Finite Group Databasebeta

← #36362783

Subgroups — #36362783

Order 22144.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C211yesmaximal, characteristic, complemented
2C2×#19149701C211yesmaximal, characteristic, complemented
3C2×#19149699C211yesmaximal, characteristic, complemented
4#26012003C211yesmaximal, characteristic, complemented
5#26011656C211yesmaximal, characteristic, complemented
6#26011654C211yesmaximal, characteristic, complemented
7#26011648C211yesmaximal, characteristic, complemented
8#26011643C211yesmaximal, characteristic, complemented
9#26011639C211yesmaximal, characteristic, complemented
10#26011637C211yesmaximal, characteristic, complemented
11#26011631C211yesmaximal, characteristic, complemented
12#26011626C211yesmaximal, characteristic, complemented
13#26011485C211yesmaximal, characteristic, complemented
14C2×#19149591C211yesmaximal, characteristic, complemented
15#416×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826—21no—
18C2×#14583826C2211yescharacteristic
19C2×#14583826C2211yescharacteristic, complemented
20C2×#14583825—21no—
21C2×#14583825C2211yescharacteristic
22#41×D173—24no—
23#41×D173—24no—
24C2×Q8×D173C2211yescharacteristic
25C2×D4×D173—21no—
26C2×D4×D173C2211yescharacteristic
27C22×C4×D173—21no—
28C22×C4×D173C2211yescharacteristic
29#19149701—24no—
30#19149699—24no—
31C2×Dic692C2211yescharacteristic, complemented
32C2×#14583816C2211yescharacteristic, complemented
33C2×#14583815C2211yescharacteristic, complemented
34C2×#14583815C2211yescharacteristic, complemented
35C2×#14583814C2211yescharacteristic, complemented
36C2×#14583813C2211yescharacteristic, complemented
37C2×D1384C2211yescharacteristic, complemented
38C2×#14583811C2211yescharacteristic, complemented
39C22×#13063436C2211yescharacteristic, complemented
40#19149663—21no—
41#19149659C2211yescharacteristic
42#19149659C2211yescharacteristic, complemented
43#19149654C2211yescharacteristic
44#19149654C2211yescharacteristic, complemented
45#19149649—21no—
46C4×D692—21no—
47C4×D692—21no—
48#19149602C2211yescharacteristic, complemented
49#19149602C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic, complemented
51#19149599C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic
54#19149591C2211yescharacteristic, complemented
55#19149591C2211yescharacteristic, complemented
56C2×#41×C173C2211yescharacteristic
57C22×C8×C173C2211yescharacteristic, complemented
58#117×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#103×C173C2211yescharacteristic, complemented
61#102×C173C2211yescharacteristic, complemented
62#102×C173C2211yescharacteristic, complemented
63C23×D173—21no—
64C23×D173D411yescharacteristic
65#14583827—22no—
66#14583827—22no—
67#14583826—28no—
68#14583826—22no—
69#14583826—22no—
70#14583826—22no—
71#14583826D414yescomplemented
72#14583825—28no—
73#14583825—22no—
74#14583825—22no—
75C2×Dic346—21no—
76C2×Dic346C2311yescharacteristic
77C2×Dic346C2311yescharacteristic
78Q8×D173—22no—
79Q8×D173—22no—
80C2×D692—21no—
81C2×D692—21no—
82C2×D692—21no—
83C2×D692C2311yescharacteristic
84C2×D692C2311yescharacteristic
85C2×C22⋊D173—21no—
86C2×C22⋊D173—21no—
87C2×C22⋊D173—21no—
88C2×C22⋊D173—21no—
89D4×D173—28no—
90D4×D173—22no—
91D4×D173—22no—
92C22×Dic173—21no—
93C22×Dic173D411yescharacteristic
94C2×C4×D173—28no—
95C2×C4×D173—21no—
96C2×C4×D173—21no—
97C2×C4×D173—21no—
98C2×C4×D173—22no—
99C2×C4×D173—22no—
100C2×C4×D173—21no—
101C2×C4×D173D411yescharacteristic
102C2×C4×D173D411yescharacteristic
103C2×C4×D173C2311yescharacteristic
104C2×C4×D173C2311yescharacteristic
105Dic692—22no—
106#14583816—22no—
107#14583815—22no—
108#14583815—22no—
109#14583814—22no—
110#14583813—22no—
111D1384—22no—
112#14583811—22no—
113C2×#13063436—22no—
114C2×#13063436C2311yescharacteristic
115C2×#13063436C2311yescharacteristic
116C4⋊Dic173—21no—
117C4⋊Dic173—21no—
118#14583803—21no—
119#14583803—21no—
120#14583803—21no—
121#14583803—21no—
122#41×C173—22no—
123#41×C173D414yescomplemented
124C2×Q8×C173C2311yescharacteristic
125C2×D4×C173—21no—
126C2×D4×C173C2311yescharacteristic
127C22×C4×C173—21no—
128C22×C4×C173C2311yescharacteristic
129C2×C8×C173—22no—
130C2×C8×C173C2311yescharacteristic
131C2×C8×C173C2311yescharacteristic
132C4⋊C4×C173C2311yescharacteristic
133C4⋊C4×C173C2311yescharacteristic
134C22⋊C4×C173—21no—
135C42×C173—21no—
136C22×D173—24no—
137C22×D173—22no—
138C22×D173—21no—
139C22×D173—22no—
140C22×D173—21no—
141C22×D173—41no—
142C22×D173—42no—
143C22×D173—42no—
144C22×D173C2×D411yescharacteristic
145C22×D173C2×D411yescharacteristic
146Dic346—24no—
147Dic346—21no—
148Dic346—21no—
149Dic346—21no—
150Dic346C2×D412yes—
151D692—24no—
152D692—21no—
153D692—21no—
154D692—21no—
155D692—42no—
156D692—42no—
157D692C2×D412yes—
158C22⋊D173—24no—
159C22⋊D173—24no—
160C22⋊D173—24no—
161C22⋊D173—42no—
162C2×Dic173—24no—
163C2×Dic173—22no—
164C2×Dic173—21no—
165C2×Dic173—21no—
166C2×Dic173C2×D411yescharacteristic
167C2×Dic173C2×D411yescharacteristic
168C4×D173—24no—
169C4×D173—24no—
170C4×D173—22no—
171C4×D173—24no—
172C4×D173—22no—
173C4×D173—21no—
174C4×D173—21no—
175C4×D173—21no—
176C4×D173—21no—
177C4×D173—42no—
178#13063436—21no—
179#13063436—21no—
180C23×C173—21no—
181C23×C173C2×D411yescharacteristic
182Q8×C173—21no—
183Q8×C173C2×D412yes—
184D4×C173—24no—
185D4×C173—21no—
186D4×C173C2×D412yes—
187C2×C4×C173—21no—
188C2×C4×C173—24no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173C2×D412yes—
192C2×C4×C173C2×D411yescharacteristic
193C2×C4×C173C2×D412yes—
194C2×C4×C173C2411yescharacteristic
195C8×C173—21no—
196C8×C173—21no—
197C22×C173—21no—
198C22×C173—21no—
199C22×C173—22no—
200C22×C173—41no—
201C22×C173—41no—
202C22×C173#12012yes—
203C22×C173C22×D411yescharacteristic
204C2×D173—21no—
205C2×D173—21no—
206C2×D173—21no—
207C2×D173—22no—
208C2×D173—21no—
209C2×D173—21no—
210C2×D173—44no—
211C2×D173—48no—
212C2×D173—41no—
213C2×D173—41no—
214C2×D173—41no—
215C2×D173—41no—
216C2×D173—41no—
217C2×D173#13512yescomplemented
218C4×C173—22no—
219C4×C173—41no—
220C4×C173—41no—
221C4×C173#12012yes—
222C4×C173C22×D411yescharacteristic
223C4×C173C22×D411yescharacteristic
224Dic173—21no—
225Dic173—22no—
226Dic173—21no—
227Dic173—41no—
228C2×C173—22no—
229C2×C173—42no—
230C2×C173C2×#12011yescharacteristic
231C2×C173C2×#13511yescharacteristic
232C2×C173#57511yescharacteristic
233D173—22no—
234D173—41no—
235D173—42no—
236D173—81no—
237C173#300211yescharacteristic, complemented, Sylow
238#3002—1731nomaximal, Sylow
239C22×#41—1731no—
240C2×#135—1731no—
241C2×#135—1731no—
242#525—1731no—
243#452—1731no—
244#452—1731no—
245#451—1731no—
246#450—1731no—
247#449—1731no—
248#449—1731no—
249#448—1731no—
250#447—1731no—
251#416—1731no—
252#416D17311yescharacteristic, complemented
253C2×C22⋊C8—1731no—
254C2×#41—3461no—
255C2×#41—3464no—
256C2×#41—3464no—
257C2×#41—3461no—
258C2×#41—1731no—
259C2×#41—1731no—
260C2×#41—1731no—
261C2×#41—1731no—
262C2×#41C2×D17311yescharacteristic
263C22×Q8—1731no—
264C22×D4—3461no—
265C22×D4—1731no—
266C23×C4—3461no—
267C23×C4—1731no—
268#135—3464no—
269#135—3464no—
270C2×Q16—1731no—
271C2×Q16—1731no—
272C2×SD16—1731no—
273C2×SD16—1731no—
274C2×SD16—1731no—
275C2×SD16—1731no—
276C2×D8—1731no—
277C2×D8—1731no—
278C22×C8—1731no—
279C22×C8C2×D17311yescharacteristic, complemented
280#123—3461no—
281C22⋊Q8—1731no—
282C22⋊Q8—1731no—
283#121—1731no—
284#121—1731no—
285#120—3461no—
286C4×D4—3461no—
287C4×D4—3461no—
288#117C2×D17311yescharacteristic, complemented
289#103—1731no—
290#103—1731no—
291#103C2×D17311yescharacteristic, complemented
292#103C2×D17311yescharacteristic, complemented
293#102—1731no—
294#102—1731no—
295#102C2×D17311yescharacteristic, complemented
296#102C2×D17311yescharacteristic, complemented
297C22⋊C8—1731no—
298C22⋊C8—1731no—
299C22⋊C8—1731no—
300C22⋊C8—1731no—
301C24—3461no—
302C24—1731no—
303#41—22no—
304#41—3468no—
305#41—3468no—
306#41—3462no—
307#41—3462no—
308#41—3462no—
309#41—3462no—
310#41—3462no—
311#41—3462no—
312#41—3462no—
313#41—1734no—
314#41D69214yescomplemented
315C2×Q8—3461no—
316C2×Q8—3462no—
317C2×Q8—3462no—
318C2×Q8—1731no—
319C2×Q8—1731no—
320C2×Q8C22×D17311yescharacteristic
321C2×D4—21no—
322C2×D4—3461no—
323C2×D4—3461no—
324C2×D4—3461no—
325C2×D4—3468no—
326C2×D4—3462no—
327C2×D4—3461no—
328C2×D4—3461no—
329C2×D4—3462no—
330C2×D4—3461no—
331C2×D4—3461no—
332C2×D4—1731no—
333C2×D4—1731no—
334C2×D4C22×D17311yescharacteristic
335C22×C4—21no—
336C22×C4—3468no—
337C22×C4—3461no—
338C22×C4—3461no—
339C22×C4—3461no—
340C22×C4—3462no—
341C22×C4—3461no—
342C22×C4—3462no—
343C22×C4—3461no—
344C22×C4—1731no—
345C22×C4—1731no—
346C22×C4—1731no—
347C22×C4—1731no—
348C22×C4—1731no—
349C22×C4C22×D17311yescharacteristic
350Q16—3462no—
351Q16—3462no—
352SD16—3462no—
353SD16—3462no—
354SD16—3462no—
355SD16—3462no—
356D8—3462no—
357D8—3462no—
358C2×C8—22no—
359C2×C8—3462no—
360C2×C8—1731no—
361C2×C8—1731no—
362C2×C8C22×D17311yescharacteristic
363C2×C8C22×D17311yescharacteristic
364C4⋊C4—3461no—
365C4⋊C4—3461no—
366C4⋊C4C22×D17311yescharacteristic
367C4⋊C4C22×D17311yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—3461no—
370C22⋊C4—3461no—
371C22⋊C4—3461no—
372C22⋊C4—3461no—
373C42—21no—
374C23—21no—
375C23—3464no—
376C23—3462no—
377C23—3462no—
378C23—3461no—
379C23—3461no—
380C23—1731no—
381C23—1731no—
382C23—6921no—
383C23—6922no—
384C23—6922no—
385C23D4×D17311yescharacteristic
386Q8—21no—
387Q8—3464no—
388Q8—3461no—
389Q8—3461no—
390Q8—3461no—
391Q8—1732no—
392Q8C2×D69212yes—
393D4—24no—
394D4—21no—
395D4—3464no—
396D4—3464no—
397D4—3464no—
398D4—3464no—
399D4—3461no—
400D4—3461no—
401D4—3461no—
402D4—1732no—
403D4—6922no—
404D4—6922no—
405D4—6922no—
406D4C2×D69212yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—3464no—
412C2×C4—3464no—
413C2×C4—3464no—
414C2×C4—3462no—
415C2×C4—3461no—
416C2×C4—3462no—
417C2×C4—3464no—
418C2×C4—3462no—
419C2×C4—3461no—
420C2×C4—3461no—
421C2×C4—3461no—
422C2×C4—3461no—
423C2×C4—3461no—
424C2×C4—1731no—
425C2×C4—1731no—
426C2×C4—6922no—
427C2×C4D4×D17312yes—
428C2×C4D4×D17311yescharacteristic
429C2×C4C2×D69212yes—
430C2×C4C23×D17311yescharacteristic
431C8—21no—
432C8—21no—
433C8—3461no—
434C8—3461no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—3461no—
441C22—3461no—
442C22—3461no—
443C22—3462no—
444C22—3461no—
445C22—3461no—
446C22—1732no—
447C22—6924no—
448C22—6928no—
449C22—6921no—
450C22—6921no—
451C22—6921no—
452C22—6921no—
453C22—6921no—
454C22#1914964912yes—
455C22C2×D4×D17311yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—3461no—
460C4—3462no—
461C4—3461no—
462C4—6921no—
463C4#1914964912yes—
464C4C2×D4×D17311yescharacteristic
465C4C22×D69211yescharacteristic
466C2—22no—
467C2—42no—
468C2—3462no—
469C2—13841no—
470C2—6921no—
471C2—6922no—
472C2C2×#1914964911yescharacteristic
473C2#135×D17311yescharacteristic
474C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
hypercenter#416D173
hypocenterC173#3002
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D173C2semidirect
2. C2×#19149701C2semidirect
3. C2×#19149699C2semidirect
4. #26012003C2semidirect
5. #26011656C2semidirect
6. #26011654C2semidirect
7. #26011648C2semidirect
8. #26011643C2semidirect
9. #26011639C2semidirect
10. #26011637C2semidirect
11. #26011631C2semidirect
12. #26011626C2semidirect
13. #26011485C2semidirect
14. C2×#19149591C2semidirect
15. #416×C173C2semidirect
19. C2×#14583826C22semidirect
31. C2×Dic692C22semidirect
32. C2×#14583816C22semidirect
33. C2×#14583815C22semidirect
34. C2×#14583815C22semidirect
35. C2×#14583814C22semidirect
36. C2×#14583813C22semidirect
37. C2×D1384C22semidirect
38. C2×#14583811C22semidirect
39. C22×#13063436C22semidirect
42. #19149659C22semidirect
44. #19149654C22semidirect
48. #19149602C22semidirect
49. #19149602C22semidirect
50. #19149599C22semidirect
51. #19149599C22semidirect
54. #19149591C22semidirect
55. #19149591C22semidirect
57. C22×C8×C173C22semidirect
58. #117×C173C22semidirect
59. #103×C173C22semidirect
60. #103×C173C22semidirect
61. #102×C173C22semidirect
62. #102×C173C22semidirect
71. #14583826D4semidirect
123. #41×C173D4semidirect
217. C2×D173#135semidirect
237. C173#3002semidirect
252. #416D173semidirect
279. C22×C8C2×D173semidirect
288. #117C2×D173semidirect
291. #103C2×D173semidirect
292. #103C2×D173semidirect
295. #102C2×D173semidirect
296. #102C2×D173semidirect
314. #41D692semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1732
C2×#191497012
C2×#191496992
#260120032
#260116562
#260116542
#260116482
#260116432
#260116392
#260116372
#260116312
#260116262
#260114852
C2×#191495912
#416×C1732
#3002173