Finite Group Databasebeta

← #452×D173 #36362768

Subgroups — #452×D173 #36362768

Order 22144.

760 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C211yesmaximal, characteristic
2C2×Q16×D173C211yesmaximal, characteristic, complemented
3C2×SD16×D173C211yesmaximal, characteristic, complemented
4C22⋊Q8×D173C211yesmaximal, characteristic, complemented
5#26011656C211yesmaximal, characteristic, complemented
6#26011655C211yesmaximal, characteristic, complemented
7#26011654C211yesmaximal, characteristic, complemented
8#26011653C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011650C211yesmaximal, characteristic, complemented
12#103×D173C211yesmaximal, characteristic, complemented
13#102×D173C211yesmaximal, characteristic, complemented
14C22⋊C8×D173C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826—21no—
18C2×#14583826C2211yescharacteristic
19C2×#14583825—21no—
20C2×#14583825C2211yescharacteristic
21#41×D173—24no—
22#41×D173—24no—
23C2×Q8×D173C2211yescharacteristic
24C2×Q8×D173C2211yescharacteristic
25C2×D4×D173—21no—
26C2×D4×D173C2211yescharacteristic
27C22×C4×D173—21no—
28C22×C4×D173C2211yescharacteristic
29C2×Dic692C2211yescharacteristic, complemented
30C2×#14583816C2211yescharacteristic, complemented
31C2×#14583816C2211yescharacteristic, complemented
32Q16×D173—24no—
33C2×#14583815C2211yescharacteristic, complemented
34C2×#14583814C2211yescharacteristic, complemented
35C2×#14583813C2211yescharacteristic, complemented
36SD16×D173—24no—
37C2×C8×D173C2211yescharacteristic
38C2×C8×D173C2211yescharacteristic
39#19149659C2211yescharacteristic, complemented
40#19149658C2211yescharacteristic, complemented
41#19149657—21no—
42#19149656—21no—
43#19149655C2211yescharacteristic, complemented
44C4⋊C4×D173—21no—
45C4⋊C4×D173C2211yescharacteristic
46C22⋊C4×D173—21no—
47#19149604C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149602C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149599C2211yescharacteristic, complemented
53C22⋊#13063436C2211yescharacteristic, complemented
54#19149591C2211yescharacteristic, complemented
55#19149591C2211yescharacteristic, complemented
56C2×#41×C173C2211yescharacteristic
57C2×Q16×C173C2211yescharacteristic, complemented
58C2×SD16×C173C2211yescharacteristic, complemented
59C22⋊Q8×C173C2211yescharacteristic, complemented
60#103×C173C2211yescharacteristic, complemented
61#102×C173C2211yescharacteristic, complemented
62C22⋊C8×C173C2211yescharacteristic, complemented
63C23×D173—21no—
64C23×D173D411yescharacteristic
65#14583827—22no—
66#14583827—22no—
67#14583826—28no—
68#14583826—22no—
69#14583826—22no—
70#14583825—28no—
71#14583825—22no—
72#14583825—22no—
73C2×Dic346—21no—
74C2×Dic346—21no—
75C2×Dic346C2311yescharacteristic
76C2×Dic346C2311yescharacteristic
77Q8×D173—22no—
78Q8×D173—22no—
79Q8×D173—22no—
80Q8×D173D414yes—
81C2×D692—21no—
82C2×D692C2311yescharacteristic
83C2×C22⋊D173—21no—
84C2×C22⋊D173—21no—
85C2×C22⋊D173—21no—
86D4×D173—28no—
87D4×D173—22no—
88D4×D173D414yes—
89C22×Dic173—21no—
90C22×Dic173D411yescharacteristic
91C2×C4×D173—28no—
92C2×C4×D173—21no—
93C2×C4×D173—21no—
94C2×C4×D173—22no—
95C2×C4×D173—22no—
96C2×C4×D173—21no—
97C2×C4×D173—21no—
98C2×C4×D173—21no—
99C2×C4×D173D411yescharacteristic
100C2×C4×D173D411yescharacteristic
101C2×C4×D173C2311yescharacteristic
102C2×C4×D173C2311yescharacteristic
103Dic692—22no—
104#14583816—22no—
105#14583816—22no—
106#14583815—22no—
107#14583814—22no—
108#14583813—22no—
109C2×#13063436C2311yescharacteristic
110C2×#13063436C2311yescharacteristic
111C8×D173—22no—
112C8×D173—22no—
113C4⋊Dic173—21no—
114C4⋊Dic173C2311yescharacteristic
115#14583805—21no—
116#14583805—21no—
117#14583805—21no—
118C22⋊Dic173—21no—
119#14583803—21no—
120#14583803—21no—
121#41×C173—22no—
122#41×C173—22no—
123C2×Q8×C173C2311yescharacteristic
124C2×Q8×C173C2311yescharacteristic
125C2×D4×C173—21no—
126C2×D4×C173C2311yescharacteristic
127C22×C4×C173—21no—
128C22×C4×C173C2311yescharacteristic
129Q16×C173—22no—
130SD16×C173—22no—
131C2×C8×C173C2311yescharacteristic
132C2×C8×C173C2311yescharacteristic
133C4⋊C4×C173—21no—
134C4⋊C4×C173C2311yescharacteristic
135C22⋊C4×C173—21no—
136C22×D173—21no—
137C22×D173—22no—
138C22×D173—24no—
139C22×D173—22no—
140C22×D173—22no—
141C22×D173—42no—
142C22×D173—42no—
143C22×D173C2×D411yescharacteristic
144C22×D173C2×D411yescharacteristic
145Dic346—21no—
146Dic346—24no—
147Dic346—21no—
148Dic346—21no—
149Dic346—42no—
150Dic346C2×D412yes—
151D692—24no—
152D692—21no—
153D692C2×D412yes—
154C22⋊D173—24no—
155C22⋊D173—24no—
156C22⋊D173—24no—
157C2×Dic173—24no—
158C2×Dic173—21no—
159C2×Dic173—22no—
160C2×Dic173—21no—
161C2×Dic173—21no—
162C2×Dic173C2×D411yescharacteristic
163C2×Dic173C2×D411yescharacteristic
164C4×D173—24no—
165C4×D173—24no—
166C4×D173—22no—
167C4×D173—24no—
168C4×D173—21no—
169C4×D173—21no—
170C4×D173—22no—
171C4×D173—42no—
172C4×D173—42no—
173C4×D173C2×D412yes—
174C4×D173C2×D412yes—
175#13063436—21no—
176#13063436—21no—
177C23×C173—21no—
178C23×C173C2×D411yescharacteristic
179Q8×C173—21no—
180Q8×C173—21no—
181Q8×C173—21no—
182Q8×C173C2×D412yes—
183D4×C173—24no—
184D4×C173—21no—
185D4×C173C2×D412yes—
186C2×C4×C173—24no—
187C2×C4×C173—21no—
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173C2×D411yescharacteristic
193C2×C4×C173C2411yescharacteristic
194C8×C173—21no—
195C8×C173—21no—
196C22×C173—21no—
197C22×C173—22no—
198C22×C173—21no—
199C22×C173—21no—
200C22×C173—41no—
201C22×C173—41no—
202C22×C173C22×D411yescharacteristic
203C2×D173—21no—
204C2×D173—22no—
205C2×D173—21no—
206C2×D173—21no—
207C2×D173—41no—
208C2×D173—41no—
209C2×D173—44no—
210C2×D173—48no—
211C2×D173#12012yes—
212C2×D173#13512yes—
213C2×D173#13712yes—
214C4×C173—21no—
215C4×C173—22no—
216C4×C173—41no—
217C4×C173—41no—
218C4×C173C22×D411yescharacteristic
219C4×C173C22×D411yescharacteristic
220Dic173—21no—
221Dic173—22no—
222Dic173—41no—
223Dic173—41no—
224Dic173#12012yes—
225C2×C173—41no—
226C2×C173—42no—
227C2×C173C2×#12011yescharacteristic
228C2×C173C2×#13511yescharacteristic
229C2×C173C2×#13711yescharacteristic
230D173—41no—
231D173—42no—
232D173#45214yesdirect product factor
233C173C2×#45211yescharacteristic, complemented, Sylow
234C2×#452—1731nomaximal, Sylow
235C22×#41—1731no—
236C22×Q16—1731no—
237C22×SD16—1731no—
238C2×C22⋊Q8—1731no—
239#452—1731no—
240#452—1731no—
241#452—1731no—
242#452—1731no—
243#452—1731no—
244#452—1731no—
245#452—1731no—
246#452D17311yescharacteristic, direct product factor
247C2×#103—1731no—
248C2×#102—1731no—
249C2×C22⋊C8—1731no—
250C2×#41—3461no—
251C2×#41—3464no—
252C2×#41—3464no—
253C2×#41—3461no—
254C2×#41—1731no—
255C2×#41—1731no—
256C2×#41—1731no—
257C2×#41C2×D17311yescharacteristic
258C22×Q8—1731no—
259C22×Q8—1731no—
260C22×D4—3461no—
261C22×D4—1731no—
262C23×C4—3461no—
263C23×C4—1731no—
264C2×Q16—3464no—
265C2×Q16—1731no—
266C2×Q16—1731no—
267C2×Q16—1731no—
268C2×Q16C2×D17311yescharacteristic, complemented
269C2×SD16—3464no—
270C2×SD16—1731no—
271C2×SD16—1731no—
272C2×SD16—1731no—
273C2×SD16C2×D17311yescharacteristic, complemented
274C22×C8—1731no—
275C22×C8—1731no—
276C22⋊Q8—3461no—
277C22⋊Q8—3461no—
278C22⋊Q8—1731no—
279C22⋊Q8—1731no—
280C22⋊Q8—1731no—
281C22⋊Q8C2×D17311yescharacteristic, complemented
282C2×C4⋊C4—3461no—
283C2×C4⋊C4—1731no—
284C2×C22⋊C4—3461no—
285#103—1731no—
286#103—1731no—
287#103—1731no—
288#103C2×D17311yescharacteristic, complemented
289#102—1731no—
290#102—1731no—
291#102—1731no—
292#102C2×D17311yescharacteristic, complemented
293C22⋊C8—1731no—
294C22⋊C8—1731no—
295C22⋊C8—1731no—
296C22⋊C8C2×D17311yescharacteristic, complemented
297C24—3461no—
298C24—1731no—
299#41—22no—
300#41—22no—
301#41—3468no—
302#41—3462no—
303#41—3462no—
304#41—3468no—
305#41—3462no—
306#41—3462no—
307#41—3462no—
308#41—3462no—
309C2×Q8—3461no—
310C2×Q8—3461no—
311C2×Q8—3462no—
312C2×Q8—3462no—
313C2×Q8—3462no—
314C2×Q8—1734no—
315C2×Q8—1731no—
316C2×Q8—1731no—
317C2×Q8C22×D17311yescharacteristic
318C2×Q8C22×D17311yescharacteristic
319C2×D4—21no—
320C2×D4—3461no—
321C2×D4—3468no—
322C2×D4—3461no—
323C2×D4—3461no—
324C2×D4—3462no—
325C2×D4—3461no—
326C2×D4—1734no—
327C2×D4—1731no—
328C2×D4C22×D17311yescharacteristic
329C22×C4—21no—
330C22×C4—3468no—
331C22×C4—3461no—
332C22×C4—3461no—
333C22×C4—3462no—
334C22×C4—3462no—
335C22×C4—3461no—
336C22×C4—3461no—
337C22×C4—3461no—
338C22×C4—3461no—
339C22×C4—1731no—
340C22×C4—1731no—
341C22×C4—1731no—
342C22×C4—1731no—
343C22×C4—1731no—
344C22×C4C22×D17311yescharacteristic
345Q16—22no—
346Q16—3462no—
347Q16—3462no—
348Q16—3462no—
349SD16—22no—
350SD16—3462no—
351SD16—3462no—
352SD16—3462no—
353C2×C8—3462no—
354C2×C8—3462no—
355C2×C8—1731no—
356C2×C8—1731no—
357C2×C8C22×D17311yescharacteristic
358C2×C8C22×D17311yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—3461no—
361C4⋊C4—3461no—
362C4⋊C4—3461no—
363C4⋊C4—3461no—
364C4⋊C4—1731no—
365C4⋊C4C22×D17311yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—3461no—
368C22⋊C4—3461no—
369C22⋊C4—3461no—
370C23—21no—
371C23—3462no—
372C23—3464no—
373C23—3462no—
374C23—3462no—
375C23—3461no—
376C23—1731no—
377C23—1731no—
378C23—6922no—
379C23—6922no—
380C23D4×D17311yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—3464no—
385Q8—3461no—
386Q8—3461no—
387Q8—3461no—
388Q8—1732no—
389Q8—6922no—
390Q8D4×D17312yes—
391D4—24no—
392D4—21no—
393D4—3464no—
394D4—3464no—
395D4—3464no—
396D4—3464no—
397D4—3461no—
398D4—1732no—
399D4D4×D17312yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—3464no—
407C2×C4—3464no—
408C2×C4—3462no—
409C2×C4—3464no—
410C2×C4—3462no—
411C2×C4—3464no—
412C2×C4—3461no—
413C2×C4—3462no—
414C2×C4—3461no—
415C2×C4—3461no—
416C2×C4—3461no—
417C2×C4—3461no—
418C2×C4—1731no—
419C2×C4—1731no—
420C2×C4—1732no—
421C2×C4—1732no—
422C2×C4—6922no—
423C2×C4—6922no—
424C2×C4D4×D17311yescharacteristic
425C2×C4C23×D17311yescharacteristic
426C8—21no—
427C8—21no—
428C8—3461no—
429C8—3461no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—3461no—
437C22—3462no—
438C22—3461no—
439C22—3461no—
440C22—1732no—
441C22—1732no—
442C22—1732no—
443C22—6924no—
444C22—6928no—
445C22—6921no—
446C22—6921no—
447C22C2×D4×D17311yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—3461no—
453C4—3462no—
454C4—1732no—
455C4—6921no—
456C4—6921no—
457C4C2×D4×D17311yescharacteristic
458C4C2×D4×D17311yescharacteristic
459C2—41no—
460C2—42no—
461C2—1734no—
462C2—6921no—
463C2—6922no—
464C2#120×D17311yescharacteristic
465C2#135×D17311yescharacteristic
466C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
prime radicalC2×#41×D173C2
hypercenter#452D173
hypocenterC173C2×#452
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D173C2semidirect
3. C2×SD16×D173C2semidirect
4. C22⋊Q8×D173C2semidirect
5. #26011656C2semidirect
6. #26011655C2semidirect
7. #26011654C2semidirect
8. #26011653C2semidirect
9. #26011652C2semidirect
10. #26011651C2semidirect
11. #26011650C2semidirect
12. #103×D173C2semidirect
13. #102×D173C2semidirect
14. C22⋊C8×D173C2semidirect
15. #452×C173C2semidirect
29. C2×Dic692C22semidirect
30. C2×#14583816C22semidirect
31. C2×#14583816C22semidirect
33. C2×#14583815C22semidirect
34. C2×#14583814C22semidirect
35. C2×#14583813C22semidirect
39. #19149659C22semidirect
40. #19149658C22semidirect
43. #19149655C22semidirect
47. #19149604C22semidirect
48. #19149603C22semidirect
49. #19149602C22semidirect
50. #19149601C22semidirect
51. #19149600C22semidirect
52. #19149599C22semidirect
53. C22⋊#13063436C22semidirect
54. #19149591C22semidirect
55. #19149591C22semidirect
57. C2×Q16×C173C22semidirect
58. C2×SD16×C173C22semidirect
59. C22⋊Q8×C173C22semidirect
60. #103×C173C22semidirect
61. #102×C173C22semidirect
62. C22⋊C8×C173C22semidirect
232. D173#452direct (N × H)
233. C173C2×#452semidirect
246. #452D173direct (N × H)
268. C2×Q16C2×D173semidirect
273. C2×SD16C2×D173semidirect
281. C22⋊Q8C2×D173semidirect
288. #103C2×D173semidirect
292. #102C2×D173semidirect
296. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1732
C2×Q16×D1732
C2×SD16×D1732
C22⋊Q8×D1732
#260116562
#260116552
#260116542
#260116532
#260116522
#260116512
#260116502
#103×D1732
#102×D1732
C22⋊C8×D1732
#452×C1732
C2×#452173