Finite Group Databasebeta

← #36363278

Subgroups — #36363278

Order 22144.

452 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012290C211yesmaximal, characteristic, complemented
2#26012287C211yesmaximal, characteristic, complemented
3C2×#19149700C211yesmaximal, characteristic, complemented
4#131×D173C211yesmaximal, characteristic, complemented
5#26011765C211yesmaximal, characteristic, complemented
6#26011764C211yesmaximal, characteristic, complemented
7#26011760C211yesmaximal, characteristic, complemented
8#26011759C211yesmaximal, characteristic, complemented
9#26011755C211yesmaximal, characteristic, complemented
10#26011754C211yesmaximal, characteristic, complemented
11#26011751C211yesmaximal, characteristic, complemented
12#108×D173C211yesmaximal, characteristic, complemented
13#26011474C211yesmaximal, characteristic, complemented
14#26011472C211yesmaximal, characteristic, complemented
15#471×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18#41×D173C2211yescharacteristic
19#19149712C2211yescharacteristic
20#19149711—21no—
21#19149710C2211yescharacteristic
22#19149709C2211yescharacteristic
23#19149708—21no—
24#19149707C2211yescharacteristic
25#19149706—21no—
26#19149705C2211yescharacteristic
27#19149704C2211yescharacteristic
28#19149703—21no—
29#19149702—21no—
30#19149700—21no—
31#19149700—21no—
32#19149700—21no—
33#19149700—21no—
34#19149700—21no—
35#19149700—21no—
36#19149698—21no—
37C2×#14583816C2211yescharacteristic, complemented
38C2×#14583815C2211yescharacteristic, complemented
39C2×#14583811C2211yescharacteristic, complemented
40#19149687—21no—
41#19149687C2211yescharacteristic
42#19149686C2211yescharacteristic
43#19149685—21no—
44#19149685C2211yescharacteristic
45M16×D173—21no—
46M16×D173C2211yescharacteristic
47M16×D173C2211yescharacteristic
48M16×D173C2211yescharacteristic
49C2×C8×D173—21no—
50C2×C8×D173C2211yescharacteristic
51#19149615C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149614C2211yescharacteristic, complemented
54#19149598C2211yescharacteristic, complemented
55#19149597C2211yescharacteristic, complemented
56#19149597C2211yescharacteristic, complemented
57#19149596C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59#19149595C2211yescharacteristic, complemented
60#137×C173C2211yescharacteristic
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62C2×SD16×C173C2211yescharacteristic, complemented
63#131×C173C2211yescharacteristic
64#108×C173C2211yescharacteristic, complemented
65#101×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583827—21no—
69#14583827—21no—
70#14583827—21no—
71#14583827D411yescharacteristic
72#14583826—21no—
73#14583826C2311yescharacteristic
74#14583825—21no—
75#14583825—21no—
76#14583825—21no—
77#14583825—21no—
78#14583825—21no—
79#14583825D411yescharacteristic
80C2×Dic346C2311yescharacteristic
81Q8×D173D411yescharacteristic
82C2×D692—21no—
83C2×D692C2311yescharacteristic
84C2×C22⋊D173—21no—
85D4×D173—21no—
86D4×D173D411yescharacteristic
87C22×Dic173—21no—
88C2×C4×D173—21no—
89C2×C4×D173—21no—
90C2×C4×D173C2311yescharacteristic
91Dic692—21no—
92#14583816—21no—
93#14583816—21no—
94#14583816—21no—
95#14583816—21no—
96#14583815—21no—
97#14583815—21no—
98#14583815—21no—
99#14583815—21no—
100#14583814—21no—
101#14583814—21no—
102#14583813—21no—
103#14583813—21no—
104D1384—21no—
105#14583811—21no—
106#14583811—21no—
107#14583811—21no—
108#14583811—21no—
109#14583810—21no—
110#14583810C2311yescharacteristic
111#14583810C2311yescharacteristic
112#14583810C2311yescharacteristic
113#14583809—21no—
114#14583809—21no—
115#14583809—21no—
116#14583809—21no—
117#14583809—21no—
118C2×#13063436—21no—
119C2×#13063436C2311yescharacteristic
120C8×D173—21no—
121C8×D173—21no—
122C8×D173—21no—
123C8×D173—21no—
124C8×D173—21no—
125C8×D173D411yescharacteristic
126C8×D173D411yescharacteristic
127C8×D173D411yescharacteristic
128C8×D173D411yescharacteristic
129#41×C173C2311yescharacteristic
130C2×Q8×C173C2311yescharacteristic
131C2×D4×C173C2311yescharacteristic
132Q16×C173—21no—
133SD16×C173—21no—
134SD16×C173—21no—
135SD16×C173—21no—
136SD16×C173—21no—
137D8×C173—21no—
138M16×C173—21no—
139M16×C173C2311yescharacteristic
140M16×C173C2311yescharacteristic
141M16×C173C2311yescharacteristic
142C2×C8×C173—21no—
143C2×C8×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—21no—
146C22×D173#4111yescharacteristic
147Dic346—21no—
148Dic346—21no—
149Dic346—21no—
150Dic346C2×D411yescharacteristic
151D692—21no—
152D692—21no—
153D692—21no—
154D692—41no—
155D692—41no—
156D692C2×D411yescharacteristic
157C22⋊D173—21no—
158C22⋊D173—21no—
159C22⋊D173—21no—
160C22⋊D173—41no—
161C22⋊D173—41no—
162C2×Dic173—21no—
163C2×Dic173—21no—
164C2×Dic173—41no—
165C2×Dic173—41no—
166C2×Dic173#4111yescharacteristic
167C4×D173—21no—
168C4×D173—21no—
169C4×D173—21no—
170C4×D173—41no—
171C4×D173—41no—
172C4×D173C2×D411yescharacteristic
173C4×D173C2×D411yescharacteristic
174C4×D173C2×D411yescharacteristic
175C4×D173C2×D411yescharacteristic
176#13063436—21no—
177#13063436—21no—
178#13063436—21no—
179#13063436C2×D411yescharacteristic
180#13063436C2×D411yescharacteristic
181C23×C173—21no—
182Q8×C173—21no—
183Q8×C173—21no—
184Q8×C173C2×D411yescharacteristic
185D4×C173—21no—
186D4×C173—21no—
187D4×C173—21no—
188D4×C173C2×D411yescharacteristic
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173C2411yescharacteristic
192C8×C173—21no—
193C8×C173—21no—
194C8×C173—21no—
195C8×C173C2×D411yescharacteristic
196C8×C173C2×D411yescharacteristic
197C22×C173—21no—
198C22×C173—41no—
199C22×C173—81no—
200C22×C173C2×#4111yescharacteristic
201C2×D173—21no—
202C2×D173—41no—
203C2×D173—42no—
204C2×D173—41no—
205C2×D173—81no—
206C2×D173#12111yescharacteristic
207C2×D173#12111yescharacteristic
208C4×C173—21no—
209C4×C173—41no—
210C4×C173C22×D411yescharacteristic
211C4×C173C22×D411yescharacteristic
212Dic173—21no—
213Dic173—41no—
214Dic173#12111yescharacteristic
215Dic173#12111yescharacteristic
216C2×C173—21no—
217C2×C173—41no—
218C2×C173—81no—
219C2×C173C2×#12111yescharacteristic
220D173—21no—
221D173—21no—
222D173—41no—
223D173—81no—
224C173#306711yescharacteristic, complemented, Sylow
225#3067—1731nomaximal, Sylow
226#575—1731no—
227#575—1731no—
228C2×#135—1731no—
229C2×#131—1731no—
230#473—1731no—
231#473—1731no—
232#472—1731no—
233#472—1731no—
234#471—1731no—
235#471—1731no—
236#471—1731no—
237#471D17311yescharacteristic, complemented
238C2×#108—1731no—
239#413—1731no—
240#413—1731no—
241C2×#41—1731no—
242C2×#41—1731no—
243C2×#41—1731no—
244#137—3461no—
245#137—3461no—
246#137—1731no—
247#137—1731no—
248#137—1731no—
249#137C2×D17311yescharacteristic
250AGL(1,ℤ/8)—3461no—
251AGL(1,ℤ/8)—3461no—
252AGL(1,ℤ/8)—1731no—
253AGL(1,ℤ/8)—1731no—
254AGL(1,ℤ/8)—1731no—
255AGL(1,ℤ/8)C2×D17311yescharacteristic
256#135—3461no—
257#135—3461no—
258#135—3461no—
259#135—3461no—
260#135—3461no—
261#135—3461no—
262#135—3461no—
263#135—3461no—
264C2×Q16—1731no—
265C2×SD16—1731no—
266C2×SD16C2×D17311yescharacteristic, complemented
267C2×D8—1731no—
268#131—3461no—
269#131—3461no—
270#131—1731no—
271#131—1731no—
272#131—1731no—
273#131C2×D17311yescharacteristic
274C2×M16—3461no—
275C2×M16—1731no—
276C2×M16—1731no—
277C2×M16—1731no—
278C22×C8—3461no—
279C22×C8—1731no—
280#108—1731no—
281#108—1731no—
282#108—1731no—
283#108C2×D17311yescharacteristic, complemented
284#101—1731no—
285#101—1731no—
286#101—1731no—
287#101C2×D17311yescharacteristic, complemented
288#100—1731no—
289#100—1731no—
290#100—1731no—
291#100C2×D17311yescharacteristic, complemented
292#41—3461no—
293#41—3461no—
294#41—3461no—
295#41—3461no—
296#41—3461no—
297#41—3461no—
298#41—3461no—
299#41—3461no—
300#41—3461no—
301#41—3461no—
302#41—1731no—
303#41—1731no—
304#41—1731no—
305#41C22×D17311yescharacteristic
306C2×Q8—1731no—
307C2×Q8—1731no—
308C2×Q8C22×D17311yescharacteristic
309C2×D4—3461no—
310C2×D4—3461no—
311C2×D4—3461no—
312C2×D4—1731no—
313C2×D4—1731no—
314C2×D4C22×D17311yescharacteristic
315C22×C4—3461no—
316C22×C4—3461no—
317C22×C4—3461no—
318C22×C4—1731no—
319Q16—21no—
320Q16—3461no—
321Q16—3461no—
322Q16—3461no—
323Q16—3461no—
324Q16—3461no—
325SD16—21no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—3461no—
330SD16—3461no—
331SD16—3461no—
332SD16—3461no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337D8—21no—
338D8—3461no—
339D8—3461no—
340D8—3461no—
341D8—3461no—
342D8—3461no—
343M16—21no—
344M16—3461no—
345M16—3461no—
346M16—3461no—
347M16—3461no—
348M16—3461no—
349M16—3461no—
350M16—1731no—
351M16—1731no—
352M16—1731no—
353M16C22×D17311yescharacteristic
354M16C22×D17311yescharacteristic
355M16C22×D17311yescharacteristic
356C2×C8—21no—
357C2×C8—3461no—
358C2×C8—3461no—
359C2×C8—3461no—
360C2×C8—3461no—
361C2×C8—3461no—
362C2×C8—3461no—
363C2×C8—1731no—
364C2×C8—1731no—
365C2×C8—1731no—
366C2×C8—1731no—
367C2×C8—1731no—
368C2×C8C22×D17311yescharacteristic
369C23—21no—
370C23—3461no—
371C23—3461no—
372C23—1731no—
373Q8—21no—
374Q8—21no—
375Q8—3461no—
376Q8—3461no—
377Q8—3461no—
378Q8—1731no—
379Q8D4×D17311yescharacteristic
380D4—21no—
381D4—21no—
382D4—21no—
383D4—3461no—
384D4—3461no—
385D4—3461no—
386D4—3461no—
387D4—3461no—
388D4—3461no—
389D4—1731no—
390D4—6921no—
391D4—6921no—
392D4—6921no—
393D4—6921no—
394D4D4×D17311yescharacteristic
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—3461no—
398C2×C4—3461no—
399C2×C4—3461no—
400C2×C4—3461no—
401C2×C4—3461no—
402C2×C4—1731no—
403C2×C4—1731no—
404C2×C4—1731no—
405C2×C4—1731no—
406C2×C4—1731no—
407C2×C4—6921no—
408C2×C4—6921no—
409C2×C4—6921no—
410C2×C4—6921no—
411C2×C4C23×D17311yescharacteristic
412C8—21no—
413C8—21no—
414C8—21no—
415C8—3461no—
416C8—3461no—
417C8—3461no—
418C8—1731no—
419C8—1731no—
420C8D4×D17311yescharacteristic
421C8D4×D17311yescharacteristic
422C22—21no—
423C22—41no—
424C22—81no—
425C22—3461no—
426C22—13841no—
427C22—1731no—
428C22—1731no—
429C22—6921no—
430C22—6922no—
431C22—6921no—
432C22#41×D17311yescharacteristic
433C4—21no—
434C4—41no—
435C4—3461no—
436C4—1731no—
437C4—1731no—
438C4—6921no—
439C4C2×D4×D17311yescharacteristic
440C4C2×D4×D17311yescharacteristic
441C2—21no—
442C2—41no—
443C2—81no—
444C2—3461no—
445C2—3461no—
446C2—13841no—
447C2—6921no—
448C2#121×D17311yescharacteristic
subgroupG/H
centerC2#121×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#471×C173C2
socleC2×C173C2×#121
hypercenter#471D173
hypocenterC173#3067
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #26012290C2semidirect
2. #26012287C2semidirect
3. C2×#19149700C2semidirect
4. #131×D173C2semidirect
5. #26011765C2semidirect
6. #26011764C2semidirect
7. #26011760C2semidirect
8. #26011759C2semidirect
9. #26011755C2semidirect
10. #26011754C2semidirect
11. #26011751C2semidirect
12. #108×D173C2semidirect
13. #26011474C2semidirect
14. #26011472C2semidirect
15. #471×C173C2semidirect
37. C2×#14583816C22semidirect
38. C2×#14583815C22semidirect
39. C2×#14583811C22semidirect
51. #19149615C22semidirect
52. #19149614C22semidirect
53. #19149614C22semidirect
54. #19149598C22semidirect
55. #19149597C22semidirect
56. #19149597C22semidirect
57. #19149596C22semidirect
58. #19149595C22semidirect
59. #19149595C22semidirect
62. C2×SD16×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
66. #100×C173C22semidirect
224. C173#3067semidirect
237. #471D173semidirect
266. C2×SD16C2×D173semidirect
283. #108C2×D173semidirect
287. #101C2×D173semidirect
291. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122902
#260122872
C2×#191497002
#131×D1732
#260117652
#260117642
#260117602
#260117592
#260117552
#260117542
#260117512
#108×D1732
#260114742
#260114722
#471×C1732
#3067173