97×C173C22semidirect
95. Dic692D4semidirect
105. #14583815D4semidirect
106. #14583815D4semidirect
118. D1384D4semidirect
131. Q16×C173D4semidirect
132. SD16×C173D4semidirect
133. SD16×C173D4semidirect
134. D8×C173D4semidirect
230. C173#3288semidirect
246. #444D173semidirect
283. #131C2×D173semidirect
284. #131C2×D173semidirect
288. #108C2×D173semidirect
291. #104C2×D173semidirect
292. #104C2×D173semidirect
297. #97C2×D173semidirect
330. Q16D692semidirect
345. SD16D692semidirect
346. SD16D692semidirect
358. D8D692semidirect

Maximal subgroups

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

subgroupindex
#260123032
#260122982
#260122962
#260122932
#260118802
#260117622
#260117612
#260117552
#260117542
#260116702
#260116652
#260116602
#260116142
#444×C1732
#3288173